diff --git a/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/2301.11602v1.pdf.txt b/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/2301.11602v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2b290d244d29c4d3a8b971b6b7b69a9215dc405 --- /dev/null +++ b/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/2301.11602v1.pdf.txt @@ -0,0 +1,2070 @@ +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +ABSTRACT. Given a (finite) simplicial complex, we define its i-th Laplacian polytope as the +convex hull of the columns of its i-th Laplacian matrix. This extends Laplacian simplices of +finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of +these polytopes, we focus on the d-th Laplacian polytope of the boundary of a pd ` 1q-simplex +Bpσd`1q. If d is odd, then as for graphs, the d-th Laplacian polytope turns out to be a pd ` 1q- +simplex in this case. If d is even, we show that the d-th Laplacian polytope of Bpσd`1q is +combinatorially equivalent to a d-dimensional cyclic polytope on d ` 2 vertices. Moreover, we +provide an explicit regular unimodular triangulation for the d-th Laplacian polytope of Bpσd`1q. +This enables us to to compute the normalized volume and to show that the h˚-polynomial is +real-rooted and unimodal, if d is odd and even, respectively. +1. INTRODUCTION +Over decades, several lattice polytopes arising from graphs have been studied, extensively. +Prominent examples include matching polytopes, cut polytopes, edge polytopes, adjacency +polytopes of several types, among which are symmetric edge polytopes (see e.g., [23, 4, 19, +21, 28, 10]). Following this line of research, in 2017, Braun and Meyer [6] initiated the study +of Laplacian simplices that are defined as the convex hull of the columns of the classical +Laplacian matrix of a simple graph (see also [24, 3]). Since each simple graph can be seen +as a 1-dimensional simplicial complex and since to each simplicial complex, we can associate +Laplacian matrices, defined via their boundary maps in simplicial homology, it is natural to +extend the definition of Laplacian simplices to arbitrary simplicial complexes and their Lapla- +cians. More precisely, given a simplicial complex ∆ (with a fixed ordering of the vertex set) and +its i-th Laplacian matrix Lip∆q :“ Bi`1B⊺ +i`1 ` B⊺ +i Bi, we define the i-th Laplacian polytope Ppiq +∆ +of ∆ as the convex hull of the columns of Lip∆q. Here, Bi and Bi`1 denote boundary maps in +simplicial homology. +We initiate the study of Laplacian polytopes by establishing first some general combinatorial +and geometric properties and then by focusing on a particular case. More precisely, we consider +the situation that the underlying simplicial complex ∆ is the boundary of the pd ` 1q-simplex, +denoted by Bpσd`1q, and that we take its highest Laplacian LdpBpσd`1qq. For simplicity, we +set PBpσd`1q :“ Ppdq +Bpσd`1q. If d is even, it is easily seen, that, as for graphs, PBpσd`1q is a pd ` 1q- +simplex. If d is odd, the situation is more complicated. By deriving a complete facet description +of PBpσd`1q in this case, we are able to show that PBpσd`1q is combinatorially equivalent to a d- +dimensional cyclic polytope on d `2 vertices. +It was shown in [6] that Laplacian simplices have unimodal h˚-vectors for certain classes of +graphs, including trees, odd cycles and complete graphs. Inspired by these results, we study +properties of the h˚-vectors of general Laplacian polytopes. This is further motivated by the +general question under which conditions a lattice polytope has a unimodal h˚-vector. It was +conjectured by Hibi and Ohsugi that this is true for reflexive lattice polytopes that have the +integer decomposition property (IDP) [27], and, recently, Adiprasito, Papadakis, Petrotou and +Steinmeyer could confirm this conjecture in the positive [1]. However, it is still mysterious what +happens if the polytope is not reflexive. We consider this question for the Laplacian polytope +1 +arXiv:2301.11602v1 [math.CO] 27 Jan 2023 + +2 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +PBpσd`1q of the boundary of the pd `1q-simplex. Even in this seemingly most simple situation, +Ppdq +∆ +turns out to be not reflexive and hence the mentioned results towards unimodality do not +apply. However, the following result shows that Ppdq +∆ +has at least the integer decomposition +property. +Theorem A. PBpσd`1q has a regular unimodular triangulation for every integer d ě 0. +We note that, combined with [2, Theorem 1.3], this result implies that the h˚-vector of +PBpσd`1q is decreasing in its second half which is obviously implied by but weaker than uni- +modality. The main ingredient for Theorem A is the so-called interior polytope of PBpσd`1q, +that is defined as the convex hull of the interior lattice points of PBpσd`1q. Indeed, this polytope +turns out to be reflexive (after translation to the origin) and miraculously, PBpσd`1q happens to +be the second dilation of it (after translating both polytopes to the origin). Using edgewise +subdivisions, we provide an explicit construction of a regular unimodular triangulation for the +interior polytope which then extends to such a triangulation of PBpσd`1q by [18, Theorem 4.8]. +As a byproduct, we can also compute the normalized volume of PBpσd`1q (see Corollary 6.7). +Theorem A combined with the results on the interior polytope enables us to show the following +statement: +Theorem B. +(a) h˚ ´ +PBpσd`1q;t +¯ +has only real roots if d P N is odd. +(b) h˚ ´ +PBpσd`1q +¯ +is unimodal with peak in the middle for every d P N. +We note that if d is odd, then the statement in pbq is just an easy consequence of the one in +paq. We conjecture paq to be true also if d is even. +The paper is organized as follows. Section 2 provides necessary background on simplicial +complexes, their Laplacian matrices and lattice polytopes. Section 3 collects basic properties of +the Laplacian matrix LdpBpσd`1qq of the boundary of a simplex. In Section 4, we introduce the +i-th Laplacian polytope Ppiq +∆ of a simplicial complex ∆. Among others, we compute its number +of vertices (Proposition 4.4), the dimension of PBpσd`1q (Lemma 4.6) and show that PBpσd`1q is +always simplicial (Theorem 4.8). The goal of Section 5 is to derive a complete facet description +of PBpσd`1q and to show that it is combinatorially equivalent to a d-dimensional cyclic polytope +on d `2 vertices if d is even (Theorem 5.3 and Theorem 5.4). Section 6 is devoted to the proofs +of Theorems A and B, including the construction and study of the interior polytope of PBpσd`1q. +Finally, in Section 7 we state some open problems and possible future directions. +2. PRELIMINARIES +In this section, we provide the necessary background on simplicial complexes, Laplacian +matrices and polytopes. For more information on these topics we refer to [30, 17, 26, 11, 18, 15]. +Moreover, we assume the reader to have basic knowledge about graphs (see e.g., [12]). +2.1. Simplicial complexes and Laplacian matrices. Given a finite set V, a simplicial com- +plex ∆ on vertex set V is a collection of subsets of V that is closed under inclusion. Elements +of ∆ are called faces and inclusion-wise maximal faces are called facets. The dimension of a +face F is dimpFq :“ |F| ´ 1 and we use Fip∆q to denote the set of i-dimensional faces of ∆. +The dimension of ∆ is defined as dimp∆q :“ maxpi : Fip∆q ‰ Hq. If all facets have the same +dimension, ∆ is called pure. 0-dimensional and 1-dimensional faces of ∆ are called vertices and +edges, respectively. The sets of vertices and edges of ∆ induce a graph in a natural way, which +we call the 1-skeleton or graph of ∆. Given a pd ´ 1q-dimensional simplicial complex ∆, its + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +3 +f-vector fp∆q “ pf´1p∆q, f0p∆q,..., fd´1p∆qq is defined by fip∆q :“ |t f P ∆ : dimpFq “ iu| for +´1 ď i ď d ´1 and its h-vector hp∆q “ ph0p∆q,h1p∆q,...,hdp∆qq by the polynomial identity +(2.1) +dÿ +k“0 +hkp∆qtd´k “ +dÿ +k“0 +fk´1p∆qpt ´1qd´k. +The polynomials fp∆;tq :“ řd´1 +i“´1 fip∆qti and hp∆;tq :“ řd +i“0 hip∆qti are called the f- and h- +polynomial of ∆, respectively. +In order to introduce general Laplacian matrices of a simplicial complex ∆, we need to recall +basic notions from simplicial homology. For this purpose, let ∆ be a pd ´ 1q-dimensional +simplicial complex on vertex set V and assume that the vertices are ordered. Without loss +of generality, assume V “ rns “ t1,...,nu endowed with the natural ordering induced by N. We +denote by Cip∆q the Q-vector space with basis teσ : σ P Fip∆qu and set Cip∆q “ t0u for i ď ´1 +and i ą d ´1. The i-th boundary map is the linear map Bi : Cip∆q Ñ Ci´1p∆q defined by +(2.2) +Bipeσq :“ +i`1 +ÿ +k“1 +p´1qk´1eσztjku, +where σ “ t j1 ă ¨¨¨ ă ji`1u P Fip∆q. By abuse of notation, we will use Bi to denote both, the map +and its corresponding matrix. The i-th Laplacian matrix of ∆ is defined as Lip∆q :“ Bi`1B⊺ +i`1 ` +B⊺ +i Bi. +Note that Lip∆q provides an endomorphism of Cip∆q which depends on the chosen +ordering of the vertices. We recall that Hip∆;Qq :“ kerpBiq{ImpBi`1q is the i-th (simplicial) +homology group of ∆. +To provide an explicit description of Lip∆q, we need some further notation. Faces F,G P Fip∆q +are called lower adjacent if F XG P Fi´1p∆q. If, additionally, eFXG appears with the same sign +in BipeFq and BipeGq, we call F XG the similar common lower simplex of F and G. Otherwise, +F XG is referred to as the dissimilar common lower simplex of F and G. The upper degree of +F P Fip∆q, denoted degUpFq, is the number of pi`1q-faces of ∆ containing F. We will use the +following description of Lip∆q from [15, Theorem 3.3.4]: +Theorem 2.1. Let ∆ be a simplicial complex on vertex set rns, ordered 1 ă ¨¨¨ ă n, and let i P N +with 0 ď i ď dimp∆q. For F,G P Fip∆q, let ℓF,G denote the entry of Lip∆q in row and column +corresponding to F and G, respectively. Then, Lip∆q is symmetric. Moreover: +(i) If i “ 0, then ℓF,G “ degUpFq if F “ G, ℓF,G “ ´1 if F Y G P Fi`1p∆q, and ℓF,G “ 0, +otherwise. +(ii) If i ą 0, then +ℓF,G “ +$ +’ +’ +’ +& +’ +’ +’ +% +degUpFq`i`1, +if F “ G, +1, +if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q similar +´1, +if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q dissimilar +0, +otherwise. +Note that if i “ 0 in the previous theorem, then L0p∆q coincides with the classical Laplacian +matrix of the graph of ∆ (from graph theory). +2.2. (Lattice) polytopes. A polytope P is the convex hull of finitely many points in Rd. If +dimP “ k, we call P a k-polytope. A linear inequality a⊺x ď b for a P Rd and b P R is called a +valid inequality for P if a⊺y ď b for all y P P. A (proper) face of P is a (non-empty) set of the +form PXtx P Rd : a⊺x “ bu for some valid inequality a⊺x ď b with a ‰ 0. Faces of dimension +0, dimP´2 and dimP´1 are called vertices, ridges and facets, respectively. We use V pPq and +FpPq to denote the set of vertices and facets of P, respectively. A valid inequality a⊺x ď b is + +4 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +facet-defining if F “ PXtx P Rd : a⊺x “ bu for some F P FpPq. The facet-ridge graph GpPq +of P is the graph on vertex set FpPq where tF,Gu is an edge if and only if F and G intersect +in a ridge. If V pPq Ď Zd, P is called a lattice polytope. Two lattice polytopes P, Q Ď Rd +are unimodular equivalent, denoted as P – Q, if there exist a unimodular matrix U P Rdˆd +and a vector b P Zd such that U ¨ P ` b “ Q. We use ∆d to denote the standard d-simplex, +i.e., ∆d “ convtt0u Y tei P Rd : i P rdsuu, where e1,...,ed denote the standard unit vectors. +A polytope P is simplicial if all of its facets are simplices. The normalized volume of a d- +dimensional lattice polytope P Ď Rd is given by nvolpPq “ d!¨volpPq, where volpPq denotes the +usual Euclidean volume. A lattice d-simplex ∆ with normalized volume 1 is called unimodular. +In this case, ∆ – ∆d. A polytope P is reflexive if P “ tx P Rd : Ax ď 1u for an integral matrix +A, where 1 denotes the all ones vector. In this case, 0 is the unique interior lattice point of P. +A triangulation T of a lattice d-polytope P is a subdivision into lattice simplices of dimension +at most d. We denote the set of vertices in T by V pT q. A triangulation is unimodular if all its +simplices are. T is called regular if there exists a height function ωP : V pT q Ñ R such that T +is the projection of the lower envelope of the convex hull of tpv,ωPpvqq : v P V pT qu Ď Rd`1 +to the first d coordinates. We note that every triangulation is in particular a simplicial complex. +Let P Ď Rd be a lattice d-polytope. Ehrhart [14] proved that the number of lattice points in the +n-th dilation of P, i.e., |nPXZd| is given by a polynomial EPpnq of degree d in n for all integers +n ě 0. The Ehrhart series of P is +ÿ +ně0 +EPpnqtn “ +h˚pP;tq +p1´tqd`1 “ h˚ +0pPq`h˚ +1pPqt `¨¨¨`h˚ +s pPqts +p1´tqd`1 +, +where h˚pP;tq P Zrts is a polynomial of degree at most d, called h˚-polynomial of P. The vector +h˚pPq “ ph˚ +0pPq,...,h˚ +s pPqq is called h˚-vector of P. We will often omit P from the notation and +just write h˚ “ ph˚ +0,...,h˚ +s q if P is clear from the context. By [29, Theorem 2.1], it is well- +known that h˚ +i pPq is non-negative for all i. If P admits a unimodular triangulation T , then +h˚pPq “ hpT q [29, Corollary 2.5]. Moreover, if T is a regular unimodular triangulation of P, +then +h˚ +tpd`1q{2upPq ě ¨¨¨ ě h˚ +d´1pPq ě h˚ +dpPq +[2, Theorem 1.3]. It was shown by Hibi in [20] that a lattice d-polytope P Ď Rd is reflexive (up +to unimodular equivalence) if and only if P contains a unique interior lattice point, and h˚pPq is +palindromic, i.e., h˚ +i pPq “ h˚ +d´ipPq for all 0 ď i ď td{2u. +3. LAPLACIAN MATRICES OF BOUNDARIES OF SIMPLICES +In this section we investigate basic properties of the Laplacian matrix of the boundary of a +simplex that will be useful for deriving properties of the corresponding Laplacian polytopes in +Section 4. +We start with an easy general statement. +Lemma 3.1. Let ∆ be a d-dimensional simplicial complex. Then +rankLdp∆q “ fdp∆q´dimQ Hdp∆;Qq. +Proof. We have the following chain of equalities: +rankLdp∆q “ rankpB⊺ +dBdq “ fdp∆q´dimQ kerpB⊺ +dBdq “ fdp∆q´dimQ kerpBdq, +where the last equality follows from the fact that kerpBdq “ kerpB⊺ +dBdq. Since dim∆ “ d, we also +have Hdp∆;Qq “ kerpBdq, which shows the claim. +□ + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +5 +In the following, we let σd`1 “ 2rd`2s be the pd `1q-simplex and we use Bpσd`1q to denote +its boundary, i.e., Bpσd`1q “ σd`1ztrd ` 2su. Let Fi “ rd ` 2sztd ` 3 ´ iu for 1 ď i ď d ` 2 +and order the columns and rows of LdpBpσd`1qq according to F1,...,Fd`2. We first provide an +explicit description of the d-th Laplacian matrix in this case. +Theorem 3.2. Let ∆ “ Bpσd`1q. Then, Ldp∆q P Zpd`2qˆpd`2q, L0p∆q “ +ˆ +0 +0 +0 +0 +˙ +and, for +d ě 1, 1 ď i, j ď d `2, we have +Ldp∆qi j “ +# +d `1, +if i “ j, +p´1qi` j´1, +otherwise. +Proof. Since fdpBpσd`1qq “ d `2, we have Ldp∆q P Zpd`2qˆpd`2q. +Assume d “ 0. As B0 is the zero map, the statement is immediate. +Now let d ě 1. Since dim∆ “ d, it follows that degUpFq “ 0 for any d-face F of ∆. Using +Theorem 2.1 this implies that Ldp∆qii “ d `1 for all 1 ď i ď d `2. +Now, let i ‰ j. +Since Ldp∆q is symmetric, we can assume that i ă j. +Fi and Fj have +the common lower simplex Fi X Fj “ rd ` 2sztd ` 3 ´ i,d ` 3 ´ ju ‰ H. By Equation (2.2), +eFiXFj appears with sign p´1qd`3´ j in Bdperd`2sztd`3´iuq and it appears with sign p´1qd`2´i +in Bdperd`2sztd`3´ juq. These signs coincide, meaning that Fi X Fj is a similar common lower +simplex of Fi and Fj, if and only if i` j is odd. The claim follows from Theorem 2.1. +□ +The next lemma will be crucial for determining the dimension of the Laplacian polytope of +Bpσd`1q in Lemma 4.6. +Lemma 3.3. Let ∆ “ Bpσd`1q. Then, Ldp∆q has rank d ` 1 and every pd ` 1q-element subset +of the columns (resp. rows) of Ldp∆q is linearly independent. +Proof. The first statement follows from Lemma 3.1 and the fact that Hdp∆;Qq “ Q. Let 1 ď i ď +d `2. Let Ai be the pd `1qˆpd `1q-matrix obtained from Ldp∆q by removing the i-th row and +column. By definition, Ai “ Ldp∆ztFiuq. Since Hdp∆ztFiuq,Qq “ 0, this matrix has full rank. +As adding any extra row or column to Ai does not change the rank, the claim follows. +□ +Lemma 3.4. Let ∆ “ Bpσd`1q. Then +rank +ˆ +Ldp∆q +1¨¨¨1 +˙ +“ +# +d `1, +if d is even, +d `2, +if d is odd. +Proof. First assume that d is even. We define λ “ pλ1,...,λd`2q⊺ P Rd`2 by +λj “ +# +0, +if j is odd, +2 +d`2, +if j is even. +Using Theorem 3.2 it is straight-forward to verify that Ldp∆q ¨ λ “ 1 which, combined with +Lemma 3.3 shows the claim. +Now let d be odd and assume by contradiction that rank +ˆ +Ldp∆q +1¨¨¨1 +˙ +ă d `2. Lemma 3.1 and +Lemma 3.3 imply that rank +ˆ +Ldp∆q +1¨¨¨1 +˙ +“ rankLdp∆q. Hence, there exists λ “ pλ1,...,λd`2q⊺ P +Rd`2, such that Ldp∆q¨λ “ 1. Let Ldp∆qrd`1s be the matrix obtained from Ldp∆q by deleting +the last row. Then, we also have Ldp∆qrd`1s ¨ λ “ 1 and it follows from Lemma 3.3 that, up + +6 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +to the choice of the last coordinate λd`2, the vector λ is unique. Indeed, a direct computation +shows that, if λd`2 “ µ for some µ P R, then we must have +(3.1) +λ j “ +$ +’ +& +’ +% +pd`2q¨µ`1 +d`2 +, +if j is odd, +´pd`2q¨µ´1 +d`2 +, +if j is even. +However, denoting by rd`2 the last row of Ldp∆q, it holds that rd`2 ¨λ “ 0 ‰ 1, which yields a +contradiction. +□ +4. GENERAL PROPERTIES OF LAPLACIAN POLYTOPES +The goal of this section is to generalize Laplacian simplices – as introduced and studied in +[6, 24] – that are associated to a graph to arbitrary simplicial complexes and their Laplacian +matrices. After stating some basic general properties of what we call Laplacian polytopes, we +focus on boundaries of simplices and their highest Laplacians. +In the following, given a matrix M, we use convpMq to denote the polytope given by the +convex hull of the columns of M. +Definition 4.1. Let ∆ be a d-dimensional simplicial complex on rns, ordered 1 ă ¨¨¨ ă n, and +let 0 ď k ď d. The k-th Laplacian polytope of ∆ is defined as the convex hull of the columns of +Lkp∆q, i.e., +Ppkq +∆ +– convpLkp∆qq Ď R fkp∆q. +We want to remark that the 0-th Laplacian polytope of a simplicial complex coincides with +the Laplacian simplex of its 1-skeleton, as defined in [6]. The next example shows that different +orderings of the vertex set of ∆ may result in polytopes of different dimensions. +Example 4.2. Let G be the 4-cycle on r4s with EpGq “ t12,23,34,14u. If the vertices of G are +ordered 1 ă 2 ă 3 ă 4, then Pp1q +G +is a 3-simplex. If the vertices of G are ordered 1 ă 2 ă 4 ă 3, +then Pp1q +G +is a 2-dimensional rectangle. +Example 4.3. Pp2q +Bpσ3q is given by the convex hull of the columns of the following matrix: +L2pBpσ3qq +“ +¨ +˚ +˚ +˝ +3 +1 +´1 +1 +1 +3 +1 +´1 +´1 +1 +3 +1 +1 +´1 +1 +3 +˛ +‹‹‚. +It will follow from Lemma 4.10 that Pp2q +Bpσ3q is unimodular equivalent to the square in R2 with +vertices p1,´1q,p´1,1q,p3,1q and p1,3q. +We start by showing that every column of Lkp∆q yields a vertex of Ppkq +∆ . +Proposition 4.4. Let ∆ be a d-dimensional simplical complex and 0 ď k ď d an integer. Then, +Ppkq +∆ +has fkp∆q many vertices. +Proof. Set m :“ fkp∆q and let vpiq denote the i-th column of Lkp∆q. We assume by contradiction +that there exists 1 ď i ď m, a set S Ď rmsztiu and λj P R with λj ą 0 and ř +jPS λ j “ 1 such that +vpiq “ ř +jPS λ jvpjq. Setting λ j “ 0 if j R S Y tiu and λi “ ´1, we see that λ :“ pλ1,...,λmq⊺ P +kerpLkp∆qq and hence λ P kerpBkq by [25, Corollary 1.3.1]. Let wpℓq denote the ℓ-th column of +Bk. If wpiq +ℓ “ 1, then, since wpjq +ℓ +P t´1,0,1u, λ j ą 0 and ř +jPS λj “ 1, we must have wpjq +ℓ +“ 1 + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +7 +for all j P S. By the same reasoning, it follows that wpjq +ℓ +“ ´1 for all j P S if wpiq +ℓ “ ´1. As all +columns of Bk have the same number of non-zero entries, we conclude wpiq “ wpℓq for all ℓ P S, +which is a contradiction. +□ +The next proposition gives a sufficient criterion for Ppdim∆q +∆ +being a simplex. +Proposition 4.5. Let ∆ be a d-dimensional simplicial complex. If Hdp∆;Qq “ 0, then Ppdim∆q +∆ +is +an pfdp∆q´1q-simplex. +Proof. Lemma 3.1 implies that rankLdp∆q “ fdp∆q. Consequently, the columns of Ldp∆q are +linearly independent which shows the claim. +□ +In the following, we focus on the d-th Laplacian polytope of Bpσd`1q. To simplify notation, +we set PBpσd`1q “ Ppdq +Bpσd`1q. We use spiq to denote the i-th column of LdpBpσd`1qq. Moreover, +given a subset S Ď rd ` 2s, we denote by LdpSq the matrix obtained from LdpBpσd`1qq by +deleting the rows with indices in S. +Combining Lemma 3.4 and [17, p. 4] the following formula for the dimension of PBpσd`1q is +immediate. +Lemma 4.6. Let ∆ “ Bpσd`1q. Then, +dimP∆ “ +# +d, +if d is even, +d `1, +if d is odd. +The previous statement together with Proposition 4.4 allows us to conclude: +Corollary 4.7. Let d P N with d ě 1 and ∆ “ Bpσd`1q. Then, P∆ has d`2 vertices. In particular, +P∆ is a pd `1q-simplex, if d is odd. +Corollary 4.7 trivially implies that PBpσd`1q is a simplicial polytope if d is odd. The same +statement turns out to be true for d even. +Theorem 4.8. PBpσd`1q is simplicial for every d P N. +Proof. Let ∆ “ Bpσd`1q. If d is odd, then the claim is trivially true by Corollary 4.7. +Now, let d be even. If d “ 0, then P∆ is just the origin and as such simplicial. Let d ě 2 and +let F be the vertices of a facet of P∆. Combining Lemma 4.6 and Corollary 4.7 it follows that +d ď |F| ď d `1. If, by contradiction, |F| “ d `1, then Lemma 3.3 implies that the convex hull +of F is d-dimensional, i.e., F cannot be a facet. Consequently, F is a simplex, which finishes +the proof. +□ +As, by Lemma 4.6, the Laplacian polytope of Bpσd`1q is never full-dimensional, our next +goal is to construct a polytope that is unimodular equivalent to PBpσd`1q and full-dimensional +with respect to its ambient space. We first need to introduce some further notation. +We let 1even and 1odd denote the 0´1-vectors in Rd`2 whose even and odd entries are equal +to 1, respectively. Given these definitions, we can easily compute the affine hull of PBpσd`1q. +Lemma 4.9. Let d P N with d ě 1 and ∆ “ Bpσd`1q. +affpP∆q “ +#␣ +x P Rd`2 : p1odd ´1evenq⊺ ¨x “ 0 +( +, +if d is odd, +␣ +x P Rd`2 : 1⊺ +odd ¨x “ 1⊺ +even ¨x “ d`2 +2 +( +, +if d is even. +Proof. By Lemma 4.6, it is enough to show that all vertices of P∆ lie in the specified subspaces +of dimension d `1 and d, respectively. This can be seen by a direct computation. +□ + +8 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +The next lemma gives the desired unimodular equivalent polytopes. +Lemma 4.10. Let d P N. The polytope PBpσd`1q is unimodular equivalent to convpLdpt1uqq and +convpLdpt1,2uqq if d is odd and even, respectively. +Proof. Define matrices A,B P Zpd`2qˆpd`2q as follows +A “ +¨ +˚ +˚ +˝ +1⊺ +odd ´1⊺ +even +0 +... +Ed`1 +0 +˛ +‹‹‚ +and +B “ +¨ +˚ +˚ +˚ +˚ +˝ +1⊺ +odd +1⊺ +even +0 +0 +... +... +Ed +0 +0 +˛ +‹‹‹‹‚ +, +where Ed and Ed`1 denote identity matrices. Note that A and B are unimodular. By Lemma 4.9, +we conclude that +A¨PBpσd`1q “ t0uˆconvpLdpt1uqq, +if d is odd and +B¨PBpσd`1q “ tppd `2q{2,pd `2q{2quˆconvpLdpt1,2uqq, +if d is even. This finishes the proof. +□ +In the following, we use rPBpσd`1q to denote the unimodular equivalent polytope to PBpσd`1q +as constructed in Lemma 4.10. By abuse of notation, we will also refer to rPBpσd`1q as d-th +Laplacian polytope of Bpσd`1q. We also want to remark that, if d is odd, we have the following, +easy-to-show containment relation: rPBpσd`1q Ď rPBpσd`2q. +5. THE FACET DESCRIPTION AND THE COMBINATORIAL TYPE OF PBpσd`1q +While, for odd d, we have already seen that rPBpσd`1q is a simplex, the goal of this section is +to determine the combinatorial type of PBpσd`1q if d is even. To reach this goal, we will first +provide a complete irredundant facet description of PBpσd`1q. +We fix some notation. Let bpℓq denote the vertex of rPBpσd`1q, that is given by the ℓ-th column of +Ldpt1,2uq. By Theorem 3.2, we have bpℓq +k +“ d `1 if k “ ℓ´2 and bpℓq +k +“ p´1qk`ℓ´1, otherwise. +Proposition 5.1. Let d ě 2 be even. Then the following inequalities are facet-defining and +irredundant for rPBpσd`1q +(i) 1⊺ ¨x ď d `2, +(ii) 1⊺ +odd ¨x´xi ď d`2 +2 , where i P rds is even, +(iii) 1⊺ +even ¨x´xj ď d`2 +2 , where j P rds is odd, +(iv) xi `xj ě 0, where 1 ď i ă j ď d such that i` j is odd. +Moreover, the vertices, that attain equality in (i)–(iv), are given by the sets tbpℓq : 3 ď ℓ ď d`2u, +tbpℓq : ℓ P rd`2szt1,i`2uu, tbpℓq : ℓ P rd`2szt2, j`2uu and tbpℓq : ℓ P rd`2szti`2, j`2uu, +respectively. +Proof. We first consider the inequality in (i). If ℓ P t1,2u, then bpℓq P t´1,1ud with alternating +entries and hence 1⊺ ¨ bpℓq ă d ` 2. Let 3 ď ℓ ď d ` 2. As d is even, it follows from above +that bpℓq has one entry equal to d ` 1, d +2 entries equal to 1 and d +2 ´ 1 entries equal to ´1. This +implies 1⊺ ¨bpℓq “ d `2. Hence, the inequality in (i) defines a facet, whose vertices are given by +tbpℓq : 3 ď ℓ ď d `2u, where we use that the affine hull of the latter set is pd ´1q-dimensional +by Lemma 3.3. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +9 +Similarly, it is straightforward to verify that the inequalities in (ii)–(iv) are valid for rPBpσd`1q +and that the given sets of vertices are the ones attaining equality. As those all differ and their +affine hulls all have dimension d ´1, it follows that the inequalities are irredundant. +□ +For the sake of completeness we add the description of the facets of rPBpσd`1q for d odd. +Remark 5.2. If d ě 3 is odd, using Theorem 3.2 it is not hard to see, that the following +inequalities are facet-defining for rPBpσd`1q +(i) 1⊺ ¨x ď d `2, +(ii) 2¨1⊺ +odd ¨x´xi ď d`2 +2 , where i P rd `1s is even, +(iii) 2¨1⊺ +odd ¨x`xj ď d `2, where j P rds is odd. +It is easy to verify that these inequalities are irredundant and as, by Lemma 4.6, rPBpσd`1q is a +simplex, they provide the complete facet description of rPBpσd`1q. We omit an explicit proof since +this description will not be needed. +We state the first main result of this section. +Theorem 5.3. For d even, rPBpσd`1q is completely described by the inequalities in Proposition 5.1. +Moreover, this description is irredundant. In particular, rPBpσd`1q has pd`2q2 +4 +many facets. +Proof. We let Ă +F denote the set of facets of rPBpσd`1q, provided by Proposition 5.1 and we write +G Ă +F for the subgraph of the facet-ridge graph of rPBpσd`1q that is induced on vertex set Ă +F. It +follows from Theorem 4.8, that the facet-ridge graph of rPBpσd`1q is d-regular and connected. +Since any d-regular subgraph does not have a proper d-regular subgraph, for the first statement, +it suffices to show that G Ă +F is d-regular. +Since G Ă +F is a subgraph of GprPBpσd`1qq, its maximal degree is at most d. Hence, to show the +claim, it suffices to show that |EpG Ă +Fq| “ +d¨|VpG Ă +F q| +2 +. +We first count the vertices of G. Using Proposition 5.1, we get that +(5.1) +|VpG Ă +Fq| “ 1` d +2 ` d +2 ` +ˆd +2 +˙2 +“ pd `2q2 +4 +, +Here, the last term in the middle comes from the fact that the inequalities in (iv) are indexed by +sets ti, ju where i P t2ℓ : ℓ P rd +2su and j P t2ℓ´1 : ℓ P rd +2su. +It remains to count the number of edges of G Ă +F. In the following, we identify a facet in Ă +F +with its set of vertices. Given this, we use the following short hand notation for the different +types of facets in Ă +F. +(i) F “ tbpℓq : 3 ď ℓ ď d `2u; +(ii) Ei “ tbpℓq : ℓ P rd `2szt1,i`2uu, where i P rds is even; +(iii) Oj “ tbpℓq : ℓ P rd `2szt2, j `2uu, where j P rds is odd; +(iv) Fk,m “ tbpℓq : ℓ P rd `2sztk `2,m`2uu for 1 ď k ă m ď d such that k `m is odd. +We immediately get that +(a) |F XEi| “ |F XOj| “ d ´1 for all even i P rds and all odd j P rds; +(b) |F XFk,m| “ d ´2 for all 1 ď k ă m ď d; +(c) |Ei XE j| “ d ´1 for all odd i, j P rds with i ‰ j; +(d) |Ei XO j| “ d ´2 for all even i P rds and all odd j P rds; +(e) |Ei XFk,m| “ d ´1 iff i P tk,mu, i even, k `m odd, and |Ei XFk,m| “ d ´2, otherwise; +(f) |Oi XO j| “ d ´1 for all even i, j P rds with i ‰ j; + +10 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +(g) |O j XFk,m| “ d ´1 iff j P tk,mu, j odd, k `m odd, and |Oj XFk,m| “ d ´2, otherwise. +(h) |Fi, j XFk,m| “ d ´1 iff |ti, j,k,mu| “ 3 and |Fi,j XFk,m| “ d ´2, otherwise. +Since edges of G Ă +F are given by tuples of facets intersecting in d ´1 vertices, we get d edges in +(a), 0 edges in (b) and (d), +`d{2 +2 +˘ +edges in each of (c) and (f), +`d +2 +˘2 edges in each of (e) and (g) +and 2¨ d +2 ¨ +`d{2 +2 +˘ +edges in (h). This yields +|EpG Ă +Fq| “ d `2¨ +ˆd{2 +2 +˙ +`2¨ +ˆd +2 +˙2 +`d ¨ +ˆd{2 +2 +˙ +“ dpd `2q2 +8 +“ +d ¨|VpG Ă +Fq| +2 +. +It follows that G Ă +F is d-regular. The In particular-statement follows from (5.1). +□ +The previous theorem allows us to determine the combinatorial type of rPBpσd`1q if d is even. It +is well-known (see e.g., [17, Section 6.1]) that there are only finitely many combinatorial types +of simplicial d-polytopes with d ` 2 vertices. More precisely, any simplicial d-polytope with +d ` 2 vertices is obtained as the convex hull of a d-simplex T d and a vertex v that is beyond +k facets of T d, where 1 ď k ď d ´ 1. It is easily seen that the combinatorial type of such a +polytope only depends on k. Following Gr¨unbaum, we use T d +k to denote the corresponding +combinatorial type. Given that we know the number of facets of rPBpσd`1q (see Theorem 5.3), we +can immediately determine its combinatorial type. +Theorem 5.4. Let d be even. Then rPBpσd`1q is of combinatorial type T d +d +2 . In particular, rPBpσd`1q +is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices. +Proof. By Theorem 5.3, rPBpσd`1q has pd`2q2 +4 +facets. Using [17, Section 6.1, Theorem 2], it +follows that this number has to be equal to +ˆd `2 +2 +˙ +´ +ˆk `1 +2 +˙ +´ +ˆd `1´k +2 +˙ +, +where rPBpσd`1q is of combinatorial type T d +k . Solving for k yields k “ d +2. The second statement +follows from [17, Section 6.1, Theorem 1]. +□ +We remark that from the previous theorem, we also get a precise formula for the f- and +h-vector of rPBpσd`1q (see, e.g., [17]). +Remark 5.5. Given the precise description of the facets from the proof of Theorem 5.3, it is not +hard to write down a shelling order for rPBpσd`1q (d even). Namely, one particular shelling is +given by +F,E2,E4,...,Ed,O1,O3,...,Od´1,F1,2,F1,4,...,F1,d,F2,3,F2,5,...,F2,d´1,...,Fd´1,d. +6. REGULAR UNIMODULAR TRIANGULATIONS AND h˚-VECTORS +This section is divided into two parts. The goal of the first is to prove Theorem A, namely, +that rPBpσd`1q admits a regular unimodular triangulation . +As a byproduct we will also be +able to compute the normalized volume rPBpσd`1q. In the second part, we provide the proof +of Theorem B. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +11 +FIGURE 1. rPBpσ3q and its interior polytope QBpσ3q translated to the origin. +6.1. Triangulations through interior polytopes. If d is even, one of our main tools towards +the formulated goal is the so-called interior polytope QBpσd`1q of rPBpσd`1q, defined as follows: +QBpσd`1q “ conv +´ +rPBpσd`1qzB +´ +rPBpσd`1q +¯ +XZd¯ +. +Figure 1 depicts rPBpσ3q and its interior polytope QBpσ3q, both translated to the origin. +Surprisingly, it turns out that PBpσd`1q and its interior polytope are combinatorially equivalent. +More precisely, the following stronger statement is true: +Theorem 6.1. Let d P N be even. Then the following statements hold: +(a) The complete and irredundant facet description of QBpσd`1q is given by: +(i) 1⊺ ¨x ď d `1, +(ii) 1⊺ +odd ¨x´xi ď d +2 for even i P rds, +(iii) 1⊺ +even ¨x´xj ď d +2 for odd j P rds, +(iv) xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd. +(b) QBpσd`1q ´1 is reflexive. In particular, 1 is the unique interior lattice point of QBpσd`1q. +(c) 2¨ +´ +QBpσd`1q ´1 +¯ +“ rPBpσd`1q ´1. +Proof. We let Q “ rPBpσd`1q ´1. The vertices of Q are given by upℓq :“ bpℓq `1 for 1 ď ℓ ď d `2. +It is immediate that all coordinates of upℓq are divisible by 2. Hence, 1 +2Q is a lattice polytope. +Using Theorem 5.3, it follows that the facets of 1 +2Q are given by +‚ 1⊺ ¨x ď 1, +‚ 1⊺ +odd ¨x´xi ď 1 for even i P rds, +‚ 1⊺ +even ¨x´xj ď 1 for odd j P rds, +‚ xi `xj ě ´1 for 1 ď i ă j ď d such that i` j is odd, +which shows that 1 +2Q is reflexive. It remains to show that 1 +2Q`1 “ QBpσd`1q. Since 1 +2Q`1 is +a lattice polytope, it follows that 1 +2Q ` 1 Ď QBpσd`1q. For the other inclusion it suffices to note +that the facets of 1 +2Q and rPBpσd`1q are parallel and that they have distance +1 +? +d, +? +2 +? +d`2, +? +2 +? +d`2 and +1 +? +2 to each other for facets of the form in (i), (ii), (iii) and (iv), respectively. This implies that +there is no lattice point in rPBpσd`1qzpp1 +2Q`1qYB rPBpσd`1qq and hence QBpσd`1q Ď 1 +2Q`1. +□ + +m +312 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +We define vectors cp1q,...,cpd`2q P Rd +by cpℓq +k +“ d`2 +2 +if k “ ℓ ´ 2 and cpℓq +k +“ +maxp0,p´1qk`ℓ´1q, otherwise. Combining Proposition 5.1 and Theorem 6.1 (c), we get the +following description of the vertices of QBpσd`1q and its facets. +Corollary 6.2. The vertices of QBpσd`1q are the vectors cp1q,...,cpd`2q. Moreover, the vertices, +that attain equality in Theorem 6.1 (i)–(iv), are given by the sets tcpℓq : 3 ď ℓ ď d ` 2u, +tcpℓq : ℓ P rd `2szt1,i`2uu, tcpℓq : ℓ P rd `2szt2, j`2uu and tcpℓq : ℓ P rd `2szti`2, j`2uu, +respectively. +We now recall several definitions and facts concerning regular unimodular triangulations (see +[18, Subsection 2.3.2.] for more on these topics). +Given full-dimensional polytopes P Ď Rd and P1 Ď Rd1 of positive dimension, their join P˚P1 +is the pd `d1 `1q-dimensional polytope defined by +Pˆt0d1uˆt0u Y t0duˆP1 ˆt1u. +The next statement, which is well-known, will be crucial for the construction of a regular +unimodular triangulation of rPBpσd`1q. +Theorem 6.3. Let P Ď Rd and P1 Ď Rd1 be polytopes of dimension d and d1, respectively. Let +S “ tSi : i P rnsu and S1 “ tS1 +j : j P rmsu be triangulations of P and P1, respectively, where Si +and S1 +j denote the full-dimensional cells. If both S and S1 are regular and unimodular, then +T “ +␣ +Si ˚S1 +j : i P rns, j P rms +( +is a regular unimodular triangulation of P˚P1. +We will also make use of the following statement, see [18, Theorem 4.8]. +Theorem 6.4. If P has a (regular) unimodular triangulation T , then so has any dilation cP, +where c is a positive integer. +A well-studied subdivision, which is related to the Veronese construction in algebra but also +appears in topology [7, 13, 16, 8], is the so-called rth edgewise subdivision of a simplicial +complex. In the following, we review this definition for the special case that ∆ is the pn ´ +1q-dimensional simplex on vertex set V “ te1,e2,...,enu Ď Rn. For a positive integer r, let +Ωr “ tpi1,...,inq P Nn : i1 ` i2 ` ¨¨¨ ` in “ ru denote the set of lattice points r∆ X Zn. For +x “ px1,...,xnq P Zn, we define +ϕpxq :“ px1,x1 `x2,...,x1 `¨¨¨`xnq P Rn. +The rth edgewise subdivision of ∆ is the simplicial complex esdrp∆q on vertex set Ωr, for which +F Ď Ωr is a face if for all x,y P F +ϕpxq´ϕpyq P t0,1un +or +ϕpyq´ϕpxq P t0,1un. +By definition, the geometric realization of the rth edgewise subdivision of ∆ gives a lattice +triangulation of r∆. It is known that this triangulation is regular [8, Proposition 6.4.], which +is also unimodular since all maximal simplices have normalized volume 1. In the following, +we will use esdrp∆q to denote both, the triangulation as a simplicial complex and its geometric +realization. Given any pn ´ 1q-dimensional unimodular simplex Γ Ď Rn, esdrp∆q, naturally +induces a regular unimodular triangulation of rΓ (by applying the corresponding unimodular +transformation). +Slightly abusing notation, we will refer to this triangulation as edgewise +subdivision of Γ or even of rΓ, denoted esdrpΓq. Moreover, the restriction of esdrpΓq to any +face F P Γ equals esdrpFq as a simplicial complex and as geometric realization. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +13 +Example 6.5. Figure 2 depicts the 3rd edgewise subdivision of the 2-dimensional simplex ∆2 :“ +convpp0,0q,p1,0q,p0,1qq as triangulation of 3∆2. The vertex labels correspond to the vertex +labels from the original definition and not the lattice points. +FIGURE 2. The triangulation of 3¨∆2 given by esd3p∆2q. +We now outline our strategy to show that rPBpσd`1q has a regular unimodular triangulation. +We first prove that rPBpσd`1q and the facets of QBpσd`1q, if d is odd and even, respectively, are +unimodular equivalent to joins of dilated standard simplices. If d is odd and even, we can hence +triangulate rPBpσd`1q and facets of QBpσd`1q as join of edgewise subdivisions. If d is odd, the claim +follows by Theorem 6.3. If d is even, we next show that these triangulations are consistent on +intersections of facets. By coning with 1, we get a unimodular triangulation of QBpσd`1q (see +Theorem 6.1 (d)) and hence of rPBpσd`1q by Theorem 6.4. The regularity follows by using that +the triangulation is regular on single facets and that each facet is triangulated in the same way. +The next statement yields the first step in the outlined strategy. +Proposition 6.6. +(a) Let d ě 2 be even and F P F +´ +QBpσd`1q +¯ +. Then +F – +ˆd `2 +2 +∆ d´2 +2 ´1 d´2 +2 +˙ +˚ +ˆd `2 +2 +∆ d´2 +2 ´1 d´2 +2 +˙ +. +(b) Let d ě 1 be an odd integer. Then +rPBpσd`1q – +` +pd `2q∆ d`1 +2 ´2¨1 +˘ +˚ +` +pd `2q∆ d´1 +2 ´2¨1 +˘ +. +Proof. The proof of (a) is divided into four cases, according to the four classes of facets from +Theorem 6.1 (a). +Let F “ tx P Rd : 1⊺ ¨ x ď d ` 1u. By Corollary 6.2, the vertices of F are cp3q,...,cpd`2q. +We now consider the matrix A, whose ℓ-th column equals cp2ℓ`1q if 1 ď ℓ ď d +2 and cp2ℓ`2´dq if +d +2 `1 ď ℓ ď d. If we reorder the rows of A, by taking first the rows with odd index and then the +ones with even index, increasingly, we obtain a matrix S, which looks as follows: +S “ +˜ d`2 +2 ¨E d +2 +1 d +2 ˆ d +2 +1 d +2 ˆ d +2 +d`2 +2 ¨E d +2 +¸ +, + +4 +(0, 0, 3) +(1, 0,2) +(0,1,2) +(2, 0, 1) +(1,1,1 +(0, 2, 1) +(3, 0, 0) +(2,1,0) +(1, 2, 0) +(0,3, 0) +-114 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +where 1kˆk denotes the pk ˆ kq-matrix with all entries equal to 1. Clearly, F – convpSq. Let +E1 +k P Zpk´1qˆk be the pk ˆkq-identity matrix with its first row removed and let +U “ +¨ +˚ +˚ +˚ +˝ +E1 +d +2 +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +E1 +d +2 +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +˛ +‹‹‹‚P Zdˆd. +It is easily seen that U is unimodular and a direct computation shows that +U ¨pS´1dˆdq “ +¨ +˚ +˚ +˚ +˚ +˝ +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +˛ +‹‹‹‹‚ +, +where 0kˆk denotes the pk ˆkq-matrix with all entries equal to 0 and M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +denotes the matrix whose columns are the vertices of d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 in the obvious order. +Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates +and by the definition of the join. We also note that the vertices of F corresponding to the vertices +of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d +2u and tcp2ℓq : 2 ď ℓ ď d +2 `1u. +Similarly, one can show that for the facets defined by +‚ 1⊺ +odd ¨x´xi ď d +2, where i P rds is even, +‚ 1⊺ +even ¨x´xj ď d +2, where j P rds is odd, +‚ xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd, +respectively, the vertices +‚ tcp2ℓ`1q : 1 ď ℓ ď d +2u and tcp2ℓq : 1 ď ℓ ď d +2 `1,ℓ ‰ i`2 +2 u, +‚ tcp2ℓ`1q : 0 ď ℓ ď d +2,ℓ ‰ j`1 +2 u and tcp2ℓq : 2 ď ℓ ď d +2 `1,ℓ ‰ i`2 +2 u, +‚ tcp2ℓ`1q : 0 ď ℓ ď d +2,ℓ ‰ j`1 +2 u and tcp2ℓq : 1 ď ℓ ď d +2 `1,ℓ ‰ i`2 +2 u, +respectively, correspond to the vertices of the dilated simplices. The rather technical proofs can +be found in the appendix. Similarly, (b) will be shown in the appendix. +□ +We recall and prove Theorem A. +Theorem A. PBpσd`1q has a regular unimodular triangulation for every integer d ě 0. +Proof. Since rPBpσd`1q and PBpσd`1q are unimodular equivalent, it suffices to show the statement +for rPBpσd`1q. First assume that d is odd. By Proposition 6.6 (b), we know that +rPBpσd`1q – +` +pd `2q∆ d`1 +2 ´2¨1 +˘ +˚ +` +pd `2q∆ d´1 +2 ´2¨1 +˘ +. +Since the pd ` 2qnd edgewise subdivision is a regular unimodular triangulation of the pd ` +2qnd dilation of any unimodular simplex (as well as of any translation), we conclude with +Theorem 6.3 that rPBpσd`1q has a regular unimodular triangulation. +Next assume that d is even. If d “ 0, rPBpσd`1q is just a point and there is nothing to show. +Let d ě 2. We construct a regular unimodular triangulation of the interior polytope QBpσd`1q. +By Proposition 6.6 every facet F of QBpσd`1q is unimodular equivalent to +(6.1) +ˆd `2 +2 +∆ d´2 +2 ´1 d´2 +2 +˙ +˚ +ˆd `2 +2 +∆ d´2 +2 ´1 d´2 +2 +˙ +. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +15 +By the same reasoning as for d odd, we can triangulate F as join of edgewise subdivisions of +unimodular simplices. In this way, we obtain regular unimodular triangulations of each facet +of QBpσd`1q. We now show that the union of these triangulations, yields a triangulation of the +boundary of QBpσd`1q. For this aim, let F and G be facets of QBpσd`1q and let T pFq and T pGq +be the considered triangulations. Let us further denote by Fi and Gi, where i P r2s, the vertex +sets corresponding to the vertex sets of the dilated (and translated) simplices in (6.1). It follows +from the end of the proof of Proposition 6.6 that (after possible renumbering) +pF1 YF2qXpG1 YG2q “ pF1 XF2qYpG1 XG2q. +This directly yields that the restrictions of T pFq and T pGq to F X G coincide: Indeed, they +are given as the join of the edgewise subdivisions of the dilated (and translated) simplices on +vertex sets F1 XF2 and G1 XG2. This shows that the union of the triangulations of the facets is +indeed a triangulation of the boundary of QBpσd`1q, which is, in particular, unimodular. Since, +by Theorem 6.1 (b), QBpσd`1q ´1 is reflexive, we can extend this triangulation to a unimodular +triangulation of QBpσd`1q by coning over the unique interior lattice point 1. In the following, we +call this triangulation T . +It remains to show that T is a regular triangulation. The previous paragraph implies that +the induced triangulations on facets QBpσd`1q are all regular and unimodular equivalent to each +other. In particular, there exists a simultaneous lifting function h yielding the triangulation of +an arbitrary facet. Fix a facet F and let T pFq be the induced triangulation on F. Since F is a +simplex, we can assume that hpvq “ 1 for any vertex v P F. Moreover, for any lattice point u in F, +that is not a vertex, we have hpuq ă 1, since otherwise u would not be a vertex of T pFq. Hence, +there exists a non-negative function g, whose values are bounded by 1, that vanishes on the +vertices of F such that h “ 1´g. Moreover, for any ε ą 0, hε “ 1´εg is also a lifting function +for F yielding T pFq. Finally, ignoring 1 and lifting all other lattice points in QBpσd`1q according +to the simultaneous lifting function hε, gives a lifting function such that the projection of the +lower envelope yields T on the boundary of QBpσd`1q and potentially additional faces in the +interior. Lifting 1 at height 0, gives a lifting of all lattice points of QBpσd`1q. If ε is sufficiently +small, one can guarantee that the triangulation obtained as the lower envelope is the cone with +1 over the boundary of the previous triangulation (ignoring 1) since potential interior faces that +we had seen before, do no longer lie in the lower envelope. +The claim follows by Theorem 6.1 (c) and Theorem 6.4. +□ +Analyzing the proof of Theorem A, we can compute the normalized volume of PBpσd`1q: +Corollary 6.7. The normalized volume of PBpσd`1q is pd `2qd. +Proof. We compute the normalized volume of rPBpσd`1q, which equals the one of PBpσd`1q, by +counting the number of maximal simplices in the unimodular triangulation T constructed in +the proof of Theorem A. +First assume that d is odd. We have seen that T is unimodular equivalent to +esdd`2 +´ +∆ d`1 +2 +¯ +˚esdd`2 +´ +∆ d´1 +2 +¯ +. +Since the rth edgewise subdivision of an m-simplex, has rm maximal simplices, it follows that +the number of maximal simplices in the constructed unimodular triangulation of rPBpσd`1q equals +pd `2q +d´1 +2 ¨pd `2q +d`1 +2 “ pd `2qd. +Let d be even. We first compute the normalized volume of QBpσd`1q. Combining Theorem 5.3 +and Theorem 6.1, it follows that QBpσd`1q has exactly pd`2q2 +4 +facets. By the proof of Theorem A + +16 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +each of these has a unimodular triangulation that is unimodular equivalent to +esd d`2 +2 +´ +∆ d´2 +2 +¯ +˚esd d`2 +2 +´ +∆ d´2 +2 +¯ +. +As in the case that d is odd, we conclude that each facet is triangulated into +`d`2 +2 +˘ d´2 +2 ¨ +`d`2 +2 +˘ d´2 +2 +“ +`d`2 +2 +˘d´2 many maximal simplices and hence QBpσd`1q has normalized volume pd`2qd +2d +. Since, +by Theorem 6.1 (c), rPBpσd`1q `1 “ 2¨QBpσd`1q, the claim follows. +□ +6.2. Unimodality and real-rootedness. The goal of this subsection is to prove Theorem B. +If d is even, then by the proof of Theorem A, QBpσd`1q has a regular unimodular triangulation. +Since it is also reflexive (after translation) by Theorem 6.1 (b), the next statement is immediate +from [9, Theorem 1] (see also [2, Theorem 1.3]): +Lemma 6.8. Let d be an even positive integer. Then h˚pQBpσd`1qq is symmetric and unimodal. +To show unimodality of h˚prPBpσd`1qq, if d is even, we need to analyze the change of the h˚- +vector under the second dilation of a polytope (cf., Theorem 6.1 (c)). Given a d-dimensional +lattice polytope P, it follows, e.g., from [7, Theorem 1.1] (see also [5, 22]) that +(6.2) +h˚ +i p2Pq “ +dÿ +j“0 +ˆd `1 +2i´ j +˙ +h˚ +jpPq. +We need the following technical but crucial lemma. +Lemma 6.9. Let i P N and rj :“ +` d`1 +2i`2´ j +˘ +´ +`d`1 +2i´ j +˘ +. Then for k P N, we have +´rr2i`2´ d`3 +2 s´k “ rt2i`2´ d`3 +2 u`k. +Proof. We set aj “ +` d`1 +2i`2´ j +˘ +and bj “ +`d`1 +2i´ j +˘ +. The claim follows if both +ar2i`2´ d`3 +2 s´k “ bt2i`2´ d`3 +2 u`k +and +br2i`2´ d`3 +2 s´k “ at2i`2´ d`3 +2 u`k +hold. Due to the symmetry of the binomial coefficient it suffices to show that +(i) p2i`2´r2i`2´ d`3 +2 s`kq`p2i´t2i`2´ d`3 +2 u´kq “ d `1 +(ii) p2i´r2i`2´ d`3 +2 s`kq`p2i`2´t2i`2´ d`3 +2 u´kq “ d `1. +It is obvious that (i) and (ii) are equivalent. The claim follows from direct computations. +□ +The next statement will be the key ingredient to show that h˚prPBpσd`1qq is unimodal. +Proposition 6.10. Let b “ pb0,...,bdq be a symmetric and unimodal sequence of non-negative +reals. Let c “ pc0,...,cdq be defined by +ci “ +dÿ +j“0 +ˆd `1 +2i´ j +˙ +b j. +Then, +c0 ď c1 ď ¨¨¨ ď ct d`1 +2 u. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +17 +Proof. We define r j as in Lemma 6.9. Note that rj ě 0 if and only if j ě 2i ` 2 ´ d`3 +2 . For +0 ď i ă d`1 +2 , we have +ci`1 ´ci “ +dÿ +j“0 +„ˆ +d `1 +2i`2´ j +˙ +´ +ˆd `1 +2i´ j +˙ȷ +b j “ +dÿ +j“0 +r jbj +“ +2p2i`2´ d`3 +2 q +ÿ +j“0 +r jbj ` +dÿ +j“2p2i`2´ d`3 +2 q`1 +r jbj +“ +r2i`2´ d`3 +2 s +ÿ +j“1 +rt2i`2´ d`3 +2 u` j +´ +bt2i`2´ d`3 +2 u` j ´br2i`2´ d`3 +2 s´ j +¯ +`r2i`2´ d`3 +2 b2i`2´ d`3 +2 ` +dÿ +j“2p2i`2´ d`3 +2 q`1 +r jbj, +where for the last equality, we use Lemma 6.9 and we set `r2i`2´ d`3 +2 b2i`2´ d`3 +2 +“ 0 if d is +even. Since b j ě 0 and rj ě 0 for j ě 2i` 2´ d`3 +2 , it follows that the single summand and the +sum in the last line of the above computation are both non-negative. Concerning the first sum, +the coefficients r2i`2´ d`3 +2 ` j are non-negative and therefore, in order to show non-negativity of +ci`1 ´ci, it suffices to show that for 1 ď j ď r2i`2´ d`3 +2 s, we have +bt2i`2´ d`3 +2 u` j ě br2i`2´ d`3 +2 s´ j. +This directly follows from the unimodality and symmetry of the sequence b if 2i`2´ d`3 +2 ` j ď +d`1 +2 . Assume 2i`2´ d`3 +2 ` j ą d`1 +2 . Since i ď d +2, we have +d `1 +2 +ă 2i`2´ d `3 +2 +` j ď d `2´ d `3 +2 +` j “ d `1 +2 +` j ď +Zd `1 +2 +^ +` j. +Using that b is symmetric and unimodal it follows that +bt2i`2´ d`3 +2 u` j ě bt d`1 +2 u` j “ bd´t d +2u´ j ě br2i`2´ d`3 +2 s´ j. +This shows the claim. +□ +We now recall and prove Theorem B: +Theorem B. +(a) h˚ ´ +PBpσd`1q;t +¯ +has only real roots if d P N is odd. +(b) h˚ ´ +PBpσd`1q +¯ +is unimodal with peak in the middle for every d P N. +Proof. Since PBpσd`1q has a regular unimodular triangulation T by Theorem A, we have +h˚pPBpσd`1qq “ hpT q. If d is odd, such a triangulation is given by +esdd`2 +´ +∆ d`1 +2 +¯ +˚esdd`2 +´ +∆ d´1 +2 +¯ +and its h-polynomial equals h +´ +esdd`2 +´ +∆ d`1 +2 +¯ +;t +¯ +¨ h +´ +esdd`2 +´ +∆ d´1 +2 +¯ +;t +¯ +. Since both factors +are real-rooted by [22, Corollary 4.4], so is h˚ ´ +PBpσd`1q;t +¯ +. + +18 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +Suppose that d is even. Combining Lemma 6.8, (6.2) and Proposition 6.10, we get that +h˚pPBpσd`1qq is increasing up to the middle, i.e., +h˚ +0pPBpσd`1qq ď h˚ +1pPBpσd`1qq ď ¨¨¨ ď h˚ +d +2 pPBpσd`1qq. +Since, by Theorem A, PBpσd`1q has a regular unimodular triangulation it follows by [2, Theorem +1.3] that h˚pPBpσd`1qq is decreasing beyond the middle, i.e., +h˚ +d +2 pPBpσd`1qq ě ¨¨¨ ě h˚ +dpPBpσd`1qq. +The claim follows. +□ +We would like to remark that even though the interior polytope QBpσd`1q has a symmteric +h˚-vector, this is not true for PBpσd`1q. +7. OPEN PROBLEMS +We end this article with some obvious directions for future research. +We have initiated the study of Laplacian polytopes Ppiq +∆ by studying the special case that ∆ is +the boundary of a pd ` 1)-simplex and i “ d. It is therefore natural to consider the following +very general problem. +Problem 7.1. Study geometric and combinatorial properties of Ppiq +∆ +for (classes of) simplicial +complexes and general 0 ď i ď dim∆. In particular: What is the normalized volume? When +do these polytopes have a regular unimodular triangulation? What properties do the h˚-vector +and the h˚-polynomial have? +In view of Proposition 4.5, a good starting point might be to study Ppdq +∆ +for simplicial d-balls, +since in this case we already know that Ppdq +∆ +is a simplex. As part of this problem, it might be +useful to consider how Laplacian polytopes change under certain operations on the simplicial +complex, e.g., deletion/contraction of vertices, taking links, connected sums, joins. We want to +remark that for i “ 1 we get Laplacian simplices as studied in [6] and [24]. +We have shown that PBpσd`1q has a regular unimodular triangulation by explicitly constructing +one. However, for more general classes of simplicial complexes, a better approach might be to +compute a Gr¨obner basis of the toric ideal. This gives rise to the following problem whose +solution would also contribute to Problem 7.1: +Problem 7.2. Describe a Gr¨obner basis of the toric ideal of Ppiq +∆ in terms of the combinatorics +of ∆. When does there exist a squarefree Gr¨obner basis (giving rise to a regular unimodular +triangulation)? +We want to emphasize that the Laplacian polytope depends on the ordering of the vertices of +∆ (see Example 4.2). It is therefore natural to ask the following question: +Question 7.3. Which orderings yield (up to unimodular or combinatorial equivalence) the same +Laplacian polytope? How many equivalence classes are there? +Apart from these more general problems, there are several open questions that are directly +related to our results. In Corollary 6.7, we have computed the normalized volume of PBpσd`1q +explicitly and thereby have obtained a precise formula for the sum of the h˚-vector entries. +Using the explicit regular unimodular triangulation from Theorem A and inclusion-exclusion +we can also express the h˚-polynomial as alternating sum, where all summands are products +of h˚-polynomials of edgewise subdivisions of dilated simplices of varying dimension. Note +that for d odd, we only have one summand. However, this does not yield a direct combinatorial +interpretation of the entries of the h˚-vector. We therefore propose the following problem: + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +19 +Problem 7.4. Find a combinatorial interpretation of the entries of the h˚-vector of PBpσd`1q (see +Table 1 for the h˚-vectors if 1 ď d ď 8). +d +h˚ ` +PBσd`1 +˘ +1 +p1,2,0q +2 +p1,10,5q +3 +p1,22,78,24,0q +4 +p1,131,726,419,19q +5 +p1,149,4049,8558,3750,300,0q +6 +p1,1478,38179,126372,85623,10422,69q +7 +p1,926,157566,1135846,2188310,1150800,145600,3920,0q +8 +p1,17617,1581403,6864069,43252570,31729319,6314903,239867,251q +TABLE 1. The h˚-vectors of PBσd`1 for d “ 1,...,8. +Finally, in view of Theorem B (a), we have the following conjecture: +Conjecture 7.5. Let d be even. Then, h˚ ´ +PBpσd`1q;x +¯ +is real-rooted. +We have verified this conjecture computationally up to d “ 10. For this problem, we suspect +that an approach via interlacing sequences might be helpful, but we have not been able to carry +it out so far. +8. APPENDIX +We provide the missing parts of the proof of Proposition 6.6. We recall some notation. We +denote by E1 +k P Zpk´1qˆk the pk ˆkq-identity matrix with its first row removed and by 1mˆn and +0mˆn the pm ˆ nq-matrices whose entries are all equal to 1 and 0, respectively. Moreover, we +denote by M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +the matrix whose columns are the vertices of d`2 +2 ∆ d´2 +2 ´ +1 d´2 +2 ˆ d +2 in the obvious order. +Proof of Proposition 6.6 (a). Let d ě 2 and for a fixed even integer i P rds, consider the facet +F “ tx P Rd : 1⊺ +odd ¨x´xi ď d +2u of QBpσd`1q. By Corollary 6.2, the vertices of F are tcpℓq : ℓ P +rd ` 2szt1,i ` 2uu. We now consider the matrix B P Zdˆd whose ℓ-th column equals cp2ℓ`1q if +1 ď ℓ ď d +2 and cp2ℓ´dq if d +2 ` 1 ď ℓ ď i`d +2 +and cp2ℓ`2´dq if i`d +2 ` 1 ď ℓ ď d. If we reorder the +rows of B, by taking first the rows with odd index, increasingly, followed by the row with index +i and then the remaining rows with even index, increasingly, we obtain a matrix S, which looks +as follows: +S “ +¨ +˚ +˝ +E d +2 ¨ d`2 +2 +1 d +2 ˆ d +2 +1 +¨¨¨ +1 +0 +¨¨¨ +0 +1 d´2 +2 ˆ d +2 +E1 +d +2 ¨ d`2 +2 +˛ +‹‚. +Clearly, F – convpTq. Let +U “ +¨ +˚ +˚ +˚ +˝ +E1 +d +2 +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +E1 +d +2 +0 +¨¨¨ +0 +´1 0 +¨¨¨ +0 +1 +¨¨¨ +1 +´1 0 +¨¨¨ +0 +˛ +‹‹‹‚P Zdˆd. + +20 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +It is easy to see that U is unimodular and a direct computation shows that +U ¨pS´1dˆdq “ +¨ +˚ +˚ +˚ +˚ +˝ +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +˛ +‹‹‹‹‚ +. +Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates +and by the definition of the join. We also note that the vertices of F corresponding to the vertices +of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d +2u and tcp2ℓq : 2 ď ℓ ď d +2 `1, ℓ ‰ i`2 +2 u. +For a fixed odd integer j P rds, consider the facet G “ tx P Rd : 1⊺ +even ¨ x ´ xj ď d +2u of +QBpσd`1q. By Corollary 6.2, the vertices of F are tcpℓq : ℓ P rd ` 2szt2, j ` 2uu. We now +consider the matrix C P Zdˆd whose ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď j`1 +2 +and cp2ℓ`1q if +j`3 +2 ď ℓ ď d +2 and cp2ℓ`2´dq if d +2 `1 ď ℓ ď d. If we reorder the rows of C by taking first the rows +with odd index k P rdszt ju, increasingly, followed by row j and then the rows with even index, +increasingly, we obtain a matrix S, which looks as follows: +S “ +¨ +˚ +˝ +E1 +d +2 ¨ d`2 +2 +1 d´2 +2 ˆ d +2 +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 d +2 ˆ d +2 +E d +2 ¨ d`2 +2 +˛ +‹‚. +Clearly, G – convpSq. Let +U “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +E d +2 ´1 +ˇˇˇˇˇ +0 d´2 +2 ˆ d`2 +2 +0 d´2 +2 ˆ d +2 +ˇˇˇˇˇ +E1 +d +2 +0 +¨¨¨ +0 +ˇˇ +1 +¨¨¨ +1 +0¨¨¨ +0 +´1 +ˇˇˇ +1 +¨¨¨ +1 +˛ +‹‹‹‹‹‹‹‹‹‚ +P Zdˆd. +It is easy to see that U is unimodular and a direct computation shows that +U ¨pS´1dˆdq “ +¨ +˚ +˚ +˚ +˚ +˝ +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +˛ +‹‹‹‹‚ +. +Since G – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates +and by the definition of the join. We also note that the vertices of G corresponding to the vertices +of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d +2,ℓ ‰ j`1 +2 u and tcp2ℓq : 2 ď ℓ ď d +2 `1,ℓ ‰ i`2 +2 u. +For fixed integers 1 ď i ă j ď d of different parity consider the facet H “ tx P Rd : xi`xj ě 1u +of QBpσd`1q. Without loss of generality assume that i is odd j is even. By Corollary 6.2, the +vertices of F are tcpℓq : ℓ P rd `2szti`2, j`2uu. We now consider the matrix D P Zdˆd whose +ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď i`1 +2 , cp2ℓ`1q if i`3 +2 ď ℓ ď d +2, cp2ℓ´dq if d +2 `1 ď ℓ ď j`d +2 +and +cp2ℓ`2´dq if j`d +2 `1 ď ℓ ď d. If we reorder the rows of D by taking first the rows with odd index + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +21 +k P rdsztiu, increasingly, followed by row i, followed by the rows with even index ℓ P rdszt ju, +increasingly, followed by row j as the last row, we obtain a matrix S, which looks as follows: +S “ +¨ +˚ +˚ +˚ +˝ +d`2 +2 ¨E1 +d +2 +11 +d +2 +0 +¨¨¨ +0 +1 +¨¨¨ +1 +11 +d +2 +d`2 +2 ¨E1 +d +2 +1 +¨¨¨ +1 +0 +¨¨¨ +0 +˛ +‹‹‹‚. +Clearly, H – convpSq. Let +U “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +E d +2 ´1 +ˇˇˇˇˇ +0p d +2 ´1qˆp d +2 `1q +01 +d +2 +ˇˇˇˇˇE d +2 ´1 +ˇˇˇˇˇ0p d +2 ´1qˆ1 +´e⊺ +d +´pe d +2 `edq⊺ +˛ +‹‹‹‹‹‹‹‹‚ +P Zdˆd. +It is easy to see that U is unimodular and a direct computation shows that +U ¨pS´1dˆdq “ +¨ +˚ +˚ +˚ +˚ +˝ +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 d´2 +2 ˆ d +2 +0 d´2 +2 ˆ d +2 +M +´ +d`2 +2 ∆ d´2 +2 ´1 d´2 +2 ˆ d +2 +¯ +0 +¨¨¨ +0 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +1 +¨¨¨ +1 +˛ +‹‹‹‹‚ +. +Since H – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates +and by the definition of the join. We also note that the vertices of H corresponding to the vertices +of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d +2,ℓ ‰ i`1 +2 u and tcp2ℓq : 1 ď ℓ ď d +2 ` 1,ℓ ‰ +j`2 +2 u. +□ +Proof of Proposition 6.6 (ii). Let d ě 1 be an odd integer. +We define vectors up1q,..., +ud`2 P Rd`1 by upℓq +k +“ d ` 1 if k “ ℓ ´ 1 and upℓq +k +“ p´1qk`ℓ´1q, otherwise. By Lemma 4.10, +up1q,...,upd`2q are the vertices of rPBpσd`1q. We now consider the matrix E P Zpd`1qˆpd`2q whose +ℓ-th column equals bp2ℓ´1q if 1 ď ℓ ď d`3 +2 +and up2ℓ´pd`3qq if d`3 +2 `1 ď ℓ ď d `2. If we reorder +the rows of E, by taking first the rows with even index and then the ones with odd index, +increasingly, we obtain a matrix Q “ pqk,ℓq P Zpd`1qˆpd`2q with +‚ qk,k`1 “ d `1 for k P rd `1s, +‚ qk,ℓ “ 1 if k ď d`1 +2 +and ℓ ą d`3 +2 , or k ą d`1 +2 +and ℓ ď d`3 +2 +‚ qk,ℓ “ ´1, otherwise. +Clearly, rPBpσd`1q – convpQq. Let +U “ +¨ +˚ +˚ +˚ +˚ +˚ +˝ +E d`1 +2 +0 d`1 +2 ˆ d`1 +2 +0 +0 d´1 +2 ˆ d`1 +2 +... +E d´1 +2 +0 +0 +¨¨¨ +0 +1 +¨¨¨ +1 +˛ +‹‹‹‹‹‚ +P Zpd`1qˆpd`1q. + +22 +MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE +It is easy to see that U is unimodular and a direct computation shows that +U ¨pQ´1pd`1qˆpd`2qq “ +¨ +˚ +˚ +˝ +M +´ +pd `2q∆ d`1 +2 ´2¨1 +¯ +0 +0 +M +´ +pd `2q∆ d´1 +2 ´2¨1 +¯ +0 +¨¨¨ +0 +1 +¨¨¨ +1 +˛ +‹‹‚. +Since rPBpσd`1q – convpU ¨pQ´1pd`1qˆpd`2qqq, the claim follows by definition of the join. +□ +REFERENCES +[1] K. Adiprasito, Stavros A. Papadakis, V. Petrotou, and J. Steinmeyer. Beyond positivity in ehrhart theory, +2022. +[2] C.A. Athanasiadis. h˚-vectors, eulerian polynomials and stable polytopes of graphs. Electron. J. Combin., +11(2):Research Paper 6, 13 pp. (electronic), 2004/06. +[3] G. Balletti, T. Hibi, M. Meyer, and A. Tsuchiya. Laplacian simplices associated to digraphs. Arkiv f¨or +matematik, 56, 12 2018. +[4] F. Barahona and A. Mahjoub. On the cut polytope. Mathematical Programming, 36:157–173, 06 1986. +[5] M. Beck and A. Stapledon. On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series. +Math. Z., 264(1):195–207, 2010. +[6] B. Braun and M. Meyer. Laplacian simplices. Advances in Applied Mathematics, 114, 2017. +[7] F. Brenti and V. Welker. The veronese construction for formal power series and graded algebras. Advances in +Applied Mathematics, 42(4):545–556, 2009. +[8] M. Brun and T. R¨omer. Subdivisions of toric complexes. Journal of Algebraic Combinatorics, 21, 2004. +[9] W. Bruns and T. R¨omer. h-vectors of gorenstein polytopes. Journal of Combinatorial Theory, Series A, +114:65–76, 01 2007. +[10] Alessio D’Al`ı, Martina Juhnke-Kubitzke, Daniel K¨ohne, and Lorenzo Venturello. On the gamma-vector of +symmetric edge polytopes. Preprint arXiv: https://arxiv.org/abs/2201.09835, 2022. +[11] J.A. De Loera, J. Rambau, and F. Santos. Triangulations. Structures for algorithms and applications, +volume 25. 2010. +[12] R. Diestel. Graph Theory, volume 173. 2017. +[13] H. Edelsbrunner and D. R. Grayson. Edgewise subdivision of a simplex. In Proceedings of the Fifteenth +Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 24–30. ACM, New York, +1999. +[14] E. Ehrhart. Sur les poly`edres rationnels homoth´etiques `a n dimensions. C. R. Acad. Sci. Paris, 254:616–618, +1962. +[15] T.E. Goldberg. Combinatorial laplacians of simplicial complexes. A Senior Project submitted to The Division +of Natural Science and Mathematics of Bard College, 5 2002. +[16] D. R. Grayson. Exterior power operations on higher K-theory. K-Theory, 3(3):247–260, 1989. +[17] B. Gr¨unbaum. Convex Polytopes. Graduate Texts in Mathematics, 2003. +[18] C. Haase, A. Paffenholz, L. Piechnik, and F. Santos. Existence of unimodular triangulations - positive results. +Memoirs of the American Mathematical Society, 270, 05 2014. +[19] J. Herzog, T. Hibi, and H. Ohsugi. Edge Polytopes and Edge Rings, pages 117–140. 09 2018. +[20] T. Hibi. Dual polytopes of rational convex polytopes. Combinatorica, 2(2):237–240, 1992. +[21] A. Higashitani, K. Jochemko, and M. Michałek. Arithmetic aspects of symmetric edge polytopes. +Mathematika, 65:763–784, 05 2019. +[22] K. Jochemko. On the real-rootedness of the veronese construction for rational formal power series. +International Mathematics Research Notices, 2018:4780–4798, 2018. +[23] L. Lov´asz and M. D. Plummer. Matching theory. Annals of Discrete Mathematics, 29, 1986. +[24] M. Meyer and Pllaha T. Laplacian simplices ii: A coding theoretic approach, 2018. +[25] R. Mulas, D. Horak, and J. Jost. Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and +Topology, pages 1–58. Springer International Publishing, Cham, 2022. +[26] J.R. Munkres. Elements of Algebraic Topology. Addison Wesley Publishing Company, 1984. +[27] H. Ohsugi and T. Hibi. Special simplices and Gorenstein toric rings. J. Combin. Theory Ser. A, 113(4):718– +725, 2006. +[28] H. Ohsugi and A. Tsuchiya. Pq-type adjacency polytopes of join graphs. 03 2021. +[29] R.P. Stanley. Decompositions of rational convex polytopes. Annals of Discrete Math., 6:333–342, 1980. + +LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES +23 +[30] G. Ziegler. Elements of Algebraic Topology. Graduate Texts in Mathematics, 1995. +UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY +Email address: juhnke-kubitzke@uos.de +UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY +Email address: dakoehne@uos.de + diff --git a/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/load_file.txt b/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ce87ffb400634ab22786b848d8a075904915ee --- /dev/null +++ b/-9FJT4oBgHgl3EQfqCwO/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf,len=1041 +page_content='LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given a (finite) simplicial complex, we define its i-th Laplacian polytope as the convex hull of the columns of its i-th Laplacian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' After studying basic properties of these polytopes, we focus on the d-th Laplacian polytope of the boundary of a pd ` 1q-simplex Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd, then as for graphs, the d-th Laplacian polytope turns out to be a pd ` 1q- simplex in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is even, we show that the d-th Laplacian polytope of Bpσd`1q is combinatorially equivalent to a d-dimensional cyclic polytope on d ` 2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, we provide an explicit regular unimodular triangulation for the d-th Laplacian polytope of Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This enables us to to compute the normalized volume and to show that the h˚-polynomial is real-rooted and unimodal, if d is odd and even, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' INTRODUCTION Over decades, several lattice polytopes arising from graphs have been studied, extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Prominent examples include matching polytopes, cut polytopes, edge polytopes, adjacency polytopes of several types, among which are symmetric edge polytopes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', [23, 4, 19, 21, 28, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Following this line of research, in 2017, Braun and Meyer [6] initiated the study of Laplacian simplices that are defined as the convex hull of the columns of the classical Laplacian matrix of a simple graph (see also [24, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since each simple graph can be seen as a 1-dimensional simplicial complex and since to each simplicial complex, we can associate Laplacian matrices, defined via their boundary maps in simplicial homology, it is natural to extend the definition of Laplacian simplices to arbitrary simplicial complexes and their Lapla- cians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' More precisely, given a simplicial complex ∆ (with a fixed ordering of the vertex set) and its i-th Laplacian matrix Lip∆q :“ Bi`1B⊺ i`1 ` B⊺ i Bi, we define the i-th Laplacian polytope Ppiq ∆ of ∆ as the convex hull of the columns of Lip∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Here, Bi and Bi`1 denote boundary maps in simplicial homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We initiate the study of Laplacian polytopes by establishing first some general combinatorial and geometric properties and then by focusing on a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' More precisely, we consider the situation that the underlying simplicial complex ∆ is the boundary of the pd ` 1q-simplex, denoted by Bpσd`1q, and that we take its highest Laplacian LdpBpσd`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For simplicity, we set PBpσd`1q :“ Ppdq Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is even, it is easily seen, that, as for graphs, PBpσd`1q is a pd ` 1q- simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By deriving a complete facet description of PBpσd`1q in this case, we are able to show that PBpσd`1q is combinatorially equivalent to a d- dimensional cyclic polytope on d `2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It was shown in [6] that Laplacian simplices have unimodal h˚-vectors for certain classes of graphs, including trees, odd cycles and complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Inspired by these results, we study properties of the h˚-vectors of general Laplacian polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This is further motivated by the general question under which conditions a lattice polytope has a unimodal h˚-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It was conjectured by Hibi and Ohsugi that this is true for reflexive lattice polytopes that have the integer decomposition property (IDP) [27], and, recently, Adiprasito, Papadakis, Petrotou and Steinmeyer could confirm this conjecture in the positive [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' However, it is still mysterious what happens if the polytope is not reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We consider this question for the Laplacian polytope 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='11602v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='CO] 27 Jan 2023 2 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE PBpσd`1q of the boundary of the pd `1q-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Even in this seemingly most simple situation, Ppdq ∆ turns out to be not reflexive and hence the mentioned results towards unimodality do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' However, the following result shows that Ppdq ∆ has at least the integer decomposition property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' PBpσd`1q has a regular unimodular triangulation for every integer d ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We note that, combined with [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3], this result implies that the h˚-vector of PBpσd`1q is decreasing in its second half which is obviously implied by but weaker than uni- modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The main ingredient for Theorem A is the so-called interior polytope of PBpσd`1q, that is defined as the convex hull of the interior lattice points of PBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Indeed, this polytope turns out to be reflexive (after translation to the origin) and miraculously, PBpσd`1q happens to be the second dilation of it (after translating both polytopes to the origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using edgewise subdivisions, we provide an explicit construction of a regular unimodular triangulation for the interior polytope which then extends to such a triangulation of PBpσd`1q by [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As a byproduct, we can also compute the normalized volume of PBpσd`1q (see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem A combined with the results on the interior polytope enables us to show the following statement: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (a) h˚ ´ PBpσd`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='t ¯ has only real roots if d P N is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (b) h˚ ´ PBpσd`1q ¯ is unimodal with peak in the middle for every d P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We note that if d is odd, then the statement in pbq is just an easy consequence of the one in paq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We conjecture paq to be true also if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Section 2 provides necessary background on simplicial complexes, their Laplacian matrices and lattice polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Section 3 collects basic properties of the Laplacian matrix LdpBpσd`1qq of the boundary of a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In Section 4, we introduce the i-th Laplacian polytope Ppiq ∆ of a simplicial complex ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Among others, we compute its number of vertices (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4), the dimension of PBpσd`1q (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6) and show that PBpσd`1q is always simplicial (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The goal of Section 5 is to derive a complete facet description of PBpσd`1q and to show that it is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices if d is even (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Section 6 is devoted to the proofs of Theorems A and B, including the construction and study of the interior polytope of PBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Finally, in Section 7 we state some open problems and possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' PRELIMINARIES In this section, we provide the necessary background on simplicial complexes, Laplacian matrices and polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For more information on these topics we refer to [30, 17, 26, 11, 18, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, we assume the reader to have basic knowledge about graphs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Simplicial complexes and Laplacian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given a finite set V, a simplicial com- plex ∆ on vertex set V is a collection of subsets of V that is closed under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Elements of ∆ are called faces and inclusion-wise maximal faces are called facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The dimension of a face F is dimpFq :“ |F| ´ 1 and we use Fip∆q to denote the set of i-dimensional faces of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The dimension of ∆ is defined as dimp∆q :“ maxpi : Fip∆q ‰ Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If all facets have the same dimension, ∆ is called pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 0-dimensional and 1-dimensional faces of ∆ are called vertices and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The sets of vertices and edges of ∆ induce a graph in a natural way, which we call the 1-skeleton or graph of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given a pd ´ 1q-dimensional simplicial complex ∆, its LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 3 f-vector fp∆q “ pf´1p∆q, f0p∆q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', fd´1p∆qq is defined by fip∆q :“ |t f P ∆ : dimpFq “ iu| for ´1 ď i ď d ´1 and its h-vector hp∆q “ ph0p∆q,h1p∆q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',hdp∆qq by the polynomial identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1) dÿ k“0 hkp∆qtd´k “ dÿ k“0 fk´1p∆qpt ´1qd´k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The polynomials fp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='tq :“ řd´1 i“´1 fip∆qti and hp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='tq :“ řd i“0 hip∆qti are called the f- and h- polynomial of ∆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In order to introduce general Laplacian matrices of a simplicial complex ∆, we need to recall basic notions from simplicial homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For this purpose, let ∆ be a pd ´ 1q-dimensional simplicial complex on vertex set V and assume that the vertices are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Without loss of generality, assume V “ rns “ t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',nu endowed with the natural ordering induced by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We denote by Cip∆q the Q-vector space with basis teσ : σ P Fip∆qu and set Cip∆q “ t0u for i ď ´1 and i ą d ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The i-th boundary map is the linear map Bi : Cip∆q Ñ Ci´1p∆q defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2) Bipeσq :“ i`1 ÿ k“1 p´1qk´1eσztjku, where σ “ t j1 ă ¨¨¨ ă ji`1u P Fip∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By abuse of notation, we will use Bi to denote both, the map and its corresponding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The i-th Laplacian matrix of ∆ is defined as Lip∆q :“ Bi`1B⊺ i`1 ` B⊺ i Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Note that Lip∆q provides an endomorphism of Cip∆q which depends on the chosen ordering of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We recall that Hip∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='Qq :“ kerpBiq{ImpBi`1q is the i-th (simplicial) homology group of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' To provide an explicit description of Lip∆q, we need some further notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Faces F,G P Fip∆q are called lower adjacent if F XG P Fi´1p∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If, additionally, eFXG appears with the same sign in BipeFq and BipeGq, we call F XG the similar common lower simplex of F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Otherwise, F XG is referred to as the dissimilar common lower simplex of F and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The upper degree of F P Fip∆q, denoted degUpFq, is the number of pi`1q-faces of ∆ containing F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We will use the following description of Lip∆q from [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ be a simplicial complex on vertex set rns, ordered 1 ă ¨¨¨ ă n, and let i P N with 0 ď i ď dimp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For F,G P Fip∆q, let ℓF,G denote the entry of Lip∆q in row and column corresponding to F and G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, Lip∆q is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover: (i) If i “ 0, then ℓF,G “ degUpFq if F “ G, ℓF,G “ ´1 if F Y G P Fi`1p∆q, and ℓF,G “ 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (ii) If i ą 0, then ℓF,G “ $ ’ ’ ’ & ’ ’ ’ % degUpFq`i`1, if F “ G, 1, if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q similar ´1, if F ‰ G, F YG R Fi`1p∆q, F XG P Fi´1p∆q dissimilar 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Note that if i “ 0 in the previous theorem, then L0p∆q coincides with the classical Laplacian matrix of the graph of ∆ (from graph theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (Lattice) polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A polytope P is the convex hull of finitely many points in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If dimP “ k, we call P a k-polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A linear inequality a⊺x ď b for a P Rd and b P R is called a valid inequality for P if a⊺y ď b for all y P P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A (proper) face of P is a (non-empty) set of the form PXtx P Rd : a⊺x “ bu for some valid inequality a⊺x ď b with a ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Faces of dimension 0, dimP´2 and dimP´1 are called vertices, ridges and facets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We use V pPq and FpPq to denote the set of vertices and facets of P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A valid inequality a⊺x ď b is 4 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE facet-defining if F “ PXtx P Rd : a⊺x “ bu for some F P FpPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The facet-ridge graph GpPq of P is the graph on vertex set FpPq where tF,Gu is an edge if and only if F and G intersect in a ridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If V pPq Ď Zd, P is called a lattice polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Two lattice polytopes P, Q Ď Rd are unimodular equivalent, denoted as P – Q, if there exist a unimodular matrix U P Rdˆd and a vector b P Zd such that U ¨ P ` b “ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We use ∆d to denote the standard d-simplex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', ∆d “ convtt0u Y tei P Rd : i P rdsuu, where e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',ed denote the standard unit vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A polytope P is simplicial if all of its facets are simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The normalized volume of a d- dimensional lattice polytope P Ď Rd is given by nvolpPq “ d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='¨volpPq, where volpPq denotes the usual Euclidean volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A lattice d-simplex ∆ with normalized volume 1 is called unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In this case, ∆ – ∆d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A polytope P is reflexive if P “ tx P Rd : Ax ď 1u for an integral matrix A, where 1 denotes the all ones vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In this case, 0 is the unique interior lattice point of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A triangulation T of a lattice d-polytope P is a subdivision into lattice simplices of dimension at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We denote the set of vertices in T by V pT q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A triangulation is unimodular if all its simplices are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' T is called regular if there exists a height function ωP : V pT q Ñ R such that T is the projection of the lower envelope of the convex hull of tpv,ωPpvqq : v P V pT qu Ď Rd`1 to the first d coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We note that every triangulation is in particular a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let P Ď Rd be a lattice d-polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ehrhart [14] proved that the number of lattice points in the n-th dilation of P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', |nPXZd| is given by a polynomial EPpnq of degree d in n for all integers n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The Ehrhart series of P is ÿ ně0 EPpnqtn “ h˚pP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='tq p1´tqd`1 “ h˚ 0pPq`h˚ 1pPqt `¨¨¨`h˚ s pPqts p1´tqd`1 , where h˚pP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='tq P Zrts is a polynomial of degree at most d, called h˚-polynomial of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The vector h˚pPq “ ph˚ 0pPq,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',h˚ s pPqq is called h˚-vector of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We will often omit P from the notation and just write h˚ “ ph˚ 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',h˚ s q if P is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By [29, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1], it is well- known that h˚ i pPq is non-negative for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If P admits a unimodular triangulation T , then h˚pPq “ hpT q [29, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, if T is a regular unimodular triangulation of P, then h˚ tpd`1q{2upPq ě ¨¨¨ ě h˚ d´1pPq ě h˚ dpPq [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It was shown by Hibi in [20] that a lattice d-polytope P Ď Rd is reflexive (up to unimodular equivalence) if and only if P contains a unique interior lattice point, and h˚pPq is palindromic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', h˚ i pPq “ h˚ d´ipPq for all 0 ď i ď td{2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN MATRICES OF BOUNDARIES OF SIMPLICES In this section we investigate basic properties of the Laplacian matrix of the boundary of a simplex that will be useful for deriving properties of the corresponding Laplacian polytopes in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We start with an easy general statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ be a d-dimensional simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then rankLdp∆q “ fdp∆q´dimQ Hdp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We have the following chain of equalities: rankLdp∆q “ rankpB⊺ dBdq “ fdp∆q´dimQ kerpB⊺ dBdq “ fdp∆q´dimQ kerpBdq, where the last equality follows from the fact that kerpBdq “ kerpB⊺ dBdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since dim∆ “ d, we also have Hdp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='Qq “ kerpBdq, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 5 In the following, we let σd`1 “ 2rd`2s be the pd `1q-simplex and we use Bpσd`1q to denote its boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', Bpσd`1q “ σd`1ztrd ` 2su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let Fi “ rd ` 2sztd ` 3 ´ iu for 1 ď i ď d ` 2 and order the columns and rows of LdpBpσd`1qq according to F1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',Fd`2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first provide an explicit description of the d-th Laplacian matrix in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, Ldp∆q P Zpd`2qˆpd`2q, L0p∆q “ ˆ 0 0 0 0 ˙ and, for d ě 1, 1 ď i, j ď d `2, we have Ldp∆qi j “ # d `1, if i “ j, p´1qi` j´1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since fdpBpσd`1qq “ d `2, we have Ldp∆q P Zpd`2qˆpd`2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Assume d “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As B0 is the zero map, the statement is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Now let d ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since dim∆ “ d, it follows that degUpFq “ 0 for any d-face F of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 this implies that Ldp∆qii “ d `1 for all 1 ď i ď d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Now, let i ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since Ldp∆q is symmetric, we can assume that i ă j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Fi and Fj have the common lower simplex Fi X Fj “ rd ` 2sztd ` 3 ´ i,d ` 3 ´ ju ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2), eFiXFj appears with sign p´1qd`3´ j in Bdperd`2sztd`3´iuq and it appears with sign p´1qd`2´i in Bdperd`2sztd`3´ juq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' These signs coincide, meaning that Fi X Fj is a similar common lower simplex of Fi and Fj, if and only if i` j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The claim follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ The next lemma will be crucial for determining the dimension of the Laplacian polytope of Bpσd`1q in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, Ldp∆q has rank d ` 1 and every pd ` 1q-element subset of the columns (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' rows) of Ldp∆q is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The first statement follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 and the fact that Hdp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='Qq “ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let 1 ď i ď d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let Ai be the pd `1qˆpd `1q-matrix obtained from Ldp∆q by removing the i-th row and column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By definition, Ai “ Ldp∆ztFiuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since Hdp∆ztFiuq,Qq “ 0, this matrix has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As adding any extra row or column to Ai does not change the rank, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then rank ˆ Ldp∆q 1¨¨¨1 ˙ “ # d `1, if d is even, d `2, if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' First assume that d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We define λ “ pλ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',λd`2q⊺ P Rd`2 by λj “ # 0, if j is odd, 2 d`2, if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2 it is straight-forward to verify that Ldp∆q ¨ λ “ 1 which, combined with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Now let d be odd and assume by contradiction that rank ˆ Ldp∆q 1¨¨¨1 ˙ ă d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 imply that rank ˆ Ldp∆q 1¨¨¨1 ˙ “ rankLdp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hence, there exists λ “ pλ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',λd`2q⊺ P Rd`2, such that Ldp∆q¨λ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let Ldp∆qrd`1s be the matrix obtained from Ldp∆q by deleting the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, we also have Ldp∆qrd`1s ¨ λ “ 1 and it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 that, up 6 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE to the choice of the last coordinate λd`2, the vector λ is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Indeed, a direct computation shows that, if λd`2 “ µ for some µ P R, then we must have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1) λ j “ $ ’ & ’ % pd`2q¨µ`1 d`2 , if j is odd, ´pd`2q¨µ´1 d`2 , if j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' However, denoting by rd`2 the last row of Ldp∆q, it holds that rd`2 ¨λ “ 0 ‰ 1, which yields a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' GENERAL PROPERTIES OF LAPLACIAN POLYTOPES The goal of this section is to generalize Laplacian simplices – as introduced and studied in [6, 24] – that are associated to a graph to arbitrary simplicial complexes and their Laplacian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' After stating some basic general properties of what we call Laplacian polytopes, we focus on boundaries of simplices and their highest Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the following, given a matrix M, we use convpMq to denote the polytope given by the convex hull of the columns of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ be a d-dimensional simplicial complex on rns, ordered 1 ă ¨¨¨ ă n, and let 0 ď k ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The k-th Laplacian polytope of ∆ is defined as the convex hull of the columns of Lkp∆q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', Ppkq ∆ – convpLkp∆qq Ď R fkp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We want to remark that the 0-th Laplacian polytope of a simplicial complex coincides with the Laplacian simplex of its 1-skeleton, as defined in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The next example shows that different orderings of the vertex set of ∆ may result in polytopes of different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let G be the 4-cycle on r4s with EpGq “ t12,23,34,14u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If the vertices of G are ordered 1 ă 2 ă 3 ă 4, then Pp1q G is a 3-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If the vertices of G are ordered 1 ă 2 ă 4 ă 3, then Pp1q G is a 2-dimensional rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Pp2q Bpσ3q is given by the convex hull of the columns of the following matrix: L2pBpσ3qq “ ¨ ˚ ˚ ˝ 3 1 ´1 1 1 3 1 ´1 ´1 1 3 1 1 ´1 1 3 ˛ ‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It will follow from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10 that Pp2q Bpσ3q is unimodular equivalent to the square in R2 with vertices p1,´1q,p´1,1q,p3,1q and p1,3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We start by showing that every column of Lkp∆q yields a vertex of Ppkq ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ be a d-dimensional simplical complex and 0 ď k ď d an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, Ppkq ∆ has fkp∆q many vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Set m :“ fkp∆q and let vpiq denote the i-th column of Lkp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We assume by contradiction that there exists 1 ď i ď m, a set S Ď rmsztiu and λj P R with λj ą 0 and ř jPS λ j “ 1 such that vpiq “ ř jPS λ jvpjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Setting λ j “ 0 if j R S Y tiu and λi “ ´1, we see that λ :“ pλ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',λmq⊺ P kerpLkp∆qq and hence λ P kerpBkq by [25, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let wpℓq denote the ℓ-th column of Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If wpiq ℓ “ 1, then, since wpjq ℓ P t´1,0,1u, λ j ą 0 and ř jPS λj “ 1, we must have wpjq ℓ “ 1 LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 7 for all j P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By the same reasoning, it follows that wpjq ℓ “ ´1 for all j P S if wpiq ℓ “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As all columns of Bk have the same number of non-zero entries, we conclude wpiq “ wpℓq for all ℓ P S, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ The next proposition gives a sufficient criterion for Ppdim∆q ∆ being a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ be a d-dimensional simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If Hdp∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='Qq “ 0, then Ppdim∆q ∆ is an pfdp∆q´1q-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 implies that rankLdp∆q “ fdp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Consequently, the columns of Ldp∆q are linearly independent which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ In the following, we focus on the d-th Laplacian polytope of Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' To simplify notation, we set PBpσd`1q “ Ppdq Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We use spiq to denote the i-th column of LdpBpσd`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, given a subset S Ď rd ` 2s, we denote by LdpSq the matrix obtained from LdpBpσd`1qq by deleting the rows with indices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4 and [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 4] the following formula for the dimension of PBpσd`1q is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, dimP∆ “ # d, if d is even, d `1, if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The previous statement together with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4 allows us to conclude: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d P N with d ě 1 and ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, P∆ has d`2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular, P∆ is a pd `1q-simplex, if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7 trivially implies that PBpσd`1q is a simplicial polytope if d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The same statement turns out to be true for d even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' PBpσd`1q is simplicial for every d P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd, then the claim is trivially true by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Now, let d be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d “ 0, then P∆ is just the origin and as such simplicial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d ě 2 and let F be the vertices of a facet of P∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combining Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7 it follows that d ď |F| ď d `1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If, by contradiction, |F| “ d `1, then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 implies that the convex hull of F is d-dimensional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', F cannot be a facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Consequently, F is a simplex, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ As, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6, the Laplacian polytope of Bpσd`1q is never full-dimensional, our next goal is to construct a polytope that is unimodular equivalent to PBpσd`1q and full-dimensional with respect to its ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first need to introduce some further notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We let 1even and 1odd denote the 0´1-vectors in Rd`2 whose even and odd entries are equal to 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given these definitions, we can easily compute the affine hull of PBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d P N with d ě 1 and ∆ “ Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' affpP∆q “ #␣ x P Rd`2 : p1odd ´1evenq⊺ ¨x “ 0 ( , if d is odd, ␣ x P Rd`2 : 1⊺ odd ¨x “ 1⊺ even ¨x “ d`2 2 ( , if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6, it is enough to show that all vertices of P∆ lie in the specified subspaces of dimension d `1 and d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This can be seen by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ 8 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE The next lemma gives the desired unimodular equivalent polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The polytope PBpσd`1q is unimodular equivalent to convpLdpt1uqq and convpLdpt1,2uqq if d is odd and even, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Define matrices A,B P Zpd`2qˆpd`2q as follows A “ ¨ ˚ ˚ ˝ 1⊺ odd ´1⊺ even 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ed`1 0 ˛ ‹‹‚ and B “ ¨ ˚ ˚ ˚ ˚ ˝ 1⊺ odd 1⊺ even 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ed 0 0 ˛ ‹‹‹‹‚ , where Ed and Ed`1 denote identity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Note that A and B are unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='9, we conclude that A¨PBpσd`1q “ t0uˆconvpLdpt1uqq, if d is odd and B¨PBpσd`1q “ tppd `2q{2,pd `2q{2quˆconvpLdpt1,2uqq, if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ In the following, we use rPBpσd`1q to denote the unimodular equivalent polytope to PBpσd`1q as constructed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By abuse of notation, we will also refer to rPBpσd`1q as d-th Laplacian polytope of Bpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We also want to remark that, if d is odd, we have the following, easy-to-show containment relation: rPBpσd`1q Ď rPBpσd`2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' THE FACET DESCRIPTION AND THE COMBINATORIAL TYPE OF PBpσd`1q While, for odd d, we have already seen that rPBpσd`1q is a simplex, the goal of this section is to determine the combinatorial type of PBpσd`1q if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' To reach this goal, we will first provide a complete irredundant facet description of PBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let bpℓq denote the vertex of rPBpσd`1q, that is given by the ℓ-th column of Ldpt1,2uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2, we have bpℓq k “ d `1 if k “ ℓ´2 and bpℓq k “ p´1qk`ℓ´1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d ě 2 be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then the following inequalities are facet-defining and irredundant for rPBpσd`1q (i) 1⊺ ¨x ď d `2, (ii) 1⊺ odd ¨x´xi ď d`2 2 , where i P rds is even, (iii) 1⊺ even ¨x´xj ď d`2 2 , where j P rds is odd, (iv) xi `xj ě 0, where 1 ď i ă j ď d such that i` j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, the vertices, that attain equality in (i)–(iv), are given by the sets tbpℓq : 3 ď ℓ ď d`2u, tbpℓq : ℓ P rd`2szt1,i`2uu, tbpℓq : ℓ P rd`2szt2, j`2uu and tbpℓq : ℓ P rd`2szti`2, j`2uu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first consider the inequality in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If ℓ P t1,2u, then bpℓq P t´1,1ud with alternating entries and hence 1⊺ ¨ bpℓq ă d ` 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let 3 ď ℓ ď d ` 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As d is even, it follows from above that bpℓq has one entry equal to d ` 1, d 2 entries equal to 1 and d 2 ´ 1 entries equal to ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This implies 1⊺ ¨bpℓq “ d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hence, the inequality in (i) defines a facet, whose vertices are given by tbpℓq : 3 ď ℓ ď d `2u, where we use that the affine hull of the latter set is pd ´1q-dimensional by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 9 Similarly, it is straightforward to verify that the inequalities in (ii)–(iv) are valid for rPBpσd`1q and that the given sets of vertices are the ones attaining equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As those all differ and their affine hulls all have dimension d ´1, it follows that the inequalities are irredundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ For the sake of completeness we add the description of the facets of rPBpσd`1q for d odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d ě 3 is odd, using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2 it is not hard to see, that the following inequalities are facet-defining for rPBpσd`1q (i) 1⊺ ¨x ď d `2, (ii) 2¨1⊺ odd ¨x´xi ď d`2 2 , where i P rd `1s is even, (iii) 2¨1⊺ odd ¨x`xj ď d `2, where j P rds is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is easy to verify that these inequalities are irredundant and as, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6, rPBpσd`1q is a simplex, they provide the complete facet description of rPBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We omit an explicit proof since this description will not be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We state the first main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For d even, rPBpσd`1q is completely described by the inequalities in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, this description is irredundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular, rPBpσd`1q has pd`2q2 4 many facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We let Ă F denote the set of facets of rPBpσd`1q, provided by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 and we write G Ă F for the subgraph of the facet-ridge graph of rPBpσd`1q that is induced on vertex set Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8, that the facet-ridge graph of rPBpσd`1q is d-regular and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since any d-regular subgraph does not have a proper d-regular subgraph, for the first statement, it suffices to show that G Ă F is d-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since G Ă F is a subgraph of GprPBpσd`1qq, its maximal degree is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hence, to show the claim, it suffices to show that |EpG Ă Fq| “ d¨|VpG Ă F q| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first count the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1, we get that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1) |VpG Ă Fq| “ 1` d 2 ` d 2 ` ˆd 2 ˙2 “ pd `2q2 4 , Here, the last term in the middle comes from the fact that the inequalities in (iv) are indexed by sets ti, ju where i P t2ℓ : ℓ P rd 2su and j P t2ℓ´1 : ℓ P rd 2su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It remains to count the number of edges of G Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the following, we identify a facet in Ă F with its set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given this, we use the following short hand notation for the different types of facets in Ă F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (i) F “ tbpℓq : 3 ď ℓ ď d `2u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (ii) Ei “ tbpℓq : ℓ P rd `2szt1,i`2uu, where i P rds is even;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (iii) Oj “ tbpℓq : ℓ P rd `2szt2, j `2uu, where j P rds is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (iv) Fk,m “ tbpℓq : ℓ P rd `2sztk `2,m`2uu for 1 ď k ă m ď d such that k `m is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We immediately get that (a) |F XEi| “ |F XOj| “ d ´1 for all even i P rds and all odd j P rds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (b) |F XFk,m| “ d ´2 for all 1 ď k ă m ď d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (c) |Ei XE j| “ d ´1 for all odd i, j P rds with i ‰ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (d) |Ei XO j| “ d ´2 for all even i P rds and all odd j P rds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (e) |Ei XFk,m| “ d ´1 iff i P tk,mu, i even, k `m odd, and |Ei XFk,m| “ d ´2, otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (f) |Oi XO j| “ d ´1 for all even i, j P rds with i ‰ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 10 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE (g) |O j XFk,m| “ d ´1 iff j P tk,mu, j odd, k `m odd, and |Oj XFk,m| “ d ´2, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (h) |Fi, j XFk,m| “ d ´1 iff |ti, j,k,mu| “ 3 and |Fi,j XFk,m| “ d ´2, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since edges of G Ă F are given by tuples of facets intersecting in d ´1 vertices, we get d edges in (a), 0 edges in (b) and (d), `d{2 2 ˘ edges in each of (c) and (f), `d 2 ˘2 edges in each of (e) and (g) and 2¨ d 2 ¨ `d{2 2 ˘ edges in (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This yields |EpG Ă Fq| “ d `2¨ ˆd{2 2 ˙ `2¨ ˆd 2 ˙2 `d ¨ ˆd{2 2 ˙ “ dpd `2q2 8 “ d ¨|VpG Ă Fq| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It follows that G Ă F is d-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The In particular-statement follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ The previous theorem allows us to determine the combinatorial type of rPBpσd`1q if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is well-known (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', [17, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1]) that there are only finitely many combinatorial types of simplicial d-polytopes with d ` 2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' More precisely, any simplicial d-polytope with d ` 2 vertices is obtained as the convex hull of a d-simplex T d and a vertex v that is beyond k facets of T d, where 1 ď k ď d ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is easily seen that the combinatorial type of such a polytope only depends on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Following Gr¨unbaum, we use T d k to denote the corresponding combinatorial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given that we know the number of facets of rPBpσd`1q (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3), we can immediately determine its combinatorial type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then rPBpσd`1q is of combinatorial type T d d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular, rPBpσd`1q is combinatorially equivalent to a d-dimensional cyclic polytope on d `2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3, rPBpσd`1q has pd`2q2 4 facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using [17, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1, Theorem 2], it follows that this number has to be equal to ˆd `2 2 ˙ ´ ˆk `1 2 ˙ ´ ˆd `1´k 2 ˙ , where rPBpσd`1q is of combinatorial type T d k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Solving for k yields k “ d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The second statement follows from [17, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ We remark that from the previous theorem, we also get a precise formula for the f- and h-vector of rPBpσd`1q (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given the precise description of the facets from the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3, it is not hard to write down a shelling order for rPBpσd`1q (d even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Namely, one particular shelling is given by F,E2,E4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',Ed,O1,O3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',Od´1,F1,2,F1,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',F1,d,F2,3,F2,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',F2,d´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',Fd´1,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' REGULAR UNIMODULAR TRIANGULATIONS AND h˚-VECTORS This section is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The goal of the first is to prove Theorem A, namely, that rPBpσd`1q admits a regular unimodular triangulation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As a byproduct we will also be able to compute the normalized volume rPBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the second part, we provide the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 11 FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' rPBpσ3q and its interior polytope QBpσ3q translated to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Triangulations through interior polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is even, one of our main tools towards the formulated goal is the so-called interior polytope QBpσd`1q of rPBpσd`1q, defined as follows: QBpσd`1q “ conv ´ rPBpσd`1qzB ´ rPBpσd`1q ¯ XZd¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Figure 1 depicts rPBpσ3q and its interior polytope QBpσ3q, both translated to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Surprisingly, it turns out that PBpσd`1q and its interior polytope are combinatorially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' More precisely, the following stronger statement is true: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d P N be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then the following statements hold: (a) The complete and irredundant facet description of QBpσd`1q is given by: (i) 1⊺ ¨x ď d `1, (ii) 1⊺ odd ¨x´xi ď d 2 for even i P rds, (iii) 1⊺ even ¨x´xj ď d 2 for odd j P rds, (iv) xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (b) QBpσd`1q ´1 is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular, 1 is the unique interior lattice point of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (c) 2¨ ´ QBpσd`1q ´1 ¯ “ rPBpσd`1q ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We let Q “ rPBpσd`1q ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The vertices of Q are given by upℓq :“ bpℓq `1 for 1 ď ℓ ď d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is immediate that all coordinates of upℓq are divisible by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hence, 1 2Q is a lattice polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3, it follows that the facets of 1 2Q are given by ‚ 1⊺ ¨x ď 1, ‚ 1⊺ odd ¨x´xi ď 1 for even i P rds, ‚ 1⊺ even ¨x´xj ď 1 for odd j P rds, ‚ xi `xj ě ´1 for 1 ď i ă j ď d such that i` j is odd, which shows that 1 2Q is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It remains to show that 1 2Q`1 “ QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since 1 2Q`1 is a lattice polytope, it follows that 1 2Q ` 1 Ď QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For the other inclusion it suffices to note that the facets of 1 2Q and rPBpσd`1q are parallel and that they have distance 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' d, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' d`2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' d`2 and 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2 to each other for facets of the form in (i), (ii), (iii) and (iv), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This implies that there is no lattice point in rPBpσd`1qzpp1 2Q`1qYB rPBpσd`1qq and hence QBpσd`1q Ď 1 2Q`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ m 312 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE We define vectors cp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',cpd`2q P Rd by cpℓq k “ d`2 2 if k “ ℓ ´ 2 and cpℓq k “ maxp0,p´1qk`ℓ´1q, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combining Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (c), we get the following description of the vertices of QBpσd`1q and its facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The vertices of QBpσd`1q are the vectors cp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',cpd`2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, the vertices, that attain equality in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (i)–(iv), are given by the sets tcpℓq : 3 ď ℓ ď d ` 2u, tcpℓq : ℓ P rd `2szt1,i`2uu, tcpℓq : ℓ P rd `2szt2, j`2uu and tcpℓq : ℓ P rd `2szti`2, j`2uu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now recall several definitions and facts concerning regular unimodular triangulations (see [18, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='] for more on these topics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given full-dimensional polytopes P Ď Rd and P1 Ď Rd1 of positive dimension, their join P˚P1 is the pd `d1 `1q-dimensional polytope defined by Pˆt0d1uˆt0u Y t0duˆP1 ˆt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The next statement, which is well-known, will be crucial for the construction of a regular unimodular triangulation of rPBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let P Ď Rd and P1 Ď Rd1 be polytopes of dimension d and d1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let S “ tSi : i P rnsu and S1 “ tS1 j : j P rmsu be triangulations of P and P1, respectively, where Si and S1 j denote the full-dimensional cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If both S and S1 are regular and unimodular, then T “ ␣ Si ˚S1 j : i P rns, j P rms ( is a regular unimodular triangulation of P˚P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We will also make use of the following statement, see [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If P has a (regular) unimodular triangulation T , then so has any dilation cP, where c is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A well-studied subdivision, which is related to the Veronese construction in algebra but also appears in topology [7, 13, 16, 8], is the so-called rth edgewise subdivision of a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the following, we review this definition for the special case that ∆ is the pn ´ 1q-dimensional simplex on vertex set V “ te1,e2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',enu Ď Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For a positive integer r, let Ωr “ tpi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',inq P Nn : i1 ` i2 ` ¨¨¨ ` in “ ru denote the set of lattice points r∆ X Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For x “ px1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',xnq P Zn, we define ϕpxq :“ px1,x1 `x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',x1 `¨¨¨`xnq P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The rth edgewise subdivision of ∆ is the simplicial complex esdrp∆q on vertex set Ωr, for which F Ď Ωr is a face if for all x,y P F ϕpxq´ϕpyq P t0,1un or ϕpyq´ϕpxq P t0,1un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By definition, the geometric realization of the rth edgewise subdivision of ∆ gives a lattice triangulation of r∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is known that this triangulation is regular [8, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ], which is also unimodular since all maximal simplices have normalized volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the following, we will use esdrp∆q to denote both, the triangulation as a simplicial complex and its geometric realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given any pn ´ 1q-dimensional unimodular simplex Γ Ď Rn, esdrp∆q, naturally induces a regular unimodular triangulation of rΓ (by applying the corresponding unimodular transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Slightly abusing notation, we will refer to this triangulation as edgewise subdivision of Γ or even of rΓ, denoted esdrpΓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, the restriction of esdrpΓq to any face F P Γ equals esdrpFq as a simplicial complex and as geometric realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 13 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Figure 2 depicts the 3rd edgewise subdivision of the 2-dimensional simplex ∆2 :“ convpp0,0q,p1,0q,p0,1qq as triangulation of 3∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The vertex labels correspond to the vertex labels from the original definition and not the lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The triangulation of 3¨∆2 given by esd3p∆2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now outline our strategy to show that rPBpσd`1q has a regular unimodular triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first prove that rPBpσd`1q and the facets of QBpσd`1q, if d is odd and even, respectively, are unimodular equivalent to joins of dilated standard simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd and even, we can hence triangulate rPBpσd`1q and facets of QBpσd`1q as join of edgewise subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd, the claim follows by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is even, we next show that these triangulations are consistent on intersections of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By coning with 1, we get a unimodular triangulation of QBpσd`1q (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (d)) and hence of rPBpσd`1q by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The regularity follows by using that the triangulation is regular on single facets and that each facet is triangulated in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The next statement yields the first step in the outlined strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (a) Let d ě 2 be even and F P F ´ QBpσd`1q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then F – ˆd `2 2 ∆ d´2 2 ´1 d´2 2 ˙ ˚ ˆd `2 2 ∆ d´2 2 ´1 d´2 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (b) Let d ě 1 be an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then rPBpσd`1q – ` pd `2q∆ d`1 2 ´2¨1 ˘ ˚ ` pd `2q∆ d´1 2 ´2¨1 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The proof of (a) is divided into four cases, according to the four classes of facets from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let F “ tx P Rd : 1⊺ ¨ x ď d ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2, the vertices of F are cp3q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',cpd`2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now consider the matrix A, whose ℓ-th column equals cp2ℓ`1q if 1 ď ℓ ď d 2 and cp2ℓ`2´dq if d 2 `1 ď ℓ ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If we reorder the rows of A, by taking first the rows with odd index and then the ones with even index, increasingly, we obtain a matrix S, which looks as follows: S “ ˜ d`2 2 ¨E d 2 1 d 2 ˆ d 2 1 d 2 ˆ d 2 d`2 2 ¨E d 2 ¸ , 4 (0, 0, 3) (1, 0,2) (0,1,2) (2, 0, 1) (1,1,1 (0, 2, 1) (3, 0, 0) (2,1,0) (1, 2, 0) (0,3, 0) 114 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE where 1kˆk denotes the pk ˆ kq-matrix with all entries equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Clearly, F – convpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let E1 k P Zpk´1qˆk be the pk ˆkq-identity matrix with its first row removed and let U “ ¨ ˚ ˚ ˚ ˝ E1 d 2 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 E1 d 2 0 ¨¨¨ 0 1 ¨¨¨ 1 1 ¨¨¨ 1 1 ¨¨¨ 1 ˛ ‹‹‹‚P Zdˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is easily seen that U is unimodular and a direct computation shows that U ¨pS´1dˆdq “ ¨ ˚ ˚ ˚ ˚ ˝ M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 ¨¨¨ 0 1 ¨¨¨ 1 1 ¨¨¨ 1 1 ¨¨¨ 1 ˛ ‹‹‹‹‚ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' where 0kˆk denotes the pk ˆkq-matrix with all entries equal to 0 and M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ denotes the matrix whose columns are the vertices of d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 in the obvious order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates and by the definition of the join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We also note that the vertices of F corresponding to the vertices of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d 2u and tcp2ℓq : 2 ď ℓ ď d 2 `1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' one can show that for the facets defined by ‚ 1⊺ odd ¨x´xi ď d 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' where i P rds is even,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ‚ 1⊺ even ¨x´xj ď d 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' where j P rds is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ‚ xi `xj ě 1 for 1 ď i ă j ď d such that i` j is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' the vertices ‚ tcp2ℓ`1q : 1 ď ℓ ď d 2u and tcp2ℓq : 1 ď ℓ ď d 2 `1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='ℓ ‰ i`2 2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ‚ tcp2ℓ`1q : 0 ď ℓ ď d 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='ℓ ‰ j`1 2 u and tcp2ℓq : 2 ď ℓ ď d 2 `1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='ℓ ‰ i`2 2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ‚ tcp2ℓ`1q : 0 ď ℓ ď d 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='ℓ ‰ j`1 2 u and tcp2ℓq : 1 ď ℓ ď d 2 `1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='ℓ ‰ i`2 2 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' correspond to the vertices of the dilated simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The rather technical proofs can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Similarly, (b) will be shown in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ We recall and prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' PBpσd`1q has a regular unimodular triangulation for every integer d ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since rPBpσd`1q and PBpσd`1q are unimodular equivalent, it suffices to show the statement for rPBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' First assume that d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 (b), we know that rPBpσd`1q – ` pd `2q∆ d`1 2 ´2¨1 ˘ ˚ ` pd `2q∆ d´1 2 ´2¨1 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since the pd ` 2qnd edgewise subdivision is a regular unimodular triangulation of the pd ` 2qnd dilation of any unimodular simplex (as well as of any translation), we conclude with Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 that rPBpσd`1q has a regular unimodular triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Next assume that d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d “ 0, rPBpσd`1q is just a point and there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We construct a regular unimodular triangulation of the interior polytope QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 every facet F of QBpσd`1q is unimodular equivalent to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1) ˆd `2 2 ∆ d´2 2 ´1 d´2 2 ˙ ˚ ˆd `2 2 ∆ d´2 2 ´1 d´2 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 15 By the same reasoning as for d odd, we can triangulate F as join of edgewise subdivisions of unimodular simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In this way, we obtain regular unimodular triangulations of each facet of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now show that the union of these triangulations, yields a triangulation of the boundary of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For this aim, let F and G be facets of QBpσd`1q and let T pFq and T pGq be the considered triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let us further denote by Fi and Gi, where i P r2s, the vertex sets corresponding to the vertex sets of the dilated (and translated) simplices in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It follows from the end of the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 that (after possible renumbering) pF1 YF2qXpG1 YG2q “ pF1 XF2qYpG1 XG2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This directly yields that the restrictions of T pFq and T pGq to F X G coincide: Indeed, they are given as the join of the edgewise subdivisions of the dilated (and translated) simplices on vertex sets F1 XF2 and G1 XG2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This shows that the union of the triangulations of the facets is indeed a triangulation of the boundary of QBpσd`1q, which is, in particular, unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (b), QBpσd`1q ´1 is reflexive, we can extend this triangulation to a unimodular triangulation of QBpσd`1q by coning over the unique interior lattice point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In the following, we call this triangulation T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It remains to show that T is a regular triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The previous paragraph implies that the induced triangulations on facets QBpσd`1q are all regular and unimodular equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular, there exists a simultaneous lifting function h yielding the triangulation of an arbitrary facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Fix a facet F and let T pFq be the induced triangulation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since F is a simplex, we can assume that hpvq “ 1 for any vertex v P F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, for any lattice point u in F, that is not a vertex, we have hpuq ă 1, since otherwise u would not be a vertex of T pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hence, there exists a non-negative function g, whose values are bounded by 1, that vanishes on the vertices of F such that h “ 1´g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, for any ε ą 0, hε “ 1´εg is also a lifting function for F yielding T pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Finally, ignoring 1 and lifting all other lattice points in QBpσd`1q according to the simultaneous lifting function hε, gives a lifting function such that the projection of the lower envelope yields T on the boundary of QBpσd`1q and potentially additional faces in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lifting 1 at height 0, gives a lifting of all lattice points of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If ε is sufficiently small, one can guarantee that the triangulation obtained as the lower envelope is the cone with 1 over the boundary of the previous triangulation (ignoring 1) since potential interior faces that we had seen before, do no longer lie in the lower envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The claim follows by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (c) and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ Analyzing the proof of Theorem A, we can compute the normalized volume of PBpσd`1q: Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The normalized volume of PBpσd`1q is pd `2qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We compute the normalized volume of rPBpσd`1q, which equals the one of PBpσd`1q, by counting the number of maximal simplices in the unimodular triangulation T constructed in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' First assume that d is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We have seen that T is unimodular equivalent to esdd`2 ´ ∆ d`1 2 ¯ ˚esdd`2 ´ ∆ d´1 2 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since the rth edgewise subdivision of an m-simplex, has rm maximal simplices, it follows that the number of maximal simplices in the constructed unimodular triangulation of rPBpσd`1q equals pd `2q d´1 2 ¨pd `2q d`1 2 “ pd `2qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We first compute the normalized volume of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combining Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1, it follows that QBpσd`1q has exactly pd`2q2 4 facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By the proof of Theorem A 16 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE each of these has a unimodular triangulation that is unimodular equivalent to esd d`2 2 ´ ∆ d´2 2 ¯ ˚esd d`2 2 ´ ∆ d´2 2 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As in the case that d is odd, we conclude that each facet is triangulated into `d`2 2 ˘ d´2 2 ¨ `d`2 2 ˘ d´2 2 “ `d`2 2 ˘d´2 many maximal simplices and hence QBpσd`1q has normalized volume pd`2qd 2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (c), rPBpσd`1q `1 “ 2¨QBpσd`1q, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Unimodality and real-rootedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The goal of this subsection is to prove Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is even, then by the proof of Theorem A, QBpσd`1q has a regular unimodular triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since it is also reflexive (after translation) by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (b), the next statement is immediate from [9, Theorem 1] (see also [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3]): Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d be an even positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then h˚pQBpσd`1qq is symmetric and unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' To show unimodality of h˚prPBpσd`1qq, if d is even, we need to analyze the change of the h˚- vector under the second dilation of a polytope (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Given a d-dimensional lattice polytope P, it follows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', from [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1] (see also [5, 22]) that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2) h˚ i p2Pq “ dÿ j“0 ˆd `1 2i´ j ˙ h˚ jpPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We need the following technical but crucial lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let i P N and rj :“ ` d`1 2i`2´ j ˘ ´ `d`1 2i´ j ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then for k P N, we have ´rr2i`2´ d`3 2 s´k “ rt2i`2´ d`3 2 u`k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We set aj “ ` d`1 2i`2´ j ˘ and bj “ `d`1 2i´ j ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The claim follows if both ar2i`2´ d`3 2 s´k “ bt2i`2´ d`3 2 u`k and br2i`2´ d`3 2 s´k “ at2i`2´ d`3 2 u`k hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Due to the symmetry of the binomial coefficient it suffices to show that (i) p2i`2´r2i`2´ d`3 2 s`kq`p2i´t2i`2´ d`3 2 u´kq “ d `1 (ii) p2i´r2i`2´ d`3 2 s`kq`p2i`2´t2i`2´ d`3 2 u´kq “ d `1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is obvious that (i) and (ii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The claim follows from direct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ The next statement will be the key ingredient to show that h˚prPBpσd`1qq is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let b “ pb0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',bdq be a symmetric and unimodal sequence of non-negative reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let c “ pc0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',cdq be defined by ci “ dÿ j“0 ˆd `1 2i´ j ˙ b j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, c0 ď c1 ď ¨¨¨ ď ct d`1 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We define r j as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Note that rj ě 0 if and only if j ě 2i ` 2 ´ d`3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For 0 ď i ă d`1 2 , we have ci`1 ´ci “ dÿ j“0 „ˆ d `1 2i`2´ j ˙ ´ ˆd `1 2i´ j ˙ȷ b j “ dÿ j“0 r jbj “ 2p2i`2´ d`3 2 q ÿ j“0 r jbj ` dÿ j“2p2i`2´ d`3 2 q`1 r jbj “ r2i`2´ d`3 2 s ÿ j“1 rt2i`2´ d`3 2 u` j ´ bt2i`2´ d`3 2 u` j ´br2i`2´ d`3 2 s´ j ¯ `r2i`2´ d`3 2 b2i`2´ d`3 2 ` dÿ j“2p2i`2´ d`3 2 q`1 r jbj, where for the last equality, we use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='9 and we set `r2i`2´ d`3 2 b2i`2´ d`3 2 “ 0 if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since b j ě 0 and rj ě 0 for j ě 2i` 2´ d`3 2 , it follows that the single summand and the sum in the last line of the above computation are both non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Concerning the first sum, the coefficients r2i`2´ d`3 2 ` j are non-negative and therefore, in order to show non-negativity of ci`1 ´ci, it suffices to show that for 1 ď j ď r2i`2´ d`3 2 s, we have bt2i`2´ d`3 2 u` j ě br2i`2´ d`3 2 s´ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This directly follows from the unimodality and symmetry of the sequence b if 2i`2´ d`3 2 ` j ď d`1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Assume 2i`2´ d`3 2 ` j ą d`1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since i ď d 2, we have d `1 2 ă 2i`2´ d `3 2 ` j ď d `2´ d `3 2 ` j “ d `1 2 ` j ď Zd `1 2 ^ ` j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using that b is symmetric and unimodal it follows that bt2i`2´ d`3 2 u` j ě bt d`1 2 u` j “ bd´t d 2u´ j ě br2i`2´ d`3 2 s´ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ We now recall and prove Theorem B: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (a) h˚ ´ PBpσd`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='t ¯ has only real roots if d P N is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (b) h˚ ´ PBpσd`1q ¯ is unimodal with peak in the middle for every d P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since PBpσd`1q has a regular unimodular triangulation T by Theorem A, we have h˚pPBpσd`1qq “ hpT q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If d is odd, such a triangulation is given by esdd`2 ´ ∆ d`1 2 ¯ ˚esdd`2 ´ ∆ d´1 2 ¯ and its h-polynomial equals h ´ esdd`2 ´ ∆ d`1 2 ¯ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='t ¯ ¨ h ´ esdd`2 ´ ∆ d´1 2 ¯ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='t ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since both factors are real-rooted by [22, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4], so is h˚ ´ PBpσd`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='t ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 18 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE Suppose that d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='8, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2) and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10, we get that h˚pPBpσd`1qq is increasing up to the middle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', h˚ 0pPBpσd`1qq ď h˚ 1pPBpσd`1qq ď ¨¨¨ ď h˚ d 2 pPBpσd`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since, by Theorem A, PBpσd`1q has a regular unimodular triangulation it follows by [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3] that h˚pPBpσd`1qq is decreasing beyond the middle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', h˚ d 2 pPBpσd`1qq ě ¨¨¨ ě h˚ dpPBpσd`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ We would like to remark that even though the interior polytope QBpσd`1q has a symmteric h˚-vector, this is not true for PBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' OPEN PROBLEMS We end this article with some obvious directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We have initiated the study of Laplacian polytopes Ppiq ∆ by studying the special case that ∆ is the boundary of a pd ` 1)-simplex and i “ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is therefore natural to consider the following very general problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Study geometric and combinatorial properties of Ppiq ∆ for (classes of) simplicial complexes and general 0 ď i ď dim∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In particular: What is the normalized volume?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' When do these polytopes have a regular unimodular triangulation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' What properties do the h˚-vector and the h˚-polynomial have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In view of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5, a good starting point might be to study Ppdq ∆ for simplicial d-balls, since in this case we already know that Ppdq ∆ is a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' As part of this problem, it might be useful to consider how Laplacian polytopes change under certain operations on the simplicial complex, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', deletion/contraction of vertices, taking links, connected sums, joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We want to remark that for i “ 1 we get Laplacian simplices as studied in [6] and [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We have shown that PBpσd`1q has a regular unimodular triangulation by explicitly constructing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' However, for more general classes of simplicial complexes, a better approach might be to compute a Gr¨obner basis of the toric ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' This gives rise to the following problem whose solution would also contribute to Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='1: Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Describe a Gr¨obner basis of the toric ideal of Ppiq ∆ in terms of the combinatorics of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' When does there exist a squarefree Gr¨obner basis (giving rise to a regular unimodular triangulation)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We want to emphasize that the Laplacian polytope depends on the ordering of the vertices of ∆ (see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is therefore natural to ask the following question: Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Which orderings yield (up to unimodular or combinatorial equivalence) the same Laplacian polytope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' How many equivalence classes are there?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Apart from these more general problems, there are several open questions that are directly related to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='7, we have computed the normalized volume of PBpσd`1q explicitly and thereby have obtained a precise formula for the sum of the h˚-vector entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Using the explicit regular unimodular triangulation from Theorem A and inclusion-exclusion we can also express the h˚-polynomial as alternating sum, where all summands are products of h˚-polynomials of edgewise subdivisions of dilated simplices of varying dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Note that for d odd, we only have one summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' However, this does not yield a direct combinatorial interpretation of the entries of the h˚-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We therefore propose the following problem: LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 19 Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Find a combinatorial interpretation of the entries of the h˚-vector of PBpσd`1q (see Table 1 for the h˚-vectors if 1 ď d ď 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' d h˚ ` PBσd`1 ˘ 1 p1,2,0q 2 p1,10,5q 3 p1,22,78,24,0q 4 p1,131,726,419,19q 5 p1,149,4049,8558,3750,300,0q 6 p1,1478,38179,126372,85623,10422,69q 7 p1,926,157566,1135846,2188310,1150800,145600,3920,0q 8 p1,17617,1581403,6864069,43252570,31729319,6314903,239867,251q TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The h˚-vectors of PBσd`1 for d “ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Finally, in view of Theorem B (a), we have the following conjecture: Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Then, h˚ ´ PBpσd`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='x ¯ is real-rooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We have verified this conjecture computationally up to d “ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For this problem, we suspect that an approach via interlacing sequences might be helpful, but we have not been able to carry it out so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' APPENDIX We provide the missing parts of the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We recall some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We denote by E1 k P Zpk´1qˆk the pk ˆkq-identity matrix with its first row removed and by 1mˆn and 0mˆn the pm ˆ nq-matrices whose entries are all equal to 1 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Moreover, we denote by M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ the matrix whose columns are the vertices of d`2 2 ∆ d´2 2 ´ 1 d´2 2 ˆ d 2 in the obvious order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d ě 2 and for a fixed even integer i P rds, consider the facet F “ tx P Rd : 1⊺ odd ¨x´xi ď d 2u of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2, the vertices of F are tcpℓq : ℓ P rd ` 2szt1,i ` 2uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now consider the matrix B P Zdˆd whose ℓ-th column equals cp2ℓ`1q if 1 ď ℓ ď d 2 and cp2ℓ´dq if d 2 ` 1 ď ℓ ď i`d 2 and cp2ℓ`2´dq if i`d 2 ` 1 ď ℓ ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If we reorder the rows of B, by taking first the rows with odd index, increasingly, followed by the row with index i and then the remaining rows with even index, increasingly, we obtain a matrix S, which looks as follows: S “ ¨ ˚ ˝ E d 2 ¨ d`2 2 1 d 2 ˆ d 2 1 ¨¨¨ 1 0 ¨¨¨ 0 1 d´2 2 ˆ d 2 E1 d 2 ¨ d`2 2 ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Clearly, F – convpTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let U “ ¨ ˚ ˚ ˚ ˝ E1 d 2 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 E1 d 2 0 ¨¨¨ 0 ´1 0 ¨¨¨ 0 1 ¨¨¨ 1 ´1 0 ¨¨¨ 0 ˛ ‹‹‹‚P Zdˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 20 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE It is easy to see that U is unimodular and a direct computation shows that U ¨pS´1dˆdq “ ¨ ˚ ˚ ˚ ˚ ˝ M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 ¨¨¨ 0 1 ¨¨¨ 1 1 ¨¨¨ 1 1 ¨¨¨ 1 ˛ ‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since F – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates and by the definition of the join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We also note that the vertices of F corresponding to the vertices of the dilated simplices are tcp2ℓ`1q : 1 ď ℓ ď d 2u and tcp2ℓq : 2 ď ℓ ď d 2 `1, ℓ ‰ i`2 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For a fixed odd integer j P rds, consider the facet G “ tx P Rd : 1⊺ even ¨ x ´ xj ď d 2u of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2, the vertices of F are tcpℓq : ℓ P rd ` 2szt2, j ` 2uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now consider the matrix C P Zdˆd whose ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď j`1 2 and cp2ℓ`1q if j`3 2 ď ℓ ď d 2 and cp2ℓ`2´dq if d 2 `1 ď ℓ ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If we reorder the rows of C by taking first the rows with odd index k P rdszt ju, increasingly, followed by row j and then the rows with even index, increasingly, we obtain a matrix S, which looks as follows: S “ ¨ ˚ ˝ E1 d 2 ¨ d`2 2 1 d´2 2 ˆ d 2 0 ¨¨¨ 0 1 ¨¨¨ 1 1 d 2 ˆ d 2 E d 2 ¨ d`2 2 ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Clearly, G – convpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let U “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ E d 2 ´1 ˇˇˇˇˇ 0 d´2 2 ˆ d`2 2 0 d´2 2 ˆ d 2 ˇˇˇˇˇ E1 d 2 0 ¨¨¨ 0 ˇˇ 1 ¨¨¨ 1 0¨¨¨ 0 ´1 ˇˇˇ 1 ¨¨¨ 1 ˛ ‹‹‹‹‹‹‹‹‹‚ P Zdˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is easy to see that U is unimodular and a direct computation shows that U ¨pS´1dˆdq “ ¨ ˚ ˚ ˚ ˚ ˝ M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 ¨¨¨ 0 1 ¨¨¨ 1 1 ¨¨¨ 1 1 ¨¨¨ 1 ˛ ‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since G – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates and by the definition of the join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We also note that the vertices of G corresponding to the vertices of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d 2,ℓ ‰ j`1 2 u and tcp2ℓq : 2 ď ℓ ď d 2 `1,ℓ ‰ i`2 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' For fixed integers 1 ď i ă j ď d of different parity consider the facet H “ tx P Rd : xi`xj ě 1u of QBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Without loss of generality assume that i is odd j is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='2, the vertices of F are tcpℓq : ℓ P rd `2szti`2, j`2uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now consider the matrix D P Zdˆd whose ℓ-th column equals cp2ℓ´1q if 1 ď ℓ ď i`1 2 , cp2ℓ`1q if i`3 2 ď ℓ ď d 2, cp2ℓ´dq if d 2 `1 ď ℓ ď j`d 2 and cp2ℓ`2´dq if j`d 2 `1 ď ℓ ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If we reorder the rows of D by taking first the rows with odd index LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 21 k P rdsztiu, increasingly, followed by row i, followed by the rows with even index ℓ P rdszt ju, increasingly, followed by row j as the last row, we obtain a matrix S, which looks as follows: S “ ¨ ˚ ˚ ˚ ˝ d`2 2 ¨E1 d 2 11 d 2 0 ¨¨¨ 0 1 ¨¨¨ 1 11 d 2 d`2 2 ¨E1 d 2 1 ¨¨¨ 1 0 ¨¨¨ 0 ˛ ‹‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Clearly, H – convpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let U “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ E d 2 ´1 ˇˇˇˇˇ 0p d 2 ´1qˆp d 2 `1q 01 d 2 ˇˇˇˇˇE d 2 ´1 ˇˇˇˇˇ0p d 2 ´1qˆ1 ´e⊺ d ´pe d 2 `edq⊺ ˛ ‹‹‹‹‹‹‹‹‚ P Zdˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' It is easy to see that U is unimodular and a direct computation shows that U ¨pS´1dˆdq “ ¨ ˚ ˚ ˚ ˚ ˝ M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 d´2 2 ˆ d 2 0 d´2 2 ˆ d 2 M ´ d`2 2 ∆ d´2 2 ´1 d´2 2 ˆ d 2 ¯ 0 ¨¨¨ 0 1 ¨¨¨ 1 1 ¨¨¨ 1 1 ¨¨¨ 1 ˛ ‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since H – convpU ¨pS´1dˆdqq, the claim follows after projection on the first d ´1 coordinates and by the definition of the join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We also note that the vertices of H corresponding to the vertices of the dilated simplices are tcp2ℓ`1q : 0 ď ℓ ď d 2,ℓ ‰ i`1 2 u and tcp2ℓq : 1 ď ℓ ď d 2 ` 1,ℓ ‰ j`2 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='6 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let d ě 1 be an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We define vectors up1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', ud`2 P Rd`1 by upℓq k “ d ` 1 if k “ ℓ ´ 1 and upℓq k “ p´1qk`ℓ´1q, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='10, up1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=',upd`2q are the vertices of rPBpσd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' We now consider the matrix E P Zpd`1qˆpd`2q whose ℓ-th column equals bp2ℓ´1q if 1 ď ℓ ď d`3 2 and up2ℓ´pd`3qq if d`3 2 `1 ď ℓ ď d `2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' If we reorder the rows of E, by taking first the rows with even index and then the ones with odd index, increasingly, we obtain a matrix Q “ pqk,ℓq P Zpd`1qˆpd`2q with ‚ qk,k`1 “ d `1 for k P rd `1s, ‚ qk,ℓ “ 1 if k ď d`1 2 and ℓ ą d`3 2 , or k ą d`1 2 and ℓ ď d`3 2 ‚ qk,ℓ “ ´1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Clearly, rPBpσd`1q – convpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Let U “ ¨ ˚ ˚ ˚ ˚ ˚ ˝ E d`1 2 0 d`1 2 ˆ d`1 2 0 0 d´1 2 ˆ d`1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' E d´1 2 0 0 ¨¨¨ 0 1 ¨¨¨ 1 ˛ ‹‹‹‹‹‚ P Zpd`1qˆpd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 22 MARTINA JUHNKE-KUBITZKE AND DANIEL K ¨OHNE It is easy to see that U is unimodular and a direct computation shows that U ¨pQ´1pd`1qˆpd`2qq “ ¨ ˚ ˚ ˝ M ´ pd `2q∆ d`1 2 ´2¨1 ¯ 0 0 M ´ pd `2q∆ d´1 2 ´2¨1 ¯ 0 ¨¨¨ 0 1 ¨¨¨ 1 ˛ ‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Since rPBpσd`1q – convpU ¨pQ´1pd`1qˆpd`2qqq, the claim follows by definition of the join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' □ REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Adiprasito, Stavros A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Papadakis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Petrotou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Steinmeyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Beyond positivity in ehrhart theory, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Athanasiadis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' h˚-vectors, eulerian polynomials and stable polytopes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', 11(2):Research Paper 6, 13 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' (electronic), 2004/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Balletti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hibi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Meyer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Tsuchiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Laplacian simplices associated to digraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Arkiv f¨or matematik, 56, 12 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Barahona and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Mahjoub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' On the cut polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Mathematical Programming, 36:157–173, 06 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Beck and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Stapledon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', 264(1):195–207, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Braun and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Meyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Laplacian simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Advances in Applied Mathematics, 114, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Brenti and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Welker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' The veronese construction for formal power series and graded algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Advances in Applied Mathematics, 42(4):545–556, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Brun and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' R¨omer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Subdivisions of toric complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Journal of Algebraic Combinatorics, 21, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Bruns and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' R¨omer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' h-vectors of gorenstein polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series A, 114:65–76, 01 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [10] Alessio D’Al`ı, Martina Juhnke-Kubitzke, Daniel K¨ohne, and Lorenzo Venturello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' On the gamma-vector of symmetric edge polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Preprint arXiv: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='09835, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' De Loera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Rambau, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Structures for algorithms and applications, volume 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Diestel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Graph Theory, volume 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Edelsbrunner and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Grayson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Edgewise subdivision of a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' In Proceedings of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 24–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' ACM, New York, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ehrhart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Sur les poly`edres rationnels homoth´etiques `a n dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Paris, 254:616–618, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Goldberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combinatorial laplacians of simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A Senior Project submitted to The Division of Natural Science and Mathematics of Bard College, 5 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Grayson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Exterior power operations on higher K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' K-Theory, 3(3):247–260, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Gr¨unbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Convex Polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Graduate Texts in Mathematics, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Haase, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Paffenholz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Piechnik, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Existence of unimodular triangulations - positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Memoirs of the American Mathematical Society, 270, 05 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Herzog, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hibi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ohsugi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Edge Polytopes and Edge Rings, pages 117–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 09 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Dual polytopes of rational convex polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combinatorica, 2(2):237–240, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Higashitani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Jochemko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Michałek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Arithmetic aspects of symmetric edge polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Mathematika, 65:763–784, 05 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Jochemko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' On the real-rootedness of the veronese construction for rational formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' International Mathematics Research Notices, 2018:4780–4798, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Lov´asz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Plummer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Matching theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Annals of Discrete Mathematics, 29, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Meyer and Pllaha T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Laplacian simplices ii: A coding theoretic approach, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Mulas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Horak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Jost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology, pages 1–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Springer International Publishing, Cham, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Munkres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Elements of Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Addison Wesley Publishing Company, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ohsugi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Hibi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Special simplices and Gorenstein toric rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' A, 113(4):718– 725, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ohsugi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Tsuchiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Pq-type adjacency polytopes of join graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' 03 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Decompositions of rational convex polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Annals of Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=', 6:333–342, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' LAPLACIAN POLYTOPES OF SIMPLICAL COMPLEXES 23 [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Ziegler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Elements of Algebraic Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' Graduate Texts in Mathematics, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content=' UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY Email address: juhnke-kubitzke@uos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='de UNIVERSIT ¨AT OSNABR ¨UCK, INSTITUT F ¨UR MATHEMATIK, 49069 OSNABR ¨UCK, GERMANY Email address: dakoehne@uos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FJT4oBgHgl3EQfqCwO/content/2301.11602v1.pdf'} diff --git a/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/2301.01012v1.pdf.txt b/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/2301.01012v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4756574f1523d358e16a4042f233c6d113d12af --- /dev/null +++ b/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/2301.01012v1.pdf.txt @@ -0,0 +1,2638 @@ +arXiv:2301.01012v1 [math.AP] 3 Jan 2023 +NOVEL SPATIAL PROFILES OF POPULATION DISTRIBUTION OF +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION +INFECTION MECHANISM AND SMALL MOVEMENT RATE FOR +THE INFECTED INDIVIDUALS +RUI PENG, ZHI-AN WANG, GUANGHUI ZHANG AND MAOLIN ZHOU +Abstract. In this paper, we are concerned with two SIS epidemic reaction-diffusion +models with mass action infection mechanism of the form SI, and study the spatial profile +of population distribution as the movement rate of the infected individuals is restricted to +be small. For the model with a constant total population number, our results show that +the susceptible population always converges to a positive constant which is indeed the +minimum of the associated risk function, and the infected population either concentrates +at the isolated highest-risk points or aggregates only on the highest-risk intervals once the +highest-risk locations contain at least one interval. In sharp contrast, for the model with +a varying total population number which is caused by the recruitment of the susceptible +individuals and death of the infected individuals, our results reveal that the susceptible +population converges to a positive function which is non-constant unless the associated risk +function is constant, and the infected population may concentrate only at some isolated +highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or +occupy the whole habitat, depending on the behavior of the associated risk function and +even its smoothness at the highest-risk locations. Numerical simulations are performed to +support and complement our theoretical findings. +1. Introduction and existing results +The outbreak of the novel coronavirus disease 2019 (COVID-19) continues to spread +rapidly around the world, and it has caused tremendous impacts on public health and +the global economy. As it is commonly recognized, population movement is a significant +factor in the spread of many reported infectious diseases including COVID-19 [5, 9, 25], +Date: January 4, 2023. +2010 Mathematics Subject Classification. 35J57, 35B40, 35Q92, 92D30. +Key words and phrases. Reaction-diffusion SIS epidemic model; mass action infection mechanism; spa- +tial profile; small movement rate; heterogeneous environment. +R. Peng: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China. +Email: pengrui seu@163.com. +Z.-A. Wang: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung +Hom, Kowloon, Hong Kong. Email: mawza@polyu.edu.hk. +G. Zhang: School of Mathematics and Statistics, Huazhong University of Science and Technology, +Wuhan, 430074, China. Email: guanghuizhang@hust.edu.cn. +M. Zhou: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China. +Email: zhouml123@nankai.edu.cn. +R. Peng was partially supported by NSF of China (Nos. 12271486, 12171176), Z.-A. Wang was partially +supported by the Hong Kong Scholars Program (Project ID P0031250) and an internal grant from the +Hong Kong Polytechnic University (Project ID P0031013), G. Zhang was partially supported by NSF of +China (No. 12171176, 11971187) and the Fundamental Research Funds for the Central Universities (No. +5003011008), and M. Zhou was partially supported by the Nankai Zhide Foundation and NSF of China +(No. 11971498). +1 + +2 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +and the lockdown and quarantine has turned out to be one of the most effective measures +to reduce or even eliminate the infection [30, 60]. On the other hand, the importance of the +population heterogeneity has also been observed in the complicated dynamical behaviour +of the transmission of COVID-19 [7, 8, 17]. +To gain a deeper understanding of the impact of population movement and heterogeneity +on the transmission of epidemic diseases from a mathematically theoretical viewpoint, in +the present work we are concerned with two SIS reaction-diffusion systems with mass action +infection mechanism in a heterogeneous environment. We aim to study the spatial profile of +population distribution as the movement rate of the infected individuals is controlled to be +sufficiently small. Such kind of information may be useful for decision-makers to predict the +pattern of disease occurrence and henceforth to conduct more effective strategies of disease +eradication. The mass action infection mechanism was first proposed in the seminal work +of Kermack and McKendrick [26], in which the disease transmission was assumed to be +governed by a bilinear incidence function SI (one may also refer to [27–29] or [54]). The +systems under consideration in this paper are possibly the simplest yet basic SIS epidemic +models. +The first model we will deal with in this work is the following coupled reaction-diffusion +equations in one-dimensional space: + + + + + + + + + + + +St − dSSxx = −β(x)SI + γ(x)I, +0 < x < L, +t > 0, +It − dIIxx = β(x)SI − γ(x)I, +0 < x < L, +t > 0, +Sx = Ix = 0, +x = 0, L, +t > 0, +S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0, +0 < x < L. +(1.1) +Here, S(x, t) and I(x, t) are respectively the population density of the susceptible and in- +fected individuals at position x ∈ [0, L] and time t; the homogeneous Neumann boundary +condition means that no population flux crosses the boundary x = 0, L; dS and dI are pos- +itive constants measuring the motility of susceptible and infected individuals, respectively; +and the functions β and γ are H¨older continuous positive functions in [0, L] representing +the disease transmission rate and the disease recovery rate, respectively. +Integrating the sum of the equations of (1.1), combined with the homogeneous Neumann +boundary value conditions, we observe that +� L +0 +(S(x, t) + I(x, t)) dx = +� L +0 +(S0(x) + I0(x)) dx =: N, +∀t ≥ 0. +Thus, the total population number in (1.1) is conserved all the time. +The system (1.1) was investigated in the recent works [16, 65, 68]; in particular, when +the movement of either the susceptible or infected population is restricted to be slow, the +authors explored the profile of the spatial distribution of the disease modelled by (1.1). The +understanding of such a profile amounts to determine the behavior of the so-called endemic +equilibrium with respect to the small diffusion rate dS or dI. The endemic equilibrium of + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +3 +(1.1) is a positive steady state solution, which satisfies the following elliptic system: + + + + + + + + + + + + + + + +−dSSxx = −β(x)SI + γ(x)I, +0 < x < L, +−dIIxx = β(x)SI − γ(x)I, +0 < x < L, +Sx = Ix = 0, +x = 0, L, +� L +0 +(S(x) + I(x)) dx = N. +(1.2) +According to [16, 65, 68], if minx∈[0,L] +γ(x) +β(x) < N +L , for any small dI > 0, (1.2) admits at least +one positive solution (S, I), which is called an endemic equilibrium (EE for abbreviation) +in terms of epidemiology; moreover, (S, I) satisfies S, I ∈ C2([0, L]) and S, I > 0 on [0, L]. +As remarked in [68], it is a challenging problem to study the spatial profile of EE of +(1.2) with respect to the small movement rate dI of the infected population; in [65], the +authors provided a first result in this research direction. Indeed, they proved the following +conclusion. +Theorem 1.1. [65, Theorem B] Assume that minx∈[0,L] +γ(x) +β(x) < N +L . Then as dI → 0, the +EE (S, I) of (1.2) satisfies (up to a sequence of dI) that S → ˆS uniformly on [0, L], where +ˆS ∈ C([0, L]) with min[0,L] +γ(x) +β(x) ≤ ˆS(x) ≤ max[0,L] +γ(x) +β(x), and I → µ weakly for some Radon +measure µ with nonempty support in the sense of +� L +0 +I(x)ζ(x)dx −→ +� +[0,L] +ζ(x)µ(dx), +∀ζ ∈ C([0, L]). +(1.3) +Obviously, Theorem 1.1 does not give a precise description for ˆS and µ and hence the +spatial profile of the susceptible and infected populations remains obscure. From the aspect +of disease control, it becomes imperative to know an informative behavior of µ. In this +paper, we manage to give a satisfactory result on the profile of ˆS and µ. +In (1.1), some important factors such as the death and recruitment rates of population +are ignored so that the total population number is a constant. In order to take into account +the death and recruitment rates of population, the following reaction-diffusion epidemic +system was proposed in [40]: + + + + + + + +St − dSSxx = Λ(x) − S − β(x)SI + γ(x)I, +0 < x < L, t > 0, +It − dIIxx = β(x)SI − [γ(x) + η(x)] I, +0 < x < L, t > 0, +Sx = Ix = 0, +x = 0, L, t > 0, +S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0, +0 < x < L. +(1.4) +The recruitment term of the susceptible population is represented by the function Λ(x)−S +so that the susceptible is subject to the linear growth/death ([4, 24]); η(x) accounts for +the death rate of the infected. Here, Λ, η are assumed to be positive H¨older continuous +functions on [0, L]. All other parameters have the same interpretation as in (1.1). +It is easily seen that the following elliptic problem +−dSSxx = Λ(x) − S, +0 < x < L; +Sx(0) = Sx(L) = 0 +(1.5) + +4 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +admits a unique positive solution ˜S. Then ( ˜S, 0) is a unique disease-free equilibrium of +(1.4). An EE of (1.4) satisfies the following ODE system: + + + + + + + +−dSSxx = Λ(x) − S − β(x)SI + γ(x)I, +0 < x < L, +−dIIxx = β(x)SI − [γ(x) + η(x)] I, +0 < x < L, +Sx = Ix = 0, +x = 0, L. +(1.6) +As one of the main results of [40], the following conclusion on the profile of EE of (1.6) +with respect to small dI was established. +Theorem 1.2. [40, Theorem 3.2] Assume that the set {x ∈ [0, L] : β(x) ˜S(x) > γ(x) + +η(x)} is non-empty. As dI → 0, then any EE (S, I) of (1.6) satisfies (up to a subsequence +of dI) that S → ˆS +uniformly on [0, L], where ˆS ∈ C([0, L]) and ˆS > 0 on [0, L], and +� L +0 Idx → ˆI for some positive constant ˆI. +As in Theorem 1.1, Theorem 1.2 does not characterize the precise distribution of the +susceptible and infected populations. In this paper, we will also provide a clear picture of +the population distributions for (1.6) as the movement rate dI tends to zero. It turns out +that the spatial profiles of the disease distribution modelled by (1.2) and (1.6) are rather +different. +The rest of paper is organized as follows. In section 2, we state the main theoretical +results, and section 3 is devoted to their proofs. In section 4, we carry out the numerical +simulations and discuss the implications of our results in terms of disease control. In the +appendix, we recall some known facts which will be used in the paper. +2. Statement of main results +In this section, we state the main findings of this paper on models (1.2) and (1.6). +To proceed, we underline some terminologies frequently used throughout the paper. For +model (1.2), we call γ(x) +β(x) the risk function, and call each element of the set +� +x ∈ [0, L] : +γ(x) +β(x) = minx∈[0,L] +γ(x) +β(x) +� +the highest-risk point (or location). Similarly, for model (1.6), we +call γ(x)+η(x) +β(x) +the risk function, and call each element of the set +� +x ∈ [0, L] : +γ(x)+η(x) +β(x) += +minx∈[0,L] +γ(x)+η(x) +β(x) +� +the highest-risk point (or location). +2.1. Results for model (1.2). For the sake of convenience, we set +k(x) = γ(x) +β(x), +kmin = min +x∈[0,L] k(x), +and +Θk = +� +x ∈ [0, L] : k(x) = kmin +� +. +We note that when the risk function k(x) = k is a positive constant, it follows from [65] +that S(x) ≡ k is a constant, and in turn by the equation of I, we immediately see that +I = N +L − k is also a positive constant provided that k < N +L . In what follows, we do not +consider such a trivial case and assume that k(x) is non-constant on [0, L]. +We now state our main result on the asymptotic behavior of any EE (S, I) of (1.2) as +dI → 0 as follows. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +5 +Theorem 2.1. Assume that k(x) is non-constant and kmin < N +L . Then as dI → 0, the EE +(S, I) of (1.2) satisfies +S(x) → kmin +uniformly for x ∈ [0, L]. +(2.7) +The following assertions hold for the asymptotic behavior of I. +(i) If Θk = {x0}, then we have +I(x) → (N − Lkmin)δ(x0) weakly in the sense of (1.3), +where δ(x0) is the Dirac measure centered at x0. Moreover, I(x) → 0 locally uni- +formly in [0, L] \ {x0}. +(ii) If Θk = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L, then we have +I(x) → 0 +uniformly on [0, ̺1] ∪ [̺2, L], +and +I(x) → ˆI(x) +uniformly for x ∈ [̺1, ̺2], +where ˆI ∈ C2([̺1, ̺2]), ˆI > 0 in (̺1, ̺2), and ˆI is the unique positive solution of + + + + + + + + + +−ˆIxx = β(x) +dS (ˆa − ˆI)ˆI, +̺1 < x < ̺2, +ˆI = 0, +x = ̺1, ̺2, +� ̺2 +̺1 +ˆI dx = N − Lkmin, +(2.8) +where the positive constant ˆa is uniquely determined by the integral constraint in +(2.8). +Regarding Theorem 2.1, we would like to make some comments in order as follows. +Remark 2.1. In addition to the two cases treated in Theorem 2.1, we can handle some +more general cases. In particular, we would like to make the following comments. +(i) If the set Θk contains only finitely many isolated points, say {xi}j +i=1 for some j ≥ 2, +then one can slightly modify the proof of Theorem 2.1(i) to show that S → kmin +uniformly on [0, L], and I → 0 locally uniformly in [0, L] \ ({xi}j +i=1), and +I(x) → +j +� +i=1 +ciδ(xi) weakly in the sense of (1.3), +where δ(xi) is the Dirac measure centered at xi and the nonnegative constants ci +fulfill �j +i=1 ci = N − Lkmin. Nevertheless, we can not determine the exact values +of ci; in other words, as dI → 0, it is unclear to us whether I concentrates at all +xi (1 ≤ i ≤ j) or only some of them. The numerical results suggest that the former +alternative holds; see Figure 1 in section 4. +(ii) If the set Θk contains at least one proper interval of [0, L], by adapting the argument +of Theorem 2.1(ii), we can show that S → kmin uniformly on [0, L], and I → ˆI +uniformly on [0, L] with +ˆI = 0 +on [0, L] \ Θk, +� +Θk +ˆI dx = N − Lkmin. + +6 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +In particular, if Θk = +� �j∗ +i=1[̺i, ̺i] +� � � �{xi}j∗ +i=0 +� +for some j∗ ≥ 1, j∗ ≥ 0, then +we can prove that +ˆI = 0 +on [0, L] \ ( +j∗ +� +i=1 +(̺i, ̺i)), +and in (̺i, ̺i) (1 ≤ i ≤ j∗), either ˆI = 0 or ˆI > 0. Without loss of generality, +assuming that ˆI(x) > 0 for x ∈ � ˆj∗ +i=1(̺i, ̺i) for some 1 ≤ ˆj∗ ≤ j∗, then in each +such (̺i, ̺i), we can conclude that ˆI solves +� +−ˆIxx = β(x) +dS (ˆa − ˆI)ˆI, +̺i < x < ̺i, +ˆI = 0, +x = ̺i, ̺i, +where the positive constant ˆa is uniquely determined by +ˆj∗ +� +i=1 +� ̺i +̺i +ˆI dx = N − Lkmin. +However, it seems rather challenging to prove whether ˆI is positive on all intervals +(̺i, ̺i) (1 ≤ i ≤ j∗) or only on some of them. Our numerical results suggest that +the former alternative holds; see Figure 2 in section 4. +(iii) The assertion in (ii) above suggests that if the highest-risk locations contain at +least one interval, then the disease can not stay on any possible isolated highest-risk +points once the infected individuals move slowly. +Remark 2.2. In the case (ii) of Theorem 2.1, if ̺1 = 0 (or ̺2 = L), the results of Theorem +2.1 still hold true if we replace the Dirichlet boundary condition of ˆI in (2.8) at ̺1 = 0 (or +̺2 = L) by the Neumann boundary condition ˆIx(0) = 0 (or ˆIx(L) = 0). A similar remark +applies to the case discussed in Remark 2.1(ii) above. +Remark 2.3. After this paper was finished, we noticed the work [10] in which the authors +derived (2.7) and the convergence of the I-component in the case (i) of Theorem 2.1 in +any spatial dimension in a more general setting; see Theorem 2.5(i) there. However, their +result does not establish the convergence of the I-component within Θk in the case (ii) of +Theorem 2.1 nor in the more general case mentioned by Remark 2.1; on the other hand, +our proof of (2.7) and the convergence of the I-component outside of Θk is rather different +from that of [10]. +2.2. Results for model (1.6). We now turn to system (1.6). For the sake of simplicity, +we assume that Λ in (1.6) is a positive constant, and also denote +h(x) = γ(x) + η(x) +β(x) +, +hmin = min +x∈[0,L] h(x), +and +Θh = +� +x ∈ [0, L] : h(x) = hmin +� +. +Clearly, ˜S(x) = Λ. We also enhance the existence condition of EE of (1.6) in Theorem 1.2 +by imposing the following condition: +Λ > h(x) +for all x ∈ [0, L]. +(2.9) + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +7 +Now we can state our main findings on the asymptotic behavior of any EE (S, I) of (1.6) +as dI → 0. The first result reads as follows. +Theorem 2.2. Assume that (2.9) holds. As dI → 0, then any EE (S, I) of (1.6) satisfies +(up to a subsequence of dI) that S → ˆS +uniformly on [0, L], and I → µ weakly in the +sense of (1.3), where µ is some Radon measure and ˆS solves weakly in W 1,2(0, L) the free +boundary problem: +−dS ˆSxx = Λ − ˆS − η(x)µ({x}) +�� +{x∈[0, L]: ˆS(x)=h(x)}, +x ∈ (0, L). +(2.10) +Here, µ({x}) +�� +{x∈[0, L]: ˆS(x)=h(x)} is the restriction of µ on the set {x ∈ [0, L] : ˆS(x) = h(x)}; +otherwise, µ({x}) = 0. Moreover we have the following properties for µ and ˆS. +(i) The Radon measure µ satisfies +µ({x ∈ [0, L] : ˆS(x) ̸= h(x)}) = 0, +µ({x ∈ [0, L] : ˆS(x) = h(x)}) > 0. +(2.11) +(ii) The function ˆS ∈ C([0, L]) satisfies +hmin ≤ ˆS(x) ≤ h(x), +∀x ∈ [0, L], +(2.12) +Θh ⊂ +� +x ∈ [0, L] : +ˆS(x) = h(x) +� +; +(2.13) +If x1, x2 ∈ Θh with x1 < x2 and (x1, x2) ∩ Θh = ∅, then +hmin < ˆS(x), +∀x ∈ (x1, x2). +(2.14) +Theorem 2.2 asserts that ˆS touches h at all highest-risk points. In what follows, our goal +is to examine the properties ˆS for some specific risk function h, which in turn provides us +with a more precise description of the profile of µ. Indeed, we can obtain the following +result for (1.6). +Theorem 2.3. Let ˆS and µ be given as in Theorem 2.2. Assume that h ∈ C2([0, L]) and +(2.9) holds. The following assertions hold. +(i) If −dShxx ≤ Λ − h in (0, L), hx(0) ≥ 0 and hx(L) ≤ 0, then we have +ˆS(x) = h(x), +∀x ∈ [0, L], +(2.15) +µ({x}) = Λ − h(x) + dShxx(x) +η(x) +, +a.e. for x ∈ (0, L). +(2.16) +(ii) If hx is non-decreasing on [0, L] and Θh = {τ0} for some 0 ≤ τ0 ≤ L, then the +following assertions hold. +(a) When 0 < τ0 < L, we have +ˆS(x) = h(x), +∀x ∈ [τ1, τ2], +(2.17) +and in [0, τ1) ∪ (τ2, L], ˆS < h satisfies + + + + + + + +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (0, τ1) ∪ (τ2, L), +ˆSx(0) = 0, +ˆSx(L) = 0, +ˆS(τ1) = h(τ1), +ˆS(τ2) = h(τ2), +(2.18) +and µ satisfies +µ({x}) = Λ − h(x) + dShxx(x) +η(x) +, +a.e. for x ∈ (τ1, τ2), +(2.19) + +8 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +µ({x}) = 0, +∀x ∈ [0, τ1) ∪ (τ2, L], +(2.20) +where the numbers τ1, τ2 with 0 < τ1 < τ0 < τ2 < L are uniquely determined +by +e2d−1/2 +S +τ1 − 1 +e2d−1/2 +S +τ1 + 1 += −d1/2 +S hx(τ1) +Λ − h(τ1) , +e2d−1/2 +S +(τ2−L) − 1 +e2d−1/2 +S +(τ2−L) + 1 += −d1/2 +S hx(τ2) +Λ − h(τ2) . +(2.21) +(b) When τ0 = L, then we have the following assertions. +(b-1) If +e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +> − +d1/2 +S +hx(L) +Λ−h(L) , then (2.17) and (2.19) hold with [τ1, τ2] re- +placed by [τ1, L], µ([0, τ1)) = 0, and on [0, τ1], ˆS satisfies +� +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (0, τ1), +ˆSx(0) = 0, +ˆS(τ1) = h(τ1), +(2.22) +where 0 < τ1 < L is uniquely determined by the first equation in (2.21). +(b-2) If e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +≤ − +d1/2 +S +hx(L) +Λ−h(L) , then ˆS is the unique positive solution of +� +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (0, L), +ˆSx(0) = 0, +ˆS(L) = h(L), +(2.23) +and µ satisfies +µ([0, L)) = 0, +µ({L}) = ΛL − +� L +0 ˆS(x)dx +η(L) +. +(2.24) +(c) When τ0 = 0, then we have the following assertions. +(c-1) If e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +> +d1/2 +S +hx(0) +Λ−h(0) , then (2.17) and (2.19) hold with [τ1, τ2] replaced +by [0, τ2], µ((τ2, L]) = 0, and on [τ2, L], ˆS satisfies +� +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (τ2, L), +ˆSx(L) = 0, +ˆS(τ2) = h(τ2), +(2.25) +where 0 < τ2 < L is uniquely determined by the second equation in (2.21). +(c-2) If e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +≤ +d1/2 +S +hx(0) +Λ−h(0) , then ˆS is the unique positive solution of +� +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (0, L), +ˆSx(L) = 0, +ˆS(0) = h(0), +(2.26) +and µ satisfies +µ((0, L]) = 0, +µ({0}) = ΛL − +� L +0 ˆS(x)dx +η(0) +. +(2.27) +(iii) If hx is non-decreasing on [0, ̺1]∪[̺2, L] and Θh = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L, +then all the assertions in (ii)-(a) above hold, where the numbers τ1, τ2 satisfying +0 < τ1 < ̺1 < ̺2 < τ2 < L are uniquely determined by (2.21). + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +9 +For model (1.2), our result shows that the infected population concentrates or aggregates +only at the highest-risk locations. In sharp contrast, for model (1.6), our result suggests +that the disease will occupy a neighborhood of the interior highest-risk locations or even +occupy the whole habitat [0, L], or concentrates only at the boundary highest-risk location, +depending on the risk function h. More detailed discussions on the implications of our +theoretical results, along with numerical simulations, will be given in section 4. +We would like to make some remarks on Theorem 2.3 as follows. +Remark 2.4. It is worth mentioning that all the statements in Theorem 2.3 except the +expression (2.19) for the Radon measure µ remain true provided that the risk function +h ∈ C1([0, L]). Such a comment also applies to Lemmas 3.1-3.4 in the forthcoming section. +Remark 2.5. +(i) It is clear that Theorem 2.3(i) holds if h < Λ is a constant or more +generally h is a unique solution to the following problem: +� +−dShxx = Λ − h, +x ∈ (0, L), +h(0) = σ1, +h(L) = σ2, +where 0 < σ1, σ2 < Λ. +When hx(0) > 0, the change of the derivatives from Sx(0) = 0 to ˆSx(0) = hx(0) > +0 would suggest that I should experience the concentration phenomenon at x = 0 +(that is, I(0) → ∞) as dI → 0. The same remark applies to the case of hx(L) < 0. +(ii) In contrast to Theorem 2.3(i), it is easily seen that ˆS ̸≡ h on [0, L] provided that +−dShxx(x∗) > Λ − h(x∗) for some x∗ ∈ (0, L). +(iii) Clearly, the assertions of Theorem 2.3(ii)-(b1) hold if hx(L) = 0 and the assertions +of Theorem 2.3(ii)-(c1) hold if hx(0) = 0. +(iv) In a general case that Θh contains an interior isolated point and hx is non-decreasing +in a neighbourhood of such a point, we can conclude that (2.15) and (2.16) hold in +some neighbourhood of this point; if Θh contains an interval, a similar conclusion +also holds. See Lemma 3.1 and Lemma 3.3 below. +3. Proof of main results: Theorems 2.1, 2.2 and 2.3 +This section is devoted to the proof of Theorems 2.1, 2.2 and 2.3. +3.1. Proof of Theorem 2.1. In this subsection, we present the proof of Theorem 2.1. +Proof of Theorem 2.1. First of all, we recall that for any EE (S, I) of (1.2), from [65] (see +(3.3) there), the following holds: +kmin ≤ S(x) ≤ max +[0,L] k(x), +∀x ∈ [0, L]. +(3.1) +By the positivity of I and the uniqueness of the principal eigenvalue, it is clear from the +equation of I that +λ1(dI, γ − βS) = 0, +∀dI > 0, +where λ1(dI, γ − βS) is defined as in the appendix. Using Theorem 1.1, as dI → 0 (up to a +subsequence), we see that S → ˆS uniformly on [0, L] for some positive function ˆS. Hence, +by Lemma 5.1 in the appendix and the continuous dependence of the principal eigenvalue +on the weight function γ − βS, we have +0 = lim +dI→0 λ1(dI, γ − βS) = min +x∈[0,L][γ(x) − β(x) ˆS(x)]. + +10 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +This obviously implies that +ˆS(x) ≤ k(x), +∀x ∈ [0, L] and +ˆS(y0) = k(y0) +(3.2) +for some y0 ∈ [0, L]. +From Theorem 1.1, we recall that I → µ weakly for some Radon measure µ with +µ([0, L]) > 0 in the following sense +� L +0 +I(x)ζ(x)dx → +� L +0 +ζ(x)µ(dx), +∀ζ ∈ C([0, L]), +as dI → 0. +(3.3) +We now integrate the first equation in (1.2) by parts over [0, L] and use the boundary +conditions to deduce that +� L +0 +[β(x)S(x) − γ(x)]I(x)dx = 0, +∀dI > 0. +(3.4) +Letting dI → 0 in (3.4), combined with (3.3) and the fact that S → ˆS uniformly on [0, L] +as dI → 0, we infer that +� +[0,L] +[β(x) ˆS(x) − γ(x)]µ(dx) = 0, +(3.5) +which, together with (3.2), gives +� +{x∈[0,L]: ˆS(x) 0. +(3.7) +In view of (3.4) and +� L +0 (S(x) + I(x)) dx = N, for any dI > 0 we have +� L +0 +S(x)I(x)dx ≤ +1 +min[0,L] β(x) +� L +0 +γ(x)I(x)dx ≤ max[0,L] γ(x) +min[0,L] β(x) N, +∀dI > 0. +(3.8) +Then, applying the L1-theory for elliptic equation (see Lemma 5.2 in the appendix) to the +S-equation, one sees that for any 1 ≤ r < ∞, +∥S∥W 1,r(0,L) ≤ C, +∀dI > 0. +(3.9) +Hereafter, C or C(ǫ) is a positive constant independent of dI > 0 but may be different +from place to place. Taking r = 2 in (3.9), we note that W 1,2(0, L) is a Hilbert space and +W 1,2(0, L) is compactly embedded to C([0, L]). Thus, we may assume that S → ˆS weakly +in W 1,2(0, L) and S → ˆS uniformly on [0, L] as dI → 0. Now, for any ζ ∈ W 1,2(0, L) (and +so ζ ∈ C([0, L])), we get from the S-equation that +dS +� L +0 +Sx(x)ζx(x)dx = +� L +0 +[−β(x)S(x) + γ(x)]I(x)ζ(x)dx, +∀dI > 0. +(3.10) +By virtue of (3.3), (3.6) and (3.7), we can send dI → 0 in (3.10) to obtain +dS +� L +0 +ˆSx(x)ζx(x)dx = 0, +∀ζ ∈ W 1,2(0, L). + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +11 +This means that ˆS is a weak (and then a classical) solution of +−uxx(x) = 0, +x ∈ (0, L); +ux(0) = ux(L). +Consequently, ˆS must be a positive constant. It then follows from (3.2) that ˆS = kmin, +and so S(x) → kmin uniformly on [0, L]. +In the sequel, we are going to determine the limit of I. +We first consider case (i): +Θk = {x0} is a singleton. By what was proved above, it is easily seen that +I(x) → (N − Lkmin)δ(x0) +weakly in the sense of (1.3), +where δ(x0) is the Dirac measure centered at x0. +It remains to show I(x) → 0 locally uniformly in [0, L] \ {x0}. We only consider the +case of x0 ∈ (0, L), and the case x0 = 0 or L can be handled similarly. Since S(x) → kmin +uniformly on [0, L], by the definition of kmin, we know from the I-equation that, given small +ǫ > 0, Ixx > 0 on [0, x0 −ǫ]∪[x0 +ǫ, L] as long as dI is small enough. As Ix(0) = Ix(L) = 0, +I is increasing in [0, x0 − ǫ] while is decreasing in [x0 + ǫ, L]. Thus, due to the arbitrariness +of ǫ, it readily follows from (3.6) that I(x) → 0 locally uniformly in [0, x0) ∪ (x0, L], as +claimed. +We next consider case (ii): Θk = [̺1, ̺2] ⊂ (0, L). +First of all, we can assert that +I(x) → 0 locally uniformly in [0, L] \ [̺1, ̺2] by a similar argument as in case (i). In what +follows, we will analyze the limiting behavior of I in the interval [̺1, ̺2]. To this end, let +us introduce the following function +w(x) = S(x) − kmin +dI +, +x ∈ [0, L]. +Due to (3.1), w ≥ 0 on [0, L]. In addition, by our assumption, one notices that w solves +−dSwxx(x) = −β(x)Iw, +x ∈ [̺1, ̺2], +(3.11) +and I satisfies +−Ixx(x) = β(x)wI, +x ∈ [̺1, ̺2]. +(3.12) +Since +� L +0 I(x)dx ≤ N, for any small ǫ > 0, Lemma 5.3(b) in the appendix can be applied +to (3.11) to assert that +max +x∈[̺1+ǫ,̺2−ǫ] w(x) ≤ C(ǫ) +min +x∈[̺1+ǫ,̺2−ǫ] w(x). +(3.13) +We now claim that w is uniformly bounded on [̺1 +ǫ, ̺2 −ǫ] for all small dI > 0. Other- +wise, there is a sequence of dI, labelled by itself for simplicity, such that the corresponding +solution sequence {(w, I)} satisfies +max +x∈[̺1+ǫ,̺2−ǫ] w(x) → ∞, +as dI → 0. +(3.14) +By (3.13), w → ∞ uniformly on [̺1 + ǫ, ̺2 − ǫ] as dI → 0. To produce a contradiction, +let us denote λD +1 to be the principal eigenvalue of the following eigenvalue problem with +Dirichlet boundary conditions: +� +−ϕxx = λϕ, +x ∈ (̺1 + ǫ, ̺2 − ǫ) +ϕ(̺1 + ǫ) = ϕ(̺2 − ǫ) = 0. +(3.15) +Apparently, λD +1 > 0. For all small dI > 0, by (3.14) we may assume that +β(x)w(x) > 2λD +1 +on [̺1 + ǫ, ̺2 − ǫ]. + +12 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +Thus, it follows from (3.12) that I ∈ C2([0, L]) is a positive and strict supersolution of the +following operator in the sense of [57, Definition 2.1]: +� +Lu := −uxx − 2λD +1 u, +x ∈ (̺1 + ǫ, ̺2 − ǫ), +∀u ∈ C2([0, L]), +u(̺1 + ǫ) = u(̺2 − ǫ) = 0. +By means of [57, Proposition 2.1], the principal eigenvalue, denoted by ˜λD +1 , of the eigenvalue +problem +� +Lϕxx = λϕ, +x ∈ (̺1 + ǫ, ̺2 − ǫ), +ϕ(̺1 + ǫ) = ϕ(̺2 − ǫ) = 0 +satisfies ˜λD +1 > 0. +On the other hand, the uniqueness of the principal eigenvalue of problem (3.15) implies +˜λD +1 + 2λD +1 = λD +1 , and so ˜λD +1 = −λD +1 < 0, leading to a contradiction. The previous claim +is thus verified. Due to the arbitrariness of ǫ, we have shown that w is locally uniformly +bounded in (̺1, ̺2) with respect to all small dI > 0. +Furthermore, by Lemma 5.2 in the appendix, it is easy to see from (3.12) that I is locally +uniformly bounded in (̺1, ̺2) independent of all small dI > 0. The standard regularity +theory for elliptic equations can be applied to (3.11) and (3.12), respectively to deduce that +w and I are locally bounded (independent of small dI) in (̺1, ̺2) in the usual C2+α-norm +for some α ∈ (0, 1). Then, by a diagonal argument, we may assume that +(w, I) → ( ˆw, ˆI) +in C2 +loc(̺1, ̺2), +as dI → 0. +Clearly, by (3.12), ( ˆw, ˆI) satisfies +−ˆIxx(x) = β(x) ˆw ˆI, +x ∈ (̺1, ̺2). +(3.16) +Furthermore, by adding (3.11) and (3.12), one easily sees that ( ˆw, ˆI) solves +−(dS ˆw + ˆI)xx = 0 +in (̺1, ̺2). +This indicates that +dS ˆw(x) + ˆI(x) = ˆa + ˆbx, +x ∈ (̺1, ̺2) +(3.17) +for some constants ˆa, ˆb. +In what follows, we aim to determine ˆa and ˆb. By a simple observation, (w, I) satisfies +� +−(dSw + I)xx = 0, +x ∈ (0, L), +(dSw + I)x = 0, +x = 0, L. +Thus, dSw + I = cdI is a positive constant on [0, L] for any dI > 0. Recall that w, I are +locally uniformly bounded in (̺1, ̺2). Hence, as dI → 0, we may assume that +dSw + I = cdI → ˆc ∈ [0, ∞) +uniformly on [0, L]. +From (3.17) it follows that ˆc = ˆa and ˆb = 0. In addition, our analysis indicates that w and +I are uniformly bounded on [0, L]. Precisely, it holds that +w(x), +I(x) ≤ C, +∀x ∈ [0, L]. +(3.18) + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +13 +We now use the equation of I, together with the fact of w, I ≥ 0 and the definition of +k, to find that +−Ixx = β(x) [S − k(x)] I +dI += β(x) +�S − kmin +dI ++ kmin − k(x) +dI +� +I +(3.19) +≤ β(x)wI, +x ∈ (0, L). +Multiplying both sides in (3.19) by I and integrating over (0, L), we obtain +� L +0 +(Ix)2dx ≤ +� L +0 +βwI2dx ≤ C +due to (3.18). This and (3.18) imply that ∥I∥W 1,2(0,L) ≤ C. Since W 1,2(0, L) is compactly +embedded to C([0, L]), we can assume that I → ˆI uniformly on [0, L]. By what was proved +before, ˆI = 0 on [0, ̺1] ∪ [̺2, L], and by (3.16) and (3.17), on [̺1, ̺2], ˆI solves +� +−ˆIxx = β(x) +dS (ˆa − ˆI)ˆI, +̺1 < x < ̺2, +ˆI = 0, +x = ̺1, ̺2. +(3.20) +Because of +� L +0 (S(x) + I(x)) dx = N and S → kmin uniformly on [0, L] as dI → 0, it is +easily seen that +� ̺2 +̺1 +ˆI dx = N − Lkmin > 0. +(3.21) +Thanks to the Harnack inequality (see Lemma 5.3(b)) and (3.21), we have from (3.20) that +ˆI > 0 in (̺1, ̺2). By (3.17) and the fact of ˆb = 0, clearly ˆa > 0. +It is well known that given ˆa > 0, the positive solution of problem (3.20), if it exists, +must be unique, denoted by ˆIˆa; moreover, if 0 < ˆa1 < ˆa2, then ˆIˆa1(x) < ˆIˆa2(x) for all +x ∈ (̺1, ̺2). With these facts, one can check that the positive constant ˆa is uniquely +determined by (3.21) in an implicit manner. Therefore, all the assertions in case (ii) have +been verified. The proof is thus complete. +□ +3.2. Proof of Theorem 2.2. We are now in a position to give the proof of Theorem 2.2. +Proof of Theorem 2.2. First of all, one can follow the analysis of Theorem 2.1, combined +with the result of Theorem 1.2 and its proof (see [40, Theorem 3.2]), to show that as +dI → 0, any EE (S, I) of (1.6) satisfies (up to a subsequence of dI) that S → ˆS weakly in +W 1,2(0, L) and uniformly on [0, L], and I → µ weakly in the sense of (1.3) for some Radon +measure µ and positive function ˆS ∈ W 1,2(0, L), and +0 < ˆS(x) ≤ h(x), +∀x ∈ [0, L], +(3.22) +and (2.11) hold. +For any ζ ∈ W 1,2(0, L) (and so ζ ∈ C([0, L])), we use the S-equation to obtain +dS +� L +0 +Sxζxdx = +� L +0 +[Λ − S − β(x)SI + γ(x)I]ζdx += +� L +0 +[Λ − S − η(x)I]ζdx − +� L +0 +[β(x)S − (γ(x) + η(x))]Iζdx +(3.23) + +14 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +for all dI > 0. In view of (3.22) and (2.11), we send dI → 0 to infer that +� L +0 +[β(x)S − (γ(x) + η(x)]Iζdx → +� +[0,L] +β(x)[ ˆS − h(x)]ζµ(dx) = 0. +Thus, by letting dI → 0, it follows from (3.23) that +dS +� L +0 +ˆSxζxdx = +� L +0 +(Λ − ˆS)ζdx − +� L +0 +η(x)ζµ(dx), +∀ζ ∈ W 1,2(0, L). +(3.24) +Together with (2.11), this means that ˆS ∈ W 1,2(0, L) is a weak solution of (2.10). +In what follows, for a general positive H¨older continuous function h, we will prove three +claims: +Claim 1. If the minimum of h is attained at x = 0 (resp. at x = L), then ˆS must touch +h at this point; that is, ˆS(0) = h(0) = hmin (resp. ˆS(L) = h(L) = hmin). +We only handle the case that hmin is attained at x = 0, and the other case can be treated +similarly. Since ˆS ≤ h on [0, L], we suppose that ˆS(0) < h(0) and so ˆS(x) < h(x) on [0, ǫ0] +for some small ǫ0 > 0. Thus, from (2.10), we have −dS ˆSxx = Λ − ˆS, ∀x ∈ (0, ǫ0]. A simple +analysis shows that +ˆS(x) = c1ed−1/2 +S +x + c2e−d−1/2 +S +x + Λ, x ∈ (0, ǫ0] +(3.25) +for some constants c1, c2. On the other hand, using the S-equation, we integrate on [0, x] +to deduce +−Sx(x) = 1 +dS +� x +0 +[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy, x ∈ [0, ǫ0]. +(3.26) +From the proof of [40, Theorem 3.2], we know that +� L +0 +S(x)I(x)dx ≤ C, +� L +0 +I(x)dx ≤ C, and S(x) ≤ C, +∀x ∈ [0, L], +(3.27) +for some positive constant C, independent of dI > 0. +In the sequel, the constant C allows to vary from line to line but does not depend +on dI > 0. It immediately follows from (3.26) that Sx is uniformly bounded on [0, ǫ0], +independent of dI > 0. Note that µ([0, ǫ0]) = 0 due to (2.11), and I → µ weakly in the +sense of (1.3). Given any small ǫ > 0, we can find a small ρ > 0 so that for all 0 < dI ≤ ρ, +� ǫ0 +0 +I(x)dx ≤ ǫ + +� +[0,ǫ0] +µ(dx) = ǫ. +Now, for any x1, x2 ∈ [0, ǫ0] satisfying |x1 − x2| < ǫ, we have +��Sx(x1) − Sx(x2) +�� = 1 +dS +��� +� x2 +x1 +[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy +��� +≤ C|x1 − x2| + C +� x2 +x1 +I(y)dy +≤ C|x1 − x2| + C +� ǫ0 +0 +I(y)dy ≤ Cǫ +provided that 0 < dI ≤ ρ. This shows that Sx is equi-continuous on [0, ǫ0] once 0 < dI ≤ ρ. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +15 +Hence, we can apply the well-known Ascoli-Arzel`a theorem, up to a further subsequence +of dI, to conclude that Sx is uniformly convergent on [0, ǫ0] as dI → 0. As +S(x) − S(0) = +� x +0 +Sx(y)dy, +S → ˆS uniformly on [0, ǫ0], +it is easily seen that S → ˆS in C1([0, ǫ0]). Thus, ˆSx(0) = 0, and in turn we get from (3.25) +that c1 = c2. Because of ˆS ≤ h on [0, L] and the condition (2.9), we have c1 = c2 < 0, and +so +ˆSx(x) = c1[ed−1/2 +S +x − e−d−1/2 +S +x] < 0, +∀x ∈ (0, ǫ0]. +This means that ˆS is decreasing on [0, ǫ0]. +By virtue of h(0) ≤ h(x) for all x ∈ [0, L] and (2.11), one can extend the above analysis to +assert that ˆS is decreasing on [0, L] and so ˆS < h on [0, L]. This clearly gives µ([0, L]) = 0, +a contradiction with µ([0, L]) > 0 due to (2.11) again. Hence, we must have ˆS(0) = h(0) = +hmin. +Claim 2. If ˆS attains its local minimum at some x0 ∈ (0, L), then ˆS must touch h at +this point; that is, ˆS(x0) = h(x0). +Suppose that ˆS(x0) < h(x0) due to ˆS ≤ h. Thus, there is a small ǫ0 > 0 such that +ˆS(x) < h(x) for all x ∈ [x0 −ǫ0, x0 + ǫ0] ⊂ (0, L). By (2.11), µ([x0 −ǫ0, x0 + ǫ0]) = 0 and so +−dS ˆSxx = Λ − ˆS +on [x0 − ǫ0, x0 + ǫ0]. +As before, ˆS takes the form of (3.25) on [x0−ǫ0, x0+ǫ0] for some constants c1, c2. Obviously, +ˆSx(x0) = 0, which leads to c2 = c1e2d−1/2 +S +x0, and so c1 < 0. Thus, it holds that +ˆS(x) = c1[ed−1/2 +S +x + ed−1/2 +S +(2x0−x)] + Λ, +x ∈ [x0 − ǫ0, x0 + ǫ0] +(3.28) +for some constant c1 < 0. In view of (3.28), basic computation gives that ˆS is increasing +on [x0 −ǫ0, x0] while is decreasing on [x0, x0 + ǫ0]. This implies that x0 is a local maximum +of ˆS, a contradiction with our assumption. As a result, ˆS must touch h at x = x0. +Claim 3. If the minimum of h is attained at some point y0 ∈ (0, L), then ˆS must touch +h at this point; that is, ˆS(y0) = h(y0) = hmin. +Suppose that ˆS(y0) < h(y0) = hmin. There are two possible cases to happen in the +interval [0, y0): Case 1. ˆS never touches h in [0, y0), that is, ˆS < h in [0, y0); Case 2. ˆS +touches h somewhere in [0, y0). +When Case 1 occurs, by (2.11), we know that ˆS must touch h in (y0, L]. Let y1 be the +first point (from the left side) at which ˆS touches h. That is, y1 ∈ (y0, L], and +ˆS(x) < h(x), +∀x ∈ (y0, y1), +ˆS(y1) = h(y1) ≥ hmin. +On the other hand, since ˆS < h in [0, y0), we can follow the analysis used in Claim 1 to +show that ˆS is decreasing on [0, y1]. This is an obvious contradiction with ˆS(y0) < hmin ≤ +ˆS(y1). +When Case 2 occurs, we denote by y2 ∈ [0, y0) the first point from the right side such +that ˆS touches h in [0, y0). That is, +ˆS(x) < h(x), +∀x ∈ (y2, y0), +ˆS(y2) = h(y2) ≥ hmin. +If ˆS does not touch h in (y0, L]. By a similar argument to the proof of Claim 1 and +appealing to the fact of Sx(L) = 0, one sees that ˆS is increasing in (y2, L], leading to + +16 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +ˆS(y2) < ˆS(y0), which contradicts with ˆS(y2) ≥ hmin > ˆS(y0). +Hence, it is necessary +that ˆS touches h in (y0, L]. Let y3 be the first point where ˆS touches h in (y0, L]. Thus, +ˆS(x) < h(x) for all x ∈ (y0, y3) and ˆS(y3) = h(y3) ≥ hmin. Therefore, ˆS(x) < h(x) in +the interval (y2, y3), ˆS(y0) < h(y0) = hmin and ˆS(y2), ˆS(y3) ≥ hmin. This implies that on +[y2, y3], ˆS must attain its minimum at some y4 ∈ (y2, y3). By Claim 2, we can conclude +that ˆS(y4) = h(y4), a contradiction again. So far, we have verified Claim 3. +A similar reasoning as that of proving Claim 3 yields ˆS ≥ hmin on [0, L]. Thus (2.12) +holds. Thanks to Claim 1 and Claim 3, (2.13) is true. It is also apparent that Claim 2 +implies (2.14). The proof is now complete. +□ +3.3. Proof of Theorem 2.3. This subsection is devoted to the proof of Theorem 2.3. We +begin with some lemmas as follows. +Lemma 3.1. Assume that h ∈ C2([0, L]) and hx is non-decreasing in some neighborhood +of ̺0 ∈ Θh. Let ˆS and µ be given as in Theorem 2.2. Then there exists a small ǫ0 > 0 such +that +ˆS(x) = h(x), +∀x ∈ (̺0 − ǫ0, ̺0 + ǫ0) ∩ (0, L) +and +µ({x}) = Λ − h + dShxx +η(x) +, +a.e. for x ∈ (̺0 − ǫ0, ̺0 + ǫ0) ∩ (0, L). +Proof. By Theorem 2.2, we know that ̺0 ∈ {x ∈ [0, L] : +ˆS(x) = h(x)}. In the sequel, we +only consider the case of ̺0 ∈ (0, L), and the case of ̺0 = 0 or L can be treated similarly. +There are three possibilities we have to distinguish: +(1) ̺0 is an isolated point in the set {x ∈ [0, L] : ˆS(x) = h(x)}; +(2) ̺0 is an accumulation point in {x ∈ [0, L] : ˆS(x) = h(x)}; +(3) there is a small ǫ0 > 0 such that (̺0 − ǫ0, ̺0 + ǫ0) ⊂ {x ∈ [0, L] : ˆS(x) = h(x)}. +In what follows, we will exclude (1) and (2). If (1) happens, then +ˆS(̺0) = h(̺0) = hmin and +ˆS < h +in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0} +for some small ǫ1 > 0. +Note that µ([0, L]) < ∞. In view of this fact, one can apply the interior regularity theory +for elliptic equations to (2.10) and assert that ˆS ∈ C1(0, L). Clearly, hx(̺0) = 0. Since +ˆS(̺0) = h(̺0) = hmin and ˆS ≥ hmin due to (2.12), we infer that ˆSx(̺0) = 0. +On the other hand, by (2.10), ˆS satisfies +−dS ˆSxx = Λ − ˆS +in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0}. +(3.29) +By using ˆSx(̺0) = 0 and (3.29), one can easily see that ˆS is increasing in (̺0 −ǫ1, ̺0) while +ˆS is decreasing in (̺0, ̺0 + ǫ1). This implies that ˆS < hmin in (̺0 − ǫ1, ̺0 + ǫ1) \ {̺0}, +contradicting against (2.12). Thus, (1) is impossible. +If (2) happens, without loss of generality, we can find two points, say z1, z2 with ̺0 < +z1 < z2 < ̺0 + ǫ2 for some small ǫ2 > 0 such that +ˆS(z1) = h(z1), +ˆS(z2) = h(z2) and +ˆS < h in (z1, z2). +(3.30) +By taking ǫ2 to be smaller if necessary, we may assume that hx(z1) ≤ hx(z2) due to the +monotonicity of hx. Then, ˆS solves (3.29) in (z1, z2). By means of (3.30), we have +ˆSx(z1) ≤ hx(z1), +ˆSx(z2) ≥ hx(z2), + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +17 +leading to ˆSx(z1) ≤ ˆSx(z2). However, it follows from (3.29) that ˆSxx < 0 in (z1, z2), which +gives ˆSx(z1) > ˆSx(z2), a contradiction. Hence, the possibility (2) has been ruled out. +The above argument shows that (3) must hold. Now, since ˆS = h on [̺0 − ǫ0, ̺0 + ǫ0], +we can multiply both sides of (2.11) by any function ζ ∈ C2([0, L]) with compact support +on [̺0 − ǫ0, ̺0 + ǫ0] and integrate to conclude that +dShxx + Λ − h − η(x)µ({x}) = 0, +a.e. for x ∈ (̺0 − ǫ0, ̺0 + ǫ0), +which yields the expression of µ({x}). +□ +Lemma 3.2. Assume that h ∈ C2([0, L]), hx is non-decreasing on [0, L], and Θh = {τ0} +for some τ0 ∈ (0, L). Then there exist two numbers τ1, τ2 with 0 < τ1 < τ0 < τ2 < L such +that +ˆS(x) = h(x), +∀x ∈ [τ1, τ2], +(3.31) +and on [0, τ1) ∪ (τ2, L], ˆS satisfies + + + + + + + +−dS ˆSxx(x) = Λ − ˆS, +x ∈ (0, τ1) ∪ (τ2, L), +ˆSx(0) = ˆSx(L) = 0, +ˆS(τ1) = h(τ1), +ˆS(τ2) = h(τ2), +(3.32) +and µ satisfies +µ({x}) = Λ − h + dShxx +η(x) +, +a.e. for x ∈ (τ1, τ2), +(3.33) +µ({x}) = 0, +∀x ∈ [0, τ1) ∪ (τ2, L]. +(3.34) +Proof. Let us denote +τ1 = inf{τ ∈ [0, τ0) : +ˆS(x) = h(x), ∀x ∈ [τ, τ0]}, +τ2 = sup{τ ∈ (τ0, L] : +ˆS(x) = h(x), ∀x ∈ [τ0, τ]}. +Lemma 3.1 implies that τ1 and τ2 are well defined, and 0 ≤ τ1 < τ0 and τ0 < τ2 ≤ L. In +addition, (3.31) and (3.33) hold. +In light of the monotonicity of hx on [0, L], it is easily seen from the proof of Lemma +3.1 that if τ1 > 0, then ˆS can not touch h in (0, τ1) and in turn µ([0, τ1)) = 0; similarly, if +τ2 < L, ˆS can not touch h in (τ2, L) and so µ((τ2, L]) = 0. +If τ1 > 0 and τ2 < L, we can use the analysis as in the proof of Claim 1 of Theorem 2.2 to +conclude that ˆSx(0) = ˆSx(L) = 0. As µ([0, τ1) ∪ (τ2, L]) = 0, by (2.10) and the continuity +of ˆS, a standard compactness argument of elliptic equations yields that ˆS solves (3.32) in +the classical sense. Clearly, the solution of (3.32) is unique. +It remains to prove τ1 > 0 and τ2 < L. Note that the monotonicity of hx, Θh = {τ0} +and hx(τ0) = 0 ensure hx(0) < 0 and hx(L) > 0. Suppose that τ1 = 0, and so (3.31) holds +on [0, τ2]. Now, given τ ∈ (0, τ0], integrating the S-equation over [0, τ] and using (3.31), + +18 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +we infer that +−dSSx(τ −) = +� τ +0 +[Λ − S(y) − β(y)S(y)I(y) + γ(y)I(y)]dy += +� τ +0 +[Λ − S(y) − η(y)I(y)]dy + +� τ +0 +[γ(y) + η(y) − β(y)S(y)]I(y)dy +→ +� +[0,τ] +[Λ − h(y) − η(y)µ](dy) = +� τ +0 +[−dShxx(y)]dy += −dShx(τ) + dShx(0), +as dI → 0. +That is, for any τ ∈ (0, τ0], it holds that +Sx(τ −) → hx(τ) − hx(0), +as dI → 0. +Since hx is non-decreasing on [0, τ0] and hx(0) < 0, there exists a small ǫ0 > 0 such that +for all x ∈ [τ0 − ǫ0, τ0], +Sx(x−) ≥ 1 +2[hx(τ0) − hx(0)] = −1 +2hx(0) > 0 +for all small dI > 0. This implies that S is increasing on [τ0 − ǫ0, τ0] for all such small +dI > 0. In view of S → h uniformly on [τ0 − ǫ0, τ0] as dI → 0, h must be non-decreasing +on [τ0 − ǫ0, τ0], which is a contradiction against our assumption. Hence, τ1 > 0. Similarly, +we have τ2 < L by using hx(L) > 0. As a consequence, we deduce (3.34). The proof is +complete. +□ +Similar to the argument of Lemma 3.1, we can conclude the following result. +Lemma 3.3. Assume that h ∈ C2([0, L]), [̺1, ̺2] ⊂ Θh and hx is non-decreasing in some +neighborhood of ̺1, ̺2. Let ˆS and µ be given as in Theorem 2.2. Then there exists a small +ǫ0 > 0 such that +ˆS(x) = h(x), +∀x ∈ (̺1 − ǫ0, ̺2 + ǫ0) ∩ (0, L) +and +µ({x}) = Λ − h + dShxx +η(x) +, +a.e. for x ∈ (̺1 − ǫ0, ̺2 + ǫ0) ∩ (0, L). +Based upon Lemma 3.3, we can deduce the following result. +Lemma 3.4. Assume that h ∈ C2([0, L]), Θh = [̺1, ̺2] and hx is non-decreasing on +[0, ̺1] ∪ [̺2, L]. Let ˆS and µ be given as in Theorem 2.2. Then there exist two numbers +τ1, τ2 with 0 < τ1 < ̺1 < ̺2 < τ2 < L such that all the assertions in Lemma 3.2 hold. +With the aid of Lemmas 3.1-3.4, we are now in a position to prove Theorem 2.3. +Proof of Theorem 2.3. We first prove (i). We proceed indirectly and suppose that ˆS ̸≡ h +on [0, L]. Since ˆS touches h at least at the highest-risk point due to Theorem 2.2, we can +find an interval, denoted by [ℓ1, ℓ2] ⊂ [0, L], such that ˆS < h in (ℓ1, ℓ2) and at the boundary +point x = ℓi for i = 1, 2, either ˆS touches h (and so ˆS(ℓi) = h(ℓi)) or ˆS(ℓi) < h(ℓi). In +the latter case, it is necessary that ℓi = 0 or L, and the analysis to deduce Claim 1 in the +proof of Theorem 2.2 shows that ˆSx(ℓi) = 0. In any case, clearly ˆS satisfies +� +−dS ˆSxx = Λ − ˆS, +x ∈ (ℓ1, ℓ2), +ˆS(ℓi) = h(ℓi) or +ˆSx(ℓi) = 0, +i = 1, 2. +(3.35) + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +19 +Thus, by our assumption, h is a sub-solution to problem (3.35), and max{Λ, maxx∈[0,L] h(x)} +is a super-solution to (3.35). The well-known technique of sub-supersolution iteration, com- +bined with the uniqueness of solutions to problem (3.35), allows us to conclude that ˆS ≥ h +on [ℓ1, ℓ2], which leads to a contradiction. Hence, (2.15) holds, and (2.16) follows from +(2.10) by using a test-function argument similarly as before. Therefore, (i) is proved. +We next prove (ii). First of all, let us consider the case of τ0 ∈ (0, L). In this case, +the assertions (2.17)-(2.20) follow from Lemma 3.2, and it remains to show that τ1, τ2 are +uniquely determined by (2.21). As ˆS < h in [0, τ1), we have +ˆS(x) = c1[ed−1/2 +S +x + e−d−1/2 +S +x] + Λ, +∀x ∈ [0, τ1] +for some c1 < 0. It then follows from ˆS(τ1) = h(τ1) that +ˆS(x) = − +Λ − h(τ1) +ed−1/2 +S +τ1 + e−d−1/2 +S +τ1 (ed−1/2 +S +x + e−d−1/2 +S +x) + Λ, +∀x ∈ [0, τ1]. +Note that ˆS is convex while h is concave in the interval [0, τ1), and moreover, ˆS ∈ C1([0, L]) +as shown before. Hence, ˆS must be tangent to h at x = τ1, which in turn implies that τ1 is +the unique solution to ˆSx(τ1) = hx(τ1). Thus, τ1 is uniquely determined by the following +equation: +ed−1/2 +S +τ1 − e−d−1/2 +S +τ1 +ed−1/2 +S +τ1 + e−d−1/2 +S +τ1 = −d1/2 +S hx(τ1) +Λ − h(τ1) . +Similarly, τ2 is uniquely determined by the second equation of (2.21). The assertions in +(ii)-(a) have been verified. +We now consider the case of τ0 = L. In view of our assumption, clearly hx(0) < 0, +hx(L) ≤ 0, and ˆS(L) = h(L). +Assume that e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +> − +d1/2 +S +hx(L) +Λ−h(L) . In order to deduce the desired conclusion in (ii)- +(b1), one can follow the analysis of Lemmas 3.1 and 3.2. By checking the analysis there, +one just needs to show that τ1 defined in the assertion (ii)-(a) satisfies τ1 > 0. It turns +out that this amounts to rule out the situation that ˆS < h in [0, L). Suppose that ˆS < h +in [0, L). Then, arguing as before, we see that ˆS satisfies −dS ˆSxx = Λ − ˆS in (0, L) and +ˆSx(0) = 0. Solving this problem, we get ˆS(x) = c1[ed−1/2 +S +x + e−d−1/2 +S +x] + Λ for some c1 < 0. +It then follows from ˆS(L) = h(L) that +c1 = − +Λ − h(L) +ed−1/2 +S +L + e−d−1/2 +S +L. +Thus, we get +ˆSx(L) = −d−1/2 +S +(Λ − h(L))e2Ld−1/2 +S +− 1 +e2Ld−1/2 +S ++ 1 +. +By means of ˆS < h in [0, L) and ˆS(L) = h(L), it is necessary that ˆSx(L) ≥ hx(L), which +leads to +e2Ld−1/2 +S +− 1 +e2Ld−1/2 +S ++ 1 +≤ −d1/2 +S hx(L) +Λ − h(L) , +contradicting with our assumption. Therefore, τ1 > 0 must hold, and (ii)-(b1) is proved. + +20 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +Assume that e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +≤ − +d1/2 +S +hx(L) +Λ−h(L) . We first show that τ1 > 0 is impossible. On the +contrary, we suppose that τ1 > 0, and by the above analysis, τ1 must solve the first equation +of (2.21). Let us consider the following auxiliary problem: +f(τ) = e2τd−1/2 +S +− 1 +e2τd−1/2 +S ++ 1 ++ d1/2 +S hx(τ) +Λ − h(τ) , +τ ∈ [0, L]. +Since hx(τ) is non-decreasing, hx(τ) ≤ 0 on [0, L], h(τ) is non-increasing and h(τ) > Λ +on [0, L], it is easy to check that +d1/2 +S +hx(τ) +Λ−h(τ) is non-decreasing on [0, L]. Clearly, e2τd−1/2 +S +−1 +e2τd−1/2 +S ++1 +is +increasing on [0, L]. Therefore, f(τ) is increasing on [0, L]. Observe that f(L) = e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 ++ +d1/2 +S +hx(L) +Λ−h(L) ≤ 0 due to our assumption. This implies that the first equation of (2.21) has no +solution with respect to τ1 in [0, L), arriving at a contradiction. Hence, ˆS < h in [0, L) and +µ([0, L)) = 0, and so ˆS solves (2.23). It remains to prove (2.24). Indeed, by integrating +the sum of (1.6), we obtain +ΛL − +� L +0 +S(x)dx = +� L +0 +η(x)I(x)dx, +∀dI > 0. +Letting dI → 0 yields +ΛL − +� L +0 +ˆS(x)dx = +� +[0,L] +η(x)µ(dx) = η(L)µ({L}). +Here we used the fact of µ([0, L)) = 0. This gives (2.24), and thus the assertions in (ii)-(b2) +hold true. +The case of τ0 = 0 can be treated similarly as above. In view of Lemma 3.4 and the +analysis above, the assertions in (iii) follow immediately. The proof is completed. +□ +4. Discussions and numerical simulations +In recent years, many reaction-diffusion models have been proposed to investigate the +transmission dynamics of infectious diseases in a heterogeneous environment. For example, +models associated with (1.1) have been studied in [2, 16, 18, 19, 35, 36, 39, 40, 49–52, 55, +56, 59, 61, 68]. When the random diffusion is not present, such kind of models have been +explored in [1, 3, 20, 21, 38, 42, 62, 66, 67] and the references therein. One may also refer +to [14, 22, 23, 32, 33, 37, 41, 58, 63, 64, 70, 71] for relevant studies on the effect of random +diffusion on the dynamics of infectious diseases. +In this paper, we have investigated the steady state solution (namely, EE) of the SIS +epidemic reaction-diffusion models (1.2) and (1.6), in which the disease transmission is +governed by the well-known mass action infection mechanism, due to Kermack and McK- +endrick [26]. In model (1.2), the total population number of the susceptible and infected +populations is a constant, while in model (1.6), the total population number is varying, +which results from the inclusion of the recruitment for the susceptible population and the +death of the infected population. Our purpose is to determine the spatial profile of EE +as the movement rate dI of the infected individuals tends to zero. Such kind of informa- +tion may be useful for decision-makers to predict the pattern of disease occurrence and +henceforth to develop effective disease control strategies. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +21 +The previous works [39, 65] derived partial results regarding the spatial profile of EE +for (1.2) and (1.6) as dI → 0; however, a precise characterization for the distribution of +susceptible and infected populations is lacking. In the present work, we have provided a +comprehensive understanding on this issue. Below we shall summarize the main theoretical +findings of this paper, which will also be supported or complemented by our numerical +simulation results. +4.1. Profile of EE of model (1.2) as dI → 0. As pointed out before, when the risk +function k(x) = γ(x) +β(x) is a constant on the entire habitat [0, L], then (k, N +L −k) is the unique +EE of (1.2) provided that k < N +L , while ( N +L , 0) is the unique disease-free equilibrium of +(1.2) provided that k ≥ N +L . Indeed, in such a trivial case, one can follow the same analysis +as in [16, Theorem 4.1] to conclude that (k, N +L − k) is a global attractor of (1.1) if k < N +L +and ( N +L , 0) is a global attractor of (1.4) if k ≥ N +L . Thus, unless otherwise specified, we +always assume below that the risk function k(x) = γ(x) +β(x) is non-constant on [0, L]. +According to Theorem 2.1, for model (1.2), one finds that the susceptible population S +converges to the positive constant kmin as dI → 0, which means that the susceptible will +always distribute homogeneously on the entire habitat once the movement of the infected +individuals is restricted to be sufficiently small. Nevertheless, the profile of the infected +population I as dI → 0 crucially depends on the distribution behavior of the highest-risk +set Θk of the risk function k(x). More precisely, concerning the profile of I for model (1.2), +we have the following findings. +(i) If Θk consists of a single point, then I must concentrate only at such a highest-risk +point. +(ii) If Θk contains only multiple isolated points, it follows from Remark 2.1 that I will +also concentrate at least at one of those highest-risk points, and the disease will vanish +elsewhere. As shown in Figure 1(a)-(b)-(c) for three typical cases, our simulation results +suggest that I should concentrate at all such highest-risk points, though the population +number of I at each such highest-risk point may vary, depending on the functions β, γ. +(iii) If Θk contains at least one proper interval, then no concentration phenomenon +occurs for the disease distribution, and the infected population will aggregate only on such +intervals consisting of highest-risk points, regardless of whether there are isolated highest- +risk points or not (see Figure 2(a)-(b)). Indeed, our numerical results indicate that the +infected population will aggregate on all such intervals consisting of highest-risk points (see +Figure 2(c)); however the population number of I at each such interval may be different, +depending on the functions β, γ. +4.2. Profile of EE of model (1.6) as dI → 0. For model (1.6), for the general H¨older +continuous risk function h, under the condition (2.9), as dI → 0, we know from Theorem +2.2 that the susceptible population S converges to a positive function ˆS, which is non- +constant unless h is constant. The infected population I converges to a positive Radon +measure µ, whose support is contained in the region in which ˆS touches h; in other words, +the disease stays only within the place where the susceptible population distributes along +the risk function. If the risk function h is of C2, we see from Lemma 3.1 and Lemma +3.3 that the infected population aggregates at least in a neighborhood of the highest-risk +locations. +Furthermore, when h ∈ C2([0, L]), in light of Theorem 2.3, one can draw the following +conclusions concerning the asymptotic profile of I. + +22 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +0 +0.2 +0.4 +0.6 +0.8 +1 +−5 +0 +5 +10 +15 +20 +25 +30 +35 +40 +x + + +k(x) +S(x) +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +80 +x + + +S(x) +I(x) +0 +0.5 +1 +0.4 +0.6 +0.8 +1 +x + + +k(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +−10 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +x + + +S(x) +I(x) +0 +0.5 +1 +0.4 +0.6 +0.8 +1 +x + + +k(x) +(a) Θk = {1 +2} +(b) Θk = {1 +8, 1 +2} +(c) Θk = {1 +8, 3 +8, 1} +Figure 1. Numerical simulations of the solution profile of model (1.2), +where L = 1, N = 2, dS = 1, dI = 10−7, β(x) = 1 + 1 +2 sin(2πx), γ(x) = +k(x)β(x), kmin += +1 +2 and k(x) is chosen as follows. +In (a), k(x) = +1 + 1 +2 cos(2πx). +In (b), k(x) = 1 − 4x, 0 ≤ x < +1 +8; k(x) = 4x, +1 +8 ≤ +x < +1 +4; k(x) = +3 +2 − 2x, +1 +4 ≤ x < +1 +2; k(x) = x, +1 +2 ≤ x ≤ 1. +In (c), +k(x) = 1 − 4x, 0 ≤ x < 1 +8; k(x) = 4x, +1 +8 ≤ x < 1 +4; k(x) = 2 − 4x, +1 +4 ≤ x < +3 +8; k(x) = 4x − 1, +3 +8 ≤ x < 1 +2; k(x) = 3 +2 − x, +1 +2 ≤ x ≤ 1. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +x + + +k(x) +S(x) +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +x + + +k(x) +S(x) +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +x + + +k(x) +S(x) +I(x) +(a) Θk = [ 1 +4, 3 +4] +(b) Θk = [ 1 +4, 1 +2] ∪ { 7 +8} +(c) Θk = [0, 1 +16] ∪ { 3 +8} ∪ [ 5 +8, 3 +4] +Figure 2. Numerical simulations of the solution profile of model (1.2), +where L = 1, N = 2, dS = 1, dI = 10−5, β(x) = 1, γ(x) = k(x)β(x), kmin = 1 +2 +and k(x) is chosen as follows. In (a), Θk = [ 1 +4, 3 +4], k(x) = 1 +2 + 5(x − 1 +4)2, 0 ≤ +x < 1 +4; k(x) = 1 +2, +1 +4 ≤ x < 3 +4; k(x) = 1 +2 + 5(x − 3 +4)2, +3 +4 ≤ x ≤ 1. +In (b), +Θk = [ 1 +4, 1 +2] ∪ { 7 +8}, k(x) = 1 +2 + 4(x − 1 +4)2, 0 ≤ x < 1 +4; k(x) = 1 +2, +1 +4 ≤ x < +1 +2; k(x) = 1 +2 + 4(x − 1 +2)2, +1 +2 ≤ x < 3 +4; k(x) = 1 +2 + 16(x − 7 +8)2, +3 +4 ≤ x ≤ 1. In +(c), Θk = [0, 1 +16] ∪ { 3 +8} ∪ [ 5 +8, 3 +4], k(x) = 1 +2, 0 ≤ x < +1 +16; k(x) = 8x, +1 +16 ≤ x < +1 +8; k(x) = 1, +1 +8 ≤ x < 1 +4; k(x) = 2 − 4x, +1 +4 ≤ x < 3 +8; k(x) = 8x − 5 +2, +3 +8 ≤ x < +1 +2; k(x) = 11 +2 − 8x, +1 +2 ≤ x < 5 +8; k(x) = 1 +2, +5 +8 ≤ x < 3 +4; k(x) = 2 +3x, +3 +4 ≤ x ≤ 1. +(i) For any risk function h satisfying −dShxx ≤ Λ − h in (0, L), hx(0) ≥ 0, hx(L) ≤ 0, +and condition (2.9) (for instance, h < Λ is a positive constant), the infected population +must occupy the entire habitat, and it also forms the concentration phenomenon at the +boundary point x = 0 (or x = 1) if hx(0) > 0 (or hx(1) < 0), which is also the highest-risk +location; see Theorem 2.3(i) and the numerical illustrations in Figure 3(a)-(b)-(c). +(ii) For any convex risk function h (i.e., hxx ≥, ̸≡ 0 on [0, L]) fulfilling (2.9), the infected +population usually stays only in part of the habitat. In particular, by Theorem 2.3(ii)(iii), +we can observe the following behaviors. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +23 +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +x + + +h(x) +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +x + + +S(x) +0 +0.5 +1 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +x + + +I(x) +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +x + + +h(x) +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +x + + +S(x) +0 +0.5 +1 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +x + + +I(x) +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +x + + +h(x) +0 +0.5 +1 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +x + + +S(x) +0 +0.5 +1 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +x +I(x) + + +I(x) +(a) h(x) = 1 + 5x2(1 − x)2 +(b) h(x) = 1 + x2(1 − x) +(c) h(x) = 1 + x(1 − x) +Figure 3. Numerical simulations of the solution profile of model (1.6), +where β(x) = 1 + 1 +2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), dS = 1, dI = +10−8, Λ = 10. In (a), h(x) = 1 + 5x2(1 − x)2, in (b), h(x) = 1 + x2(1 − x), +and in (c), h(x) = 1 + x(1 − x). +(ii-a) If the highest-risk set Θh contains only one point, denoted by τ0, then the distri- +bution behavior of the infected population is affected by whether τ0 is a boundary point or +an interior point. More precisely, when τ0 is an interior point, then the infected population +resides in a certain left neighborhood of τ0, staying away from the boundary points x = 0 +and x = 1. In fact, such a neighborhood can be calculated through the formula (2.21). +One may further refer to Figure 4(a). +However, if τ0 is a boundary point, say τ0 = L, then the infected population stays in a +certain neighborhood of L provided e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +> − +d1/2 +S +hx(L) +Λ−h(L) , while the infected population +concentrates only at L provided e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 +≤ − +d1/2 +S +hx(L) +Λ−h(L) . Since hx(L) ≤ 0 in this situation, +the infected population stays in a certain neighborhood of L provided for all dS > 0 if +hx(L) = 0. If hx(L) < 0, it should be noted that the function q(dS) = d−1/2 +S +e2Ld−1/2 +S +−1 +e2Ld−1/2 +S ++1 ++ +hx(L) +Λ−h(L) deceases in dS ∈ (0, ∞), limdS→0 q(dS) = ∞ and limdS→∞ q(dS) = +hx(L) +Λ−h(L) < 0. As a +result, there is a unique d∗ +S > 0 such that q(d∗ +S) = 0, and in turn the infected population +stays in a left neighborhood of L for 0 < dS < d∗ +S , and the infected population concentrates +only at L for all dS ≥ d∗ +S. +(ii-b) If the highest-risk set Θh contains only an interval, then the infected population +resides in a certain neighborhood of such an interval. Again, such a neighborhood can be +calculated through the formula (2.21). See the numerical simulation in Figure 4(b). +(ii-c) For a general H¨older continuous risk function h, we can conclude that the disease +must exist in all isolated highest-risk point(s) and a neighborhood of each highest-risk +interval if exists; nevertheless, it is challenging to give a precise characterization for the +distribution behavior of the susceptible and infected populations, due to the mathematical +difficulties on the analysis of the free boundary problem (2.10). We have performed the +numerical simulations in Figure 5(a)-(b) as an illustration. +In what follows, we would like to make some more discussions on (ii-a) above in the case +that τ0 is a boundary point. For example, we take τ0 = L, and also assume that hx(L) < 0. +On the one hand, by fixing hx(L), we have known from (ii-b) that large diffusion rate dS +can result in the disease concentration only at the location L and small diffusion rate dS + +24 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +2 +4 +6 +8 +10 +12 +x + + +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +1 +1.1 +1.2 +1.3 +x + + +τ1 + τ2 +h(x) +S(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +25 +30 +x + + +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0.5 +0.6 +0.7 +0.8 +x + + +τ1 + τ2 +h(x) +S(x) +(a) Θh = {1 +2} +(b) Θh = [1 +4, 3 +4] +Figure 4. Numerical simulations of the solution profile of model (1.6), +where β(x) = 1 + 1 +2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), dS = +1, dI = 10−10, Λ = 10, and h(x) = 1 + (x − 1 +2)2 in (a), while in (b), h(x) = +1 +2 + 5(x − 1 +4)2, 0 ≤ x < 1 +4; h(x) = 1 +2, +1 +4 ≤ x < 3 +4; h(x) = 1 +2 + 5(x − 3 +4)2, +3 +4 +≤ x < 1. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +25 +30 +35 +x + + +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0.5 +0.6 +0.7 +0.8 +x + + +h(x) +S(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +x + + +I(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.5 +1 +1.5 +x + + +h(x) +S(x) +(a) Θh = [1 +4, 1 +2] ∪ {7 +8} +(b) Θh = [0, 1 +16] ∪ {3 +8} ∪ [5 +8, 3 +4] +Figure 5. Numerical simulations of the solution profile of model (1.6), +where dS = 1, dI = 10−5, β(x) = 1 + 1 +2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − +η(x), Λ = 10. In (a) and (b), h(x) is chosen to be the same as k(x) in Figure +2(b) and Figure 2(c), respectively. +will cause the disease to distribute in a left neighborhood of L. On the other hand, once +dS is fixed, the concentration phenomenon happens only if −hx(L) is properly large. This +motivates us to see whether a similar concentration phenomenon could occur at an interior +isolated highest-risk point if the risk function h is merely H¨older continuous. To illustrate +this phenomenon, let us consider the following risk function whose curve is the connection +of two segments: +h(x) = +� +a1 +� +x − L +2 +� ++ Λ +4 , +x ∈ +� +0, L +2 +� +, +a2 +� +x − L +2 +� ++ Λ +4 , +x ∈ +� L +2 , L +� +, +(4.1) +with a1 < 0, a2 > 0. Obviously, h is merely Lipschitz continuous at x = L +2 . Our numer- +ical simulation results demonstrate that if the slopes |a1|, a2 are properly large, then the +infected population will concentrate at x = L +2 (Figure 6(a)); if |a1|, a2 are small, then the + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +25 +infected population will aggregate in a neighborhood of x = L +2 (Figure 6(b)); and if |a1| is +small while a2 is large, then the infected population will aggregate in a left-neighborhood +of x = +L +2 (Figure 6(b)). These profiles behave rather differently from that in Theorem +2.3(ii) for h ∈ C2([0, L]), as shown by Figure 4(a). Therefore, the numerical results reveal +that the smoothness of h may have a substantial effect on the spatial distribution of the +disease. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +50 +100 +150 +200 +250 +x + + +I(x) +0 +0.5 +1 +2 +3 +4 +5 +6 +7 +8 +x + + +h(x) +S(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +x + + +I(x) + + +0 +0.5 +1 +2 +3 +4 +5 +6 +7 +8 +x + + +h(x) +S(x) +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +20 +40 +60 +80 +100 +120 +x + + +I(x) +0 +0.5 +1 +2 +3 +4 +5 +6 +7 +8 +x + + +h(x) +S(x) +(a) a1 = −10, a2 = 10 +(b) a1 = −1, a2 = 1 +(c) a1 = −1, a2 = 10 +Figure 6. Numerical simulations of the solution profile of model (1.6), +where β(x) = 1 + 1 +2 sin(2πx), η(x) = 1, γ(x) = h(x)β(x) − η(x), L = 1, dS = +1, dI = 10−5, Λ = 10 and h(x) is given by (4.1). +4.3. Conclusion. The discussions in the above two subsections, together with the numer- +ical simulations, show that the spatial profile of the susceptible and infected populations +of (1.2) and (1.6) with respect to small movement rate of the infected individuals are +rather different. This is caused by the presence of the recruitment term for the suscep- +tible population and the death rate for the infected population. On the other hand, we +would like to mention that the recent works [11–14, 31, 34, 69] studied various kinds of +reaction-diffusion-advection SIS epidemic models, in which the advection term represents +some passive movement in a certain direction, e.g., due to external environmental forces +such as water flow [46–48, 57], wind [15] and so on. In particular, if an advection is present +in (1.2) and stands for, for instance, the water flow, it was proved in [13, Theorem 1.4] +that, as dI → 0, the susceptible population converges to a positive function while the in- +fected population concentrates only at the downstream of the water flow; a similar result +can be shown to hold for the corresponding system (1.6). Such a distribution behavior is +essentially different from that of (1.2) and (1.6) with small dI. +In summary, our results here, combined with those of [13, 31, 40], suggest that the re- +cruitment term for the susceptible population, the death rate for the infected population +(even the smoothness of the associated risk function) as well as the advection can lead +to significant impacts on the disease transmission and thus decision-makers should attach +great importance to these factors when taking measures such as the lockdown and quaran- +tine to control the movement or immigration of the infected individuals so as to eliminate +the disease infection. + +26 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +5. Appendix +In this appendix, we always let Ω be a smooth and bounded domain in Rn (n ≥ 1). Given +f ∈ C(Ω), consider the following eigenvalue problem with Neumann boundary condition: +� +−D∆φ + f(x)φ = λφ +in Ω, +∂φ +∂ν = 0 +on ∂Ω, +(5.2) +where ν(x) is the unit exterior normal vector of ∂Ω at x, and the coefficient D is a positive +constant. +We start with a well-known fact concerning the asymptotic behavior of the principal +eigenvalue of (5.2) with respect to small diffusion; one may refer to, for example, [45, +Lemma 3.1]. +Lemma 5.1. Let λ1(D, f) be the principal eigenvalue of (5.2). Then it holds that +lim +D→0 λ1(D, f) = min +x∈Ω f(x). +We next recall the L1-estimate for the weak solution (due to [6]) of the following linear +elliptic problem: +−∆w + c(x)w = g +in Ω, +∂w +∂ν = 0 on ∂Ω. +(5.3) +Lemma 5.2. (a) (Global estimates) +Assume that c ∈ L∞(Ω), g ∈ L1(Ω) and let w ∈ +W 1,1(Ω) be a weak solution of (5.3). Then, for any r ∈ [1, n/(n − 1)), we have w ∈ W 1,r(Ω) +and the following estimate +∥w∥W 1,r(Ω) ≤ C∥g∥L1(Ω), +where the positive constant C is independent of w. +(b) (Interior estimates) Assume that Ω′ ⊂⊂ Ω is a smooth domain, c ∈ L∞(Ω), g ∈ +L1(Ω), and let w ∈ W 1,1(Ω) be a weak solution to the equation −∆w + c(x)w = g. Then, +for any r ∈ [1, n/(n − 1)), we have w ∈ W 1,r(Ω′) and the following estimate +∥w∥W 1,r(Ω′) ≤ C∥g∥L1(Ω), +where the positive constant C is independent of w. +At last, we state a Harnack-type inequality for weak solutions (see, e.g., [43] or [53]), +whose strong form was obtained in [44]. +Lemma 5.3. (a) (Global Harnack inequality) +Let c ∈ Lr(Ω) for some r > n/2. +If +w ∈ W 1,2(Ω) is a non-negative weak solution of the boundary value problem +−∆w + c(x)w = 0 in Ω, +∂w +∂ν = 0 on ∂Ω, +then there is a constant C, determined only by ∥c∥r, r and Ω such that +sup +Ω +w ≤ C inf +Ω w. +(b) (Local Harnack inequality) Let Ω′ ⊂⊂ Ω be a smooth domain and c ∈ Lr(Ω) for some +r > n/2. If w ∈ W 1,2(Ω) is a non-negative weak solution of the equation −∆w+c(x)w = 0, +then there is a constant C, determined only by ∥c∥r, r, Ω and Ω′, such that +sup +Ω′ w ≤ C inf +Ω′ w. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +27 +References +[1] L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS +epidemic patch model, SIAM J. Appl. Math., 67(2007), 1283-1309. +[2] L.J.S. Allen, B.M. Bolker, Y.Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS +epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21(2008), 1-20. +[3] L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Spatial patterns in a discrete-time SIS patch model, +J. Math. Biol., 58(2009), 339-375. +[4] R.M. Anderson, R.M. May, Population biology of infectious diseases, Nature, 280(1979), 361-367. +[5] D. Balcan, et al., Multiscale mobility networks and the spatial spreading of infectious diseases, Proc. +Natl Acad. Sci. USA, 106(2009), 21484-21489. +[6] H. Brezis, W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Jpn., +25(1973), 565-590. +[7] T. Britton, F. Ball, P. Trapman, A mathematical model reveals the influence of population hetero- +geneity on herd immunity to SARS-CoV-2, Science, 369(2020), 846-849. +[8] T. Britton, F. Ball, P. Trapman, The disease-induced herd immunity level for Covid-19 is substantially +lower than the classical herd immunity level, preprint, arXiv:2005.03085. +[9] D. Brockmann, D. Helbing, The hidden geometry of complex, network-driven contagion phenomena, +Science, 342(2013), 1337-1342. +[10] K. Castellano, R.B. Salako, On the effect of lowering population’s movement to control the spread of +an infectious disease, J. Differential Equations, 316(2022), 1-27. +[11] R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic +model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 26(2021), 2997-3022. +[12] R. Cui, K.-Y. Lam, Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model +in advective environments, J. Differential Equations, 263(2017), 2343-2373. +[13] R. Cui, H. Li, R. Peng, M. Zhou, Concentration behavior of endemic equilibrium for a reaction- +diffusion-advection SIS epidemic model with mass action infection mechanism, Calc. Var. Partial +Differential Equations, 60(2021), paper no. 184, 38 pp. +[14] R. Cui, Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equa- +tions, 261(2016), 3305-3343. +[15] K.A. Dahmen, D.R. Nelson, N.M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41(2000), +1-23. +[16] K. Deng, Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh +Sect. A, 146(2016), 929-946. +[17] F. Di Lauro, et al., The impact of network properties and mixing on control measures and +disease-induced herd immunity in epidemic models: +a mean-field model perspective, preprint, +arXiv:2007.06975. +[18] Z. Du, R. Peng, A priori L∞-estimate for solutions of a class of reaction-diffusion systems, J. Math. +Biol., 72(2016), 429-1439. +[19] D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79(2019), 1581-1606. +[20] D. Gao, C-P. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148(2020), +1709-1722. +[21] D. Gao, S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., +232(2011), 110-115. +[22] J. Ge, K. Kim, Z. Lin, H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk +domain, J. Differential Equations, 259(2015), 5486-5509. +[23] S. Han, C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear +incidence, Appl. Math. Lett., 98(2019), 114-120. +[24] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42(2000), 599-653. +[25] J.S. Jia, et al., Population flow drives spatio-temporal distribution of COVID-19 in China, Nature, +582(2020), 389-394. +[26] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–I, Proc. +Roy. Soc. London Ser. A, 115(1927), 700-721. +[27] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–I, Bull. +Math. Biol., 53(1991), 33-55. + +28 +R. PENG, Z.-A. WANG, G. ZHANG AND M. ZHOU +[28] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–II. The +problem of endemicity, Bull. Math. Biol., 53(1991), 57-87. +[29] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics–III. Fur- +ther studies of the problem of endemicity, Bull. Math. Biol., 53(1991), 89-118. +[30] M.U.G. Kraemer, et al., The effect of human mobility and control measures on the COVID-19 epi- +demic in China, Science, 368(2020), 493-497. +[31] K. Kuto, H. Matsuzawa, R. Peng, Concentration profile of the endemic equilibria of a reaction- +diffusion-advection SIS epidemic model, Calc. Var. Partial Differential Equations, 56(2017), paper +no. 112, 28 pp. +[32] C. Lei, F. Li, J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence +in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23(2018), 4499-4517. +[33] C. Lei, J. Xiong, X. Zhou, Qualitative analysis on an SIS epidemic reaction-diffusion model with +mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete +Contin. Dyn. Syst. Ser. B, 25(2020), 81-98. +[34] C. Lei, X. Zhou, Concentration phenomenon of the endemic equilibrium of a reaction-diffusion- +advection SIS epidemic model with spontaneous infection, Discrete Contin. Dyn. Syst. Ser. B, to +appear. +[35] B. Li, Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. +Appl., 475(2019), 1910-1926. +[36] B. Li, H. Li, Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. +Math. Phys., 68(2017), no. 4, Art. 96, 25 pp. +[37] B. Li, J. Zhou, X. Zhou, Asymptotic profiles of endemic equilibrium of a diffusive SIS epidemic +system with nonlinear incidence function in a heterogeneous environment, Proc. Amer. Math. Soc., +148(2020), 4445-4453. +[38] H. Li, R. Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch +models, J. Math. Biol., 79(2019), 1279-1317. +[39] H. Li, R. Peng, F.-B. Wang, Vary total population enhances disease persistence: qualitative analysis +on a diffusive SIS epidemic model, J. Differential Equations, 262(2017), 885-913. +[40] H. Li, R. Peng, Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with +mass action mechanism and birth-death effect: analysis, simulations, and comparison with other +mechanisms, SIAM J. Appl. Math., 78(2018), 2129-2153. +[41] H. Li, R. Peng, T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency- +dependent SIS epidemic models with cross-diffusion, European J. Appl. Math., 31(2018), 26-56. +[42] M.Y. Li, Z. Shuai, Global stability of an epidemic model in a patchy environment, Can. Appl. Math. +Q., 17(2009), 175-187. +[43] G.M. Lieberman, Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimen- +sions, SIAM J. Math. Anal., 36(2005), 1400-1406. +[44] C.S. Lin, W.M. Ni, I. Takagi, +Large amplitude stationary solutions to a chemotaxis system, J. +Differential Equations, 72(1988), 1-27. +[45] Y. Lou, T. Nagylaki, Evolution of a semilinear parabolic system for migration and selection without +dominance, J. Differential Equations, 225(2006), 624-665. +[46] F. Lutscher, M.A. Lewis, E. McCauley, Effects of heterogeneity on spread and persistence in rivers, +Bull. Math. Biol., 68(2006), 2129-2160. +[47] F. Lutscher, E. McCauley, M.A. Lewis, Spatial patterns and coexistence mechanisms in systems with +unidirectional flow, Theor. Popul. Biol., 71(2007), 267-277. +[48] F. Lutscher, E. Pachepsky, M.A. Lewis, The effect of dispersal patterns on stream populations, SIAM +Rev., 47(2005), 749-772. +[49] P. Magal, G. Webb, Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity, +31(2018), 5589-5614. +[50] P. Magal, G. Webb, Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, +SIAM J. Appl. Math., 79(2019), 284-304. +[51] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. +Part I, J. Differential Equations, 247(2009), 1096-1119. +[52] R. Peng, S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, +Nonlinear Anal., 71(2009), 239-247. + +TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM +29 +[53] R. Peng, J. Shi, M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis +and saturation law, Nonlinearity, 21(2008), 1471-1488. +[54] R. Peng, Y. Wu, Global L∞-bounds and long-time behavior of a diffusive epidemic system in a +heterogeneous environment, SIAM J. Math. Anal., 53(2021), 2776-2810. +[55] R. Peng, F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion +model: Effects of epidemic risk and population movement, Phys. D. 259(2013), 8-25. +[56] R. Peng, X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Non- +linearity, 25(2012), 1451-1471. +[57] R. Peng, X.-Q. Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic- +parabolic problem with applications, Calc. Var. Partial Differential Equations, 54(2015), 1611-1642. +[58] P. Song, Y. Lou, Y. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, +J. Differential Equations, 267(2019), 5084-5114. +[59] J. Suo, B. Li, Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent +incidence function in a heterogeneous environment, Math. Biosci. Eng., 17(2019), 418-441. +[60] H. Tian, et al., An investigation of transmission control measures during the first 50 days of the +COVID-19 epidemic in China, Science, 368(2020), 638-642. +[61] Y. Tong, C. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially +heterogeneous environment, Nonlinear Anal. Real World Appl., 41(2018), 443-460. +[62] C. Vargas-De-Leon, A. Korobeinikov, Global stability of a population dynamics model with inhibition +and negative feedback, J. Math. Medicine and Biol., 30(2013), 65-72. +[63] J. Wang, X. Wu, Dynamics and profiles of a diffusive Cholera model with bacterial hyperinfectivity +and distinct dispersal rates, J. Dyn. Diff. Equat., 2021, https://doi.org/10.1007/s10884-021-09975-3 +[64] J. Wang, J. Wang, Analysis of a reaction-diffusion Cholera model with distinct dispersal rates in the +human population, J. Dyn. Diff. Equat., 33(2021), 549-575. +[65] X. Wen, J. Ji, B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model +with mass action infection mechanism, J. Math. Anal. Appl., 458(2018), 715-729. +[66] D. Wodarz, J.P. Christensen, A.R. Thomsen, The importance of lytic and nonlytic immune responses +in viral infections, Trends Immunol., 23(2002), 194-200. +[67] D. Wodarz, M.A. Nowak, Immune response and viral phenotype: +do replication rate and cy- +topathogenicity influence virus load? J. Theor. Med., 2(2000), 113-127. +[68] Y. Wu, X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass +action infection mechanism, J. Differential Equations, 261(2016), 4424-4447. +[69] J. Zhang, R. Cui, Asymptotic behavior of an SIS reaction-diffusion-advection model with saturation +and spontaneous infection mechanism, Z. Angew. Math. Phys., 71(2020), paper no. 150, 21 pp. +[70] S. Zhu, J. Wang, Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear +source in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 25(2020), 1999- +2019. +[71] S. Zhu, J. Wang, Asymptotic profiles of steady states for a diffusive SIS epidemic model with spon- +taneous infection and a logistic source, Commun. Pure Appl. Anal., 19(2020), 3323-3340. + diff --git a/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/load_file.txt b/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39b0f424e3e84490b3e921f2a7687529eb166f32 --- /dev/null +++ b/-dAzT4oBgHgl3EQfFfqG/content/tmp_files/load_file.txt @@ -0,0 +1,1669 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf,len=1668 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='01012v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='AP] 3 Jan 2023 NOVEL SPATIAL PROFILES OF POPULATION DISTRIBUTION OF TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM AND SMALL MOVEMENT RATE FOR THE INFECTED INDIVIDUALS RUI PENG, ZHI-AN WANG, GUANGHUI ZHANG AND MAOLIN ZHOU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In this paper, we are concerned with two SIS epidemic reaction-diffusion models with mass action infection mechanism of the form SI, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In sharp contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' and the infected population may concentrate only at some isolated highest-risk points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Numerical simulations are performed to support and complement our theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Introduction and existing results The outbreak of the novel coronavirus disease 2019 (COVID-19) continues to spread rapidly around the world, and it has caused tremendous impacts on public health and the global economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' As it is commonly recognized, population movement is a significant factor in the spread of many reported infectious diseases including COVID-19 [5, 9, 25], Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 35J57, 35B40, 35Q92, 92D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Reaction-diffusion SIS epidemic model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' mass action infection mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' spa- tial profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' small movement rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' heterogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Peng: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Email: pengrui seu@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Wang: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Email: mawza@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Zhang: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Email: guanghuizhang@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Zhou: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Email: zhouml123@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Peng was partially supported by NSF of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 12271486, 12171176), Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Wang was partially supported by the Hong Kong Scholars Program (Project ID P0031250) and an internal grant from the Hong Kong Polytechnic University (Project ID P0031013), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Zhang was partially supported by NSF of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 12171176, 11971187) and the Fundamental Research Funds for the Central Universities (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 5003011008), and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Zhou was partially supported by the Nankai Zhide Foundation and NSF of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 11971498).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 1 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' PENG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' WANG, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHANG AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHOU and the lockdown and quarantine has turned out to be one of the most effective measures to reduce or even eliminate the infection [30, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' On the other hand, the importance of the population heterogeneity has also been observed in the complicated dynamical behaviour of the transmission of COVID-19 [7, 8, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' To gain a deeper understanding of the impact of population movement and heterogeneity on the transmission of epidemic diseases from a mathematically theoretical viewpoint, in the present work we are concerned with two SIS reaction-diffusion systems with mass action infection mechanism in a heterogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We aim to study the spatial profile of population distribution as the movement rate of the infected individuals is controlled to be sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Such kind of information may be useful for decision-makers to predict the pattern of disease occurrence and henceforth to conduct more effective strategies of disease eradication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The mass action infection mechanism was first proposed in the seminal work of Kermack and McKendrick [26], in which the disease transmission was assumed to be governed by a bilinear incidence function SI (one may also refer to [27–29] or [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The systems under consideration in this paper are possibly the simplest yet basic SIS epidemic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The first model we will deal with in this work is the following coupled reaction-diffusion equations in one-dimensional space: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 St − dSSxx = −β(x)SI + γ(x)I, 0 < x < L, t > 0, It − dIIxx = β(x)SI − γ(x)I, 0 < x < L, t > 0, Sx = Ix = 0, x = 0, L, t > 0, S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0, 0 < x < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1) Here, S(x, t) and I(x, t) are respectively the population density of the susceptible and in- fected individuals at position x ∈ [0, L] and time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' the homogeneous Neumann boundary condition means that no population flux crosses the boundary x = 0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' dS and dI are pos- itive constants measuring the motility of susceptible and infected individuals, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' and the functions β and γ are H¨older continuous positive functions in [0, L] representing the disease transmission rate and the disease recovery rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Integrating the sum of the equations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1), combined with the homogeneous Neumann boundary value conditions, we observe that � L 0 (S(x, t) + I(x, t)) dx = � L 0 (S0(x) + I0(x)) dx =: N, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Thus, the total population number in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1) is conserved all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1) was investigated in the recent works [16, 65, 68];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' in particular, when the movement of either the susceptible or infected population is restricted to be slow, the authors explored the profile of the spatial distribution of the disease modelled by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The understanding of such a profile amounts to determine the behavior of the so-called endemic equilibrium with respect to the small diffusion rate dS or dI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The endemic equilibrium of TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM 3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1) is a positive steady state solution, which satisfies the following elliptic system: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −dSSxx = −β(x)SI + γ(x)I, 0 < x < L, −dIIxx = β(x)SI − γ(x)I, 0 < x < L, Sx = Ix = 0, x = 0, L, � L 0 (S(x) + I(x)) dx = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) According to [16, 65, 68], if minx∈[0,L] γ(x) β(x) < N L , for any small dI > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) admits at least one positive solution (S, I), which is called an endemic equilibrium (EE for abbreviation) in terms of epidemiology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' moreover, (S, I) satisfies S, I ∈ C2([0, L]) and S, I > 0 on [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' As remarked in [68], it is a challenging problem to study the spatial profile of EE of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) with respect to the small movement rate dI of the infected population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' in [65], the authors provided a first result in this research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Indeed, they proved the following conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' [65, Theorem B] Assume that minx∈[0,L] γ(x) β(x) < N L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Then as dI → 0, the EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) satisfies (up to a sequence of dI) that S → ˆS uniformly on [0, L], where ˆS ∈ C([0, L]) with min[0,L] γ(x) β(x) ≤ ˆS(x) ≤ max[0,L] γ(x) β(x), and I → µ weakly for some Radon measure µ with nonempty support in the sense of � L 0 I(x)ζ(x)dx −→ � [0,L] ζ(x)µ(dx), ∀ζ ∈ C([0, L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3) Obviously, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 does not give a precise description for ˆS and µ and hence the spatial profile of the susceptible and infected populations remains obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' From the aspect of disease control, it becomes imperative to know an informative behavior of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In this paper, we manage to give a satisfactory result on the profile of ˆS and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1), some important factors such as the death and recruitment rates of population are ignored so that the total population number is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In order to take into account the death and recruitment rates of population, the following reaction-diffusion epidemic system was proposed in [40]: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 St − dSSxx = Λ(x) − S − β(x)SI + γ(x)I, 0 < x < L, t > 0, It − dIIxx = β(x)SI − [γ(x) + η(x)] I, 0 < x < L, t > 0, Sx = Ix = 0, x = 0, L, t > 0, S(x, 0) = S0(x) ≥ 0, I(x, 0) = I0(x) ≥, ̸≡ 0, 0 < x < L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4) The recruitment term of the susceptible population is represented by the function Λ(x)−S so that the susceptible is subject to the linear growth/death ([4, 24]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' η(x) accounts for the death rate of the infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Here, Λ, η are assumed to be positive H¨older continuous functions on [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' All other parameters have the same interpretation as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' It is easily seen that the following elliptic problem −dSSxx = Λ(x) − S, 0 < x < L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Sx(0) = Sx(L) = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='5) 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' PENG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' WANG, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHANG AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHOU admits a unique positive solution ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Then ( ˜S, 0) is a unique disease-free equilibrium of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' An EE of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4) satisfies the following ODE system: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −dSSxx = Λ(x) − S − β(x)SI + γ(x)I, 0 < x < L, −dIIxx = β(x)SI − [γ(x) + η(x)] I, 0 < x < L, Sx = Ix = 0, x = 0, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) As one of the main results of [40], the following conclusion on the profile of EE of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) with respect to small dI was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' [40, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2] Assume that the set {x ∈ [0, L] : β(x) ˜S(x) > γ(x) + η(x)} is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' As dI → 0, then any EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) satisfies (up to a subsequence of dI) that S → ˆS uniformly on [0, L], where ˆS ∈ C([0, L]) and ˆS > 0 on [0, L], and � L 0 Idx → ˆI for some positive constant ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' As in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2 does not characterize the precise distribution of the susceptible and infected populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In this paper, we will also provide a clear picture of the population distributions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) as the movement rate dI tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' It turns out that the spatial profiles of the disease distribution modelled by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) are rather different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The rest of paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In section 2, we state the main theoretical results, and section 3 is devoted to their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In section 4, we carry out the numerical simulations and discuss the implications of our results in terms of disease control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In the appendix, we recall some known facts which will be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Statement of main results In this section, we state the main findings of this paper on models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' To proceed, we underline some terminologies frequently used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' For model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2), we call γ(x) β(x) the risk function, and call each element of the set � x ∈ [0, L] : γ(x) β(x) = minx∈[0,L] γ(x) β(x) � the highest-risk point (or location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Similarly, for model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6), we call γ(x)+η(x) β(x) the risk function, and call each element of the set � x ∈ [0, L] : γ(x)+η(x) β(x) = minx∈[0,L] γ(x)+η(x) β(x) � the highest-risk point (or location).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Results for model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' For the sake of convenience, we set k(x) = γ(x) β(x), kmin = min x∈[0,L] k(x), and Θk = � x ∈ [0, L] : k(x) = kmin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We note that when the risk function k(x) = k is a positive constant, it follows from [65] that S(x) ≡ k is a constant, and in turn by the equation of I, we immediately see that I = N L − k is also a positive constant provided that k < N L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In what follows, we do not consider such a trivial case and assume that k(x) is non-constant on [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We now state our main result on the asymptotic behavior of any EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) as dI → 0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Assume that k(x) is non-constant and kmin < N L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Then as dI → 0, the EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) satisfies S(x) → kmin uniformly for x ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='7) The following assertions hold for the asymptotic behavior of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (i) If Θk = {x0}, then we have I(x) → (N − Lkmin)δ(x0) weakly in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3), where δ(x0) is the Dirac measure centered at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Moreover, I(x) → 0 locally uni- formly in [0, L] \\ {x0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (ii) If Θk = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L, then we have I(x) → 0 uniformly on [0, ̺1] ∪ [̺2, L], and I(x) → ˆI(x) uniformly for x ∈ [̺1, ̺2], where ˆI ∈ C2([̺1, ̺2]), ˆI > 0 in (̺1, ̺2), and ˆI is the unique positive solution of \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −ˆIxx = β(x) dS (ˆa − ˆI)ˆI, ̺1 < x < ̺2, ˆI = 0, x = ̺1, ̺2, � ̺2 ̺1 ˆI dx = N − Lkmin, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='8) where the positive constant ˆa is uniquely determined by the integral constraint in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Regarding Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, we would like to make some comments in order as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In addition to the two cases treated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, we can handle some more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In particular, we would like to make the following comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (i) If the set Θk contains only finitely many isolated points, say {xi}j i=1 for some j ≥ 2, then one can slightly modify the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1(i) to show that S → kmin uniformly on [0, L], and I → 0 locally uniformly in [0, L] \\ ({xi}j i=1), and I(x) → j � i=1 ciδ(xi) weakly in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3), where δ(xi) is the Dirac measure centered at xi and the nonnegative constants ci fulfill �j i=1 ci = N − Lkmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Nevertheless, we can not determine the exact values of ci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' in other words, as dI → 0, it is unclear to us whether I concentrates at all xi (1 ≤ i ≤ j) or only some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The numerical results suggest that the former alternative holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' see Figure 1 in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (ii) If the set Θk contains at least one proper interval of [0, L], by adapting the argument of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1(ii), we can show that S → kmin uniformly on [0, L], and I → ˆI uniformly on [0, L] with ˆI = 0 on [0, L] \\ Θk, � Θk ˆI dx = N − Lkmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' PENG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' WANG, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHANG AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHOU In particular, if Θk = � �j∗ i=1[̺i, ̺i] � � � �{xi}j∗ i=0 � for some j∗ ≥ 1, j∗ ≥ 0, then we can prove that ˆI = 0 on [0, L] \\ ( j∗ � i=1 (̺i, ̺i)), and in (̺i, ̺i) (1 ≤ i ≤ j∗), either ˆI = 0 or ˆI > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Without loss of generality, assuming that ˆI(x) > 0 for x ∈ � ˆj∗ i=1(̺i, ̺i) for some 1 ≤ ˆj∗ ≤ j∗, then in each such (̺i, ̺i), we can conclude that ˆI solves � −ˆIxx = β(x) dS (ˆa − ˆI)ˆI, ̺i < x < ̺i, ˆI = 0, x = ̺i, ̺i, where the positive constant ˆa is uniquely determined by ˆj∗ � i=1 � ̺i ̺i ˆI dx = N − Lkmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' However, it seems rather challenging to prove whether ˆI is positive on all intervals (̺i, ̺i) (1 ≤ i ≤ j∗) or only on some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Our numerical results suggest that the former alternative holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' see Figure 2 in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (iii) The assertion in (ii) above suggests that if the highest-risk locations contain at least one interval, then the disease can not stay on any possible isolated highest-risk points once the infected individuals move slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In the case (ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, if ̺1 = 0 (or ̺2 = L), the results of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 still hold true if we replace the Dirichlet boundary condition of ˆI in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='8) at ̺1 = 0 (or ̺2 = L) by the Neumann boundary condition ˆIx(0) = 0 (or ˆIx(L) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' A similar remark applies to the case discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1(ii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' After this paper was finished, we noticed the work [10] in which the authors derived (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='7) and the convergence of the I-component in the case (i) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 in any spatial dimension in a more general setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='5(i) there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' However, their result does not establish the convergence of the I-component within Θk in the case (ii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 nor in the more general case mentioned by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' on the other hand, our proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='7) and the convergence of the I-component outside of Θk is rather different from that of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Results for model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We now turn to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' For the sake of simplicity, we assume that Λ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) is a positive constant, and also denote h(x) = γ(x) + η(x) β(x) , hmin = min x∈[0,L] h(x), and Θh = � x ∈ [0, L] : h(x) = hmin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Clearly, ˜S(x) = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We also enhance the existence condition of EE of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2 by imposing the following condition: Λ > h(x) for all x ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='9) TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM 7 Now we can state our main findings on the asymptotic behavior of any EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) as dI → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The first result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' As dI → 0, then any EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6) satisfies (up to a subsequence of dI) that S → ˆS uniformly on [0, L], and I → µ weakly in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3), where µ is some Radon measure and ˆS solves weakly in W 1,2(0, L) the free boundary problem: −dS ˆSxx = Λ − ˆS − η(x)µ({x}) �� {x∈[0, L]: ˆS(x)=h(x)}, x ∈ (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='10) Here, µ({x}) �� {x∈[0, L]: ˆS(x)=h(x)} is the restriction of µ on the set {x ∈ [0, L] : ˆS(x) = h(x)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' otherwise, µ({x}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Moreover we have the following properties for µ and ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (i) The Radon measure µ satisfies µ({x ∈ [0, L] : ˆS(x) ̸= h(x)}) = 0, µ({x ∈ [0, L] : ˆS(x) = h(x)}) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='11) (ii) The function ˆS ∈ C([0, L]) satisfies hmin ≤ ˆS(x) ≤ h(x), ∀x ∈ [0, L], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='12) Θh ⊂ � x ∈ [0, L] : ˆS(x) = h(x) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='13) If x1, x2 ∈ Θh with x1 < x2 and (x1, x2) ∩ Θh = ∅, then hmin < ˆS(x), ∀x ∈ (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='14) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2 asserts that ˆS touches h at all highest-risk points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In what follows, our goal is to examine the properties ˆS for some specific risk function h, which in turn provides us with a more precise description of the profile of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Indeed, we can obtain the following result for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Let ˆS and µ be given as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Assume that h ∈ C2([0, L]) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (i) If −dShxx ≤ Λ − h in (0, L), hx(0) ≥ 0 and hx(L) ≤ 0, then we have ˆS(x) = h(x), ∀x ∈ [0, L], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='15) µ({x}) = Λ − h(x) + dShxx(x) η(x) , a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' for x ∈ (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='16) (ii) If hx is non-decreasing on [0, L] and Θh = {τ0} for some 0 ≤ τ0 ≤ L, then the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (a) When 0 < τ0 < L, we have ˆS(x) = h(x), ∀x ∈ [τ1, τ2], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='17) and in [0, τ1) ∪ (τ2, L], ˆS < h satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −dS ˆSxx(x) = Λ − ˆS, x ∈ (0, τ1) ∪ (τ2, L), ˆSx(0) = 0, ˆSx(L) = 0, ˆS(τ1) = h(τ1), ˆS(τ2) = h(τ2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='18) and µ satisfies µ({x}) = Λ − h(x) + dShxx(x) η(x) , a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' for x ∈ (τ1, τ2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='19) 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' PENG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' WANG, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHANG AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHOU µ({x}) = 0, ∀x ∈ [0, τ1) ∪ (τ2, L], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='20) where the numbers τ1, τ2 with 0 < τ1 < τ0 < τ2 < L are uniquely determined by e2d−1/2 S τ1 − 1 e2d−1/2 S τ1 + 1 = −d1/2 S hx(τ1) Λ − h(τ1) , e2d−1/2 S (τ2−L) − 1 e2d−1/2 S (τ2−L) + 1 = −d1/2 S hx(τ2) Λ − h(τ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='21) (b) When τ0 = L, then we have the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (b-1) If e2Ld−1/2 S −1 e2Ld−1/2 S +1 > − d1/2 S hx(L) Λ−h(L) , then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='19) hold with [τ1, τ2] re- placed by [τ1, L], µ([0, τ1)) = 0, and on [0, τ1], ˆS satisfies � −dS ˆSxx(x) = Λ − ˆS, x ∈ (0, τ1), ˆSx(0) = 0, ˆS(τ1) = h(τ1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='22) where 0 < τ1 < L is uniquely determined by the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (b-2) If e2Ld−1/2 S −1 e2Ld−1/2 S +1 ≤ − d1/2 S hx(L) Λ−h(L) , then ˆS is the unique positive solution of � −dS ˆSxx(x) = Λ − ˆS, x ∈ (0, L), ˆSx(0) = 0, ˆS(L) = h(L), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='23) and µ satisfies µ([0, L)) = 0, µ({L}) = ΛL − � L 0 ˆS(x)dx η(L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='24) (c) When τ0 = 0, then we have the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (c-1) If e2Ld−1/2 S −1 e2Ld−1/2 S +1 > d1/2 S hx(0) Λ−h(0) , then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='19) hold with [τ1, τ2] replaced by [0, τ2], µ((τ2, L]) = 0, and on [τ2, L], ˆS satisfies � −dS ˆSxx(x) = Λ − ˆS, x ∈ (τ2, L), ˆSx(L) = 0, ˆS(τ2) = h(τ2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='25) where 0 < τ2 < L is uniquely determined by the second equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (c-2) If e2Ld−1/2 S −1 e2Ld−1/2 S +1 ≤ d1/2 S hx(0) Λ−h(0) , then ˆS is the unique positive solution of � −dS ˆSxx(x) = Λ − ˆS, x ∈ (0, L), ˆSx(L) = 0, ˆS(0) = h(0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='26) and µ satisfies µ((0, L]) = 0, µ({0}) = ΛL − � L 0 ˆS(x)dx η(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='27) (iii) If hx is non-decreasing on [0, ̺1]∪[̺2, L] and Θh = [̺1, ̺2] for some 0 < ̺1 < ̺2 < L, then all the assertions in (ii)-(a) above hold, where the numbers τ1, τ2 satisfying 0 < τ1 < ̺1 < ̺2 < τ2 < L are uniquely determined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' TWO DIFFUSIVE SIS EPIDEMIC MODELS WITH MASS ACTION INFECTION MECHANISM 9 For model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2), our result shows that the infected population concentrates or aggregates only at the highest-risk locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In sharp contrast, for model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='6), our result suggests that the disease will occupy a neighborhood of the interior highest-risk locations or even occupy the whole habitat [0, L], or concentrates only at the boundary highest-risk location, depending on the risk function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' More detailed discussions on the implications of our theoretical results, along with numerical simulations, will be given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' We would like to make some remarks on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' It is worth mentioning that all the statements in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3 except the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='19) for the Radon measure µ remain true provided that the risk function h ∈ C1([0, L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Such a comment also applies to Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4 in the forthcoming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (i) It is clear that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3(i) holds if h < Λ is a constant or more generally h is a unique solution to the following problem: � −dShxx = Λ − h, x ∈ (0, L), h(0) = σ1, h(L) = σ2, where 0 < σ1, σ2 < Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' When hx(0) > 0, the change of the derivatives from Sx(0) = 0 to ˆSx(0) = hx(0) > 0 would suggest that I should experience the concentration phenomenon at x = 0 (that is, I(0) → ∞) as dI → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' The same remark applies to the case of hx(L) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (ii) In contrast to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3(i), it is easily seen that ˆS ̸≡ h on [0, L] provided that −dShxx(x∗) > Λ − h(x∗) for some x∗ ∈ (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (iii) Clearly, the assertions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3(ii)-(b1) hold if hx(L) = 0 and the assertions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3(ii)-(c1) hold if hx(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (iv) In a general case that Θh contains an interior isolated point and hx is non-decreasing in a neighbourhood of such a point, we can conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='16) hold in some neighbourhood of this point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' if Θh contains an interval, a similar conclusion also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Proof of main results: Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3 This section is devoted to the proof of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' In this subsection, we present the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' First of all, we recall that for any EE (S, I) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2), from [65] (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3) there), the following holds: kmin ≤ S(x) ≤ max [0,L] k(x), ∀x ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1) By the positivity of I and the uniqueness of the principal eigenvalue, it is clear from the equation of I that λ1(dI, γ − βS) = 0, ∀dI > 0, where λ1(dI, γ − βS) is defined as in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, as dI → 0 (up to a subsequence), we see that S → ˆS uniformly on [0, L] for some positive function ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' Hence, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1 in the appendix and the continuous dependence of the principal eigenvalue on the weight function γ − βS, we have 0 = lim dI→0 λ1(dI, γ − βS) = min x∈[0,L][γ(x) − β(x) ˆS(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' PENG, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' WANG, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHANG AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' ZHOU This obviously implies that ˆS(x) ≤ k(x), ∀x ∈ [0, L] and ˆS(y0) = k(y0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) for some y0 ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' From Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='1, we recall that I → µ weakly for some Radon measure µ with µ([0, L]) > 0 in the following sense � L 0 I(x)ζ(x)dx → � L 0 ζ(x)µ(dx), ∀ζ ∈ C([0, L]), as dI → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3) We now integrate the first equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2) by parts over [0, L] and use the boundary conditions to deduce that � L 0 [β(x)S(x) − γ(x)]I(x)dx = 0, ∀dI > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4) Letting dI → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='4), combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='3) and the fact that S → ˆS uniformly on [0, L] as dI → 0, we infer that � [0,L] [β(x) ˆS(x) − γ(x)]µ(dx) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='5) which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQfFfqG/content/2301.01012v1.pdf'} +page_content='2), gives � {x∈[0,L]: ˆS(x)< = +) +,!"",- 𝜈Σ-<, then +we rewrite Eq. (8) as +𝜇 +!"* +(()*)($,&) +!$ ++ Σ(<𝜓) +(,9))(𝑥, 𝜇) += +) +* Σ(<𝜙) +(,9))(𝑥) − +) +* Σ(<𝛾𝑇) +(,)(𝑥) , (9) +where +𝛾 = (1 − 𝑐<) Q +?.* +?.- − +?"* +?"-R 𝜙< , (10a) +with +Σ>) = Σ() − Σ+) , (10b) + +𝑐< = +?/- +?0- . (10c) +Substituting Eqs. (7b), (7d), and (7e) into (6a) and (6b) +respectively, we obtain +𝑇) +∗(𝑥) = 𝐴Σ-<𝜙) +(,9))(𝑥) + 𝐴Σ-)𝜙<𝑇) +(,)(𝑥) , (11) +𝑇) +(,9))(𝑥) = 𝜔𝑇) +∗(𝑥) + (1 − 𝜔)𝑇) +(,)(𝑥) . (12) +Then we substitute Eq. (11) into (12) to give +𝑇) +(,9))(𝑥) += 𝜔𝐴Σ-<𝜙) +(,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙)/Σ>< +(K4)) +0.718 +0.0297 0.96 −1.99 × 104L 8.67 × 104M +Note that the temperature coefficient of the absorption +cross section is negative, whereas that of the fission cross +section is positive. For such problems, the convergence of the +unrelaxed Picard iteration is determined by the smallest error +mode 𝜉 = 𝜋/(Σ(<𝐿), and the spectral radius is given as +𝜌 = (𝑇 − 𝑇.) 6Q +?.* +?.- − +?"* +?"-R +)46- +)4D12(@) 𝜌EF(𝜉) − +?"* +?"-; . (29) +Fig. 1 shows that the spectral radius increases with the +increasing reactor core height and eventually Picard iteration +fails to converge when the reactor core height is larger than a +critical value (for the given total cross section). If the +temperature difference between the fuel and coolant increases +(e.g., the increasing thermal resistance or linear heat +generation rate 𝑞:), then the coupling becomes less stable as +the spectral radius becomes larger. + +Fig. 1. Spectral radius vs. core height. +To verify the FA results, we compute numerical +convergence rates based on a 1-D model problem, which is +the homogeneous slab with the reflective boundary on both +sides. The Gauss-Legendre S12 quadrature set is used for +angular discretization and the Diamond Difference (DD) +method is employed for spatial discretization. Note that the +angular quadrature and the mesh size used are sufficiently +fine to minimize the numerical errors. A simple heat balance +model is used to calculate the fuel and coolant temperatures +at each axial cell. Fig. 1 shows that the numerical results for +the problem are in excellent agreement with the FA results. +For the relaxed case, i.e., 0 < 𝜔 < 1, underrelaxation +can not only help to stabilize Picard iteration but improve the +convergence rate as shown in Fig. 2. + +Fig. 2. Spectral radius vs. underrelaxation. +For this case, the core height 𝐿 = 150 cm, and the +typical PWR data in Table I is used. Again, the FA +predictions are consistent with the numerical results. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1 +10 +100 +Spectral Radius +L (cm) +FA (T-Tm = 275K) +FA (T-Tm = 500K) +Numerical +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +Spectral Radius +Underrelaxation, 𝜔 +FA +Numerical + +To derive the optimal underrelaxation factor 𝜔/N(, it is +noted that when 𝜔 < 𝜔/N(, max +@ |𝜚(𝜉)| is found at 𝜉 = ∞, +where 𝜌EF(𝜉) = 0, and +𝜌 = 1 − 𝜔 61 − (𝑇 − 𝑇.) +?"* +?"-; , + (30) +while for 𝜔 > 𝜔/N(, max +@ |𝜚(𝜉)| is found at 𝜉 = 𝜋/(Σ(<𝐿), +and +𝜌 = −1 + 𝜔 '1 − (𝑇 − 𝑇!) + +"!" +"!# − , +"$" +"$# − +"!" +"!#- +#$%# +#$&%&' +' +()#*( 𝜌)* . ++ +")#,/01. (31) +Then the optimal 𝜔/N( can be obtained by equating Eqs. +(30) and (31): +𝜔/N( = +* +*4(O4O5)P* +4"* +4"-4J4.* +4.-4 +4"* +4"- +K +*3$- +*36127 +8 +40-9: +D12Q +8 +40-9RS + . (32) +For the case shown in Fig 2, the FA predicted optimal +underrelaxation factor is the same as the numerical result, +𝜔/N( = 0.66. Note that this case is unstable (𝜌 = 1.042) +unless underrelaxation is applied. It indicates that the +theoretical estimate of the optimal underrelaxation factor is +quite accurate. The optimal underrelaxation depends on +various parameters as indicated by Eq. (32). For example, it +varies with the core height (for this case, Σ(< = 0.718 cm4)) +as depicted in Fig. 3. The more underrelaxation is needed for +higher cores (or longer fuel rods). It also indicates that the +higher fuel/coolant temperature difference (i.e., larger linear +power or thermal resistance), the more underrelaxation is +necessitated for stabilizing Picard iteration. + +Fig. 3. Optimal underrelaxation vs. core height. + +CONCLUSIONS + We have presented a formal Fourier analysis to +theoretically predict the convergence properties of Picard +fixed-point +iteration +for +coupled +neutronics/thermal- +hydraulics calculations. The work provides a more rigorous +theoretical basis for applications of Picard iteration for such +calculations. The derived closed form estimate for the +spectral radius of the Picard coupling method is a function of +various reactor parameters such as the fuel and coolant +temperature difference (which instead depends on the rod +linear power and thermal resistance), fuel temperature +feedback (Doppler effect), scattering ratio, and reactor core +height. It implies that Picard iteration is more stable for +smaller reactors and lower rod linear power (or thermal +resistance). In addition, it is worth noting that for LWRs the +Doppler feedback plays a more dominant role in Picard +iteration than the moderator temperature (density) feedback. +This finding is consistent with numerical experiments +reported in Ref. 4. We will report the analysis in the future. +A long-standing issue with Picard iteration is that it +oftentimes relies on underrelaxation to stabilize the coupled +calculation. However, a priori optimal underrelaxation was +previously not available for a specific problem. We hope that +our new theoretical result can provide a valuable estimate of +underrelaxation for stabilizing coupled neutronics/thermal- +hydraulics calculations. It is now possible to determine local +optimal underrelaxation factors and apply them to different +fuel rods or assemblies in the reactor. +The relaxed Picard iteration is similar to the undamped +Anderson acceleration with depth 𝑚 = 1 (AA-1), in which +only one previous iterate is used [6,2,3]. However, it is +expected that the Picard with optimal underrelaxation will +outperform the AA-1 algorithm since the linear coefficients +of AA-1 are determined by minimizing the norm of an affine +combination of residual vectors and they are generally +different from the optimal underrelaxation factor. +Although we have only focused on the Fourier analysis +for the simple PWR model problem, the methodology +presented should be applicable for other types of reactors and +more realistic problems such as multigroup problems. In +addition, it can be also applied to other coupling methods. + +REFERENCES +1. C. T. KELLY, Iterative Methods for Linear and +Nonlinear Equations, SIAM, Philadelphia (1995). +2. A. TOTH et al., “Analysis of Anderson Acceleration on +a Simplified Neutronics/Thermal Hydraulics System,” +Proceedings of Joint International Conference on +Mathematics and Computation (M&C), Supercomputing +in Nuclear Applications (SNA) and the Monte Carlo +(MC) Method 2015, Nashville, TN, April 19-23, 2015. +3. S. HAMILTON, et al., “An assessment of coupling +algorithms for nuclear reactor core physics simulations,” +J. Comput. Phys., 311, 194 (2017). +4. D. F. GILL, et al., “Numerical Methods in Coupled +Monte Carlo and Thermal-Hydraulic Calculations,” +Nucl. Sci. Eng., 185, 722 (2019). +5. D. WANG and F. ABDULLATIF, “Neutron Transport +Problems with Nonlinear Temperature Feedback,” +Proceedings +of +International +Conference +on +Mathematics and Computational Methods Applied to +Nuclear Science and Engineering 2021 (M&C 2021), +Virtual Meeting, October 3-7, 2021, pp. 1326-1335 +(2021). +6. D. G. ANDERSON, “Iterative Procedures for Nonlinear +Integral Equations,” J. Assoc. Comput. Mach., 12, 547 +(1965). +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1 +10 +100 +𝜔opt +L (cm) +T-Tm = 275K +T-Tm = 500K + diff --git a/69AyT4oBgHgl3EQfcvd4/content/tmp_files/load_file.txt b/69AyT4oBgHgl3EQfcvd4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb654091edc429fe6ff7af60e641d5f76ed3ed8c --- /dev/null +++ b/69AyT4oBgHgl3EQfcvd4/content/tmp_files/load_file.txt @@ -0,0 +1,261 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf,len=260 +page_content='Stability Analysis of Picard Iteration for Coupled Neutronics/Thermal-Hydraulics Simulations Dean Wang Nuclear Engineering Program, The Ohio State University, Columbus, OH 43210 wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='12239@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='edu David P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' Griesheimer Bettis Atomic Power Laboratory, Bechtel Marine Propulsion Corporation, West Mifflin, PA 15122 david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='griesheimer@unnpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='gov INTRODUCTION Reactor core analysis often needs to solve a multiphysics nonlinear coupled system, including neutron transport, thermal-hydraulics, and other important physics phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' One straightforward method for solving such a coupled system is Picard fixed-point iteration [1], which alternates between solving individual physics problems separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' However, many numerical studies show that Picard iteration can be unstable, and a user-defined relaxation is usually required to achieve convergence [2-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' In this paper, we present a formal Fourier analysis (FA) of Picard iteration for the coupled neutronics/thermal hydraulics (N/TH) problem and derive theoretical predictions for the spectral radius of Picard iteration for such coupled calculations as a function of the temperature difference between the fuel and coolant, temperature coefficients of cross sections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=', Doppler feedback), scattering ratio, and core height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' An optimal underrelaxation factor is also derived based on the Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' FORMULATION AND ALGORITHM We consider the following simple one-group, planar- geometry k-eigenvalue problem on the domain 0 ≤ 𝑥 ≤ 𝐿 with reflective boundary conditions: 𝜇 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "($,&) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='$ + Σ((𝑇)𝜓(𝑥, 𝜇) = ) Σ+(𝑇)𝜙(𝑥) + ) ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='"" 𝜈Σ-(𝑇)𝜙(𝑥) , (1) and the simplified heat transfer equation for a single typical pressurized water reactor (PWR) fuel pin: 𝑇 = 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' + 𝐴Σ-(𝑇)𝜙(𝑥) , (2) with 𝐴 = 𝜋𝑟-/ 𝜅𝑅( , (3a) and 𝑅( = 6 ) 01," + ) 12#3# + ) 1,$ ln 9 2$% 2$&: + ) 12$%3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' , (3b) where 𝜓 = neutron angular flux 𝜙 = ∫ 𝜓(𝑥, 𝜇)𝑑𝜇 ) 4) , neutron scalar flux Σ( = macroscopic total cross section Σ+ = macroscopic scattering cross section Σ- = macroscopic fission cross section ν = average neutron yields per fission 𝑘5-- = effective multiplication factor 𝜅 = average energy released per fission 𝑇 = volume averaged fuel temperature 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' = bulk coolant temperature 𝑟-/ = fuel radius 𝑟67 = cladding inner radius 𝑟6/ = cladding outer radius 𝑟8 = 2$&92$% , mean radius in the gap 𝑘- = fuel thermal conductivity 𝑘6 = cladding thermal conductivity ℎ8 = effective gap conductance ℎ = coolant convection heat transfer coefficient Note that the linear heat generation rate (or linear power) of the fuel rod, 𝑞′, can be calculated by 𝑞:(𝑥) = 𝜋𝑟-/ 𝜅Σ-(𝑇)𝜙(𝑥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (4) Picard iteration is used to solve the above coupled N/TH system as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' The transport equation is solved first, then the fuel temperature is calculated using the newly obtained thermal power (neutron flux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' An underrelaxation factor is introduced in the temperature update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' Note that the transport iteration is fully converged during each TH update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' 𝜇 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "(()*)($,&) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='$ + Σ(C𝑇(,)D𝜓(,9)) = ) Σ+C𝑇(,)D𝜙(,9))(𝑥) + ) ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='"" 𝜈Σ-C𝑇(,)D𝜙(,9)) , (5) 𝑇∗ = 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' + 𝐴Σ-C𝑇(,)D𝜙(,9))(𝑥) , (6a) 𝑇(,9)) = 𝜔𝑇∗ + (1 − 𝜔)𝑇(,) , (6b) where 𝜔 is the underrelaxation factor and the superscript 𝑘 denotes the iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' LINEARIZATION To perform Fourier analysis of the coupled N/TH problem, we need to first linearize the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' We define the following linearized variables: 𝜓(𝑥, 𝜇) = 𝜓<(𝑥, 𝜇) + 𝜀𝜓)(𝑥, 𝜇) , (7a) 𝜙(𝑥) = 𝜙<(𝑥) + 𝜀𝜙)(𝑥) , (7b) 𝑘5-- = 𝑘5--,/ , (7c) 𝑇(𝑥) = 𝑇< + 𝜀𝑇)(𝑥) , (7d) Σ7(𝑇) = Σ7< + Σ7)(𝑇 − 𝑇<) = Σ7< + 𝜀Σ7)𝑇)(𝑥) , 𝑖 = 𝑡, 𝑠, 𝑓, 𝑎 (7e) Note that 𝑘5-- = 𝑘5--,/ due to the flux normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' The cross sections are assumed to be linearly dependent on the fuel temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' However, other feedback mechanisms such as thermal expansion [5] and moderator temperature feedbacks can be treated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (7a) - (7e) into (5), after some algebra we obtain by neglecting the 𝑂(𝜀*) terms 𝜇 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "* (()*)($,&) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='$ + Σ(<𝜓) (,9))(𝑥, 𝜇) + Σ()𝑇) (,)(𝑥)𝜓<(𝑥, 𝜇) = ) Σ+<𝜙) (,9))(𝑥) + ) Σ+)𝑇) (,)(𝑥)𝜙< + ) ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "",- 𝜈Σ-<𝜙) (,9))(𝑥) + ) ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "",- 𝜈Σ-)𝑇) (,)(𝑥)𝜙< .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (8) For reflective BC, 𝜓< = =- , and Σ>< = ) ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "",- 𝜈Σ-<, then we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (8) as 𝜇 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "* (()*)($,&) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='$ + Σ(<𝜓) (,9))(𝑥, 𝜇) = ) Σ(<𝜙) (,9))(𝑥) − ) Σ(<𝛾𝑇) (,)(𝑥) , (9) where 𝛾 = (1 − 𝑐<) Q ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='. * ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='.- − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "* ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' "-R 𝜙< , (10a) with Σ>) = Σ() − Σ+) , (10b) 𝑐< = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='/- ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content='0- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (10c) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (7b), (7d), and (7e) into (6a) and (6b) respectively, we obtain 𝑇) ∗(𝑥) = 𝐴Σ-<𝜙) (,9))(𝑥) + 𝐴Σ-)𝜙<𝑇) (,)(𝑥) , (11) 𝑇) (,9))(𝑥) = 𝜔𝑇) ∗(𝑥) + (1 − 𝜔)𝑇) (,)(𝑥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (12) Then we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfcvd4/content/2301.00289v1.pdf'} +page_content=' (11) into (12) to give 𝑇) (,9))(𝑥) = 𝜔𝐴Σ-<𝜙) (,9))(𝑥) + C1 − 𝜔 + 𝜔𝐴Σ-)𝜙 3, 4 +has a pair of indexes contracted with a SCHA propaga- +tor G(0)(t = 0−) (Eq. (65)). This means that the an- +harmonic vertices are renormalized by quantum-thermal +fluctuations. +C. +Anharmonic propagators +In this Section, we discuss the TDSCHA interacting +propagators. +Specifically, we present two-phonon pro- +cesses that have been neglected in previous works [1, 2]. +In TDSCHA, the one-phonon G(ω)µν, two-phonon +χ(ω)µνηλ, and the one-two phonon Γ(ω)µηλ interacting +propagators are obtained as response functions by set- +ting in χ(ω)A,B (Eq. (53)) +AG = δ �R(0) +µ +BG =δ �R(0) +ν , +(66a) +Aχ = 1 +2δ �R(0) +µ δ �R(0) +ν +Bχ =1 +2δ �R(0) +η δ �R(0) +λ , +(66b) +AΓ = δ �R(0) +µ +BΓ =1 +2δ �R(0) +η δ �R(0) +λ . +(66c) +The response and perturbation vectors (see Eqs (54)) cor- + +10 +responding to Eqs (66) are as follows +rG = +� +��� +δµ +0 +0 +� +��� +pG = +� +��� +δν +0 +0 +� +��� , +(67a) +rχ = +� +��� +0 +Sµν +Sµν +� +��� +pχ = +� +��� +0 +Sηλ +Sηλ +� +��� , +(67b) +rΓ = +� +��� +δµ +0 +0 +� +��� +pΓ = +� +��� +0 +Sηλ +Sηλ +� +��� , +(67c) +where δµ is a 3N vector with 1 in the mode index µ and +zero elsewhere and Sηλ is a 3N ×3N matrix with 1/2 on +the mode indexes η and λ and zeros elsewhere. +The choice of A/B, as in Eqs (66), is not arbitrary. +In the non-interacting case, L(ω) is diagonal, so the re- +sponse function is simply +χ(ω)A,B =ri +� +��� +G(0)(ω) +0 +0 +0 +χ(0) +− (ω) +0 +0 +0 +−χ(0) ++ (ω) +� +��� pi +i =G, χ, Γ, +(68) +From Eq. (68) we recover the one and two phonon free +propagators (Eqs (59) (60)) and no cross terms connect- +ing single to double excitations. These results are con- +sistent with the standard linear response theory, in the +non-interacting case, treated with the many-body for- +malism. This proves that with Eqs (67) we recover the +physically relevant quantities. +To get the interacting Green’s function, we plug r and +p of Eqs (67) in the expression of χ(ω)A,B, Eq. (53), and +we invert L(ω) following Refs [1, 55, 56] (see also Ap- +pendix G). To do this, we consider L(ω), Eq. (55), as +a 3 × 3 block-matrix (as represented in Fig. 2) where +each block itself is a tensor. The same applies to r and +p, Eqs (54), which are understood as 3 components vec- +tors. From the non-interacting case (Eq. (68)), we learn +that the first component of p/r controls the single-mode +propagation while the second and third the two-phonon +channel. +The inversion process mixes the matrix elements of +L(ω) adding interactions to the free propagators which +are expressed as diagrammatic series. The representation +of Fig. 2 is a graphical aid to visualize the building blocks +of these diagrams. In Appendix H we report the details +of all the results presented. +In our calculations we always reduce the inversion of +L(ω) to a 2 × 2 block-matrix with the following form +� +� A +C +C† B +� +� +(69) +which can be easily inverted (see Appendix G) +� +� +� +A − CB−1C†�−1 +� +C† − BC−1A +�−1 +� +C − AC†−1B +�−1 +B−1 − B−1C† � +C† − BC−1A +�−1 +� +� . +(70) +First, we discuss the one-phonon propagator. Because +only the first component of both pG and rG, Eq. (67a), +are non-zero, the response calculation is simplified. In +particular, we compact the 2 × 2 two-phonon sector of +L(ω) (i.e. L(ω)ij with i, j > 1) in a 1×1 block matrix. As +shown in Appendix H, L(ω) is reduced to a 2 × 2 block- +matrix L1ph(ω), so that the one-phonon propagator is +given by +G(ω) =rG · L(ω)−1 · pG = +� +L1ph(ω)−1� +11 +(71) +where +L1ph(ω) = +� +�� +G(0)(ω)−1 +− +(3) +D +− +(3) +D +χ(0)(ω)−1 − +(4) +D +� +�� . +(72) +Note that (L1ph(ω))22 represents the anharmonic two- +phonon channel in which single modes can decay through +the three-phonon vertex, (L1ph(ω))12. Graphically Eq. +(71) corresponds to += +� +� +−1 +− +− +−1 +− +� +� +−1 +11 +. +(73) +In Eq. (71) we apply the general result of Eq. (70) to +obtain the interacting Green’s function +G(ω) = G(0)(ω) + G(0)(ω) · Π(ω) · G(ω), +(74) +here A : B = �3N +µν=1 A..µνBµν.., where the self-energy +Π(ω) coincides with the one reported in Refs [1, 2, 49] +Π(ω) = +(3) +D : +� +1 − χ(0)(ω) : +(4) +D +�−1 +: χ(0)(ω) : +(3) +D. +(75) +Our definition of the non-interacting two-phonon propa- +gator χ(0)(ω) (Eq. (60)) is one reported by Ref. [49] in +Eq. (72) multiplied by −1/2 so that all the definitions +are consistent. +Physical phonon frequencies and lifetimes are given by +real and imaginary parts of G(ω + i0+), Eq. (74), as dis- +cussed in Ref. [45]. In addition, we remark that the polar- +ization vectors can also change when adding dynamical +effects so polarization-mixing is automatically included +in Eq. (74). +The lowest-order approximation for the self-energy Eq. +(75) is the bubble diagram which is included by many +SCP calculations in the improved SCP (ISCP) frame- +work [40, 57–59]. TDSCHA represents a theoretical ap- +proach that justifies this from the least action principle + +11 +and paves the way to go beyond the bubble approxima- +tion. +For the two-phonon case, we proceed as before. We use +the definition of rχ/pχ (Eq. (67b)) to reabsorb the one- +phonon sector of L(ω) (i.e. the row L(ω)1j and column +L(ω)j1 with j = 1, 2, 3) into the two-phonon sector. So +we solve +χ(ω) = rχ · L(ω)−1 · pχ = +1,2 +� +ij +� +L2ph(ω)−1� +ij +(76) +where L2ph(ω) is a 2 × 2 block-matrix +L2ph(ω)= +� +�χ(0) +− (ω)−1 − Σ(ω) +−Σ(ω) +−Σ(ω) +−χ(0) ++ (ω)−1 − Σ(ω) +� +� . +(77) +Σ(ω) is the two-phonon self-energy +Σ(ω) = Σ(ω)† = +(4) +D + +(3) +D · G(0)(ω) · +(3) +D +(78) +where single phonon excitations enter in Σ(ω) via the +three-phonon vertex. +Graphically, Eq. (76) corresponds to += +1,2 +� +ij +� +�� +−1 +− +− +− +− +−1 +− +� +�� +−1 +ij +Σ(ω) = += ++ +. +(79) +Again we use Eq. (70) to invert Eq. (77) and we end +up with the interacting two-phonon propagator +χ(ω) = χ(0)(ω) + χ(0)(ω) : Σ(ω) : χ(ω) +(80) +with Σ(ω), Eq. (78), being the TDSCHA two-phonon +self-energy. We find that in the two-phonon propagation +there is the possibility of either decay in a single phonon +through the third-order scattering vertex or in another +pair through the fourth-order scattering vertex. There +are no high-order decay processes. +For the mixed propagator, we proceed as before. +Thanks to the form of pΓ/rΓ (Eq. (67c)), single phonon +excitations can be triggered by two-phonon propagation +via the three-phonon vertex. This leads to a non-zero +one-two phonon propagation +Γ(ω) = G(0)(ω) · +(3) +D : χ(ω). +(81) +Fig. 4 summarizes the diagrammatic expressions for +the interacting propagators, Eqs (74) (80) (81). +By computing all the TDSCHA interacting propaga- +tors, we have a full comprehension of the diagrammatic +expression, Fig. 5, introduced in Ref. [1] for the TDSCHA +response function χ(ω)A,A (Eq. (53)). In fact, Eq. (53) +can be decomposed into the interacting propagators. +FIG. 4. Diagrammatic expression of the interacting one, two +and one-two phonon Green’s functions, Eqs (74) (80) (81). +Thinner solid lines represent the non-interacting SCHA prop- +agators G(0)(ω) and χ(0)(ω), defined in Eqs (74) (80). The +three/four-phonon scattering vertices are defined in Eqs (62) +(63) and represented as orange triangles and red squares. +FIG. 5. Diagrammatic expression of the processes included +in the fully interacting response, Eq. (53), if A = B. +The +interacting TDSCHA Green’s functions are reported in Eqs +(74) (80) (81). The green vertex is related to the first entry +of p/r while the blue vertex to the second and third entries +of p/r, see Eqs. (54). +One-phonon processes G(ω) are coupled to first-order +position derivatives of the perturbation, first entry of p/r +Eqs (54). +On the other hand, two-phonon excitations +χ(ω) and Γ(ω) are triggered by non-zero second-order +position derivatives of the perturbation, second and third +entries of p/r Eqs. (54). +We remark that the Lanczos algorithm includes the +effect of the third and fourth-order scattering vertex, Eqs +(62) (63), in a non-perturbative way [13] (see Appendix +F). TDSCHA evolves ab-initio all the phonon modes in a +given supercell without free parameters. This feature is +interesting for applications in non-linear phononics where +is crucial to comprehend relaxation pathways of coherent +phonon oscillations [28, 60–62]. + +Xr(w) +(3)5 +X(W)A,A ++2 +0A +OR +OROR +(0) +(0)12 +1. +Momentum Green function +In Wigner-TDSCHA, the ionic momentum is con- +trolled directly, which was not possible in the original +formulation. Here we discuss the TDSCHA momentum- +momentum Green’s function Gp(ω). In our theory, this +is computed setting +A = �Pµ +B = �Pν. +(82) +The SCHA momentum Green’s function G(0) +p (ω) is pro- +portional to G(0)(ω) (Eq. +(74)) since the equation for +position and momentum are coupled +iω �P(ω) = �R(ω) −→ G(0) +p (ω)µν = ω2 +µG(0)(ω)µν. +(83) +The free propagators are the building blocks for the inter- +acting theory. Hence the interacting momentum Green’s +function Gp(ω) satisfies a perturbative expansion that is +proportional to the one of G(ω) once the TDSCHA dia- +grams are selected, i.e. those from Eq. (74). So we have +that, see Appendix J for details, +Gp(ω) = −1 + ω2G(ω). +(84) +Thus, a Lanczos calculation also provides access to the +TDSCHA momentum Green’s function. +D. +Multiple excitations in TDSCHA +The Gaussian approximation defines a hierarchy of di- +agrams that is truncated at the two-phonon level. +In +this Section, we show that, in TDSCHA, all higher-order +phonon propagators are related to the Green’s functions +of Eqs (74) (80) (81). +For example, the three-phonon propagator is obtained +setting in Eq. (53) a tensor-like perturbation/response +functions +A = δ �R(0) +α δ �R(0) +β δ �R(0) +γ +B = δ �R(0) +µ δ �R(0) +ν δ �R(0) +η . +(85) +In +this +case, +only +the +first +entries +of +p/r, +i.e. +� +∂A/∂ � +R +� +(0), are non-zero. This means that in the case +of Eq. (85) we have a one-phonon response, as the one +obtained for G(ω) (Eq. (67a)). As computed in Appendix +H, the three-phonon response is +χαβγ +µνη (ω) = G(0)(t = 0−)βγG(0)(t = 0−)νηG(ω)αµ ++ permutations of (αβγ) and (µνη) +(86) +and in Fig. 6 we report its diagrammatic structure. This +contains the one-phonon Green’s function G(ω), Eq. (74), +and a disconnected part, G(0)(t = 0−) which comes from +the averages +� +δ � +Rδ � +R +� +(0) (Eq. (39)) in +� +∂A/∂ � +R +� +(0). +FIG. 6. Diagrammatic expression for the TDSCHA interact- +ing three-phonon Green’s function obtained as a response to a +cubic perturbation, Eq. (85). In TDSCHA, the tree-phonons +propagator is a disconnected diagram. +In this case, the SCHA correction, G(0)(t = 0−), does +not enter the phonon propagation but it dresses the in- +teraction with the external probe. This means that if we +take a scalar perturbation +A = B = 1 +3 +3N +� +αβγ=1 +Wαβγδ �R(0) +α δ �R(0) +β δ �R(0) +γ , +(87) +with Wαβγ a tensor that does not depend on atomic po- +sitions, G(0)(t = 0−) is contracted only the tensor W . +Similarly, all the higher-order propagators, i.e. those +obtained with +A ∼ +� +δ �R(0)�n−2 +B ∼ +� +δ �R(0)�m−2 +m, n > 2, +(88) +give disconnected diagrams. +In all these cases we will +get a χA,B(ω) that contains only one of the TDSCHA +propagators (Eqs (74) (80) (81)) plus a disconnected part +that depends only on G(0)(t = 0−), Eq. (39). +FIG. 7. Diagrammatic expression for the Saturn diagram with +a three SCHA phonon propagation (solid lines are the propa- +gators of Eq. (59)). The red vertex is the four-phonon scatter- +ing vertex Eq. (63) which leads to three phonon excitations. +This class of diagrams is missed by TDSCHA where we can +not connect a single SCHA line to the four-phonon scattering +vertex. +So TDSCHA can not capture processes beyond a two- +phonon mechanism: the propagators of Eqs (74) (80) +(81) are the building blocks of the response. This means +that there are general rules in the symbolic inversion of +L(ω) (see Eq. (55)). In Fig. 2 the solid line (one-phonon +SCHA propagator of Eq. (60)) is always attached to one +extremity of the orange triangle (Eq. (62)). +The dou- +ble solid line (two-phonon SCHA propagator Eq. (60)) +is connected to two extremities either of the red square +(four phonon vertex Eq. (63)) or of the orange triangle +(three phonon vertex Eq. (62)). +We do not get three or more SCHA phonon resonances. +One example is the ’Saturn’ diagram (Fig. 7) which is + +13 +missed by our method. This diagram would correspond +to a single SCHA propagator attached to the four-phonon +vertex and this is not contained in TDSCHA. +Notably, the TDSCHA diagrams emerge from the sta- +tionary action principle of quantum mechanics [1]. This +guarantees that there is no double counting and that +the theory is consistent. The inclusion of new scattering +mechanisms, as Fig. 7, must be done with extreme care +to avoid spoiling the internal coherence and overcounting +some anharmonic processes. +V. +NONLINEAR PHONON-PHOTON +COUPLING: INFRARED AND RAMAN +In this Section, we review the infrared (IR) and Raman +response in TDSCHA. In particular, we focus on the two- +phonon effect. +IR experiments are based on the absorption of infrared +light by normal modes associated with a dipole moment +variation which, in a crystal, are optical phonons. The IR +signal is proportional to the imaginary part of the dipole- +dipole response function, Im[χ(ω + i0+)pα,pβ], along two +Cartesian directions α and β hence +A(R) = pα(R), +B(R) = pβ(R), +(89) +where the dipole pα is per unit volume. +To get the response function we need the response and +perturbation vector r′ and p′ (see Section IV A). The first +component of these vectors contains equilibrium averages +of the effective charges: +�∂pα(R) +∂Ra +� +(0) += ⟨Z∗(R)a,α⟩(0) , +(90) +where a is a supercell index and Z∗(R) is the effec- +tive charges tensor. This vertex is the coupling for one +phonon process. +The second and third components of r′/p′ contain the +first derivatives of the effective charges, the second-order +dipole moment: +�∂2pα(R) +∂Ra∂Rb +� +(0) += +�∂Z∗(R)a,α +∂Rb +� +(0) +. +(91) +A Raman process consists in the scattering of light +(usually visible) by zone-center phonons that induce a +change in polarizability. The Raman cross-section con- +tains the imaginary part of polarizability-polarizability +response, Im[χαµν,αηλ(ω + i0+)], obtained with +A(R) = α(R)µν +B(R) = α(R)ηλ, +(92) +where µ, ν, η, λ are Cartesian directions. In a Raman pro- +cess phonons and photons scatter so we take into account +the quantization of the electromagnetic field by multi- +plying Im[χαµν,αηλ(ω + i0+)] by 1 + n(ω) where n(ω) is +the Bose-Einstein distribution for the photons. The first +component of r and p gives one-phonon processes and +contains: +�∂α(R)µν +∂Ra +� +(0) += ⟨Ξ(R)a,µν⟩(0) , +(93) +where Ξ(R) is the Raman tensor. As before, the other +components of r′ and p′ depends on the second-order +Raman polarizability: +�∂2α(R)µν +∂Ra∂Rb +� +(0) += +�∂Ξ(R)a,µν +∂Rb +� +(0) +. +(94) +Eq. (91) and Eq. (94) trigger second-order IR/Raman +processes exiting two phonons in the system, see Fig. 5. +In principle, higher-order processes are possible, such as +three-phonon etc. However, TDSCHA can not account +for them as we showed that the three-phonon propagator +is a disconnected diagram (see Sec. IV D). +A two-phonon process, both in IR and Raman spectra, +is a scattering mechanism between photons and phonons +that conserves both energy and momentum. The long- +wavelength electromagnetic field can either absorb or +generate two phonons. Another possibility is that one +phonon is absorbed and the other one is emitted inter- +acting with photons. This involves pairs of phonons with +opposite momentum in the Brillouin zone forming a con- +tinuum signal overlapped to the sharper peaks of one +phonon process. +This mechanism is found both in harmonic and anhar- +monic systems. The IR spectra of both Si and Ge can +only be explained with these processes since there are no +IR-active phonons due to inversion symmetry [63]. Also, +anharmonic systems, such as liquid water [64], present +features due to effective charge position modulations. +Two phonons effects play a central role in many Raman +spectra: from diamond and SiC [65, 66] to BaTiO3 [67]. +The most common approximation is +Z∗(R)a,α ≃ Z∗(R(0))a,α, +(95a) +Ξ(R)a,µν ≃ Ξ(R(0))a,µν, +(95b) +which suppresses all two phonon processes. +We use integration by parts and a Monte Carlo sam- +pling, as proposed in [1], to compute all the components +of r′ and p′ in an efficient and non-perturbative way us- +ing only effective charges and Raman tensor: +�∂2p(R)α +∂Ra∂Rb +� +(0) += − +3N +� +c=1 +G(0)(t = 0−)ac +� +δ �R(0) +c Zb,α(R) +� +(0) , +(96a) +�∂2α(R)µν +∂Ra∂Rb +� +(0) += − +3N +� +c=1 +G(0)(t = 0−)ac +� +δ �R(0) +c Ξb,µν(R) +� +(0) , +(96b) +We remark that TDSCHA is the only method that com- +putes second-order Raman tensors or effective charges +with full position dependence without the need for +higher-order DFT response. In Appendix K we report +in detail how to prepare a IR/Raman calculation. + +14 +VI. +INFRARED SPECTRA OF +HIGH-PRESSURE HYDROGEN +In this Section, we show the relevance of two phonon +effects in a strongly anharmonic system such as high- +pressure hydrogen phase III (C2c/24). +We apply our +new TDSCHA implementation on the infrared spectra +of high-pressure hydrogen at P = 250 GPa and T = 0 K +including the effect of second-order effective charges, Eq. +(91). We employ 40000 energy/forces and 2000 effective +charges calculations, on a 2×2×1 supercell, to converge +the anharmonic vertices, Eqs (62) (63), and the IR over- +tone. +Energies, forces and effective charges were com- +puted using the BLYP functional [68] on a 4×4×4 k-grid +(energy cutoff of 60 Ry and 240 Ry on the charge den- +sity) as implemented in QUANTUM ESPRESSO [69, 70], +with a plane wave basis set and a norm-conserving pseu- +dopotential from the PSEUDO DOJO library [71]. +FIG. 8. Infrared signal for high pressure hydrogen at P = 250 +GPa and T = 0 K. Panel (a) reports the IR spectra obtained +with the SCHA phonons of Eq. (27) without two-phonon ef- +fects. +These are included at the SCHA level in panel (b). +Panel (c) reports the spectra setting both +(4) +D , +(3) +D ̸=0 compared +with the data extracted from Ref. [5] at 248 GPa and 20 K. +The smearing δ is 30 cm−1. +In Figure 8 we plot the IR signal using different approx- +imations defined as +1 +3 +x,y,z +� +α +Im [χ(ω + iδ)pα,pα] +(97) +where δ is the smearing. In Fig. 9 we plot the IR signal +as a function of the Lanczos steps. +FIG. 9. Convergence of the IR signal with +(4) +D , +(3) +D ̸=0 as a func- +tion of the Lanczos steps. The smearing δ is 15 cm−1. +The convergence is achieved in Nstep = 500 (Fig. 9) steps +which are half of those employed in Ref. [1] (Nstep = +1000). This is due to a more stable Lanczos algorithm +thanks to the symmetry of L(ω) in the Wigner formalism. +Panel (a) and (b) of Figure 8 show the effect of adding +the second-order IR effects using the non-interacting +SCHA phonons. +The position modulation of effective +charges generates a signal between 2000-4000 cm−1 and +around 5000 cm−1. +In panels (c) of Figure 8 we add +all the anharmonic interactions contained in TDSCHA. +Notably, the two phonon processes at high frequency +are stable after adding the anharmonic scattering of two +phonons. This feature is in agreement with the overtone +observed in the experiments by Goncharov et al. [5], con- +firming that it is a high-order IR process. + +SCHA phonons +150 +(a) +(Z(R))(0) +125 +100 +ignal +75 +s +R +50 +25 +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +SCHA phonons +150 +(b) +aZ(R) +(Z(R))(0) +/(0) +äR +125 +100 +75 +s +R +50 +25 +0 +1000 +3000 +5000 +6000 +0 +2000 +4000 +(3) +(4) +TDSCHA phonons with D + D +150 +(Z(R))(0) +(c) +/aZ(R)/ +125 +(Z(R))(0) +(0) +aR +100 +Goncharov et al. +75 +R +50 +25 +0 +1000 +2000 +3000 +4000 +5000 +6000 +0 +w (cm-1)Nsteps=200 +(Z(R))(0) +200 +/aZ(R) +(Z(R))(0) +/(0) +aR +100 +R +50 +0 +2000 +3000 +5000 +0 +1000 +4000 +6000 +Nsteps=300 +250 +(Z(R))(0) +200 +aZ(R) +(Z(R))(0) + +/(0) +150 +sigr +R100 +50 +0 +6000 +2000 +3000 +5000 +0 +1000 +4000 +Nsteps=400 +250 +(Z(R))(0) +200 +/aZ(R) +(Z(R))(0) +(0) +äR +S. +R100 +50 +0 +2000 +0 +3000 +5000 +6000 +1000 +4000 +Nsteps=500 +250 +(Z(R)(0) +aZ(R) +200 +(Z(R))(0) + +sigr +R100 +50 +0 +1000 +3000 +4000 +6000 +2000 +5000 +w (cm-1)15 +VII. +CONCLUSIONS +The Wigner picture simplifies the TDSCHA equations +improving the physical intuition of the method. +This +allows us to discuss the equivalence of quantum and clas- +sical dynamics and rewrite the equations of motion in +terms of position and momentum correlators. +The response function is directly related to the dia- +grammatic expression of the interacting Green’s function, +through which we have built a bridge to many-body per- +turbation theory. In the context of linear response theory, +we clarified which diagrams and scattering processes are +included in the method. +The TDSCHA infrared spec- +tra of high-pressure hydrogen phase III showed that only +two phonon effects explain the overtone experimentally +observed in Ref. [5]. +ACKNOWLEDGEMENTS +The authors acknowledge support by EU under project +MORE-TEM ERC-SYN (grant agreement No 951215) +and the CINECA award under the ISCRA initiative, for +the availability of high-performance computing resources +and support. We also acknowledge PRACE for awarding +us access to Joliot-Curie Rome at TGCC, France. + +16 +Appendix A: Equations of motion +In this Appendix, we prove the Wigner-TDSCHA equations of motion Eqs (23). For compactness, we define the +mass-rescaled free parameters +�α(t)ab = +α(t)ab +√mamb +�β(t)ab = √mambβ(t)ab +�γ(t)ab = +� mb +ma +γ(t)ab +�Ra = √maRa +�R(t)a = √maR(t)a +�Pa = +Pa +√ma +�P(t)a = P(t)a +√ma +(A1) +The equation of motion for the free parameters �α(t), �β(t), �γ(t), �R(t), �P(t) are found with Eq. (20): +∂�ρ(R, P , t) +∂t += ∂H(�ρ) +∂ � +R +· ∂�ρ(R, P , t) +∂ � +P +− ∂H(�ρ) +∂ � +P +· ∂�ρ(R, P , t) +∂ � +R += += +��∂V (tot) +∂ � +R +� +�ρ(t) ++ δ � +R(t) · +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +� +· ∂�ρ(R, P , t) +∂ � +P +− +� +δ � +P (t) + �P(t) +� +· ∂�ρ(R, P , t) +∂ � +R +(A2) +with δ � +R(t) = � +R − �R(t) and δ � +P (t) = � +P − �P(t). The gradient of �ρ(R, P , t), defined in Eq. (15), is +∂ log (�ρ(R, P , t)) +∂ � +P += − �β(t) · δ � +P (t) + �γT (t) · δ � +R(t), +(A3a) +∂ log (�ρ(R, P , t)) +∂ � +R += −�α(t) · δ � +R(t) + �γ(t) · δ � +P (t). +(A3b) +The time derivative of �ρ(R, P , t) gives +∂ log (�ρ(R, P , t)) +∂t += +˙N(t) +N(t) − 1 +2δ � +R(t) · ˙�α(t) · δ � +R(t) − 1 +2δ � +P (t) · ˙�β(t) · δP (t + δ � +R(t) · ˙�γ(t) · δ � +P (t)+ ++ ˙�R(t) · α(t) · δ � +R(t) + ˙�P(t) · �β(t) · δ � +P (t) − ˙�R(t) · �γ(t) · δ � +P (t) − ˙�P(t) · �γT (t) · δ � +R(t), +(A4) +where ˙◦ denotes the time derivative. With Eqs (A3) and Eq. (A4) the Wigner-Liouville equation Eq. (A2) becomes a +polynomial in δ � +R(t) δ � +P (t) then, setting to zero the coefficients, we get the equations of motion for the free parameters +d +dt +�R(t) = �P(t) +(A5a) +d +dt +�P(t) = − +�∂V (tot) +∂ � +R +� +�ρ(t) +(A5b) +d +dt �α(t) = − +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +· �γ†(t) − �γ(t) · +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +(A5c) +d +dt +�β(t) =�γ†(t) + �γ(t) +(A5d) +d +dt �γ(t) =�α(t) − +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +· �β(t) +(A5e) +where ◦† denotes the hermitian conjugate of a matrix. The equations of motion for the tensors keep the distribution +normalized. +The equal-time position and momentum correlators can be written in terms of �α(t), �β(t), �γ(t) +� +δ � +X(t)aδ �Y (t)b +� +�ρ(t) = +� +dR +� +dP �ρ(R, P , t)δ � +X(t)aδ �Y (t)b, +δ� +X(t), δ �Y (t) = δ � +R(t), δ � +P (t), +(A6) +and using Gaussian integration we get the following expressions +� +δ � +R(t)δ � +R(t) +� +�ρ(t) = +� +�α(t) − �γ(t) · �β−1(t) · �γT (t) +�−1 +, +(A7a) +� +δ � +R(t)δ � +P (t) +� +�ρ(t) = �α−1(t) · �γ(t) · +� +�β(t) − �γT (t) · �α−1(t) · �γ(t) +�−1 +, +(A7b) +� +δ � +P (t)δ � +P (t) +� +�ρ(t) = +� +�β(t) − �γT (t) · �α−1(t) · �γ(t) +�−1 +. +(A7c) + +17 +Then deriving with respect to time Eqs. (A7) and using the equations of motion Eqs. (A5) we prove Eqs. (23). +Appendix B: Equivalence with Time-Dependent Self-consistent Harmonic Approximation +In this Appendix, we show that our method is a Wigner reformulation of TDSCHA presented in [1]. We compute +the matrix elements of the von Neumann density operator corresponding to the Wigner distribution, Eq. (15). To do +this we need the inverse of the Wigner transformation which is defined as (see Ref. [41]) +ˆρ = +� dRdR′dP dP ′ +(2πℏ)3N +ρ(R′, P ′) exp +� +− i +ℏ +� +P · +� +ˆR − R′� ++ R · +� +ˆP − P ′��� +, +(B1) +where ρ(R′, P ′) is the Wigner quasi-distribution. The ˆ◦ indicates quantum operators. +Inserting the Wigner distribution of Eq. (15) in Eq. (B1) we get a Gaussian integral for the density operator matrix +elements +⟨R| ˆ�ρ(t) |R′⟩ = +� +dP exp +� i +ℏ(R − R′) · P +� +�ρ +�R + R′ +2 +, P , t +� +=N(t) exp +� +−1 +8 (δR(t) + δR′(t)) · α(t) · (δR(t) + δR′(t)) + i +ℏ(R − R′) · P(t) +� +� +dP exp +� +−1 +2δP (t) · β(t) · δP (t) + 1 +2 +�2i +ℏ (R − R′) + (δR(t) + δR′(t)) · γ(t) +� +· δP (t) +� += +� +det +�Υ(t) +2π +� +exp +� +iQ(t) · (R − R′) +− (R − R(t)) · +�1 +4Θ(t) + iC(t) +� +· (R − R(t)) − (R′ − R(t)) · +�1 +4Θ(t) − iC(t) +� +· (R′ − R(t)) ++ (R − R(t)) · (Re A(t) + i Im A(t)) · (R′ − R(t)) +� +. +(B2) +The last line of Eq. (B2) is the trial density operator used in [1]. The free parameters used in [1] Q(t), Θ(t), C(t), +Re A(t), Im A(t) are related to the ones used in the Wigner formalism +Q(t) = 1 +ℏP(t), +(B3a) +Θ(t) = 1 +2 +� +α(t) − γ(t) · β−1(t) · γT (t) +� ++ 2 +ℏ2 β−1(t), +(B3b) +C(t) = − 1 +2ℏβ−1(t) · γT (t), +(B3c) +Re A(t) = −1 +4 +� +α(t) − γ(t) · β−1(t) · γT (t) +� ++ 1 +ℏ2 β−1(t), +(B3d) +Im A(t) = 1 +2ℏ +� +γ(t) · β−1(t) − β−1(t) · γT (t) +� +(B3e) +Υ(t) = Θ(t) − 2 Re A(t) = α(t) − γ(t) · β−1(t) · γT (t), +(B3f) +where ◦−1 denotes the inverse of a matrix. The tensor Υ(t) is a linear combination of Θ(t) and Re A(t). The same +notation for the average position R(t) is adopted. Using the relations between free parameters, Eqs (B3), it is easy +to prove that the equations of motion Eqs (A5) are equivalent to the TDSCHA ones reported in [1]. +Here, we also prove that Eq. (25) is the Wigner transform of the SCHA equilibrium density matrix ˆ�ρ +(0) [1, 45]: +⟨R| ˆ�ρ +(0) |R′⟩ = +� +det +�Υ(0) +2π +� +exp +� +−1 +4 +3N +� +ab=1 +Θ(0) +ab (Ra − R(0) +a )(Rb − R(0) +b ) − 1 +4 +3N +� +ab=1 +Θ(0) +ab (R′ +a − R(0) +a )(R′ +b − R(0) +b ) ++ +3N +� +ab=1 +A(0) +ab (Ra − R(0) +a )(R′ +b − R(0) +b ) +� +(B4) + +18 +with Υ(0) = Θ(0) − 2A(0), where Υ(0) and A(0) are defined as +Υ(0) +ab = +3N +� +µ=1 +2ωµ +ℏ(1 + 2nµ)ea +µeb +µ, +A(0) +ab = +3N +� +µ=1 +2ωµnµ(1 + nµ) +ℏ(1 + 2nµ) +ea +µeb +µ. +(B5) +where ω2 +µ and {eµ} are the auxiliary SCHA modes Eq. (27). The Wigner quasi-distribution, according to Eq. (8), is +obtained in the following way +�ρ(0)(R, P ) = +� +det +�Υ(0) +2π +� +exp +� +−1 +2 +3N +� +ab=1 +(Ra − Ra)Υ(0) +ab (Rb − Rb) +� +� +d3NR′ +(4πℏ2)3N/2 exp +� +i +3N +� +a=1 +PaR′ +a +ℏ +− 1 +8 +3N +� +ab=1 +(Θ(0) +ab + 2A(0) +ab )R′ +aR′ +b +� += +� +det +�Υ(0) +2π +�� +� +� +�det +� +1 +2πℏ2 +� +A(0) + 1 +4Υ(0) +�−1� +exp +� +−1 +2 +3N +� +ab=1 +(Ra − R(0) +a )Υ(0) +ab (Rb − R(0) +b ) − 1 +2 +3N +� +ab=1 +Pa +� +ℏ2A(0) + ℏ2 +4 Υ(0) +�−1 +ab +Pb +� +(B6) +The final result is a positive-definite Gaussian Wigner distribution which coincides with Eq. (25) +�ρ(0)(R, P ) = +� +det +�α(0) +2π +� +det +�β(0) +2π +� +exp +� +−1 +2 +3N +� +ab=1 +(Ra − R(0) +a )α(0) +ab (Rb − R(0) +b ) − 1 +2 +3N +� +ab=1 +Paβ(0) +ab Pb +� +, +(B7) +once we recognize that +⟨δRδR⟩(0) = α(0)−1 = Υ(0)−1, +(B8) +and +⟨P P ⟩(0) = β(0)−1 = ℏ2 +� +A(0) + 1 +4Υ(0) +� +. +(B9) +where the equilibrium correlators are defined in Eqs (26). +Appendix C: Energy conservation +In this Appendix, we show that the TDSCHA equations of motion Eqs (23) satisfy the energy conservation principle. +The Wigner quantum time-dependent Hamiltonian has the same form as the classical one +H(t) = +3N +� +a=1 +P 2 +a +2ma ++ V (BO)(R) + V (ext)(R, t). +(C1) +We compute the total time-derivative of ⟨H(t)⟩�ρ(t) where �ρ(t) is defined in Eq. (15): +d ⟨H(t)⟩�ρ(t) +dt += d +dt +� 3N +� +a=1 +1 +2 +�� +δ �P(t)aδ �P(t)a +� +�ρ(t) + �P(t)2 +a +� ++ +� +V (tot)� +�ρ(t) +� +. +(C2) +The time derivative of the kinetic energy gives: +d +dt +3N +� +a=1 +1 +2 +�� +δ �P(t)aδ �P(t)a +� +�ρ(t) + �P(t)2 +a +� += 1 +2 +3N +� +a=1 +d +� +δ �P(t)aδ �P(t)a +� +�ρ(t) +dt ++ +3N +� +a=1 +�P(t)a +d �P(t)a +dt += 1 +2 Tr +� +�� +d +� +δ � +P (t)δ � +P (t) +� +�ρ(t) +dt +� +�� − �P(t) · +�∂V (tot) +∂ � +R +� +�ρ(t) +. +(C3) + +19 +The derivative of the total potential average is more involved since the position probability distribution depends on +time through R(t) and +� +δ � +R(t)δ � +R(t) +� +�ρ(t). The derivative is worked out using the formulas proved in Ref. [49]: +d +� +V (tot)� +�ρ(t) +dt += +�∂V (ext) +∂t +� ++ +3N +� +a=1 +d �R(t)a +dt +� +∂V (tot) +∂ �R(t)a +� +�ρ(t) ++ 1 +2 +3N +� +ab=1 +d +� +δ �R(t)aδ �R(t)b +� +�ρ(t) +dt +�∂2V (tot) +∂ �Rb∂ �Ra +� +�ρ(t) += +�∂V (ext) +∂t +� +�ρ(t) ++ �P(t) · +�∂V (tot) +∂ � +R +� +�ρ(t) ++ 1 +2 Tr +� +�� +d +� +δ � +R(t)δ � +R(t) +� +�ρ(t) +dt +· +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +� +�� . +(C4) +Then using Eqs (23c)-(23d) and the permutation properties of the trace it is shown that: +1 +2 Tr +� d +dt +� +δ � +P (t)δ � +P (t) +�� += −1 +2 Tr +� +�� +d +� +δ � +R(t)δ � +R(t) +� +�ρ(t) +dt +· +�∂2V (tot) +∂ � +R∂ � +R +� +�ρ(t) +� +�� . +(C5) +So in the end, we found: +d ⟨H(t)⟩�ρ(t) +dt += +�∂V (ext) +∂t +� +�ρ(t) +. +(C6) +This derivation is more compact than the one presented in Ref. [1]. +Appendix D: Expansion of the probability distribution +In this Appendix, we show how to expand at first order the TDSHCA position probability distribution and how the +anharmonic vertices, Eqs (62) (63), emerge. All the free parameters are perturbed with respect to their static value +(denoted by (0)) +R(t) =R(0) + R(1)(t) +(D1a) +P(t) =P(1)(t) +(D1b) +α(t) =α(0) + α(1)(t) +(D1c) +β(t) =β(0) + β(1)(t) +(D1d) +γ(t) =γ(1)(t) +(D1e) +The first thing to do is to expand at first order the position probability distribution in the perturbative free parameters, +i.e. those denoted by the superscript (1). Before performing the expansion, we report the full position probability +distribution obtained from Eq. (15) +�ρ(R, t) = +� +� +� +� +�det +� +�α(t) − γ(t) · +−1 +β (t) · γ(t)T +2π +� +� +exp +� +−1 +2(R − R(t)) · +� +α(t) − γ(t) · +−1 +β (t) · γ(t)T +� +· (R − R(t)) +� +. +(D2) +The leading order is controlled by α(0) + α(1)(t). We define the displacements with respect the equilibrium position +as δ � +R(0) = � +R − �R +(0). The expansion gives +�ρ(R, t) = �ρ(0)(R) + �ρ(1)(R, t), +(D3) +where �ρ(0)(R) is the equilibrium probability distribution (see Eq. (25)) +�ρ(0)(R) = +� +det +�α(0) +2π +� +exp +� +−1 +2δ � +R(0) · �α(0) · δ � +R(0) +� +. +(D4) + +20 +The explicit expression for �ρ(1)(R, t) in Eq. (D3), following [1], is +�ρ(1)(R, t) = �ρ(0)(R) +�1 +2Tr +� +α(0)−1 · α(1)(t) +� +− 1 +2δR(0) · α(1)(t) · δR(0) + δR(0) · α(0) · R(1)(t) +� +. +(D5) +Next, we derive an expression for the perturbed averages of a position-dependent observable O(R). Using the expres- +sion for �ρ(1)(R, t), Eq. (D5), and integration by parts we get +⟨O⟩(1) (t) = +� +dR�ρ(1)(R, t)O(R) += −1 +2 +3N +� +ab=1 +�α(1)(t)ab +�� +δ �R(0) +a δ �R(0) +b O +� +(0) − (�α(0))−1 +ba ⟨O⟩(0) +� ++ +3N +� +a=1 +�R(1) +a (t) +� ∂O +∂ �Ra +� +(0) +. +(D6) +Note that now all the averages have to be performed on the equilibrium ensemble. +We introduce the equilibrium three and four phonon scattering vertices as in [49] +(3) +D abc = +� +∂V (BO) +∂ �Ra∂ �Rb∂ �Rc +� +(0) += +3N +� +mn=1 +�α(0) +an �α(0) +bm +� +δ �Rnδ �Rm ∂V (BO) +∂ �Rc +� +(0) +− �α(0) +ab +�∂V (BO) +∂ �Rc +� +(0) +(D7a) +(4) +D abcd = +� +∂V (BO) +∂ �Ra∂ �Rb∂ �Rc∂ �Rd +� +(0) += +3N +� +nm=1 +�α(0) +an �α(0) +bm +� +δ �Rnδ �Rm ∂2V (BO) +∂ �Rc∂ �Rd +� +(0) +− �α(0) +ab +�∂2V (BO) +∂ �Rc∂ �Rd +� +(0) +. +(D7b) +It is convenient also to introduce the potential V(R) as the difference between the BO potential and the harmonic +auxiliary potential obtained at equilibrium +V(R) = V (BO)(R) − 1 +2δ � +R(0) · +(2) +D · δ � +R(0), +(D8) +where +(2) +D defines the SCHA phonons, Eq. (27). Using Eq. (D6), we relate the perturbed averages of V(R) to the +scattering vertices of Eqs (D7) +� ∂V +∂ �Ra +� +(1) += 1 +2 +3N +� +bcde=1 +�α(1)(t)de(�α(0)−1)db(�α(0)−1)ec +(3) +D abc; +(D9a) +� +∂2V +∂ �Ra∂ �Rb +� +(1) += +3N +� +c=1 +(3) +D abc �R(1)(t)c − 1 +2 +3N +� +cdef=1 +�α(1)(t)ef(�α(0)−1)ec(�α(0)−1)fd +(4) +D abcd. +(D9b) +At this point it is straightforward to get the following perturbed averages for the BO potential: +� ∂V +∂ �Ra +� +(1) += +�∂V (BO) +∂ �Ra +� +(1) +− +3N +� +b=1 +(2) +D ab ˜R(1) +b (t), +(D10a) +� +∂2V +∂ �Ra∂ �Rb +� +(1) += +�∂2V (BO) +∂ �Ra∂ �Rb +� +(1) +. +(D10b) +Appendix E: Derivation of the linear response system +In this Appendix, we prove the linearized equations of motion discussed in Section IV A. To do this we write all +the supercell tensors in the static equilibrium polarization basis {eµ} defined in Eq. (27). So a multi-indices tensor +A(t)a1,..,aN defined in the supercell can be written in the polarization basis as +A(t)µ1,..,µN = +3N +� +a1,..,aN=1 +ea1 +µ1..eaN +µN A(t)a1,..,aN . +(E1) +From now on all the quantities are written in this basis. + +21 +All the supercell tensors are written in the equilibrium polarization basis, see Eq. (E1). The equations of motion +(Eqs (A5)) expanded at first order are +d2 +dt2 �R(1)(t)α = −ω2 +α �R(1)(t)α − +� ∂V +∂ �Rα +� +(1) +− +�∂V (ext)(t) +∂ �Rα +� +(0) +; +(E2a) +d +dt �α(1)(t)αβ = −2 +3N +� +µν=1 +Sαβµν +� +�γ(1)(t)µνω2 +ν +� +; +(E2b) +d +dt +�β(1)(t)αβ = 2 +3N +� +µν=1 +Sαβµν +� +�γ(1)(t)µν +� +; +(E2c) +d +dt�γ(1)(t)αβ = �α(1)(t)αβ − ω2 +α �β(1)(t)αβ − +� +� +� +∂2V (ext)(t) +∂ �Rα∂ �Rβ +� +(0) ++ +� +∂2V +∂ �Rα∂ �Rβ +� +(1) +� +� �β(0) +ββ , +(E2d) +where +Sαβµν = 1 +2(δαµδβν + δανδβµ) +(E3) +and V(R) is the difference between the exact BO energy surface and the SCHA auxiliary potential, Eq. (D8). The +averages of V(R) are defined in Eqs. (D9) and contain anharmonic corrections. Now we make three more steps. +First, we derive with respect to time Eqs (E2) to delete the equation for �γ(1)(t) since the perturbed averages do +not depend on this parameter, see Eq. (D6). +Secondly, we take the Fourier transform of the second order set of differential equations for �R +(1)(t) �α(1)(t) and +�β(1)(t). +The third and last step is to perform a change of variables. Instead of using the basis {�α(1)(ω), �β(1)(ω)}, we work +with {�a′(1)(ω),�b′(1)(ω)} which is defined as a linear combination of the original free parameters +� +��a′(1)(ω)µν +�b′(1)(ω)µν +� +� = Mµν +� +��α(1)(ω)µν +�β(1)(ω)µν +� +� , +(E4) +where we define M as +Mµν = +� +��� +K− +µν +ωµων +K− +µν +− +K+ +µν +ωµων +K+ +µν +� +��� . +(E5) +The coefficients in Eqs (E5) are functions of the equilibrium auxiliary frequencies {ω2 +µ} defined in Eq. (27) and +K± +µν =ℏ2nµν +2X± +µν +, +(E6a) +X± +µν = +� +±1 +2 +ℏ [ωµ ± ων] [(1 ± 1) + 2(nµ ± nν)] +4ωµων +, +(E6b) +nµν =1 +8(1 + 2nν)(1 + 2nµ). +(E6c) +Using the basis defined in Eq. (E5) we get the following equations of motion for { �R +(1)(ω), �a′(1)(ω), �b′(1)(ω)} +� +ω2 − ω2 +α +� �R(1)(ω)α − +� ∂V +∂ �Rα +� +(1) += +�∂V (ext)(ω) +∂ �Rα +� +(0) +, +(E7a) +� +ω2 − ω−2 +αβ +� +�a′(1)(ω)αβ + X− +αβ +� +∂2V +∂ �Rα∂ �Rβ +� +(1) += −X− +αβ +� +∂2V (ext)(ω) +∂ �Rα∂ �Rβ +� +(0) +, +(E7b) +� +ω2 − ω+2 +αβ +��b′(1)(ω)αβ − X+ +αβ +� +∂2V +∂ �Rα∂ �Rβ +� +(1) += X+ +αβ +� +∂2V (ext)(ω) +∂ �Rα∂ �Rβ +� +(0) +, +(E7c) + +22 +where ω2 comes from the Fourier transform of a second order derivative with respect to time. +Eqs. (E7) are written in terms of a matrix vector product in the space of the perturbative free parameters. Recalling +the definition of V (ext)(R, ω) given in Eq. (45), we write the linearized equations of motion as a matrix-vector product +(ω21 + L′) · +� +��� +�R +(1)(ω) +�a′(1)(ω) +�b′(1)(ω) +� +��� = +� +������� +� +∂B +∂ � +R +� +(0) +− +(4) +X− : +� +∂2B +∂ � +R∂ � +R +� +(0) +(4) +X+ : +� +∂2B +∂ � +R∂ � +R +� +(0) +� +������� +V(ω), +(E8) +where +(4) +X± +αβµν = X± +αβSαβµν +(E9) +with X± +αβ defined in Eq. (E6b) and Sαβµν in Eq. (E3). We define the RHS vector as the perturbation vector p′ +p′† = +�� +∂B +∂ � +R +� +(0) , − +(4) +X− : +� +∂2B +∂ � +R∂ � +R +� +(0) , +(4) +X+ : +� +∂2B +∂ � +R∂ � +R +� +(0) +� +. +(E10) +The matrix L′ acts in the space of the perturbative parameters. It is symmetric and contains two terms, the harmonic +and anharmonic contribution +L′ = L′ +harm + L′ +anh. +(E11) +The harmonic part L′ +harm is diagonal in our basis +L′ +harm · +� +��� +�R +(1)(ω) +�a′(1)(ω) +�b′(1)(ω) +� +��� = − +� +���� +(2) +D· +0 +0 +0 +(4) +ω−2 : +0 +0 +0 +(4) +ω+2 : +� +���� +� +��� +�R +(1)(ω) +�a′(1)(ω) +�b′(1)(ω) +� +��� . +(E12) +The matrix L′ +harm depends only on the equilibrium auxiliary frequencies of Eq. (27). We introduced a four indices +tensor +(4) +ω± +2 +αβµν = (ω± +αβ)2Sαβµν, +(E13) +with S defined in Eq. (E3) and +ω± +µν = ωµ ± ων. +(E14) +The RHS of Eq. (E12) should be read as a standard matrix-vector product. +The matrix element contains also +information on how to contract the indices, the operation · is defined in general as the contraction of the last and first +index of two tensors +A · B = +3N +� +µ=1 +A...µBµ... +(E15) +and : is defined as +C : D = +3N +� +µν=1 +C...µνDµν... +(E16) +For example the first line of Eq. (E12) is +− +(2) +D · �R +(1)(ω) = − +3N +� +ν=1 +(2) +D µν �R(1)(ω)ν, +(E17) + +23 +and returns a tensor of rank 1. The same holds for the other lines. As an example, consider +− +(4) +ω− +2 : �a′(1)(ω) = − +3N +� +µν=1 +(ω− +µν)2�a′(1)(ω)µν, +(E18) +The application of L′ +anh gives +L′ +anh · +� +��� +�R +(1)(ω) +�a′(1)(ω) +�b′(1)(ω) +� +��� = +� +������� +− +� +∂V +∂ � +R +� +(1) +(4) +X− : +� +∂2V +∂ � +R∂ � +R +� +(1) +− +(4) +X+ : +� +∂2V +∂ � +R∂ � +R +� +(1) +� +������� +. +(E19) +Writing the perturbed averages of V(R) in terms of the scattering tensors, as in Eq. (D9), and using the change of +variables of Eq. (E5), it is trivial to prove that in the new basis L′ +anh is symmetric and has the following form +L′ +anh = +� +������ +0 +(3) +D : +(4) +X− +− +(3) +D : +(4) +X+ +(4) +X− : +(3) +D +− +(4) +X− : +(4) +D : +(4) +X− +(4) +X− : +(4) +D : +(4) +X+ +− +(4) +X+ : +(3) +D +(4) +X+ : +(4) +D : +(4) +X− +− +(4) +X+ : +(4) +D : +(4) +X+ +� +������ +. +(E20) +Again, the matrix contains the information on how to contract the indices. This term contains information on the +anharmonicity of the system through the third and fourth phonon scattering tensors, defined in Eq. (D7). +Now that we have the linearized equations of motion, we present the general response function Eq. (52). To do this +we need the correction of a position-dependent observable A(R) in the new basis Eq. (E5) +⟨A⟩(1) (ω) = +3N +� +α=1 +∂ ⟨A⟩(0) +∂ �Rα +�R(1)(ω)α + +3N +� +αβ=1 +∂ ⟨A⟩(0) +∂�a′(0) +αβ +�a′(1)(ω)αβ + +3N +� +αβ=1 +∂ ⟨A⟩(0) +∂�b′(0) +αβ +�b′(1)(ω)αβ = += +3N +� +α=1 +� ∂A +∂ ˜Rα +� +(0) +�R(1)(ω)α − +3N +� +αβ=1 +X− +αβ +� +∂2A +∂ �Rα∂ �Rβ +� +(0) +˜a′(1)(ω)αβ + +3N +� +αβ=1 +X+ +αβ +� +∂2A +∂ �Rα �Rβ +� +(0) +˜b′(1)(ω)αβ, +(E21) +The previous expression can be demonstrated using the change of variable definition, Eq. (E5), the chain rule and the +following relations in the original basis (i.e. the one used in Appendix D) +∂ ⟨A⟩(0) +∂ �R(0) +α += +� ∂A +∂ ˜Rα +� +(0) +, +(E22a) +∂ ⟨A⟩(0) +∂�α(0) +αβ += − +1 +2�α(0) +αα�α(0) +ββ +� +∂2A +∂ �Rα∂ �Rβ +� +(0) +, +(E22b) +∂ ⟨A⟩(0) +∂ �β(0) +αβ += 0. +(E22c) +The derivative with respect to �α(0) +αβ is obtained using the formalism of [49]. +We define the response vector r′ similarly to p′ (Eq. (E10)) +r′† = +�� +∂A +∂ � +R +� +(0) , − +(4) +X− : +� +∂2A +∂ � +R∂ � +R +� +(0) , +(4) +X+ : +� +∂2A +∂ � +R∂ � +R +� +(0) +� +(E23) +so from Eq. (E21) we can extract the response formula (Eq. (52)) +⟨A⟩(1) (ω) +V(ω) += +1 +V(ω)r′ · +� +��� +�R +(1)(ω) +�a′(1)(ω) +�b′(1)(ω) +� +��� = r′ · +� +L′ + ω2�−1 · p′ = χ(ω)A,B +(E24) +where ⟨A⟩(1) (ω) is expressed as a scalar product in the space of the perturbative parameters. This is the expression +of χ(ω)A,B implemented in the code. + +24 +Appendix F: Lanczos algorithm +In this Appendix, we discuss the Lanczos implementation [1, 72] of the general response function Eq. (E24). Both +for infrared and Raman calculations, we can always work with p′ = r′ setting A = B (see Eqs (E10) (E23)) so Eq. +(E24) becomes +χ(ω)A,A = (p′ · p′)p′ · +� +L′ + ω2�−1 · p′ +(F1) +where we normalize the vector p′ (Eq. (E10)) +p′ = +p′ +√p′ · p′ . +(F2) +To get in one shot for all values of ω the response formula, Eq. (F1), we modified the Lanczos algorithm presented +in [1] exploiting that L′ = L′†. This algorithm allows to find a basis in which L′ is tridiagonal +P ′−1 · L′ · P ′ = T ′, +(F3) +where T ′ has the following form +T ′ = +� +����������� +t1 +r1 . . . +. . . +0 +r1 +t2 +... +... +... ... +... +... +... +... +rn−1 +0 +rn−1 +tn +� +����������� +(F4) +where n is the size of L′. The change of basis matrix P ′ is +P ′ = +� +p′ +1 +p′ +2 +... +p′ +n +� +(F5) +and it is unitary +P ′−1 = P ′†. +(F6) +The coefficients of T ′ can be found following this iterative procedure [1, 72] +tk =p′ +k · L′ · p′ +k +(F7a) +rkp′ +k+1 =vk = (L′ − tk) · p′ +k − rk−1p′ +k−1 +(F7b) +rk =√vk · vk +(F7c) +p′ +k+1 =vk/rk +(F7d) +with the initial vector equal to the normalized perturbation vector, p′ +1 = p′. This procedure ends when either p′ +k is a +linear combination of the previous vectors or p′ +k · p′ +k = 0. Unless the system is perfectly harmonic, this condition is +usually never reached in practical runs, and the algorithm is truncated after a maximum number of steps Nsteps. +After we build the change of variables matrix P ′ we can use it in Eq. (F1) +χ(ω)A,A = (p′ · p′)p′ · P ′ · +� +P ′−1 · +� +L′ + ω2�−1 · P ′� +· P ′−1p′ = (p′ · p′)p′ · P ′ · +� +T ′ + ω2�−1 · P ′−1p′ +(F8) +then noting that +P ′−1 · p′ = P ′† · p′ = +� +������� +1 +0 +0 +... +� +������� +(F9) + +25 +we get that the response function is given by +χ(ω)A,A = (p′ · p′) +� +T ′ + ω2�−1 +11 +(F10) +where +� +T ′ + ω2�−1 +11 can be written as a continuous fraction using the coefficients obtained up to Nsteps +� +T ′ + ω2�−1 +11 = +1 +ω2 + t1 − +r2 +1 +ω2+t2− +r2 +2 +ω2+... +. +(F11) +At each Lanczos step, we have to apply L′ to a given vector w in the space of the perturbed free parameters. As +showed in Appendix E L′ contains two terms +L′ · w = L′ +harm · w + L′ +anh · w. +(F12) +The application of the harmonic part is done using Eq. (E12), while the anharmonic part is done using Eq. (E19) +applying a reweighting procedure to compute the perturbed average as explained in [1]. +Appendix G: Symbolic inversion +In this Appendix we describe the symbolic inversion of a symmetric square super-tensor with this form +L = +� +� A +C +CT B +� +� +(G1) +where A, B, C, C† are invertible tensors. Using Gaussian reduction we get the inverse +L−1 = +� +� +D−1 +−D−1 · C · B−1 +−B−1 · C† · D−1 +B−1 + B−1 · C† · D−1C · B−1. +� +� +(G2) +where D = A − CB−1C†. It is trivial to check that LL−1 = L−1L = 1. For what follows we need C = 1 so Eq. +(G2) becomes +L−1 = +� +�A−1 − A−1 · (1 − A · B)−1 +(1 − B · A)−1 +(1 − A · B)−1 +−A · (1 − B · A)−1 . +� +� +(G3) +Again, for our purposes (see next Appendix H), we need to find a formula for the sum of entries of Eq. (G3). Summing +the coefficients of Eq. (G3) we get +L−1 +11 + L−1 +21 + L−1 +12 + L−1 +22 =A−1 − A−1 · (1 − A · B)−1 + (1 − B · A)−1 + (1 − A · B)−1 − A · (1 − B · A)−1 +=A−1 · [(1 − A · B) − (1 − A)] · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 +=A−1 · [A · (1 − B)] · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 +=(1 − B) · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 . +(G4) +Now we set A = 1 + � +A and B = 1 + � +B so we have that Eq. (G4) is +� +B · +� +� +A + � +B + � +A · � +B +�−1 ++ � +A · +� +� +A + � +B + � +B · � +A +�−1 += +� +1 + � +A · � +B−1 + � +A +�−1 ++ +� +1 + � +B · � +A−1 + � +B +�−1 += +� +� +A−1 + � +B−1 + 1 +�−1 +· � +A + +� +� +B−1 + � +A−1 + 1 +�−1 +· � +B += +� +1 + � +B−1 + � +A−1�−1 +· ( � +A−1 + � +B−1). +(G5) +We will use this formula in Appendix H. + +26 +Appendix H: Derivation of the interacting Green’s function +The easiest way to get the interacting Green’s function is to use another change of variables in Eqs (E2) +� +��a(1) +µν (ω) +�b(1) +µν (ω) +� +� = −ℏ2nµν +2 +� +� +1 +ωµων +1 +1 +ωµων +−1 +� +� +� +��α(1) +µν (ω) +�β(1) +µν (ω) +� +� +(H1) +where nµν is defined in Eq. (E6c) and {ω2 +µ} in Eq. (27). As done in Appendix E, we write Eqs (E2) in this new basis +switching to second-order time-derivatives +� +���� +� +G(0)(ω) +�−1 +· +0 +0 +0 +� +χ(0) +− (ω) +�−1 +: +0 +0 +0 +− +� +χ(0) ++ (ω) +�−1 +: +� +���� +� +��� +�R +(1)(ω) +�a(1)(ω) +�b(1)(ω) +� +��� = +� +����� +� +∂V +∂ � +R +� +(1) +� +∂2V +∂ � +R∂ � +Rβ +� +(1) +� +∂2V +∂ � +R∂ � +Rβ +� +(1) +� +����� ++ +� +����� +� +∂V (ext) +∂ � +R +� +(0) +� +∂2V (ext) +∂ � +R∂ � +R +� +(0) +� +∂2V (ext) +∂ � +R∂ � +R +� +(0) +� +����� +(H2) +where we recognize the resonant and anti-resonant terms of the two-phonon propagator +χ(0) +− (ω)µνσπ = δµσδνπ +ℏ [ωµ − ων] [nµ − nν] +4ωµων[(ωµ − ων)2 − ω2]; +χ(0) ++ (ω)µνσπ = δµσδνπ +ℏ [ωµ + ων] [1 + nµ + nν] +4ωµων[(ωµ + ων)2 − ω2] . +(H3) +The anharmonic vector in this basis is simply +� +����� +� +∂V +∂ ˜ +R +� +(1) +� +∂2V +∂ � +R∂ � +Rβ +� +(1) +� +∂2V +∂ � +R∂ � +Rβ +� +(1) +� +����� += +� +����� +0 +(3) +D· +(3) +D· +(3) +D· +(4) +D : +(4) +D : +(3) +D· +(4) +D : +(4) +D : +� +����� +� +��� +�R +(1) +�a(1) +�b(1) +� +��� . +(H4) +In a compact form, the linearized equations of motion are +L(ω) · +� +��� +�R +(1)(ω) +�a(1)(ω) +�b(1)(ω) +� +��� = +� +����� +� +∂V (ext) +∂ � +R +� +(0) +� +∂2V (ext) +∂ � +R∂ � +R +� +(0) +� +∂2V (ext) +∂ � +R∂ � +R +� +(0) +� +����� +(H5) +The tensor L(ω) describes the evolution in the linear regime and it is +L(ω) = +� +������ +� +G(0)(ω) +�−1 +− +(3) +D +− +(3) +D +− +(3) +D +� +χ(0) +− (ω) +�−1 +− +(4) +D +− +(4) +D +− +(3) +D +− +(4) +D +− +� +χ(0) ++ (ω) +�−1 +− +(4) +D +� +������ +(H6) +The correction to the average of an observable A is +⟨A⟩(1) = +� ∂A +∂ � +R +� +(0) +· �R +(1)(ω) + +� ∂2A +∂ � +R∂ � +R +� +(0) +: S : �a(1)(ω) + +� ∂2A +∂ � +R � +R +� +(0) +: S : ˜b(1)(ω) +(H7) +where S is defined in Eq. (E3). So, following the procedure described in Appendix E, the response function is +χA,B(ω) = r† · L(ω)−1 · p +(H8) +defining the response vector r as +r† = +�� +∂A +∂ � +R +� +(0) , +� +∂2A +∂ � +R∂ � +R +� +(0) , +� +∂2A +∂ � +R∂ � +R +� +(0) +� +, +(H9) + +27 +and the perturbation vector p as +p† = +�� +∂B +∂ � +R +� +(0) , +� +∂2B +∂ � +R∂ � +R +� +(0) , +� +∂2B +∂ � +R∂ � +R +� +(0) +� +. +(H10) +First we discuss the non-interacting case setting +(3) +D = 0 +(4) +D = 0. Using A and B as in Eq. (66a) we get +r† = +� +δ 0 0 +� +p† = +� +δ 0 0 +� +. +(H11) +with δ = δµ as in Eq. (67a). The free phonon propagator is +G(0)(ω)µν = +δµν +ω2 − ω2µ +. +(H12) +Then we chose A and B according to Eq. (66b) so +r† = +� +0 S S +� +p† = +� +0 S S +� +, +(H13) +where S is defined in Eq. (E3). The two-phonon free propagator is +χ(0)(ω)µνσπ = − +� +χ(0) ++ (ω)µνσπ − χ(0) +− (ω)µνσπ. +� +(H14) +We chose A and B according to Eq. (66c) +r† = +� +δ 0 0 +� +p† = +� +0 S S +� +(H15) +L(ω) is diagonal in the case +(3) +D = 0 +(4) +D = 0 so we get that the one-two phonon free propagator is zero +Γ(0)(ω)µσπ = 0. +(H16) +Now we derive the one-phonon interacting Green’s function ( +(3) +D ̸= 0 +(4) +D ̸= 0). Following [1] we chose the observables +A and B as in Eq. (66a) +G(ω) = +� +δ 0 0 +� +· L(ω)−1 · +� +��� +δ +0 +0 +� +��� = +� +L(ω)−1� +11 . +(H17) +We use Eq (G2) to get +G(ω)−1 = G(0)(ω)−1 − +� +(3) +D : +(3) +D : +� +� +����� +� +χ(0) +− (ω) : +(4) +D +�−1 +− 1 +−1 +−1 +− +� +χ(0) ++ (ω) : +(4) +D +�−1 +− 1 +� +����� +−1 +� +�� +: +(4) +D +−1 +: +(3) +D +: +(4) +D +−1 +: +(3) +D +� +�� +(H18) +now Eq. (G5) comes in help since we just need the sum of the inverse tensor’s entries +G(ω)−1 = G(0)(ω)−1 − +(3) +D : +� +1 − χ(0)(ω) : +(4) +D +�−1 +: +� +χ(0)(ω) : +(4) +D +� +: +(4) +D +−1 +: +(3) +D += G(0)(ω)−1 − Π(ω) +(H19) +where Π(ω) coincides with the one presented in [1, 2, 49]. + +28 +Now we derive the two phonons interacting Green’s function χ(ω) which is obtained by choosing A and B according +to Eq. (66b). In this basis this means computing +χ(ω) = +� +0 S S +� +· L(ω)−1 · +� +��� +0 +S +S +� +��� . +(H20) +The perturbation B chosen (Eq. (66b)) leads to the following linearized equations of motion (see Eq. (H5)) +L(ω) · +� +��� +�R +(1)(ω) +�a(1)(ω) +�b(1)(ω) +� +��� = +� +��� +0 +S +S +� +��� . +(H21) +Using the expression of L(ω), Eq. (H6), we find that the first free parameter is related to the other two +�R +(1)(ω) = G(0)(ω) · +(3) +D : +� +�a(1)(ω) + �b(1)(ω) +� +. +(H22) +So instead of having to invert the full L(ω) we reduce Eq. (H20) to +χ(ω) = +� +S : S : +� +� +����� ++ +� +χ(0) +− (ω) +�−1 +− +(4) +D − +(3) +D · G(0)(ω) · +(3) +D +− +(4) +D − +(3) +D · G(0)(ω) · +(3) +D +− +(4) +D − +(3) +D · G(0)(ω) · +(3) +D +− +� +χ(0) ++ (ω) +�−1 +− +(4) +D − +(3) +D · G(0)(ω) · +(3) +D +� +����� +−1 +� +�: S +: S +� +� += +� +S : S : +� +� +���� ++ +� +χ(0) +− (ω) +�−1 +− Σ(ω) +−Σ(ω) +−Σ(ω) +− +� +χ(0) ++ (ω) +�−1 +− Σ(ω) +� +���� +−1 � +�: S +: S +� +� += − +� +S : S : +� +� +���� ++1 − +� +χ(0) +− (ω) : Σ(ω) +�−1 +1 +1 +� +χ(0) ++ (ω) : Σ(ω)(ω) +�−1 ++ 1 +� +���� +−1 � +�: Σ(ω)−1 : S +: Σ(ω)−1 : S +� +� +(H23) +where we define the two phonon self-energy Σ(ω) as in Eq. (78) +Σ(ω) = +(4) +D + +(3) +D · G(0)(ω) · +(3) +D. +(H24) +Again we can use Eq. (G5) to get the two-phonon interacting Green’s function +χ(ω) = S : +� +1 − χ(0)(ω) : Σ(ω) +�−1 +: χ(0)(ω) : Σ(ω) : Σ−1(ω) : S += +� +1 − χ(0)(ω) : Σ(ω) +�−1 +: χ(0)(ω) += χ(0)(ω) : +� +1 − Σ(ω) : χ(0)(ω) +�−1 +(H25) +proving Eq. (80). +The last Green’s function to discuss is the one-two phonon Γ(ω) obtained setting A and B as in Eq. (66c) +Γ(ω) = +� +��� +δ +0 +0 +� +��� · L(ω)−1 · +� +��� +0 +S +S +� +��� . +(H26) + +29 +We simplify this inversion using again Eq. (H22). Now �a(1)(ω) + �b(1)(ω) are found considering the reduced linear +system extracted from Eq. (H21) +Σ(ω) : +� +�� ++ +� +χ(0) +− (ω) : Σ(ω) +�−1 +− 1 +−1 +−1 +− +� +χ(0) ++ (ω) : Σ(ω) +�−1 +− 1 +� +�� +� +�: �a(1)(ω) +: �b(1)(ω) +� +� = +� +�S +S +� +� +� +��a(1)(ω) +�b(1)(ω) +� +� = − +� +�� +1 − +� +χ(0) +− (ω) : Σ(ω) +�−1 +1 +1 +1 + +� +χ(0) ++ (ω) : Σ(ω) +�−1 +� +�� +−1 � +�: Σ−1(ω) +: Σ−1(ω) +� +� +(H27) +From Eq. (H27) we get the sum �a(1)(ω) + �b(1)(ω) +�a(1)(ω) + �b(1)(ω) = − +� +S : S : +� +� +���� +1 − +� +χ(0) +− (ω) : Σ(ω) +�−1 +1 +1 +1 + +� +χ(0) ++ (ω) : Σ(ω) +�−1 +� +���� +−1 � +��� +: S : Σ−1(ω) +: S : Σ−1(ω) +� +��� . +(H28) +Again Eq. (G5) comes in help so +�a(1)(ω) + �b(1)(ω) = +� +1 − χ(0)(ω) : Σ(ω) +�−1 +: χ(0)(ω) : Σ(ω) : Σ−1(ω) = χ(ω) +(H29) +Since the response vector r is just [δ +0 +0] the one-two phonon Green’s function Γ(ω) is given by R(1)(ω) (see Eq. +(H7)), so, with Eq. (H22), we end up with +Γ(ω) = R(1)(ω) = G(0)(ω) · +(3) +D : χ(ω). +(H30) +This proves Eq. (81). +We discuss also the three phonon Green’s function obtained with A and B as in Eq. (85) +A = δ �R(0) +α δ �R(0) +β δ �R(0) +γ +B = δ �R(0) +α′ δ �R(0) +β′ δ �R(0) +γ′ +(H31) +In this case we have +� ∂2A +∂ � +R∂ � +R +� +(0) += +� ∂2B +∂ � +R∂ � +R +� +(0) += 0 +(H32) +and +� +∂A +∂ �Rµ +� +(0) += δµα +� +�α(0)−1� +βγ + δµβ +� +�α(0)−1� +αγ + δµγ +� +�α(0)−1� +αβ +(H33) +so only the first entries of r′ and p′ are non zero. The response calculation is formally identical to the one-phonon +interacting Green’s function one Eq. (H17). Using, Eq. (H8), we get the three phonon propagator +χ3ph(ω) = +3N +� +µν=1 +� +δµα +� +�α(0)−1� +βγ + δµβ +� +�α(0)−1� +αγ + δµγ +� +�α(0)−1� +αβ +� +G(ω)µν +� +δνα′ +� +�α(0)−1� +β′γ′ + δνβ′ +� +�α(0)−1� +α′γ′ + δνγ′ +� +�α(0)−1� +α′β′ +� +(H34) +The diagrammatic interpretation is straightforward once we use Eq. (39). The three-phonon response is +χ3ph(ω) =G(0)(t = 0−)βγG(0)(t = 0−)β′γ′G(ω)αα′ ++ permutations of (αβγ) and (α′β′γ′) separately +(H35) +This proves the diagrammatic expression of Fig. 6. + +30 +Appendix I: Scattering vertices +In this Appendix, we present the diagrammatic expression of the scattering vertices in TDSCHA, Eqs (62) (63). +We consider first the three-phonon term Eq. (62) since the same holds for Eq. (63). +We average the third-derivative of the BO potential on the equilibrium SCHA distribution �ρ(0)(R) (see Eq. (B7)) +(3) +D ijk = +� +dR�ρ(0)(R)∂3V (BO)(R) +∂ �Ri∂ �Rj∂ �Rk +. +(I1) +Starting from Eq. (I1) we perform the change of variables �ua = �Ra − �R(0) +a +and we expand in u +(3) +D ijk = +� +du�ρ(0)(u) +�+∞ +� +n=0 +1 +n! +� +a1..an +(3+n) +D(0) +ijka1..an�ua1...�uan +� +, +(I2) +where +(n) +D(0) is defined in Eq. (65). Note that +(n) +D(0) differs in general from +(n) +d , see Eq. (34), since the minimum of the +Born-Oppenheimer potential RBO does not coincide with the SCHA centroid R(0). +Only even terms in Eq. (I2) are non-zero +(3) +D ijk = ++∞ +� +n=0 +1 +(2n)! +� +a1..a2n +(3+2n) +D(0) +ijka1..a2n ⟨�ua1...�ua2n⟩(0) += ++∞ +� +n=0 +1 +(2n)! +� +a1..a2n +(3+2n) +D(0) +ijka1..a2n +� +P +P +�� +�α(0)−1� +a1a2 ... +� +�α(0)−1� +a2n−1a2n +� += ++∞ +� +n=0 +1 +2nn! +� +a1..a2n +(3+2n) +D(0) +ijka1..a2n +� +�α(0)−1� +a1a2 +... +� +�α(0)−1� +a2n−1a2n +(I3) +where P denotes the permutations of the indices according to the Wick theorem. In the last line, we use the symmetry +properties of the anharmonic vertices and the fact that the number of contractions for a 2n multivariate Gaussian +expectation value is (2n − 1)!!, where !! is the double factorial. In polarization the final result is +(3) +D µνφ = ++∞ +� +n=0 +(−1)n +2nn! +� +α1..α2n +(3+2n) +D(0) +µνφα1..α2n G(0)(t = 0−)α1α2...G(0)(t = 0−)α2n−1α2n +� +�� +� +n +, +(I4) +where we use �α(0)−1 = +� +δ � +Rδ � +R +� +(0), see Eq. (B8) and Eq. (39). The same holds for the fourth-order scattering vertex +(4) +D µνφψ = ++∞ +� +n=0 +(−1)n +2nn! +� +α1..α2n +(4+2n) +D(0) +µνφψα1..α2n G(0)(t = 0−)α1α2...G(0)(t = 0−)α2n−1α2n +� +�� +� +n +. +(I5) +Eq. (I4) Eq. (I5) give a diagrammatic expression for the TDSCHA scattering vertices, see Fig. 3. +Appendix J: Momentum Green’s function +In this Appendix, we discuss the momentum Green’s function using the many-body formalism for bosons. The +interacting Green’s function with imaginary time τ ∈ [−β, +β] (β−1 = kbT with kb the Boltzmann constant) is +defined as +GAB(τ) = − +� +Tτ +� +ˆS(β, 0) ˆA(τ) ˆB(0) +�� +0 +(J1) +where only the connected diagrams are included. The average ⟨...⟩0 is performed on the harmonic system defined by +ˆHharm = +3N +� +µ=1 +ℏΩµ +� +ˆa† +µˆaµ + 1 +2 +� +, +(J2) + +31 +where {Ω2 +µ} are the harmonic frequencies, i.e. the poles of the harmonic propagator Eq. (30). The scattering matrix +is +ˆS(τ) = ˆS(τ, 0) = Tτe− +� τ +0 dτ ′ ˆ +Hanh(τ ′), +(J3) +here ˆHanh(τ) is the anharmonic part of the BO energy surface in the interacting picture. The Matsubara transform is +GAB(iΩn) = 1 +2 +� +β +−β +dτeiΩnτGAB(τ) +(J4) +with Ωn = 2πn +β +with n integer. +First, we define the harmonic (non-interacting) Green’s function for position and momentum. In the harmonic +polarization basis we have +δ ˆ�R(τ)µ = ˆ�R(τ)µ − �Rµ = +� +ℏ +2Ωµ +� +ˆa(τ)µ + ˆa†(τ)µ +� +ˆ�P(τ)µ = −i +� +ℏΩµ +2 +� +ˆa(τ)µ − ˆa†(τ)µ +� +(J5) +The Green’s functions in Matsubara frequencies are +G(0)RR +µν (iΩn) = δµν +ℏ2 +(iΩn)2 − (ℏΩµ)2 +(J6a) +G(0)P P +µν (iΩn) = δµνΩ2 +µG(0)RR +µν (iΩn) +(J6b) +G(0)P R +µν (iΩn) = iδµνℏ +iΩn +(iΩn)2 − (ℏΩµ)2 = i +ℏ(iΩn)G(0)RR +µν (iΩn) +(J6c) +G(0)RP +µν (iΩn) = G(0)P R +µν (−iΩn) = − i +ℏ(iΩn)G(0)RR +µν (iΩn). +(J6d) +Note that the analytical continuation of G(0)RR +µν (iΩn) gives +G(0)RR +µν (iΩn → ℏω + i0+) = +δµν +(ω + i0+)2 − Ω2µ +(J7) +which coincides with our definition of harmonic free propagators, see Eq. (30). The interacting momentum Green’s +function is +GP P +µν (τ) = − +� +Tτ +� +ˆS(β, 0) �P(τ)µ �Pν(0) +�� +0 = G(0)P P +µν (τ) − +� +Tτ +� +ˆSanh(β, 0) �P(τ)µ �Pν(0) +�� +0 +(J8) +where ˆSanh(β, 0) = ˆS(β, 0) − ˆ1. The anharmonic correction is proportional to terms like +� β +0 +dτ1.. +� β +0 +dτm +� +Tτ +�ˆ�P(τ)µ ˆ�R(τ1)α1 ˆ�R(τ1)α2...ˆ�R(τn)αm ˆ�P ν(0) +�� +0 +(J9) +where all the indices, except for µν, will be contracted with anharmonic vertices contained in the full BO energy +surface. Eq. (J9) is computed using the Wick theorem and contains terms that have the following form +� β +0 +dτ1.. +� β +0 +dτm +� +Tτ +�ˆ�P(τ)µ ˆ�R(τ1)α1 +�� +0 +� +Tτ +�ˆ�R(τ1)α2 ˆ�R(τ2)α3 +�� +0 ... +� +Tτ +�ˆ�R(τn)αm ˆ�P(0)µ +�� +0 += +� β +0 +dτ1.. +� β +0 +dτmG(0)P R +µα1(τ − τ1)G(0)RR +α2α3(τ1 − τ2)...G(0)RP +αmν(τm) +(J10) +When doing the contraction of the momentum variables we use G(0)P R +µν (τ) = G(0)RP +µν (−τ) and we take into account +the multiplicity of the diagrams which cancels the n! coming from the scattering matrix. +Knowing that the Matsubara frequencies are conserved in all the diagrams and the relation between G(0)RP/P R +µν +(iΩn) +and G(0)RR +µν (iΩn) (Eq. (J6)), Eq. (J10) becomes simply proportional to the anharmonic correction of the one-phonon +Green’s function +G(0)P R +µα1(iΩn)π(iΩn)α2..φm−1G(0)RP +φmν(iΩn) =(iΩn)2G(0)RR +µα1(iΩn)π(iΩn)α2..αm−1G(0)RR +αmν(iΩn) +=(iΩn)2 � +GRR +µν (iΩn) − G(0)RR +µν (iΩn) +� +(J11) + +32 +where π(iΩn)α2..αm−1 is the Matsubara transform of the terms that contain only products of G(0)RR +αiαj(τi − τj) in Eq. +(J10). +In the end, using Eqs (J6c) (J6d), we get the following result for the interacting momentum Green’s function +GP P +µν (iΩn) = G(0)P P +µν (iΩn) + (iΩn)2 +ℏ2 +� +GRR +µν (iΩn) − G(0)RR +µν (iΩn) +� += ω2 +µG(0)RR +µν (iΩn) + (iΩn)2 +ℏ2 +� +GRR +µν (iΩn) − G(0)RR +µν (iΩn) +� += −δµν + (iΩn)2 +ℏ2 +GRR +µν (iΩn) +(J12) +Performing the analytical continuation iΩn → ℏω + i0+ and using the TDSCHA one-phonon Green’s function, Eq. +(74), we prove Eq. (84). +Appendix K: Prepare IR/Raman spectra calculation +To compute IR spectra we need Eqs (90) (91) in polarization basis Eq. (27). +The first component of the re- +sponse/perturbation vector is the one phonon vertex and contains equilibrium averages of the effective charges +Zµ,aα = +�∂pµ(R) +∂Raα +� +(0) += ⟨Z∗(R)µ,aα⟩(0) , +(K1) +where µ indicates the direction of the electric field, Ra,α is the position of atom a along the α coordinate and Z∗(R) is +the effective charges tensor for a given configuration R. The second and third components of the response/perturbation +vector contain the two-phonon vertex, i.e. first derivatives of the effective charges. Integration by parts leads to +Zµ,aα,bβ = +� ∂2pµ(R) +∂Raα∂Rbβ +� +(0) += +N +� +c=1 +3 +� +γ=1 +α(0) +bβ,cγ +� +δR(0) +cγ +� +Z∗(R)µ,aα − Z∗(R(0))µ,aα +�� +(0) . +(K2) +We subtract the equilibrium effective charges to reduce the noise in the average. +To compute Raman spectra we need Eqs (93) (94). +The first component of the response/perturbation vector +contains equilibrium averages of the Raman tensor which give one-phonon processes +Ξµ,ν,aα = +�∂α(R)µν +∂Raα +� +(0) += ⟨Ξ(R)µν,aα⟩(0) , +(K3) +where µ ν indicates the photon polarization and Ξ(R) is the Raman tensor for a configurations R. The two phonon +channel depends on the Raman tensor first derivatives and using integration by parts we have +Ξµ,ν,aα,bβ = +� ∂2α(R)µν +∂Raα∂Rbβ +� +(0) += +N +� +c=1 +3 +� +γ=1 +α(0) +bβ,cγ +� +δR(0) +cγ +� +Ξ(R)µ,ν,aα − Ξ(R(0))µ,ν,aα +�� +(0) . +(K4) +So to prepare the response and perturbation vector r and p we can use a stochastic approach as in [45] since all the +averages have to be done on the equilibrium ensemble. +We can enforce symmetries both for effective charges/Raman tensors and for their second-older counterparts. To +symmetrize Z we note that the dipole p is related to the effective charge +pµ = +N +� +a=1 +3 +� +α=1 +Zµ,aαuaα +(K5) +where uaα is a displacement of atom a in the direction α. If we apply a symmetry on u (defined in the supercell), the +dipole will change according to the symmetry σ (3 × 3 unitary matrix) +3 +� +β=1 +σµνpν = +N +� +ab=1 +3 +� +αβ=1 +Zµ,aαSσ +aα,bβubβ +(K6) + +33 +where Sσ (3N × 3N matrix) is the symmetry operation associated with σ in the supercell +Sσ +aα,bβ = σαβδaσ(b). +(K7) +j = σ(i) indicates that the symmetry σ maps i into j. So using Eq. (K6) we get +Zµ′,aα′ = 1 +Ns +Ns +� +σ=1 +3 +� +µ,β=1 +� +σ† +µ′µZµ,σ(a)ασαα′ +� +(K8) +where Ns is the number of symmetries. The symmetries for the second-order dipole moment are extracted noting that +Pµ = +N +� +ab=1 +3 +� +αβ=1 +Zµ,aα,bβuaαubβ. +(K9) +Since we know how the effective charges transform under a symmetry operation we can symmetrize Z +Zν′,aα′,bβ′ = 1 +Ns +Ns +� +σ=1 +3 +� +ν=1 +3 +� +αβ=1 +� +σ† +ν′νZν,σ(a)α,σ(b)βσαα′σββ′ +� +. +(K10) +We do the same for the Raman tensors. Similarly to what we do before, the polarizability α is related to the Raman +tensors +αµ,ν = +N +� +a=1 +3 +� +α=1 +Ξµ,ν,aαua,α, +αµ,ν = +N +� +ab=1 +3 +� +αβ=1 +Ξµ,ν,aα,bβua,αub,β +(K11) +and we end up with the rules to symmetrize the averages of Raman-tensors +Ξχ,φ,mµ′ = 1 +Ns +Ns +� +σ=1 +� +� +3 +� +αβ=1 +3 +� +ν′=1 +σχασφβΞα,β,σ(m),ν′σν′µ′ +� +� , +(K12a) +Ξχ,φ,pπ,rρ = 1 +Ns +Ns +� +σ=1 +� +� +3 +� +αβ=1 +3 +� +µν=1 +σχασφβΞα,β,σ(p),µ,σ(r),νσνρσµπ +� +� . +(K12b) +[1] L. +Monacelli +and +F. +Mauri, +Time-dependent +self- +consistent harmonic approximation: Anharmonic nuclear +quantum dynamics and time correlation functions, Phys. +Rev. B 103, 104305 (2021). +[2] J.-M. Lihm and C.-H. Park, Gaussian time-dependent +variational principle for the finite-temperature anhar- +monic lattice dynamics, Phys. Rev. Research 3, L032017 +(2021). +[3] F. Rasetti, Raman spectra of crystals, Nature 127, 626 +(1931). +[4] E. Fermi and F. Rasetti, ¨Uber den ramaneffekt des stein- +salzes, Zeitschrift f¨ur Physik 71, 689 (1931). +[5] A. +F. +Goncharov, +E. +Gregoryanz, +R. +J. +Hem- +ley, +and +H. +kwang +Mao, +Spectroscopic +studies +of +the +vibrational +and +electronic +properties +of +solid +hydrogen +to +285 +gpa, +Proceedings +of +the +National +Academy +of +Sciences +98, +14234 +(2001), +https://www.pnas.org/doi/pdf/10.1073/pnas.201528198. +[6] G. Hautier, A. Jain, H. Chen, C. Moore, S. P. Ong, and +G. Ceder, Novel mixed polyanions lithium-ion battery +cathode materials predicted by high-throughput ab initio +computations, Journal of Materials Chemistry 21, 17147 +(2011). +[7] B. Lilia, R. Hennig, P. Hirschfeld, G. Profeta, A. Sanna, +E. Zurek, W. E. Pickett, M. Amsler, R. Dias, M. I. +Eremets, C. Heil, R. J. Hemley, H. Liu, Y. Ma, C. Pier- +leoni, A. N. Kolmogorov, N. Rybin, D. Novoselov, +V. Anisimov, A. R. Oganov, C. J. Pickard, T. Bi, +R. Arita, I. Errea, C. Pellegrini, R. Requist, E. K. U. +Gross, E. R. Margine, S. R. Xie, Y. Quan, A. Hire, +L. Fanfarillo, G. R. Stewart, J. J. Hamlin, V. Stanev, +R. S. Gonnelli, E. Piatti, D. Romanin, D. Daghero, and +R. Valenti, The 2021 room-temperature superconductiv- +ity roadmap, Journal of Physics: Condensed Matter 34, +183002 (2022). +[8] N. Mounet, M. Gibertini, P. Schwaller, D. Campi, +A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Ce- +pellotti, G. Pizzi, and N. Marzari, Two-dimensional ma- + +34 +terials from high-throughput computational exfoliation +of experimentally known compounds, Nature Nanotech- +nology 13, 246 (2018). +[9] I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J. +Needs, Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, +Quantum hydrogen-bond symmetrization in the super- +conducting hydrogen sulfide system, Nature 532, 81 +(2016). +[10] I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune, +T. Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri, +and J. A. Flores-Livas, Quantum crystal structure in the +250-kelvin superconducting lanthanum hydride, Nature +578, 66 (2020). +[11] M. Cherubini, L. Monacelli, and F. Mauri, The mi- +croscopic origin of the anomalous isotopic properties of +ice relies on the strong quantum anharmonic regime of +atomic vibration, The Journal of Chemical Physics 155, +184502 (2021), https://doi.org/10.1063/5.0062689. +[12] L. Monacelli, I. Errea, M. Calandra, and F. Mauri, Pres- +sure and stress tensor of complex anharmonic crystals +within the stochastic self-consistent harmonic approxi- +mation, Physical Review B 98, 024106 (2018). +[13] L. Monacelli, M. Casula, K. Nakano, S. Sorella, and +F. Mauri, Quantum phase diagram of high-pressure hy- +drogen (2022). +[14] N. D. Drummond, B. Monserrat, J. H. Lloyd-Williams, +P. L. R´ıos, C. J. Pickard, and R. J. Needs, Quantum +monte carlo study of the phase diagram of solid molecular +hydrogen at extreme pressures, Nature Communications +6, 10.1038/ncomms8794 (2015). +[15] J. S. Zhou, L. Monacelli, R. Bianco, I. Errea, F. Mauri, +and M. Calandra, Anharmonicity and doping melt the +charge density wave in single-layer TiSe2, Nano Letters +20, 4809 (2020). +[16] M. Leroux, I. Errea, M. Le Tacon, S.-M. Souliou, G. Gar- +barino, L. Cario, A. Bosak, F. Mauri, M. Calandra, and +P. Rodi`ere, Strong anharmonicity induces quantum melt- +ing of charge density wave in 2h − nbse2 under pressure, +Phys. Rev. B 92, 140303 (2015). +[17] R. Bianco, I. Errea, L. Monacelli, M. Calandra, and +F. Mauri, Quantum enhancement of charge density wave +in NbS2 in the two-dimensional limit, Nano Letters 19, +3098 (2019). +[18] J. Diego, A. H. Said, S. K. Mahatha, R. Bianco, L. Mona- +celli, M. Calandra, F. Mauri, K. Rossnagel, I. Errea, and +S. Blanco-Canosa, van der waals driven anharmonic melt- +ing of the 3d charge density wave in VSe2, Nature Com- +munications 12, 10.1038/s41467-020-20829-2 (2021). +[19] L. Monacelli, I. Errea, M. Calandra, and F. Mauri, Black +metal hydrogen above 360 gpa driven by proton quantum +fluctuations, Nature Physics 17, 63 (2021). +[20] P. Loubeyre, F. Occelli, and P. Dumas, Synchrotron in- +frared spectroscopic evidence of the probable transition +to metal hydrogen, Nature 577, 631 (2020). +[21] M. Bernasconi, P. L. Silvestrelli, and M. Parrinello, Ab +initio infrared absorption study of the hydrogen-bond +symmetrization in ice, Physical Review Letters 81, 1235 +(1998). +[22] F. Capitani, B. Langerome, J.-B. Brubach, P. Roy, +A. Drozdov, M. I. Eremets, E. J. Nicol, J. P. Carbotte, +and T. Timusk, Spectroscopic evidence of a new energy +scale for superconductivity in h3s, Nature Physics 13, +859 (2017). +[23] L. +Ranalli, +C. +Verdi, +L. +Monacelli, +M. +Calandra, +G. Kresse, and C. Franchini, Temperature-dependent an- +harmonic phonons in quantum paraelectric ktao +3 by +first principles and machine-learned force fields, arXiv +preprint arXiv:2209.12036 (2022). +[24] C. Verdi, L. Ranalli, C. Franchini, and G. Kresse, +Quantum paraelectricity and structural phase transitions +in strontium titanate beyond density-functional theory, +arXiv preprint arXiv:2211.09616 (2022). +[25] D. Juraschek, M. Fechner, and N. Spaldin, Ultrafast +structure switching through nonlinear phononics, Physi- +cal Review Letters 118, 10.1103/physrevlett.118.054101 +(2017). +[26] A. Subedi, A. Cavalleri, and A. Georges, Theory of non- +linear phononics for coherent light control of solids, Phys- +ical Review B 89, 10.1103/physrevb.89.220301 (2014). +[27] M. Rini, R. Tobey, N. Dean, J. Itatani, Y. Tomioka, +Y. Tokura, R. W. Schoenlein, and A. Cavalleri, Control +of the electronic phase of a manganite by mode-selective +vibrational excitation, Nature 449, 72 (2007). +[28] C. L. Johnson, B. E. Knighton, and J. A. Johnson, +Distinguishing nonlinear terahertz excitation pathways +with two-dimensional spectroscopy, Phys. Rev. Lett. 122, +073901 (2019). +[29] J. Cao and G. A. Voth, The formulation of quantum sta- +tistical mechanics based on the feynman path centroid +density. iv. algorithms for centroid molecular dynam- +ics, The Journal of Chemical Physics 101, 6168 (1994), +https://doi.org/10.1063/1.468399. +[30] J. A. Poulsen, G. Nyman, and P. J. Rossky, Practical +evaluation of condensed phase quantum correlation func- +tions: A feynman–kleinert variational linearized path in- +tegral method, The Journal of Chemical Physics 119, +12179 (2003), https://doi.org/10.1063/1.1626631. +[31] T. J. H. Hele, M. J. Willatt, A. Muolo, and S. C. +Althorpe, Boltzmann-conserving classical dynamics in +quantum time-correlation functions: +“matsubara dy- +namics”, The Journal of Chemical Physics 142, 134103 +(2015), https://doi.org/10.1063/1.4916311. +[32] M. Ceotto, G. Di Liberto, and R. Conte, Semiclassical +“divide-and-conquer” method for spectroscopic calcula- +tions of high dimensional molecular systems, Phys. Rev. +Lett. 119, 010401 (2017). +[33] T. Pl´e, S. Huppert, F. Finocchi, P. Depondt, and +S. Bonella, Anharmonic spectral features via trajectory- +based quantum dynamics: +A perturbative analysis of +the interplay between dynamics and sampling, The +Journal +of +Chemical +Physics +155, +104108 +(2021), +https://doi.org/10.1063/5.0056824. +[34] J. Beutier, D. Borgis, R. Vuilleumier, and S. Bonella, +Computing thermal wigner densities with the phase in- +tegration method, The Journal of Chemical Physics 141, +084102 (2014), https://doi.org/10.1063/1.4892597. +[35] T. Pl´e, S. Huppert, F. Finocchi, P. Depondt, and +S. +Bonella, +Sampling +the +thermal +wigner +den- +sity +via +a +generalized +langevin +dynamics, +The +Journal +of +Chemical +Physics +151, +114114 +(2019), +https://doi.org/10.1063/1.5099246. +[36] J. A. Poulsen, G. Nyman, and P. J. Rossky, Practical +evaluation of condensed phase quantum correlation func- +tions: A feynman–kleinert variational linearized path in- +tegral method, The Journal of Chemical Physics 119, +12179 (2003), https://doi.org/10.1063/1.1626631. +[37] Q. Shi and E. Geva, Semiclassical theory of vibra- +tional energy relaxation in the condensed phase, The + +35 +Journal of Physical Chemistry A 107, 9059 (2003), +https://doi.org/10.1021/jp030497+. +[38] E. Wigner, On the quantum correction for thermody- +namic equilibrium, Phys. Rev. 40, 749 (1932). +[39] T. Tadano and S. Tsuneyuki, First-principles lattice dy- +namics method for strongly anharmonic crystals, Jour- +nal of the Physical Society of Japan 87, 041015 (2018), +https://doi.org/10.7566/JPSJ.87.041015. +[40] T. Tadano and S. Tsuneyuki, Self-consistent phonon cal- +culations of lattice dynamical properties in cubic srtio3 +with first-principles anharmonic force constants, Phys. +Rev. B 92, 054301 (2015). +[41] K. Imre, E. ¨Ozizmir, M. Rosenbaum, and P. F. Zweifel, +Wigner method in quantum statistical mechanics, Jour- +nal of Mathematical Physics 8, 1097 (1967). +[42] J. Brogaard, Wigner function formalism in quantum me- +chanics, Signature (2015). +[43] M. Novaes, Wigner and husimi functions in the double- +well potential, Journal of Optics B: Quantum and Semi- +classical Optics 5, S342 (2003). +[44] J. A. Poulsen, S. K.-M. Svensson, and G. Nyman, Dy- +namics of gaussian wigner functions derived from a time- +dependent variational principle, AIP Advances 7, 115018 +(2017), https://doi.org/10.1063/1.5004757. +[45] L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, +I. Errea, and F. Mauri, The stochastic self-consistent har- +monic approximation: calculating vibrational properties +of materials with full quantum and anharmonic effects, +Journal of Physics: Condensed Matter 33, 363001 (2021). +[46] I. Georgescu and V. A. Mandelshtam, Self-consistent +phonons revisited. i. the role of thermal versus quantum +fluctuations on structural transitions in large lennard- +jones clusters, The Journal of Chemical Physics 137, +144106 (2012), https://doi.org/10.1063/1.4754819. +[47] S. E. Brown, +I. Georgescu, and V. A. Mandelsh- +tam, Self-consistent phonons revisited. ii. a general +and efficient method for computing free energies and +vibrational +spectra +of +molecules +and +clusters, +The +Journal +of +Chemical +Physics +138, +044317 +(2013), +https://doi.org/10.1063/1.4788977. +[48] M. +Monteferrante, +S. +Bonella, +and +G. +Cic- +cotti, +Linearized +symmetrized +quantum +time +correlation +functions +calculation +via +phase +pre- +averaging, +Molecular +Physics +109, +3015 +(2011), +https://doi.org/10.1080/00268976.2011.619506. +[49] R. Bianco, I. Errea, L. Paulatto, M. Calandra, and +F. Mauri, Second-order structural phase transitions, free +energy curvature, and temperature-dependent anhar- +monic phonons in the self-consistent harmonic approx- +imation: Theory and stochastic implementation, Phys. +Rev. B 96, 014111 (2017). +[50] A. A. Maradudin and A. E. Fein, Scattering of neutrons +by an anharmonic crystal, Phys. Rev. 128, 2589 (1962). +[51] G. D. Mahan, Many-Particle Physics (Springer US, +2000). +[52] T. Tadano, Y. Gohda, and S. Tsuneyuki, Anharmonic +force constants extracted from first-principles molecu- +lar dynamics: applications to heat transfer simulations, +Journal of Physics: Condensed Matter 26, 225402 (2014). +[53] M. Lazzeri, M. Calandra, and F. Mauri, Anharmonic +phonon frequency shift in mgb 2, Physical Review B 68, +220509 (2003). +[54] W. G¨otze and K. H. Michel, Elastic constants of nonionic +anharmonic crystals, Zeitschrift f¨ur Physik A Hadrons +and nuclei 217, 170 (1968). +[55] F. Macheda, P. Barone, and F. Mauri, Electron-phonon +interaction and longitudinal-transverse phonon splitting +in doped semiconductors, Phys. Rev. Lett. 129, 185902 +(2022). +[56] F. Macheda, +T. Sohier, +P. Barone, and F. Mauri, +Electron-phonon interaction and phonons in 2d doped +semiconductors (2022). +[57] Y. Oba, T. Tadano, R. Akashi, and S. Tsuneyuki, First- +principles study of phonon anharmonicity and negative +thermal expansion in scf3, Phys. Rev. Materials 3, 033601 +(2019). +[58] V. V. Goldman, G. K. Horton, and M. L. Klein, An im- +proved self-consistent phonon approximation, Phys. Rev. +Lett. 21, 1527 (1968). +[59] N. R. Werthamer, Self-consistent phonon formulation of +anharmonic lattice dynamics, Phys. Rev. B 1, 572 (1970). +[60] D. M. Juraschek and S. F. Maehrlein, Sum-frequency +ionic raman scattering, Phys. Rev. B 97, 174302 (2018). +[61] M. Basini, M. Udina, M. Pancaldi, V. Unikandanunni, +S. Bonetti, and L. benfatto, Terahertz ionic kerr effect +(2022). +[62] A. von Hoegen, R. Mankowsky, M. Fechner, M. F¨orst, +and A. Cavalleri, Probing the interatomic potential of +solids with strong-field nonlinear phononics, Nature 555, +79 (2018). +[63] G. Deinzer and D. Strauch, Two-phonon infrared absorp- +tion spectra of germanium and silicon calculated from +first principles, Phys. Rev. B 69, 045205 (2004). +[64] P. L. Silvestrelli, M. Bernasconi, and M. Parrinello, Ab +initio infrared spectrum of liquid water, Chemical Physics +Letters 277, 478 (1997). +[65] W. Windl, K. Karch, P. Pavone, O. Sch¨utt, D. Strauch, +W. Weber, K. Hass, and L. Rimai, Second-order raman +spectra of sic: Experimental and theoretical results from +ab initio phonon calculations, Physical Review B 49, +8764 (1994). +[66] W. Windl, P. Pavone, K. Karch, O. Sch¨utt, D. Strauch, +P. Giannozzi, and S. Baroni, Second-order raman spectra +of diamond from ab initio phonon calculations, Physical +Review B 48, 3164 (1993). +[67] M. Menahem, N. Benshalom, M. Asher, S. Aharon, +R. Korobko, S. Safran, O. Hellman, and O. Yaffe, The +disorder origin of raman scattering in perovskites single +crystals (2022). +[68] B. Miehlich, A. Savin, H. Stoll, and H. Preuss, Results +obtained with the correlation energy density functionals +of becke and lee, yang and parr, Chemical Physics Letters +157, 200 (1989). +[69] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, +C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- +cioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, +G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, +A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, +F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, +L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. +Seitsonen, A. Smogunov, P. Umari, and R. M. Wentz- +covitch, QUANTUM ESPRESSO: a modular and open- +source software project for quantum simulations of mate- +rials, Journal of Physics: Condensed Matter 21, 395502 +(2009). +[70] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, +M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, + +36 +D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, +A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, +A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, +U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawa- +mura, H.-Y. Ko, A. Kokalj, E. K¨u¸c¨ukbenli, M. Lazzeri, +M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.- +V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Ponc´e, +D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitso- +nen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, +N. Vast, X. Wu, and S. Baroni, Advanced capabilities for +materials modelling with quantum ESPRESSO, Journal +of Physics: Condensed Matter 29, 465901 (2017). +[71] D. R. Hamann, Optimized norm-conserving vanderbilt +pseudopotentials, Phys. Rev. B 88, 085117 (2013). +[72] D. Rocca, R. Gebauer, Y. Saad, and S. Baroni, Turbo +charging time-dependent density-functional theory with +lanczos chains, The Journal of Chemical Physics 128, +154105 (2008), https://doi.org/10.1063/1.2899649. + diff --git a/6NFAT4oBgHgl3EQfnR2x/content/tmp_files/load_file.txt b/6NFAT4oBgHgl3EQfnR2x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db16a5d2187cdead6181106207f7db2eedd85ec7 --- /dev/null +++ b/6NFAT4oBgHgl3EQfnR2x/content/tmp_files/load_file.txt @@ -0,0 +1,2125 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf,len=2124 +page_content='Wigner Gaussian dynamics: simulating the anharmonic and quantum ionic motion Antonio Siciliano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ∗ Lorenzo Monacelli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2 Giovanni Caldarelli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1 and Francesco Mauri1 1Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Universit`a di Roma La Sapienza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Piazzale Aldo Moro 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 00185 Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Italy 2Theory and Simulation of Materials (THEOS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' and National Centre for Computational Design and Discovery of Novel Materials (MARVEL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ´Ecole Polytechnique F´ed´erale de Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 1015 Lausanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Switzerland (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2023) The atomic motion controls important features of materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' such as thermal transport,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' phase transitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' and vibrational spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' However, the simulation of ionic dynamics is exceptionally challenging when quantum fluctuations are relevant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=', at low temperatures or with light atoms) and the energy landscape is anharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this work, we formulate the Time-Dependent Self- Consistent Harmonic Approximation (TDSCHA) [1, 2] in the Wigner framework, paving the way for the efficient computation of the nuclear motion in systems with sizable quantum and thermal anharmonic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Besides the improved numerical efficiency, the Wigner formalism unveils the classical limit of TDSCHA and provides a link with the many-body perturbation theory of Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We further extend the method to account for the non-linear couplings between phonons and pho- tons, responsible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=', for a nonvanishing Raman signal in high-symmetry Raman inactive crystals, firstly discussed by Rasetti and Fermi [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We benchmark the method in phase III of high-pressure hydrogen ab initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The nonlinear photon-phonon coupling reshapes the IR spectra and explains the high-frequency shoulder of the H2 vibron observed in experiments [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Wigner TDSCHA is computationally cheap and derived from first principles: it is unbiased by assumptions on the phonon-phonon and phonon-photon scattering and does not depend on empirical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Therefore, the method can be adopted in unsupervised high-throughput calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' INTRODUCTION The burst of computational resources accomplished during the last decades unlocked the path to material design: we can synthesize in silico new materials and measure their properties before the experimental realiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' High-throughput simulations are driving the dis- covery of new cathode materials for batteries [6], super- conductor hydrides [7], and 2D materials [8], among oth- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' However, the toolchain available for high-throughput simulation fails when anharmonicity is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In these cases, the harmonic approximation and perturbative ap- proaches are inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This happens in hydrides where quantum fluctuations alter the free energy landscape [9, 10], phase diagram of hydrogen-rich compounds like high-pressure ice [11, 12] and solid hydrogen [13, 14], and materials undergoing displacive phase transition like charge density wave (CDW) transition metal dichalco- genides [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lattice anharmonicity influences the vibrational spec- tra, optical properties, and conductivity of a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Those consist in the main experimental signatures of the atomic structure and phase transitions when diffraction is not possible due to small sample size or low cross-section, as happens in solid hydrogen [19, 20], high-pressure water [21], and hydrides superconductors [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Furthermore, accurate simulations of the ionic motion are fundamental in quantum paraelectric perovskites like KTaO3 [23] and SrTiO3 [24], in which the ferroelectric ∗ antoniosiciliano@uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='it phase transition is hindered by quantum ionic fluctua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These materials play a crucial role in applica- tions of nonlinear phononics [25, 26], where the anhar- monic coupling between phonons induces transient struc- tural changes and crystal-symmetry breaking upon light pumping [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Their theoretical and computational investigation has been limited to models assuming spe- cific patterns of phonon-phonon and phonon-photon in- teractions [25, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Moreover, the lack of an unsupervised technique prevents the systematic and high-throughput search for better materials in nonlinear phononics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Time-Dependent Self-consistent Harmonic Ap- proximation (TDSCHA) [1, 2] tackles these complex problems by providing an efficient numerical solution for the finite-temperature nuclear dynamics with quantum and anharmonic fluctuations beyond perturbative ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This method has been successfully applied to simulate Raman and IR spectra of high-pressure molecu- lar hydrogen [19] and to characterize the quantum para- electric transition in KTaO3 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' TDSCHA approximates the nuclear density matrix with the most general Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This leads to a more ef- ficient technique than path-integral (PI) methods, where classical-like trajectories of different replicas are sam- pled [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Instead, it shares some similarities with the Linearized Semi-Classical Initial Value Representa- tion method (LSC-IVR) [33], in which an approximate quantum initial condition for the ionic system [34–37] is evolved subject to classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Among all these methods, TDSCHA is the computationally most efficient for medium-sized systems (containing hundreds of ions) with ab-initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='08628v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='mtrl-sci] 20 Jan 2023 2 However, the exceptional complexity of the original TDSCHA formulation hampers its physical interpreta- tions, and many questions remain unanswered: how does it relate to other approaches employed in quantum chem- istry [33]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Which phonon scattering mechanisms is the theory able to describe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' What are the limitations of its applicability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' And how does the theory behave in the classical limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this work, we reformulate the TDSCHA in the Wigner formalism, where the density matrix is expressed as a function of position and momentum in a quantum- phase space [38] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this way, the TDSCHA equa- tions are simplified, and the nuclear evolution is governed only by the position-momentum averages and correlators at a fixed time (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The ℏ vanishes from the prop- agator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' thus, the initial condition determines whether the evolution is quantum or classical, and they share the same computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The TDSCHA linear response in the Wigner formalism (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' IV) becomes very intuitive and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We re- formulate the propagator in terms of Feynman diagrams, providing a simple link with other many-body approaches like the Self-Consistent Phonon (SCP) approach [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' V, we extend the formalism allowing for the sim- ulation of nonlinear coupling between phonons and pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Finally, we benchmark the theory on the IR spec- trum of high-pressure hydrogen phase III (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' VI), where we show that the high-frequency overtone observed in [5] is due to the aforementioned nonlinear coupling between photons and phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We start by reviewing the classical and quantum dy- namics within the Wigner formalism in the next Section (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' NUCLEAR DYNAMICS Here, we review the Wigner formalism for the exact nuclear evolution and compare the quantum and classical dynamics of N ions in a closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nuclear spins and ionic exchange are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Classical nuclear evolution In the classical limit, the nuclei behave like dimension- less particles moving according to the time-dependent Hamiltonian H(t) = 3N � a=1 P 2 a 2ma + V (tot)(R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (1) For brevity, we indicate with a = (i, α) a composite index with the atomic index i and Cartesian index α, ma = mi is the mass of atom i, Pa = Pi,α is the momentum of atom i along the α direction and R = (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='., RN) represents a configuration of atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The total potential can be divided into an internal static interaction and an external time-dependent perturbation V (tot)(R, t) = V (BO)(R) + V (ext)(R, t), (2) where V (BO)(R) is the nuclear interaction mediated by the electrons within the Born-Oppenheimer approxima- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the ground state electronic energy at fixed nu- clear configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' It can be evaluated as the total energy of an ab-initio calculation like density-functional theory (DFT), or an appropriately parametrized force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' V (ext)(R, t) is the external time-dependent poten- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' It encodes the interaction between the probe (usu- ally an electromagnetic field) and the ions, mediated by the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The dynamics of physical properties are computed as averages over the phase-space of the corresponding ob- servable with the time-dependent probability distribu- tion (normalized and positive-definite) ρcl(R, P , t): ⟨O⟩ρcl = � dR � dP O(R, P )ρcl(R, P , t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (3) In a closed system, ρcl(R, P , t) evolves according to the Liouville equation ∂ ∂tρcl(R, P , t) + iLclρcl(R, P , t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (4) The classical Liouville operator is defined as iLcl◦ = −H(t) ↔ Λ◦ (5) and ↔ Λ is the Poisson brackets operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ↔ Λ = 3N � a=1 � � ← ∂ ∂Ra → ∂ ∂Pa − ← ∂ ∂Pa → ∂ ∂Ra � � , (6) where the arrows indicate on which side the derivative is applied: A ↔ ΛB = {A, B} = 3N � a=1 � ∂A ∂Ra ∂B ∂Pa − ∂A ∂Pa ∂B ∂Ra � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (7) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (4) preserves the phase-space volume, which be- haves as an incompressible fluid, so probability can not be created nor destroyed leading to entropy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Quantum nuclear evolution The Wigner formalism describes the quantum nuclear evolution in terms of position and momentum degrees of freedom, similarly to the classical Liouville propagation discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this way, the differences between quantum and classical dynamics are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3 The quantum equivalent to the classical probability distribution in the phase-space is the Wigner quasi- distribution [41], defined as a Fourier transform of the Von Neumann density operator ˆρ(t) as ρw(R, P , t)= � dR′e− i ℏ P ·R′ (2πℏ)3N � R + R′ 2 ���� ˆρ(t) ����R − R′ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (8) We remark that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (8) is normalized, as in the classical case, but, in general, it is not positive-definite [42, 43], hence it can not be interpreted as a probability distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Still, it encodes all the information on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Similarly, the Wigner expression for an operator ˆO is defined as Ow(R, P ) = � dR′e− i ℏ P ·R′ � R + R′ 2 ���� ˆO ����R − R′ 2 � , (9) so that quantum averages in the Wigner formalism have the same expression as the classical ones (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (3)): ⟨Ow⟩ρw = � dR � dP Ow(R, P )ρw(R, P , t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (10) The Wigner-Liouville equation controls the time evolu- tion of ρw(R, P , t) ∂ ∂tρw(R, P , t) + iLρw(R, P , t) = 0, (11) where iL is unitary and contains a classical (cl) and a quantum (q) propagator: iL = iLcl + iLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (12) The classical propagator iLcl coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' On the other hand, the quantum part depends explicitly on ℏ: iLq◦ = − +∞ � n=1 (−ℏ2)n 22n(2n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='H(t) �↔ Λ �2n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (13) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (13) gives quantum corrections to the dynamics as odd powers of the Poisson brackets operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Interestingly, any quadratic potential has iLq = 0, even when its coefficients are time-dependent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' H(t) = 3N � a=1 P 2 a 2ma + 1 2 3N � ab=1 (R − R0(t))aK0(t)ab(R − R0(t))b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (14) The density matrix propagates only according to iLcl, and the dynamic is independent on ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The quan- tum/classical nature of the system is encoded only in the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the next Section, we will show that TDSCHA [1] shares the same feature since it is based on a Hamiltonian with the same form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (14), where K0(t) and R0(t) are evaluated self-consistently from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' GAUSSIAN DYNAMICS IN THE WIGNER FRAMEWORK The TDSCHA constrains the quantum density matrix as the most general time-dependent Gaussian [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Since the Wigner transformation is equivalent to a Fourier transform, also the nuclear time-dependent Wigner quasi-distribution is a Gaussian �ρ(R, P , t)=N(t) exp � −1 2 3N � ab=1 (R − R(t))aα(t)ab(R − R(t))b −1 2 3N � ab=1 (P − P(t))aβ(t)ab(P − P(t))b + 3N � ab=1 (R − R(t))aγ(t)ab(P − P(t))b � , (15) where N(t) is the normalization, defined such that � dR � dP �ρ(R, P , t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (16) R(t), P(t), α(t), β(t), γ(t) are the time-dependent free parameters of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Differently from the gen- eral Wigner distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15) is positive-definite and can be interpreted as the quantum probability distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Thanks to the Wigner transformation, it is evident that �ρ(R, P , t) is the multidimensional generalization of the one reported in [44] for the 1D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The position and momentum centroids R(t) and P(t) are 3N real vectors and represent, respectively, the instantaneous av- erage position and momentum of the ions: R(t) = ⟨R⟩�ρ(t) , P(t) = ⟨P ⟩�ρ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (17) α(t), β(t) and γ(t) are 3N × 3N real tensors (α(t) and β(t) are symmetric) and represent the instantaneous position-momentum correlators: � δ � R(t)δ � R(t) � �ρ(t)= � �α(t) − �γ(t) · �β−1(t) · �γT (t) �−1 , (18a) � δ � R(t)δ � P (t) � �ρ(t)= − � �γT (t) − �β(t) · �γ−1(t) · �α(t) �−1 , (18b) � δ � P (t)δ � P (t) � �ρ(t)= � �β(t) − �γT (t) · �α−1(t) · �γ(t) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (18c) where δ � P (t) = � P − �P(t) and the �◦ indicates a mass- rescaled variable like �Ra = √maRa, �Pa = Pa √ma .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (19) The detailed derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (18) is reported in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The dynamics of the Wigner distribution can be ob- tained transforming the time-dependent equation of the TDSCHA in the Wigner basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We prove in Appendix B 4 that this is equivalent to evolve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15) with a self- consistent Wigner-Liouville equation ∂ ∂t �ρ(R, P , t) + iLsc�ρ(R, P , t) = 0, (20) where iLsc◦ = −H(�ρ) ↔ Λ◦, (21) and H(�ρ) is a quadratic time-dependent Hamiltonian that depends self-consistently on �ρ(t): H(�ρ) = 3N � a=1 P 2 a 2ma + 3N � a=1 δR(t)a �∂V (tot)(R, t) ∂Ra � �ρ(t) + 1 2 3N � ab=1 δR(t)a �∂2V (tot)(R, t) ∂Ra∂Rb � �ρ(t) δR(t)b (22) with δR(t) = R − R(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Inserting the Wigner-TDSCHA distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15), in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (20) and substituting the expression of the propagators of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (18) (details in Ap- pendix A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' we get the equations of motion d dt � � R � �ρ(t) = � � P � �ρ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23a) d dt � � P � �ρ(t) = − �∂V (tot) ∂ � R � �ρ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23b) d dt � δ � Rδ � R � �ρ(t) = � δ � Rδ � P � �ρ(t) + � δ � P δ � R � �ρ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23c) d dt � δ � P δ � P � �ρ(t) = − �∂2V (tot) ∂ � R∂ � R � �ρ(t) � δ � Rδ � P � �ρ(t) (23d) − � δ � P δ � R � �ρ(t)· �∂2V (tot) ∂ � R∂ � R � �ρ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' d dt � δ � Rδ � P � �ρ(t) = � δ � P δ � P � �ρ(t) (23e) − � δ � Rδ � R � �ρ(t)· �∂2V (tot) ∂ � R∂ � R � �ρ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' where to compact the notation we drop the explicit time and position dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The TDSCHA dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23) are simpler than the original derivation using the standard formalism of operators in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As anticipated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' II B, Eqs (23) do not contain ℏ and the dynamics is the same for quantum and classical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nev- ertheless, quantum features are included in the initial condition and preserved during the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The clas- sical limit of the TDSCHA dynamics was not evident in [1], as the off-diagonal parameters of the density matrix are ill-defined as ℏ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This approach is exact in the case of a time-dependent harmonic oscillator since the Wigner distribution is a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Thanks to self-consistency, we go beyond har- monic/perturbative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' No approximations are made on the total potential itself so anharmonic effects are included in a non-perturbative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The dynamics of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23) satisfy the conservation of energy and entropy, as expected from a closed system (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Equilibrium SCHA in the Wigner formalism A particular solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23) is the steady state equilibrium in absence of a time-dependent perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' From Eqs (23) we get �∂V BO ∂ � R � (0) =0 (24a) � � P � (0) =0 � δ � Rδ � P � (0) = 0, (24b) � δ � P δ � P � (0) = � δ � Rδ � R � (0) · �∂2V (BO) ∂ � R∂ � R � (0) , (24c) where (0) indicates that the averages are performed on �ρ(0), the equilibrium Wigner distribution that solve self- consistently Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (24c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Since the average momentum is zero (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 24b), we take δP = P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As the mixed position-momentum correlation vanishes at equilibrium, the Wigner distribution becomes �ρ(0)(R, P ) =N (0) exp � −1 2 � P · � � P � P �−1 (0) · � P − 1 2δ � R · � δ � Rδ � R �−1 (0) · δ � R � (25) where � � P � P � (0) and � δ � Rδ � R � (0) solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (24c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (25) is a steady-state of the Wigner TDSCHA equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Among all steady-state distributions, the equilibrium one satisfy � δ �Raδ �Rb � (0) = 3N � µ=1 ℏ(1 + 2nµ) 2ωµ ea µeb µ, (26a) � �Pa �Pb � (0) = 3N � µ=1 ℏωµ(1 + 2nµ) 2 ea µeb µ, (26b) where {ωµ} and {eµ} define non-interacting phonons of a generalized dynamical matrix (2) D ab = �∂2V (BO) ∂ �Ra∂ �Rb � (0) = 3N � µ=1 ω2 µea µeb µ, (27) and nµ is the Bose-Einstein distribution nµ = 1 eβℏωµ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (28) The equilibrium solution is the one with the minimum free energy at fixed temperature and in Appendix B we show that it coincides with the Wigner transform of the Self-Consistent Harmonic Approximation (SCHA) den- sity matrix [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Our initial condition is constrained to be a Gaussian so we do not have to employ a sam- pling of the Wigner quasi-distribution [34, 35] as done in Linearized Semi-Classical Initial Value Representation methods (LSC-IVR) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These approaches compute time correlation functions approximating the quantum dynamics with a classical evolution (setting iLq = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (12)) [33], whereas the TDSCHA evolution is justi- fied from the quantum action principle [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic representation of the SCHA Here, we report the diagrammatic expansion of the SCHA in a quartic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This clarifies the anhar- monic processes included in the theory, and the differ- ences between SCHA, TDSCHA, and other approaches, such as Self-Consistent Phonon (SCP) [39, 40] and per- turbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We define the SCHA propagator through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' G(0)(ω)−1 = ω2 − (2) D ω=0 −→ − (2) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (29) Since the SCHA is a static theory and it is defined through ω = 0 quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Similarly, one can define the harmonic propagator as g(0)(ω)−1 = ω2 − ∂2V (BO) ∂ � R∂ � R ���� R=RBO ω=0 −→ −∂2V (BO) ∂ � R∂ � R ���� R=RBO , (30) where RBO is the minimum of V (BO)(R) ∂V (BO) ∂ � R ���� R=RBO = 0, (31) and the harmonic phonons are the eigenvalues of the second-order expansion of the BO potential around its minimum RBO ∂2V (BO) ∂ �Ra∂ �Rb ���� R=RBO = 3N � µ=1 Ω2 µϵa µϵb ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (32) In what follows we use g(0) and G(0) to denote the static limit (ω = 0) of the propagators introduced in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (29), (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To connect the SCHA with perturbation theory, we expand the BO energy landscape V (BO)(R) in a Taylor series around the positions RBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' All the SCHA equa- tions contain averages of all BO potential derivatives � ∂kV (BO) ∂ �Rb1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.∂ �Rbk � (0) = ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3N � a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.an=1 (k+n) d b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.bka1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.an �� δ �R + �δ � a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. � δ �R + �δ � an � (0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (33) where the anharmonic vertices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (33) are evaluated at the minimum of the BO energy landscape (n) d a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.an = ∂nV (BO) ∂ �Ra1∂ �Ra2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂ �Ran ���� R=RBO , (34) and �δ is the difference between the minimum of the BO potential and the equilibrium centroids of the SCHA �δ = �R (0) − �RBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (35) The SCHA distribution is a Gaussian so the averages in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (33) can be evaluated analytically up to any order by means of the Wick theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In a perturbative expansion, we assume that each an- harmonic vertex scales as (n) d ∼ O(λn−2) (36) where λ is the perturbative parameter and can be esti- mated as the ratio of the thermal length and the average bond distance [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In what follows we truncate the BO potential to the fourth-order, setting (n) d = 0 n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (37) By doing this we get the SCP equations as a limit case of SCHA ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Substituting the Taylor expansion into the first SCHA equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (24a), we obtain a self-consistent equation for �δ that takes into account quantum and anharmonic effects on the atomic positions shift δa = 3N � bcd=1 g(0) ab 2 �� (3) d bcd + 1 3 3N � e=1 (4) d bcdeδe � δcδd − � (3) d bcd + 3N � e=1 (4) d bcdeδe � G(0)(t = 0−)cd � , (38) where, in analogy with many-body theory, G(0)(t = 0−) is proportional to the SCHA position-position correlator (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23c)) G(0)(t = 0−)ab = − � ( �R − �R(0))a( �R − �R(0))b � (0) , (39) here t = 0− is the many-body analytical continuation in time [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Similarly, the second SCHA equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 27) results in a self-consistent expression for the self-energy G(0)−1 ab = g(0)−1 ab − Π(0) ab , (40) Π(0) ab = 3N � c=1 (3) d abcδc− 3N � cd=1 (4) d abcd 2 � G(0)(t = 0−)cd − δcδd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (41) Self-consistent phonon (SCP) methods [39, 40] solves Eqs (38) (41) (40) by fitting the anharmonic force con- stants [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Often Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (38) is ignored and �δ is assumed to be zero, which is true only if all the atomic coordinates are constrained by symmetry [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' On the contrary, the SCHA can go beyond the SCP method including all the anharmonic vertices with n ≥ 5 and polarization mix- ing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These effects are automatically incorporated thanks to a stochastic sampling of the potential [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We remark that the SCHA and SCP method are static the- ories: the self-energy is real and the phonons defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (40) are non-interacting excitations with an infinite lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression for the SCHA one-phonon propagator at lowest perturbative order λ2 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The single solid line is the static SCHA propagator G(0) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The thin dotted line is the static harmonic propagator g(0) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The scattering vertices are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (34) with n = 3 (triangle) and n = 4(square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The tadpole and loop diagram are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (44) Expressing both the SCHA propagator and the posi- tion shift as a series of O(λn) corrections G(0) = G(0)λ=0 + G(0)λ=1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. G(0)λ=n ∼ O(λn) (42a) �δ = �δλ=0 + �δλ=1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. �δλ=n ∼ O(λn) (42b) one can solve order by order in λ Eqs (38) (41) (40) to systematically get all the corrections to the SCHA prop- agator with a cubic-quartic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [49] solved Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (38), (41), (40) up to O(λ2) and showed that � G(0)�−1 ab ≃ � g(0)�−1 ab − � Π(L) ab + Π(T) ab � (43) where Π(L) and Π(T) are respectively the tadpole (T) and loop (L) diagram (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 1), Π(L) ab = − 1 2 3N � cd=1 (4) d abcdg(0)(t = 0−)cd, (44a) Π(T) ab = − 1 2 3N � cdef=1 (3) d abcg(0) cd (3) d defg(0)(t = 0−)ef , (44b) here g(0)(t = 0−) is the harmonic counterpart of G(0)(t = 0−), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The loop diagram, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (44a), comes from quantum/anharmonic fluctuations at fixed positions by setting �δ = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' On the other hand, the tadpole diagram, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (44b), comes from the renormalization of atomic positions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' LINEAR RESPONSE The static SCHA corrects the bare phonon propaga- tor with a real self-energy, thus only renormalizing the phonon frequency without introducing a finite lifetime of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this Section, we revise the fully-dynamical TDSCHA linear response within the Wigner formalism and show how new diagrams with a nonvanishing imagi- nary part emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Linearized equations of motion and general response function When the external time-dependent potential coupled to the phonons does not cause irreversible changes in the material, we are in the linear response regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This is relevant, for example, when the ionic degrees of freedom are probed with electromagnetic fields (X-ray or Raman scattering and IR absorption) or with neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In these cases the external perturbation V (ext)(R, t) is in the form V (ext)(R, t) = B(R)V(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (45) In this regime, the SCHA distribution �ρ(0)(R) (see Ap- pendix D) is perturbed by �ρ(1)(R, t) �ρ(R, t) = �ρ(0)(R) + �ρ(1)(R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (46) As the probability distribution changes, the correlators, Eqs (18), are the sum of the equilibrium ones, Eqs (26), and a time-dependent correction (denoted by the (1)) �R(t)µ = �R(0) µ + �R(1) µ , (47a) � δ �Rµδ �Rν � �ρ(t) = � δ �Rµδ �Rν � (0) + � δ �Rµδ �Rν � (1) , (47b) � δ �Pµδ �Pν � �ρ(t) = � �Pµ �Pν � (0) + � δ �Pµδ �Pν � (1) , (47c) where we express the tensors in the polarization basis {eµ} of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Appendix E, we show that the dy- namics of the average momentum �P (1) and of the mixed correlator � δ � Rδ � P � (1) can be reabsorbed in those of the variables of Eqs (47), which define unambiguously the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The linearized equations of motion are obtained by plugging the perturbed correlators, Eqs (47), into the equations of motion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23), and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (46) when computing averages of the total potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This (see Ap- pendix E for details) leads to � L′ + ω2� � ����� �R (1) � δ � Rδ � R � (1) � δ � P δ � P � (1) � ����� = p′V(ω), (48) which defines the linearized TDSCHA equations in fre- quency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (48), ω2 comes from the Fourier transform of second-order time derivatives, and L′ is the linearized Wigner-Liouville operator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In general, we can separate L′ in two terms L′ = L′ harm + L′ anh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (49) 7 The harmonic part L′ harm describes the free evolution of the SCHA phonons defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The scattering, hence the interaction between these phonons comes from the anharmonic part L′ anh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The perturbation vector p′ depends, in general, on equilibrium averages of first and second derivatives of B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The external potential modifies the non-interacting equilibrium state, defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The response function encodes how the observable A(R) changes when the system is out of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the linear response regime, this modification depends only on equilibrium quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The TDSCHA response function is obtained expand- ing ⟨A⟩�ρ(0)+�ρ(1) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (46)) in the perturbed parame- ters of Eqs (47) around the equilibrium value ⟨A⟩(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Appendix E, we show that the first-order correction in frequency domain is a simple scalar product in the space of the correlators (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (47)) ⟨A⟩(1) (ω) = r′† · � ����� �R (1) � δ � Rδ � R � (1) � δ � P δ � P � (1) � ����� , (50) where the response vector r′, similarly to p′, contains equilibrium averages of position-derivatives of A(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Finally, inverting the linearized equations of motion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (48), we get ⟨A⟩(1) (ω) = r′† · � L′ + ω2�−1 · p′V(ω), (51) where ◦−1 denotes the inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The general response function of an observable A(R) to an external pertur- bation B(R) is χ(ω)A,B = r′† · � L′ + ω2�−1 · p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52) This expression has the same form as the one presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [1] where the standard description of quantum mechanics is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Appendix F we show how to compute the general response function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52)) with the Lanczos algorithm following the original work on TDSCHA [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The algo- rithm generates a basis in which L′ + ω2 is tridiagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As shown in Appendix F, from this form it is easy to get in one shot the response function for all values of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Lanczos basis is generated from Nsteps subsequent applications of L′ to a starting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Each iteration corresponds to free propagations (L′ harm) and scattering (L′ anh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this way, we build the full anharmonic prop- agators (see next Section IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In addition, we exploit the properties of the Wigner formulation that L′ = L′† and that, if A = B, p′ = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These features imply that the calculation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52) requires a symmetric Lanczos algorithm speeding up the original code [1] by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic interpretation of linear response In this Section, we provide a physical interpretation in terms of Feynman diagrams of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To do this, we introduce a new basis (see Appendix G and Appendix H for details), a linear combination of the position and momentum correlators, Eqs (47b) (47c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this basis, the general response function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52)) takes the form χ(ω)A,B = r · L(ω)−1 · p, (53) where the response vector r and the perturbation vec- tor p have a simple expression in terms of equilibrium averages of A(R) and B(R) p = � ����� � ∂B ∂ � Rµ � (0) � ∂2B ∂ � Rµ∂ � Rν � (0) � ∂2B ∂ � Rµ∂ � Rν � (0) � ����� , r = � ����� � ∂A ∂ � Rµ � (0) � ∂2A ∂ � Rµ∂ � Rν � (0) � ∂2A ∂ � Rµ∂ � Rν � (0) � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (54) L(ω) propagates the perturbation caused by p in the system and encodes the information on how the latter af- fects r which defines how the observable changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Indeed, its multiplication by a vector representing the status of the system, as r or p (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 54), gives the anharmonic scattering processes that further dress the SCHA Green’s function and introduce a finite lifetime (see Section IV C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this basis, L(ω) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53) has a simple symmetric form L(ω)= � ����� G(0)(ω)−1 − (3) D − (3) D − (3) D χ(0) − (ω)−1 − (4) D − (4) D − (3) D − (4) D −χ(0) + (ω)−1 − (4) D � ����� , (55) L(ω) = L(ω)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (56) The harmonic and anharmonic contributions to L(ω) are Lharm(ω)= � ��� G(0)(ω)−1 0 0 0 χ(0) − (ω)−1 0 0 0 −χ(0) + (ω)−1 � ��� , (57) Lanh(ω)= � ����� 0 − (3) D − (3) D − (3) D − (4) D − (4) D − (3) D − (4) D − (4) D � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (58) We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2 a graphical expression for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To construct a diagrammatic representation, we associate each tensor in L(ω) with a symbol that possesses a num- ber of extremities equal to the rank of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 8 Notation Diagram Formula Description Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='G(0)(ω)µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='δµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ω2−ω2µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Bare one-phonon SCHA propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='g(0)(ω)µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='δµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ω2−Ω2µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Bare perturbative one-phonon propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0)(ω)µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω)µνηλ − χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ (ω)µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Bare two-phonon SCHA propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω)µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='δµηδνλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ℏ[ωµ−ων][nµ−nν] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4ωµων[(ωµ−ων)2−ω2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Resonant two-phonon SCHA propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(61a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ (ω)µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='δµηδνλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ℏ[ωµ+ων][1+nµ+nν] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4ωµων[(ωµ+ων)2−ω2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Antiresonant two-phonon SCHA propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(61b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂4V (BO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Rµ∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Rν∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Rη∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Rλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Four-phonon SCHA scattering vertex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(63) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D µνηλ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) µ ν η λ ∂4V (BO) ∂ � Rµ∂ � Rν∂ � Rη∂ � Rλ ���� R=R(0) Perturbative four-phonon scattering vertex evaluated at the SCHA equilibrium positions (65) (4) d µνηλ µ ν η λ ∂4V (BO) ∂ � Rµ∂ � Rν∂ � Rη∂ � Rλ ���� R=RBO Perturbative four-phonon scattering vertex (34) (3) D µνη µ ν η � ∂3V (BO) ∂ � Rµ∂ � Rν∂ � Rη � (0) Three-phonon SCHA scattering vertex (62) (3) D µνη ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) µ ν η ∂3V (BO) ∂ � Rµ∂ � Rν∂ � Rη ���� R=R(0) Perturbative three-phonon scattering vertex evaluated at SCHA equilibrium positions (65) (3) d µνη µ ν η ∂3V (BO) ∂ � Rµ∂ � Rν∂ � Rη ���� R=RBO Perturbative three-phonon scattering vertex (34) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Collection of symbols frequently used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' First column: the notation used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Second column: graphical expression with indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Third column: mathematical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fourth column: description of the symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fifth column: first labeled equation where the symbol appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The single solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2 represents the equilibrium SCHA Green’s function G(0)(ω) G(0)(ω)µν = δµν ω2 − ω2µ , (59) where {ωµ} are the self-consistent auxiliary frequencies defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The double single solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2 is the two-phonon SCHA propagator χ(0)(ω) χ(0)(ω)µνηλ = χ(0) − (ω)µνηλ − χ(0) + (ω)µνηλ (60) which contains a resonant χ(0) − (ω) and an anti resonant term χ(0) + (ω) χ(0) − (ω)µνηλ =δµηδνλ ℏ [ωµ − ων] [nµ − nν] 4ωµων[(ωµ − ων)2 − ω2], (61a) χ(0) + (ω)µνηλ =δµηδνλ ℏ [ωµ + ων] [1 + nµ + nν] 4ωµων[(ωµ + ων)2 − ω2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (61b) The anti resonant part χ(0) + (ω) describes the absorp- tion/emission processes of a phonon pair (double solid line with both arrows in the same directions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2) while the resonant χ(0) − (ω) the case in which one phonon is absorbed and the other one is emitted (double solid line with arrows pointing in opposite directions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The first row of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55) relates the propagation of single phonon excitation G(0)(ω) to two-phonon processes χ(0) ± (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This is mediated by the three-phonon scattering vertex (orange triangle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2) (3) D µνη = � ∂3V (BO) ∂ �Rµ∂ �Rν∂ �Rη � (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (62) The second and third rows of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55) show that double excitations interact with each other via the fourth-order 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Graphical expression for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55) in terms of the free SCHA propagators (single and double solid line Eqs (59) (60)) and the third and fourth order scattering vertex (orange triangle and red square defined in Eqs (62) (63)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' scattering vertex (red square of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2) (4) D µνηλ = � ∂4V (BO) ∂ �Rµ∂ �Rν∂ �Rη∂ �Rλ � (0) , (63) or can decay in a single phonon via the three-phonon scattering vertex, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As a guide for the reader, in Table IV A we report a summary of all the symbols used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So, by just looking at the expression of L(ω), we un- derstand that in TDSCHA only single and double exci- tation are dressed by anharmonicity and that only single phonons can decay in a higher-order phonon propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression for the SCHA scattering tensors Eqs (62) (63) as presented in Eqs (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Each SCHA propagator is contracted with a higher order derivative of the anharmonic tensor Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (65) which are evaluated at the SCHA positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' they do not coincide with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These are represented as n = 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. and n = 4, 6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. regular polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The scattering vertices, Eqs (62) (63), included in the dynamical response have an interesting diagrammatic ex- pression (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' They do not coincide in gen- eral with the derivatives of the BO potential evaluated at the equilibrium SCHA positions R(0) but they contain extra terms due to quantum-thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' All of these terms are included in the TDSCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Expanding Eqs (62) (63) in R − R(0), we get the following series, as reported in Appendix I, (3) D µνη= +∞ � n=0 (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3N � α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n=1 (3+2n) D µνηα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n ,(0) G(0)(t = 0−)α1α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.G(0)(t = 0−)α2n−1α2n, (64a) (4) D µνηλ= +∞ � n=0 (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3N � α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n=1 (4+2n) D µνηλα1α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n−1α2n ,(0) G(0)(t = 0−)α1α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.G(0)(t = 0−)α2n−1α2n, (64b) where the anharmonic vertices in the series are (n) D α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.αn ,(0) = ∂nV (BO) ∂ �Rα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.∂ �Rαn ���� R=R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (65) These differ in general from the vertices (n) d , see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (34), since the minimum of the Born-Oppenheimer potential RBO does not coincide with the SCHA centroid R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3 we report the diagrammatic expansion for Eqs (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Each anharmonic tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (65) with n > 3, 4 has a pair of indexes contracted with a SCHA propaga- tor G(0)(t = 0−) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (65)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This means that the an- harmonic vertices are renormalized by quantum-thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Anharmonic propagators In this Section, we discuss the TDSCHA interacting propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Specifically, we present two-phonon pro- cesses that have been neglected in previous works [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In TDSCHA, the one-phonon G(ω)µν, two-phonon χ(ω)µνηλ, and the one-two phonon Γ(ω)µηλ interacting propagators are obtained as response functions by set- ting in χ(ω)A,B (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53)) AG = δ �R(0) µ BG =δ �R(0) ν , (66a) Aχ = 1 2δ �R(0) µ δ �R(0) ν Bχ =1 2δ �R(0) η δ �R(0) λ , (66b) AΓ = δ �R(0) µ BΓ =1 2δ �R(0) η δ �R(0) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66c) The response and perturbation vectors (see Eqs (54)) cor- 10 responding to Eqs (66) are as follows rG = � ��� δµ 0 0 � ��� pG = � ��� δν 0 0 � ��� , (67a) rχ = � ��� 0 Sµν Sµν � ��� pχ = � ��� 0 Sηλ Sηλ � ��� , (67b) rΓ = � ��� δµ 0 0 � ��� pΓ = � ��� 0 Sηλ Sηλ � ��� , (67c) where δµ is a 3N vector with 1 in the mode index µ and zero elsewhere and Sηλ is a 3N ×3N matrix with 1/2 on the mode indexes η and λ and zeros elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The choice of A/B, as in Eqs (66), is not arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the non-interacting case, L(ω) is diagonal, so the re- sponse function is simply χ(ω)A,B =ri � ��� G(0)(ω) 0 0 0 χ(0) − (ω) 0 0 0 −χ(0) + (ω) � ��� pi i =G, χ, Γ, (68) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (68) we recover the one and two phonon free propagators (Eqs (59) (60)) and no cross terms connect- ing single to double excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These results are con- sistent with the standard linear response theory, in the non-interacting case, treated with the many-body for- malism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This proves that with Eqs (67) we recover the physically relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To get the interacting Green’s function, we plug r and p of Eqs (67) in the expression of χ(ω)A,B, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53), and we invert L(ω) following Refs [1, 55, 56] (see also Ap- pendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To do this, we consider L(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55), as a 3 × 3 block-matrix (as represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2) where each block itself is a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The same applies to r and p, Eqs (54), which are understood as 3 components vec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' From the non-interacting case (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (68)), we learn that the first component of p/r controls the single-mode propagation while the second and third the two-phonon channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The inversion process mixes the matrix elements of L(ω) adding interactions to the free propagators which are expressed as diagrammatic series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The representation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2 is a graphical aid to visualize the building blocks of these diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Appendix H we report the details of all the results presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In our calculations we always reduce the inversion of L(ω) to a 2 × 2 block-matrix with the following form � � A C C† B � � (69) which can be easily inverted (see Appendix G) � � � A − CB−1C†�−1 � C† − BC−1A �−1 � C − AC†−1B �−1 B−1 − B−1C† � C† − BC−1A �−1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (70) First, we discuss the one-phonon propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Because only the first component of both pG and rG, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (67a), are non-zero, the response calculation is simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In particular, we compact the 2 × 2 two-phonon sector of L(ω) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L(ω)ij with i, j > 1) in a 1×1 block matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As shown in Appendix H, L(ω) is reduced to a 2 × 2 block- matrix L1ph(ω), so that the one-phonon propagator is given by G(ω) =rG · L(ω)−1 · pG = � L1ph(ω)−1� 11 (71) where L1ph(ω) = � �� G(0)(ω)−1 − (3) D − (3) D χ(0)(ω)−1 − (4) D � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (72) Note that (L1ph(ω))22 represents the anharmonic two- phonon channel in which single modes can decay through the three-phonon vertex, (L1ph(ω))12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Graphically Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (71) corresponds to = � � −1 − − −1 − � � −1 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (73) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (71) we apply the general result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (70) to obtain the interacting Green’s function G(ω) = G(0)(ω) + G(0)(ω) · Π(ω) · G(ω), (74) here A : B = �3N µν=1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.µνBµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='., where the self-energy Π(ω) coincides with the one reported in Refs [1, 2, 49] Π(ω) = (3) D : � 1 − χ(0)(ω) : (4) D �−1 : χ(0)(ω) : (3) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (75) Our definition of the non-interacting two-phonon propa- gator χ(0)(ω) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (60)) is one reported by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [49] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (72) multiplied by −1/2 so that all the definitions are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Physical phonon frequencies and lifetimes are given by real and imaginary parts of G(ω + i0+), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74), as dis- cussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In addition, we remark that the polar- ization vectors can also change when adding dynamical effects so polarization-mixing is automatically included in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The lowest-order approximation for the self-energy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (75) is the bubble diagram which is included by many SCP calculations in the improved SCP (ISCP) frame- work [40, 57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' TDSCHA represents a theoretical ap- proach that justifies this from the least action principle 11 and paves the way to go beyond the bubble approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' For the two-phonon case, we proceed as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We use the definition of rχ/pχ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (67b)) to reabsorb the one- phonon sector of L(ω) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the row L(ω)1j and column L(ω)j1 with j = 1, 2, 3) into the two-phonon sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So we solve χ(ω) = rχ · L(ω)−1 · pχ = 1,2 � ij � L2ph(ω)−1� ij (76) where L2ph(ω) is a 2 × 2 block-matrix L2ph(ω)= � �χ(0) − (ω)−1 − Σ(ω) −Σ(ω) −Σ(ω) −χ(0) + (ω)−1 − Σ(ω) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (77) Σ(ω) is the two-phonon self-energy Σ(ω) = Σ(ω)† = (4) D + (3) D · G(0)(ω) · (3) D (78) where single phonon excitations enter in Σ(ω) via the three-phonon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Graphically, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (76) corresponds to = 1,2 � ij � �� −1 − − − − −1 − � �� −1 ij Σ(ω) = = + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (79) Again we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (70) to invert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (77) and we end up with the interacting two-phonon propagator χ(ω) = χ(0)(ω) + χ(0)(ω) : Σ(ω) : χ(ω) (80) with Σ(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (78), being the TDSCHA two-phonon self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We find that in the two-phonon propagation there is the possibility of either decay in a single phonon through the third-order scattering vertex or in another pair through the fourth-order scattering vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' There are no high-order decay processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' For the mixed propagator, we proceed as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Thanks to the form of pΓ/rΓ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (67c)), single phonon excitations can be triggered by two-phonon propagation via the three-phonon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This leads to a non-zero one-two phonon propagation Γ(ω) = G(0)(ω) · (3) D : χ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (81) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 4 summarizes the diagrammatic expressions for the interacting propagators, Eqs (74) (80) (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' By computing all the TDSCHA interacting propaga- tors, we have a full comprehension of the diagrammatic expression, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 5, introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [1] for the TDSCHA response function χ(ω)A,A (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53) can be decomposed into the interacting propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression of the interacting one, two and one-two phonon Green’s functions, Eqs (74) (80) (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Thinner solid lines represent the non-interacting SCHA prop- agators G(0)(ω) and χ(0)(ω), defined in Eqs (74) (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The three/four-phonon scattering vertices are defined in Eqs (62) (63) and represented as orange triangles and red squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression of the processes included in the fully interacting response, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53), if A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The interacting TDSCHA Green’s functions are reported in Eqs (74) (80) (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The green vertex is related to the first entry of p/r while the blue vertex to the second and third entries of p/r, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' One-phonon processes G(ω) are coupled to first-order position derivatives of the perturbation, first entry of p/r Eqs (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' On the other hand, two-phonon excitations χ(ω) and Γ(ω) are triggered by non-zero second-order position derivatives of the perturbation, second and third entries of p/r Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We remark that the Lanczos algorithm includes the effect of the third and fourth-order scattering vertex, Eqs (62) (63), in a non-perturbative way [13] (see Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' TDSCHA evolves ab-initio all the phonon modes in a given supercell without free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This feature is interesting for applications in non-linear phononics where is crucial to comprehend relaxation pathways of coherent phonon oscillations [28, 60–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Xr(w) (3)5 X(W)A,A +2 0A OR OROR (0) (0)12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Momentum Green function In Wigner-TDSCHA, the ionic momentum is con- trolled directly, which was not possible in the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Here we discuss the TDSCHA momentum- momentum Green’s function Gp(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In our theory, this is computed setting A = �Pµ B = �Pν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (82) The SCHA momentum Green’s function G(0) p (ω) is pro- portional to G(0)(ω) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74)) since the equation for position and momentum are coupled iω �P(ω) = �R(ω) −→ G(0) p (ω)µν = ω2 µG(0)(ω)µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (83) The free propagators are the building blocks for the inter- acting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hence the interacting momentum Green’s function Gp(ω) satisfies a perturbative expansion that is proportional to the one of G(ω) once the TDSCHA dia- grams are selected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' those from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So we have that, see Appendix J for details, Gp(ω) = −1 + ω2G(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (84) Thus, a Lanczos calculation also provides access to the TDSCHA momentum Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Multiple excitations in TDSCHA The Gaussian approximation defines a hierarchy of di- agrams that is truncated at the two-phonon level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this Section, we show that, in TDSCHA, all higher-order phonon propagators are related to the Green’s functions of Eqs (74) (80) (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' For example, the three-phonon propagator is obtained setting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (53) a tensor-like perturbation/response functions A = δ �R(0) α δ �R(0) β δ �R(0) γ B = δ �R(0) µ δ �R(0) ν δ �R(0) η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (85) In this case, only the first entries of p/r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � ∂A/∂ � R � (0), are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This means that in the case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (85) we have a one-phonon response, as the one obtained for G(ω) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (67a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As computed in Appendix H, the three-phonon response is χαβγ µνη (ω) = G(0)(t = 0−)βγG(0)(t = 0−)νηG(ω)αµ + permutations of (αβγ) and (µνη) (86) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 6 we report its diagrammatic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This contains the one-phonon Green’s function G(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74), and a disconnected part, G(0)(t = 0−) which comes from the averages � δ � Rδ � R � (0) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (39)) in � ∂A/∂ � R � (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression for the TDSCHA interact- ing three-phonon Green’s function obtained as a response to a cubic perturbation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In TDSCHA, the tree-phonons propagator is a disconnected diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this case, the SCHA correction, G(0)(t = 0−), does not enter the phonon propagation but it dresses the in- teraction with the external probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This means that if we take a scalar perturbation A = B = 1 3 3N � αβγ=1 Wαβγδ �R(0) α δ �R(0) β δ �R(0) γ , (87) with Wαβγ a tensor that does not depend on atomic po- sitions, G(0)(t = 0−) is contracted only the tensor W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Similarly, all the higher-order propagators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' those obtained with A ∼ � δ �R(0)�n−2 B ∼ � δ �R(0)�m−2 m, n > 2, (88) give disconnected diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In all these cases we will get a χA,B(ω) that contains only one of the TDSCHA propagators (Eqs (74) (80) (81)) plus a disconnected part that depends only on G(0)(t = 0−), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diagrammatic expression for the Saturn diagram with a three SCHA phonon propagation (solid lines are the propa- gators of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (59)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The red vertex is the four-phonon scatter- ing vertex Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (63) which leads to three phonon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This class of diagrams is missed by TDSCHA where we can not connect a single SCHA line to the four-phonon scattering vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So TDSCHA can not capture processes beyond a two- phonon mechanism: the propagators of Eqs (74) (80) (81) are the building blocks of the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This means that there are general rules in the symbolic inversion of L(ω) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (55)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 2 the solid line (one-phonon SCHA propagator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (60)) is always attached to one extremity of the orange triangle (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (62)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The dou- ble solid line (two-phonon SCHA propagator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (60)) is connected to two extremities either of the red square (four phonon vertex Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (63)) or of the orange triangle (three phonon vertex Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (62)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We do not get three or more SCHA phonon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' One example is the ’Saturn’ diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 7) which is 13 missed by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This diagram would correspond to a single SCHA propagator attached to the four-phonon vertex and this is not contained in TDSCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Notably, the TDSCHA diagrams emerge from the sta- tionary action principle of quantum mechanics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This guarantees that there is no double counting and that the theory is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The inclusion of new scattering mechanisms, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 7, must be done with extreme care to avoid spoiling the internal coherence and overcounting some anharmonic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' NONLINEAR PHONON-PHOTON COUPLING: INFRARED AND RAMAN In this Section, we review the infrared (IR) and Raman response in TDSCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In particular, we focus on the two- phonon effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' IR experiments are based on the absorption of infrared light by normal modes associated with a dipole moment variation which, in a crystal, are optical phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The IR signal is proportional to the imaginary part of the dipole- dipole response function, Im[χ(ω + i0+)pα,pβ], along two Cartesian directions α and β hence A(R) = pα(R), B(R) = pβ(R), (89) where the dipole pα is per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To get the response function we need the response and perturbation vector r′ and p′ (see Section IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The first component of these vectors contains equilibrium averages of the effective charges: �∂pα(R) ∂Ra � (0) = ⟨Z∗(R)a,α⟩(0) , (90) where a is a supercell index and Z∗(R) is the effec- tive charges tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This vertex is the coupling for one phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The second and third components of r′/p′ contain the first derivatives of the effective charges, the second-order dipole moment: �∂2pα(R) ∂Ra∂Rb � (0) = �∂Z∗(R)a,α ∂Rb � (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (91) A Raman process consists in the scattering of light (usually visible) by zone-center phonons that induce a change in polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Raman cross-section con- tains the imaginary part of polarizability-polarizability response, Im[χαµν,αηλ(ω + i0+)], obtained with A(R) = α(R)µν B(R) = α(R)ηλ, (92) where µ, ν, η, λ are Cartesian directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In a Raman pro- cess phonons and photons scatter so we take into account the quantization of the electromagnetic field by multi- plying Im[χαµν,αηλ(ω + i0+)] by 1 + n(ω) where n(ω) is the Bose-Einstein distribution for the photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The first component of r and p gives one-phonon processes and contains: �∂α(R)µν ∂Ra � (0) = ⟨Ξ(R)a,µν⟩(0) , (93) where Ξ(R) is the Raman tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As before, the other components of r′ and p′ depends on the second-order Raman polarizability: �∂2α(R)µν ∂Ra∂Rb � (0) = �∂Ξ(R)a,µν ∂Rb � (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (94) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (91) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (94) trigger second-order IR/Raman processes exiting two phonons in the system, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In principle, higher-order processes are possible, such as three-phonon etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' However, TDSCHA can not account for them as we showed that the three-phonon propagator is a disconnected diagram (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' IV D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A two-phonon process, both in IR and Raman spectra, is a scattering mechanism between photons and phonons that conserves both energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The long- wavelength electromagnetic field can either absorb or generate two phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Another possibility is that one phonon is absorbed and the other one is emitted inter- acting with photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This involves pairs of phonons with opposite momentum in the Brillouin zone forming a con- tinuum signal overlapped to the sharper peaks of one phonon process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This mechanism is found both in harmonic and anhar- monic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The IR spectra of both Si and Ge can only be explained with these processes since there are no IR-active phonons due to inversion symmetry [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Also, anharmonic systems, such as liquid water [64], present features due to effective charge position modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Two phonons effects play a central role in many Raman spectra: from diamond and SiC [65, 66] to BaTiO3 [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The most common approximation is Z∗(R)a,α ≃ Z∗(R(0))a,α, (95a) Ξ(R)a,µν ≃ Ξ(R(0))a,µν, (95b) which suppresses all two phonon processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We use integration by parts and a Monte Carlo sam- pling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' as proposed in [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' to compute all the components of r′ and p′ in an efficient and non-perturbative way us- ing only effective charges and Raman tensor: �∂2p(R)α ∂Ra∂Rb � (0) = − 3N � c=1 G(0)(t = 0−)ac � δ �R(0) c Zb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='α(R) � (0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (96a) �∂2α(R)µν ∂Ra∂Rb � (0) = − 3N � c=1 G(0)(t = 0−)ac � δ �R(0) c Ξb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µν(R) � (0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (96b) We remark that TDSCHA is the only method that com- putes second-order Raman tensors or effective charges with full position dependence without the need for higher-order DFT response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Appendix K we report in detail how to prepare a IR/Raman calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 14 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' INFRARED SPECTRA OF HIGH-PRESSURE HYDROGEN In this Section, we show the relevance of two phonon effects in a strongly anharmonic system such as high- pressure hydrogen phase III (C2c/24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We apply our new TDSCHA implementation on the infrared spectra of high-pressure hydrogen at P = 250 GPa and T = 0 K including the effect of second-order effective charges, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We employ 40000 energy/forces and 2000 effective charges calculations, on a 2×2×1 supercell, to converge the anharmonic vertices, Eqs (62) (63), and the IR over- tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Energies, forces and effective charges were com- puted using the BLYP functional [68] on a 4×4×4 k-grid (energy cutoff of 60 Ry and 240 Ry on the charge den- sity) as implemented in QUANTUM ESPRESSO [69, 70], with a plane wave basis set and a norm-conserving pseu- dopotential from the PSEUDO DOJO library [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Infrared signal for high pressure hydrogen at P = 250 GPa and T = 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Panel (a) reports the IR spectra obtained with the SCHA phonons of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27) without two-phonon ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' These are included at the SCHA level in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Panel (c) reports the spectra setting both (4) D , (3) D ̸=0 compared with the data extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [5] at 248 GPa and 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The smearing δ is 30 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Figure 8 we plot the IR signal using different approx- imations defined as 1 3 x,y,z � α Im [χ(ω + iδ)pα,pα] (97) where δ is the smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 9 we plot the IR signal as a function of the Lanczos steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Convergence of the IR signal with (4) D , (3) D ̸=0 as a func- tion of the Lanczos steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The smearing δ is 15 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The convergence is achieved in Nstep = 500 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 9) steps which are half of those employed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [1] (Nstep = 1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This is due to a more stable Lanczos algorithm thanks to the symmetry of L(ω) in the Wigner formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Panel (a) and (b) of Figure 8 show the effect of adding the second-order IR effects using the non-interacting SCHA phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The position modulation of effective charges generates a signal between 2000-4000 cm−1 and around 5000 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In panels (c) of Figure 8 we add all the anharmonic interactions contained in TDSCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Notably, the two phonon processes at high frequency are stable after adding the anharmonic scattering of two phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This feature is in agreement with the overtone observed in the experiments by Goncharov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [5], con- firming that it is a high-order IR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' SCHA phonons 150 (a) (Z(R))(0) 125 100 ignal 75 s R 50 25 0 0 1000 2000 3000 4000 5000 6000 SCHA phonons 150 (b) aZ(R) (Z(R))(0) /(0) äR 125 100 75 s R 50 25 0 1000 3000 5000 6000 0 2000 4000 (3) (4) TDSCHA phonons with D + D 150 (Z(R))(0) (c) /aZ(R)/ 125 (Z(R))(0) (0) aR 100 Goncharov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 75 R 50 25 0 1000 2000 3000 4000 5000 6000 0 w (cm-1)Nsteps=200 (Z(R))(0) 200 /aZ(R) (Z(R))(0) /(0) aR 100 R 50 0 2000 3000 5000 0 1000 4000 6000 Nsteps=300 250 (Z(R))(0) 200 aZ(R) (Z(R))(0) + /(0) 150 sigr R100 50 0 6000 2000 3000 5000 0 1000 4000 Nsteps=400 250 (Z(R))(0) 200 /aZ(R) (Z(R))(0) (0) äR S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R100 50 0 2000 0 3000 5000 6000 1000 4000 Nsteps=500 250 (Z(R)(0) aZ(R) 200 (Z(R))(0) + sigr R100 50 0 1000 3000 4000 6000 2000 5000 w (cm-1)15 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' CONCLUSIONS The Wigner picture simplifies the TDSCHA equations improving the physical intuition of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This allows us to discuss the equivalence of quantum and clas- sical dynamics and rewrite the equations of motion in terms of position and momentum correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The response function is directly related to the dia- grammatic expression of the interacting Green’s function, through which we have built a bridge to many-body per- turbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the context of linear response theory, we clarified which diagrams and scattering processes are included in the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The TDSCHA infrared spec- tra of high-pressure hydrogen phase III showed that only two phonon effects explain the overtone experimentally observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge support by EU under project MORE-TEM ERC-SYN (grant agreement No 951215) and the CINECA award under the ISCRA initiative, for the availability of high-performance computing resources and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We also acknowledge PRACE for awarding us access to Joliot-Curie Rome at TGCC, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 16 Appendix A: Equations of motion In this Appendix, we prove the Wigner-TDSCHA equations of motion Eqs (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' For compactness, we define the mass-rescaled free parameters �α(t)ab = α(t)ab √mamb �β(t)ab = √mambβ(t)ab �γ(t)ab = � mb ma γ(t)ab �Ra = √maRa �R(t)a = √maR(t)a �Pa = Pa √ma �P(t)a = P(t)a √ma (A1) The equation of motion for the free parameters �α(t), �β(t), �γ(t), �R(t), �P(t) are found with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (20): ∂�ρ(R, P , t) ∂t = ∂H(�ρ) ∂ � R ∂�ρ(R, P , t) ∂ � P − ∂H(�ρ) ∂ � P ∂�ρ(R, P , t) ∂ � R = = ��∂V (tot) ∂ � R � �ρ(t) + δ � R(t) · �∂2V (tot) ∂ � R∂ � R � �ρ(t) � ∂�ρ(R, P , t) ∂ � P − � δ � P (t) + �P(t) � ∂�ρ(R, P , t) ∂ � R (A2) with δ � R(t) = � R − �R(t) and δ � P (t) = � P − �P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The gradient of �ρ(R, P , t), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15), is ∂ log (�ρ(R, P , t)) ∂ � P = − �β(t) · δ � P (t) + �γT (t) · δ � R(t), (A3a) ∂ log (�ρ(R, P , t)) ∂ � R = −�α(t) · δ � R(t) + �γ(t) · δ � P (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A3b) The time derivative of �ρ(R, P , t) gives ∂ log (�ρ(R, P , t)) ∂t = ˙N(t) N(t) − 1 2δ � R(t) · ˙�α(t) · δ � R(t) − 1 2δ � P (t) · ˙�β(t) · δP (t + δ � R(t) · ˙�γ(t) · δ � P (t)+ + ˙�R(t) · α(t) · δ � R(t) + ˙�P(t) · �β(t) · δ � P (t) − ˙�R(t) · �γ(t) · δ � P (t) − ˙�P(t) · �γT (t) · δ � R(t), (A4) where ˙◦ denotes the time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' With Eqs (A3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A4) the Wigner-Liouville equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A2) becomes a polynomial in δ � R(t) δ � P (t) then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' setting to zero the coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' we get the equations of motion for the free parameters d dt �R(t) = �P(t) (A5a) d dt �P(t) = − �∂V (tot) ∂ � R � �ρ(t) (A5b) d dt �α(t) = − �∂2V (tot) ∂ � R∂ � R � �ρ(t) �γ†(t) − �γ(t) · �∂2V (tot) ∂ � R∂ � R � �ρ(t) (A5c) d dt �β(t) =�γ†(t) + �γ(t) (A5d) d dt �γ(t) =�α(t) − �∂2V (tot) ∂ � R∂ � R � �ρ(t) �β(t) (A5e) where ◦† denotes the hermitian conjugate of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The equations of motion for the tensors keep the distribution normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The equal-time position and momentum correlators can be written in terms of �α(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' �β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' �γ(t) � δ � X(t)aδ �Y (t)b � �ρ(t) = � dR � dP �ρ(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' t)δ � X(t)aδ �Y (t)b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' δ� X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' δ �Y (t) = δ � R(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' δ � P (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A6) and using Gaussian integration we get the following expressions � δ � R(t)δ � R(t) � �ρ(t) = � �α(t) − �γ(t) · �β−1(t) · �γT (t) �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A7a) � δ � R(t)δ � P (t) � �ρ(t) = �α−1(t) · �γ(t) · � �β(t) − �γT (t) · �α−1(t) · �γ(t) �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A7b) � δ � P (t)δ � P (t) � �ρ(t) = � �β(t) − �γT (t) · �α−1(t) · �γ(t) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A7c) 17 Then deriving with respect to time Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A7) and using the equations of motion Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (A5) we prove Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix B: Equivalence with Time-Dependent Self-consistent Harmonic Approximation In this Appendix, we show that our method is a Wigner reformulation of TDSCHA presented in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We compute the matrix elements of the von Neumann density operator corresponding to the Wigner distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To do this we need the inverse of the Wigner transformation which is defined as (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [41]) ˆρ = � dRdR′dP dP ′ (2πℏ)3N ρ(R′, P ′) exp � − i ℏ � P · � ˆR − R′� + R · � ˆP − P ′��� , (B1) where ρ(R′, P ′) is the Wigner quasi-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The ˆ◦ indicates quantum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Inserting the Wigner distribution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B1) we get a Gaussian integral for the density operator matrix elements ⟨R| ˆ�ρ(t) |R′⟩ = � dP exp � i ℏ(R − R′) · P � �ρ �R + R′ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' t � =N(t) exp � −1 8 (δR(t) + δR′(t)) · α(t) · (δR(t) + δR′(t)) + i ℏ(R − R′) · P(t) � � dP exp � −1 2δP (t) · β(t) · δP (t) + 1 2 �2i ℏ (R − R′) + (δR(t) + δR′(t)) · γ(t) � δP (t) � = � det �Υ(t) 2π � exp � iQ(t) · (R − R′) − (R − R(t)) · �1 4Θ(t) + iC(t) � (R − R(t)) − (R′ − R(t)) · �1 4Θ(t) − iC(t) � (R′ − R(t)) + (R − R(t)) · (Re A(t) + i Im A(t)) · (R′ − R(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B2) The last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B2) is the trial density operator used in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The free parameters used in [1] Q(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Θ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' C(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Re A(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Im A(t) are related to the ones used in the Wigner formalism Q(t) = 1 ℏP(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B3a) Θ(t) = 1 2 � α(t) − γ(t) · β−1(t) · γT (t) � + 2 ℏ2 β−1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B3b) C(t) = − 1 2ℏβ−1(t) · γT (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B3c) Re A(t) = −1 4 � α(t) − γ(t) · β−1(t) · γT (t) � + 1 ℏ2 β−1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B3d) Im A(t) = 1 2ℏ � γ(t) · β−1(t) − β−1(t) · γT (t) � (B3e) Υ(t) = Θ(t) − 2 Re A(t) = α(t) − γ(t) · β−1(t) · γT (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B3f) where ◦−1 denotes the inverse of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The tensor Υ(t) is a linear combination of Θ(t) and Re A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The same notation for the average position R(t) is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using the relations between free parameters, Eqs (B3), it is easy to prove that the equations of motion Eqs (A5) are equivalent to the TDSCHA ones reported in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Here, we also prove that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (25) is the Wigner transform of the SCHA equilibrium density matrix ˆ�ρ (0) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 45]: ⟨R| ˆ�ρ (0) |R′⟩ = � det �Υ(0) 2π � exp � −1 4 3N � ab=1 Θ(0) ab (Ra − R(0) a )(Rb − R(0) b ) − 1 4 3N � ab=1 Θ(0) ab (R′ a − R(0) a )(R′ b − R(0) b ) + 3N � ab=1 A(0) ab (Ra − R(0) a )(R′ b − R(0) b ) � (B4) 18 with Υ(0) = Θ(0) − 2A(0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' where Υ(0) and A(0) are defined as Υ(0) ab = 3N � µ=1 2ωµ ℏ(1 + 2nµ)ea µeb µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A(0) ab = 3N � µ=1 2ωµnµ(1 + nµ) ℏ(1 + 2nµ) ea µeb µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B5) where ω2 µ and {eµ} are the auxiliary SCHA modes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Wigner quasi-distribution, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' is obtained in the following way �ρ(0)(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P ) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='det ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='3N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(Ra − Ra)Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab (Rb − Rb) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='d3NR′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4πℏ2)3N/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='3N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='a=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='PaR′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ℏ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='3N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(Θ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab + 2A(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab )R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='aR′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='det ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�det ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2πℏ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='A(0) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='3N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(Ra − R(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='a )Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab (Rb − R(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='b ) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='3N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ℏ2A(0) + ℏ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4 Υ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(B6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='The final result is a positive-definite Gaussian Wigner distribution which coincides with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (25) �ρ(0)(R, P ) = � det �α(0) 2π � det �β(0) 2π � exp � −1 2 3N � ab=1 (Ra − R(0) a )α(0) ab (Rb − R(0) b ) − 1 2 3N � ab=1 Paβ(0) ab Pb � , (B7) once we recognize that ⟨δRδR⟩(0) = α(0)−1 = Υ(0)−1, (B8) and ⟨P P ⟩(0) = β(0)−1 = ℏ2 � A(0) + 1 4Υ(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B9) where the equilibrium correlators are defined in Eqs (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix C: Energy conservation In this Appendix, we show that the TDSCHA equations of motion Eqs (23) satisfy the energy conservation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Wigner quantum time-dependent Hamiltonian has the same form as the classical one H(t) = 3N � a=1 P 2 a 2ma + V (BO)(R) + V (ext)(R, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C1) We compute the total time-derivative of ⟨H(t)⟩�ρ(t) where �ρ(t) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15): d ⟨H(t)⟩�ρ(t) dt = d dt � 3N � a=1 1 2 �� δ �P(t)aδ �P(t)a � �ρ(t) + �P(t)2 a � + � V (tot)� �ρ(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C2) The time derivative of the kinetic energy gives: d dt 3N � a=1 1 2 �� δ �P(t)aδ �P(t)a � �ρ(t) + �P(t)2 a � = 1 2 3N � a=1 d � δ �P(t)aδ �P(t)a � �ρ(t) dt + 3N � a=1 �P(t)a d �P(t)a dt = 1 2 Tr � �� d � δ � P (t)δ � P (t) � �ρ(t) dt � �� − �P(t) · �∂V (tot) ∂ � R � �ρ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C3) 19 The derivative of the total potential average is more involved since the position probability distribution depends on time through R(t) and � δ � R(t)δ � R(t) � �ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The derivative is worked out using the formulas proved in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [49]: d � V (tot)� �ρ(t) dt = �∂V (ext) ∂t � + 3N � a=1 d �R(t)a dt � ∂V (tot) ∂ �R(t)a � �ρ(t) + 1 2 3N � ab=1 d � δ �R(t)aδ �R(t)b � �ρ(t) dt �∂2V (tot) ∂ �Rb∂ �Ra � �ρ(t) = �∂V (ext) ∂t � �ρ(t) + �P(t) · �∂V (tot) ∂ � R � �ρ(t) + 1 2 Tr � �� d � δ � R(t)δ � R(t) � �ρ(t) dt �∂2V (tot) ∂ � R∂ � R � �ρ(t) � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C4) Then using Eqs (23c)-(23d) and the permutation properties of the trace it is shown that: 1 2 Tr � d dt � δ � P (t)δ � P (t) �� = −1 2 Tr � �� d � δ � R(t)δ � R(t) � �ρ(t) dt �∂2V (tot) ∂ � R∂ � R � �ρ(t) � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C5) So in the end, we found: d ⟨H(t)⟩�ρ(t) dt = �∂V (ext) ∂t � �ρ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (C6) This derivation is more compact than the one presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix D: Expansion of the probability distribution In this Appendix, we show how to expand at first order the TDSHCA position probability distribution and how the anharmonic vertices, Eqs (62) (63), emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' All the free parameters are perturbed with respect to their static value (denoted by (0)) R(t) =R(0) + R(1)(t) (D1a) P(t) =P(1)(t) (D1b) α(t) =α(0) + α(1)(t) (D1c) β(t) =β(0) + β(1)(t) (D1d) γ(t) =γ(1)(t) (D1e) The first thing to do is to expand at first order the position probability distribution in the perturbative free parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' those denoted by the superscript (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Before performing the expansion, we report the full position probability distribution obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (15) �ρ(R, t) = � � � � �det � �α(t) − γ(t) · −1 β (t) · γ(t)T 2π � � exp � −1 2(R − R(t)) · � α(t) − γ(t) · −1 β (t) · γ(t)T � (R − R(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D2) The leading order is controlled by α(0) + α(1)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We define the displacements with respect the equilibrium position as δ � R(0) = � R − �R (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The expansion gives �ρ(R, t) = �ρ(0)(R) + �ρ(1)(R, t), (D3) where �ρ(0)(R) is the equilibrium probability distribution (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (25)) �ρ(0)(R) = � det �α(0) 2π � exp � −1 2δ � R(0) · �α(0) · δ � R(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D4) 20 The explicit expression for �ρ(1)(R, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D3), following [1], is �ρ(1)(R, t) = �ρ(0)(R) �1 2Tr � α(0)−1 · α(1)(t) � − 1 2δR(0) · α(1)(t) · δR(0) + δR(0) · α(0) · R(1)(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D5) Next, we derive an expression for the perturbed averages of a position-dependent observable O(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using the expres- sion for �ρ(1)(R, t), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D5), and integration by parts we get ⟨O⟩(1) (t) = � dR�ρ(1)(R, t)O(R) = −1 2 3N � ab=1 �α(1)(t)ab �� δ �R(0) a δ �R(0) b O � (0) − (�α(0))−1 ba ⟨O⟩(0) � + 3N � a=1 �R(1) a (t) � ∂O ∂ �Ra � (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D6) Note that now all the averages have to be performed on the equilibrium ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We introduce the equilibrium three and four phonon scattering vertices as in [49] (3) D abc = � ∂V (BO) ∂ �Ra∂ �Rb∂ �Rc � (0) = 3N � mn=1 �α(0) an �α(0) bm � δ �Rnδ �Rm ∂V (BO) ∂ �Rc � (0) − �α(0) ab �∂V (BO) ∂ �Rc � (0) (D7a) (4) D abcd = � ∂V (BO) ∂ �Ra∂ �Rb∂ �Rc∂ �Rd � (0) = 3N � nm=1 �α(0) an �α(0) bm � δ �Rnδ �Rm ∂2V (BO) ∂ �Rc∂ �Rd � (0) − �α(0) ab �∂2V (BO) ∂ �Rc∂ �Rd � (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D7b) It is convenient also to introduce the potential V(R) as the difference between the BO potential and the harmonic auxiliary potential obtained at equilibrium V(R) = V (BO)(R) − 1 2δ � R(0) · (2) D · δ � R(0), (D8) where (2) D defines the SCHA phonons, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D6), we relate the perturbed averages of V(R) to the scattering vertices of Eqs (D7) � ∂V ∂ �Ra � (1) = 1 2 3N � bcde=1 �α(1)(t)de(�α(0)−1)db(�α(0)−1)ec (3) D abc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D9a) � ∂2V ∂ �Ra∂ �Rb � (1) = 3N � c=1 (3) D abc �R(1)(t)c − 1 2 3N � cdef=1 �α(1)(t)ef(�α(0)−1)ec(�α(0)−1)fd (4) D abcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D9b) At this point it is straightforward to get the following perturbed averages for the BO potential: � ∂V ∂ �Ra � (1) = �∂V (BO) ∂ �Ra � (1) − 3N � b=1 (2) D ab ˜R(1) b (t), (D10a) � ∂2V ∂ �Ra∂ �Rb � (1) = �∂2V (BO) ∂ �Ra∂ �Rb � (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D10b) Appendix E: Derivation of the linear response system In this Appendix, we prove the linearized equations of motion discussed in Section IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To do this we write all the supercell tensors in the static equilibrium polarization basis {eµ} defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So a multi-indices tensor A(t)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.,aN defined in the supercell can be written in the polarization basis as A(t)µ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.,µN = 3N � a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.,aN=1 ea1 µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.eaN µN A(t)a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.,aN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E1) From now on all the quantities are written in this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 21 All the supercell tensors are written in the equilibrium polarization basis, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The equations of motion (Eqs (A5)) expanded at first order are d2 dt2 �R(1)(t)α = −ω2 α �R(1)(t)α − � ∂V ∂ �Rα � (1) − �∂V (ext)(t) ∂ �Rα � (0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E2a) d dt �α(1)(t)αβ = −2 3N � µν=1 Sαβµν � �γ(1)(t)µνω2 ν � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E2b) d dt �β(1)(t)αβ = 2 3N � µν=1 Sαβµν � �γ(1)(t)µν � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E2c) d dt�γ(1)(t)αβ = �α(1)(t)αβ − ω2 α �β(1)(t)αβ − � � � ∂2V (ext)(t) ∂ �Rα∂ �Rβ � (0) + � ∂2V ∂ �Rα∂ �Rβ � (1) � � �β(0) ββ , (E2d) where Sαβµν = 1 2(δαµδβν + δανδβµ) (E3) and V(R) is the difference between the exact BO energy surface and the SCHA auxiliary potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The averages of V(R) are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D9) and contain anharmonic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Now we make three more steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' First, we derive with respect to time Eqs (E2) to delete the equation for �γ(1)(t) since the perturbed averages do not depend on this parameter, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Secondly, we take the Fourier transform of the second order set of differential equations for �R (1)(t) �α(1)(t) and �β(1)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The third and last step is to perform a change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Instead of using the basis {�α(1)(ω), �β(1)(ω)}, we work with {�a′(1)(ω),�b′(1)(ω)} which is defined as a linear combination of the original free parameters � ��a′(1)(ω)µν �b′(1)(ω)µν � � = Mµν � ��α(1)(ω)µν �β(1)(ω)µν � � , (E4) where we define M as Mµν = � ��� K− µν ωµων K− µν − K+ µν ωµων K+ µν � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E5) The coefficients in Eqs (E5) are functions of the equilibrium auxiliary frequencies {ω2 µ} defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27) and K± µν =ℏ2nµν 2X± µν , (E6a) X± µν = � ±1 2 ℏ [ωµ ± ων] [(1 ± 1) + 2(nµ ± nν)] 4ωµων , (E6b) nµν =1 8(1 + 2nν)(1 + 2nµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E6c) Using the basis defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E5) we get the following equations of motion for { �R (1)(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' �a′(1)(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' �b′(1)(ω)} � ω2 − ω2 α � �R(1)(ω)α − � ∂V ∂ �Rα � (1) = �∂V (ext)(ω) ∂ �Rα � (0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E7a) � ω2 − ω−2 αβ � �a′(1)(ω)αβ + X− αβ � ∂2V ∂ �Rα∂ �Rβ � (1) = −X− αβ � ∂2V (ext)(ω) ∂ �Rα∂ �Rβ � (0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E7b) � ω2 − ω+2 αβ ��b′(1)(ω)αβ − X+ αβ � ∂2V ∂ �Rα∂ �Rβ � (1) = X+ αβ � ∂2V (ext)(ω) ∂ �Rα∂ �Rβ � (0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E7c) 22 where ω2 comes from the Fourier transform of a second order derivative with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E7) are written in terms of a matrix vector product in the space of the perturbative free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Recalling the definition of V (ext)(R, ω) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (45), we write the linearized equations of motion as a matrix-vector product (ω21 + L′) · � ��� �R (1)(ω) �a′(1)(ω) �b′(1)(ω) � ��� = � ������� � ∂B ∂ � R � (0) − (4) X− : � ∂2B ∂ � R∂ � R � (0) (4) X+ : � ∂2B ∂ � R∂ � R � (0) � ������� V(ω), (E8) where (4) X± αβµν = X± αβSαβµν (E9) with X± αβ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E6b) and Sαβµν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We define the RHS vector as the perturbation vector p′ p′† = �� ∂B ∂ � R � (0) , − (4) X− : � ∂2B ∂ � R∂ � R � (0) , (4) X+ : � ∂2B ∂ � R∂ � R � (0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E10) The matrix L′ acts in the space of the perturbative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' It is symmetric and contains two terms, the harmonic and anharmonic contribution L′ = L′ harm + L′ anh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E11) The harmonic part L′ harm is diagonal in our basis L′ harm · � ��� �R (1)(ω) �a′(1)(ω) �b′(1)(ω) � ��� = − � ���� (2) D· 0 0 0 (4) ω−2 : 0 0 0 (4) ω+2 : � ���� � ��� �R (1)(ω) �a′(1)(ω) �b′(1)(ω) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E12) The matrix L′ harm depends only on the equilibrium auxiliary frequencies of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We introduced a four indices tensor (4) ω± 2 αβµν = (ω± αβ)2Sαβµν, (E13) with S defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E3) and ω± µν = ωµ ± ων.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E14) The RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E12) should be read as a standard matrix-vector product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The matrix element contains also information on how to contract the indices, the operation · is defined in general as the contraction of the last and first index of two tensors A · B = 3N � µ=1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µBµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E15) and : is defined as C : D = 3N � µν=1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='µνDµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E16) For example the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E12) is − (2) D · �R (1)(ω) = − 3N � ν=1 (2) D µν �R(1)(ω)ν, (E17) 23 and returns a tensor of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The same holds for the other lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As an example, consider − (4) ω− 2 : �a′(1)(ω) = − 3N � µν=1 (ω− µν)2�a′(1)(ω)µν, (E18) The application of L′ anh gives L′ anh · � ��� �R (1)(ω) �a′(1)(ω) �b′(1)(ω) � ��� = � ������� − � ∂V ∂ � R � (1) (4) X− : � ∂2V ∂ � R∂ � R � (1) − (4) X+ : � ∂2V ∂ � R∂ � R � (1) � ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E19) Writing the perturbed averages of V(R) in terms of the scattering tensors, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D9), and using the change of variables of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E5), it is trivial to prove that in the new basis L′ anh is symmetric and has the following form L′ anh = � ������ 0 (3) D : (4) X− − (3) D : (4) X+ (4) X− : (3) D − (4) X− : (4) D : (4) X− (4) X− : (4) D : (4) X+ − (4) X+ : (3) D (4) X+ : (4) D : (4) X− − (4) X+ : (4) D : (4) X+ � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E20) Again, the matrix contains the information on how to contract the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This term contains information on the anharmonicity of the system through the third and fourth phonon scattering tensors, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (D7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Now that we have the linearized equations of motion, we present the general response function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To do this we need the correction of a position-dependent observable A(R) in the new basis Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E5) ⟨A⟩(1) (ω) = 3N � α=1 ∂ ⟨A⟩(0) ∂ �Rα �R(1)(ω)α + 3N � αβ=1 ∂ ⟨A⟩(0) ∂�a′(0) αβ �a′(1)(ω)αβ + 3N � αβ=1 ∂ ⟨A⟩(0) ∂�b′(0) αβ �b′(1)(ω)αβ = = 3N � α=1 � ∂A ∂ ˜Rα � (0) �R(1)(ω)α − 3N � αβ=1 X− αβ � ∂2A ∂ �Rα∂ �Rβ � (0) ˜a′(1)(ω)αβ + 3N � αβ=1 X+ αβ � ∂2A ∂ �Rα �Rβ � (0) ˜b′(1)(ω)αβ, (E21) The previous expression can be demonstrated using the change of variable definition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E5), the chain rule and the following relations in the original basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the one used in Appendix D) ∂ ⟨A⟩(0) ∂ �R(0) α = � ∂A ∂ ˜Rα � (0) , (E22a) ∂ ⟨A⟩(0) ∂�α(0) αβ = − 1 2�α(0) αα�α(0) ββ � ∂2A ∂ �Rα∂ �Rβ � (0) , (E22b) ∂ ⟨A⟩(0) ∂ �β(0) αβ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E22c) The derivative with respect to �α(0) αβ is obtained using the formalism of [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We define the response vector r′ similarly to p′ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E10)) r′† = �� ∂A ∂ � R � (0) , − (4) X− : � ∂2A ∂ � R∂ � R � (0) , (4) X+ : � ∂2A ∂ � R∂ � R � (0) � (E23) so from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E21) we can extract the response formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (52)) ⟨A⟩(1) (ω) V(ω) = 1 V(ω)r′ · � ��� �R (1)(ω) �a′(1)(ω) �b′(1)(ω) � ��� = r′ · � L′ + ω2�−1 · p′ = χ(ω)A,B (E24) where ⟨A⟩(1) (ω) is expressed as a scalar product in the space of the perturbative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This is the expression of χ(ω)A,B implemented in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 24 Appendix F: Lanczos algorithm In this Appendix, we discuss the Lanczos implementation [1, 72] of the general response function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Both for infrared and Raman calculations, we can always work with p′ = r′ setting A = B (see Eqs (E10) (E23)) so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E24) becomes χ(ω)A,A = (p′ · p′)p′ · � L′ + ω2�−1 · p′ (F1) where we normalize the vector p′ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E10)) p′ = p′ √p′ · p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F2) To get in one shot for all values of ω the response formula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F1), we modified the Lanczos algorithm presented in [1] exploiting that L′ = L′†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This algorithm allows to find a basis in which L′ is tridiagonal P ′−1 · L′ · P ′ = T ′, (F3) where T ′ has the following form T ′ = � ����������� t1 r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 0 r1 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' rn−1 0 rn−1 tn � ����������� (F4) where n is the size of L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The change of basis matrix P ′ is P ′ = � p′ 1 p′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' p′ n � (F5) and it is unitary P ′−1 = P ′†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F6) The coefficients of T ′ can be found following this iterative procedure [1, 72] tk =p′ k · L′ · p′ k (F7a) rkp′ k+1 =vk = (L′ − tk) · p′ k − rk−1p′ k−1 (F7b) rk =√vk · vk (F7c) p′ k+1 =vk/rk (F7d) with the initial vector equal to the normalized perturbation vector, p′ 1 = p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' This procedure ends when either p′ k is a linear combination of the previous vectors or p′ k · p′ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Unless the system is perfectly harmonic, this condition is usually never reached in practical runs, and the algorithm is truncated after a maximum number of steps Nsteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' After we build the change of variables matrix P ′ we can use it in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F1) χ(ω)A,A = (p′ · p′)p′ · P ′ · � P ′−1 · � L′ + ω2�−1 · P ′� P ′−1p′ = (p′ · p′)p′ · P ′ · � T ′ + ω2�−1 · P ′−1p′ (F8) then noting that P ′−1 · p′ = P ′† · p′ = � ������� 1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � ������� (F9) 25 we get that the response function is given by χ(ω)A,A = (p′ · p′) � T ′ + ω2�−1 11 (F10) where � T ′ + ω2�−1 11 can be written as a continuous fraction using the coefficients obtained up to Nsteps � T ′ + ω2�−1 11 = 1 ω2 + t1 − r2 1 ω2+t2− r2 2 ω2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F11) At each Lanczos step, we have to apply L′ to a given vector w in the space of the perturbed free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As showed in Appendix E L′ contains two terms L′ · w = L′ harm · w + L′ anh · w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (F12) The application of the harmonic part is done using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E12), while the anharmonic part is done using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E19) applying a reweighting procedure to compute the perturbed average as explained in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix G: Symbolic inversion In this Appendix we describe the symbolic inversion of a symmetric square super-tensor with this form L = � � A C CT B � � (G1) where A, B, C, C† are invertible tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using Gaussian reduction we get the inverse L−1 = � � D−1 −D−1 · C · B−1 −B−1 · C† · D−1 B−1 + B−1 · C† · D−1C · B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � � (G2) where D = A − CB−1C†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' It is trivial to check that LL−1 = L−1L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' For what follows we need C = 1 so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G2) becomes L−1 = � �A−1 − A−1 · (1 − A · B)−1 (1 − B · A)−1 (1 − A · B)−1 −A · (1 − B · A)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � � (G3) Again, for our purposes (see next Appendix H), we need to find a formula for the sum of entries of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Summing the coefficients of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G3) we get L−1 11 + L−1 21 + L−1 12 + L−1 22 =A−1 − A−1 · (1 − A · B)−1 + (1 − B · A)−1 + (1 − A · B)−1 − A · (1 − B · A)−1 =A−1 · [(1 − A · B) − (1 − A)] · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 =A−1 · [A · (1 − B)] · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 =(1 − B) · (1 − A · B)−1 + (1 − A) · (1 − B · A)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G4) Now we set A = 1 + � A and B = 1 + � B so we have that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G4) is � B · � � A + � B + � A · � B �−1 + � A · � � A + � B + � B · � A �−1 = � 1 + � A · � B−1 + � A �−1 + � 1 + � B · � A−1 + � B �−1 = � � A−1 + � B−1 + 1 �−1 � A + � � B−1 + � A−1 + 1 �−1 � B = � 1 + � B−1 + � A−1�−1 ( � A−1 + � B−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G5) We will use this formula in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 26 Appendix H: Derivation of the interacting Green’s function The easiest way to get the interacting Green’s function is to use another change of variables in Eqs (E2) � ��a(1) µν (ω) �b(1) µν (ω) � � = −ℏ2nµν 2 � � 1 ωµων 1 1 ωµων −1 � � � ��α(1) µν (ω) �β(1) µν (ω) � � (H1) where nµν is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E6c) and {ω2 µ} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' As done in Appendix E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' we write Eqs (E2) in this new basis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='switching to second-order time-derivatives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(H2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='where we recognize the resonant and anti-resonant terms of the two-phonon propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω)µνσπ = δµσδνπ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ℏ [ωµ − ων] [nµ − nν] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4ωµων[(ωµ − ων)2 − ω2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' χ(0) + (ω)µνσπ = δµσδνπ ℏ [ωµ + ων] [1 + nµ + nν] 4ωµων[(ωµ + ων)2 − ω2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H3) The anharmonic vector in this basis is simply � ����� � ∂V ∂ ˜ R � (1) � ∂2V ∂ � R∂ � Rβ � (1) � ∂2V ∂ � R∂ � Rβ � (1) � ����� = � ����� 0 (3) D· (3) D· (3) D· (4) D : (4) D : (3) D· (4) D : (4) D : � ����� � ��� �R (1) �a(1) �b(1) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H4) In a compact form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the linearized equations of motion are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='L(ω) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(1)(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�a(1)(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�b(1)(ω) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='The correction to the average of an observable A is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='⟨A⟩(1) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ∂A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(1)(ω) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ∂2A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='R∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=': S : �a(1)(ω) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ∂2A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='∂ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='R � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=': S : ˜b(1)(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(H7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='where S is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So, following the procedure described in Appendix E, the response function is χA,B(ω) = r† · L(ω)−1 · p (H8) defining the response vector r as r† = �� ∂A ∂ � R � (0) , � ∂2A ∂ � R∂ � R � (0) , � ∂2A ∂ � R∂ � R � (0) � , (H9) 27 and the perturbation vector p as p† = �� ∂B ∂ � R � (0) , � ∂2B ∂ � R∂ � R � (0) , � ∂2B ∂ � R∂ � R � (0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H10) First we discuss the non-interacting case setting (3) D = 0 (4) D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using A and B as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66a) we get r† = � δ 0 0 � p† = � δ 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H11) with δ = δµ as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (67a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The free phonon propagator is G(0)(ω)µν = δµν ω2 − ω2µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H12) Then we chose A and B according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66b) so r† = � 0 S S � p† = � 0 S S � , (H13) where S is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The two-phonon free propagator is χ(0)(ω)µνσπ = − � χ(0) + (ω)µνσπ − χ(0) − (ω)µνσπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � (H14) We chose A and B according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66c) r† = � δ 0 0 � p† = � 0 S S � (H15) L(ω) is diagonal in the case (3) D = 0 (4) D = 0 so we get that the one-two phonon free propagator is zero Γ(0)(ω)µσπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H16) Now we derive the one-phonon interacting Green’s function ( (3) D ̸= 0 (4) D ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Following [1] we chose the observables A and B as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66a) G(ω) = � δ 0 0 � L(ω)−1 · � ��� δ 0 0 � ��� = � L(ω)−1� 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H17) We use Eq (G2) to get G(ω)−1 = G(0)(ω)−1 − � (3) D : (3) D : � � ����� � χ(0) − (ω) : (4) D �−1 − 1 −1 −1 − � χ(0) + (ω) : (4) D �−1 − 1 � ����� −1 � �� : (4) D −1 : (3) D : (4) D −1 : (3) D � �� (H18) now Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G5) comes in help since we just need the sum of the inverse tensor’s entries G(ω)−1 = G(0)(ω)−1 − (3) D : � 1 − χ(0)(ω) : (4) D �−1 : � χ(0)(ω) : (4) D � : (4) D −1 : (3) D = G(0)(ω)−1 − Π(ω) (H19) where Π(ω) coincides with the one presented in [1, 2, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 28 Now we derive the two phonons interacting Green’s function χ(ω) which is obtained by choosing A and B according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In this basis this means computing χ(ω) = � 0 S S � L(ω)−1 · � ��� 0 S S � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H20) The perturbation B chosen (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66b)) leads to the following linearized equations of motion (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H5)) L(ω) · � ��� �R (1)(ω) �a(1)(ω) �b(1)(ω) � ��� = � ��� 0 S S � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H21) Using the expression of L(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H6), we find that the first free parameter is related to the other two �R (1)(ω) = G(0)(ω) · (3) D : � �a(1)(ω) + �b(1)(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H22) So instead of having to invert the full L(ω) we reduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H20) to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(ω) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='S : S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D · G(0)(ω) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D · G(0)(ω) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D · G(0)(ω) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ (ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D · G(0)(ω) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�: S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=': S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='S : S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− Σ(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−Σ(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−Σ(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ (ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− Σ(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�: S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=': S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='= − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='S : S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='− (ω) : Σ(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='χ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ (ω) : Σ(ω)(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�: Σ(ω)−1 : S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=': Σ(ω)−1 : S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='(H23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='where we define the two phonon self-energy Σ(ω) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (78) Σ(ω) = (4) D + (3) D · G(0)(ω) · (3) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H24) Again we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G5) to get the two-phonon interacting Green’s function χ(ω) = S : � 1 − χ(0)(ω) : Σ(ω) �−1 : χ(0)(ω) : Σ(ω) : Σ−1(ω) : S = � 1 − χ(0)(ω) : Σ(ω) �−1 : χ(0)(ω) = χ(0)(ω) : � 1 − Σ(ω) : χ(0)(ω) �−1 (H25) proving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The last Green’s function to discuss is the one-two phonon Γ(ω) obtained setting A and B as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (66c) Γ(ω) = � ��� δ 0 0 � ��� · L(ω)−1 · � ��� 0 S S � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H26) 29 We simplify this inversion using again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Now �a(1)(ω) + �b(1)(ω) are found considering the reduced linear system extracted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H21) Σ(ω) : � �� + � χ(0) − (ω) : Σ(ω) �−1 − 1 −1 −1 − � χ(0) + (ω) : Σ(ω) �−1 − 1 � �� � �: �a(1)(ω) : �b(1)(ω) � � = � �S S � � � ��a(1)(ω) �b(1)(ω) � � = − � �� 1 − � χ(0) − (ω) : Σ(ω) �−1 1 1 1 + � χ(0) + (ω) : Σ(ω) �−1 � �� −1 � �: Σ−1(ω) : Σ−1(ω) � � (H27) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H27) we get the sum �a(1)(ω) + �b(1)(ω) �a(1)(ω) + �b(1)(ω) = − � S : S : � � ���� 1 − � χ(0) − (ω) : Σ(ω) �−1 1 1 1 + � χ(0) + (ω) : Σ(ω) �−1 � ���� −1 � ��� : S : Σ−1(ω) : S : Σ−1(ω) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H28) Again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (G5) comes in help so �a(1)(ω) + �b(1)(ω) = � 1 − χ(0)(ω) : Σ(ω) �−1 : χ(0)(ω) : Σ(ω) : Σ−1(ω) = χ(ω) (H29) Since the response vector r is just [δ 0 0] the one-two phonon Green’s function Γ(ω) is given by R(1)(ω) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H7)), so, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H22), we end up with Γ(ω) = R(1)(ω) = G(0)(ω) · (3) D : χ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H30) This proves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We discuss also the three phonon Green’s function obtained with A and B as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (85) A = δ �R(0) α δ �R(0) β δ �R(0) γ B = δ �R(0) α′ δ �R(0) β′ δ �R(0) γ′ (H31) In this case we have � ∂2A ∂ � R∂ � R � (0) = � ∂2B ∂ � R∂ � R � (0) = 0 (H32) and � ∂A ∂ �Rµ � (0) = δµα � �α(0)−1� βγ + δµβ � �α(0)−1� αγ + δµγ � �α(0)−1� αβ (H33) so only the first entries of r′ and p′ are non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The response calculation is formally identical to the one-phonon interacting Green’s function one Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Using, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (H8), we get the three phonon propagator χ3ph(ω) = 3N � µν=1 � δµα � �α(0)−1� βγ + δµβ � �α(0)−1� αγ + δµγ � �α(0)−1� αβ � G(ω)µν � δνα′ � �α(0)−1� β′γ′ + δνβ′ � �α(0)−1� α′γ′ + δνγ′ � �α(0)−1� α′β′ � (H34) The diagrammatic interpretation is straightforward once we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The three-phonon response is χ3ph(ω) =G(0)(t = 0−)βγG(0)(t = 0−)β′γ′G(ω)αα′ + permutations of (αβγ) and (α′β′γ′) separately (H35) This proves the diagrammatic expression of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 30 Appendix I: Scattering vertices In this Appendix, we present the diagrammatic expression of the scattering vertices in TDSCHA, Eqs (62) (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We consider first the three-phonon term Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (62) since the same holds for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We average the third-derivative of the BO potential on the equilibrium SCHA distribution �ρ(0)(R) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B7)) (3) D ijk = � dR�ρ(0)(R)∂3V (BO)(R) ∂ �Ri∂ �Rj∂ �Rk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I1) Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I1) we perform the change of variables �ua = �Ra − �R(0) a and we expand in u (3) D ijk = � du�ρ(0)(u) �+∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.an (3+n) D(0) ijka1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.an�ua1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�uan � , (I2) where (n) D(0) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Note that (n) D(0) differs in general from (n) d , see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (34), since the minimum of the Born-Oppenheimer potential RBO does not coincide with the SCHA centroid R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Only even terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I2) are non-zero (3) D ijk = +∞ � n=0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n (3+2n) D(0) ijka1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n ⟨�ua1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='�ua2n⟩(0) = +∞ � n=0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n (3+2n) D(0) ijka1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n � P P �� �α(0)−1� a1a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � �α(0)−1� a2n−1a2n � = +∞ � n=0 1 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n (3+2n) D(0) ijka1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.a2n � �α(0)−1� a1a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � �α(0)−1� a2n−1a2n (I3) where P denotes the permutations of the indices according to the Wick theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the last line, we use the symmetry properties of the anharmonic vertices and the fact that the number of contractions for a 2n multivariate Gaussian expectation value is (2n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=', where !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' is the double factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In polarization the final result is (3) D µνφ = +∞ � n=0 (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n (3+2n) D(0) µνφα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n G(0)(t = 0−)α1α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='G(0)(t = 0−)α2n−1α2n � �� � n , (I4) where we use �α(0)−1 = � δ � Rδ � R � (0), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (B8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The same holds for the fourth-order scattering vertex (4) D µνφψ = +∞ � n=0 (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n (4+2n) D(0) µνφψα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.α2n G(0)(t = 0−)α1α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='G(0)(t = 0−)α2n−1α2n � �� � n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I5) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (I5) give a diagrammatic expression for the TDSCHA scattering vertices, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix J: Momentum Green’s function In this Appendix, we discuss the momentum Green’s function using the many-body formalism for bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The interacting Green’s function with imaginary time τ ∈ [−β, +β] (β−1 = kbT with kb the Boltzmann constant) is defined as GAB(τ) = − � Tτ � ˆS(β, 0) ˆA(τ) ˆB(0) �� 0 (J1) where only the connected diagrams are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The average ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='⟩0 is performed on the harmonic system defined by ˆHharm = 3N � µ=1 ℏΩµ � ˆa† µˆaµ + 1 2 � , (J2) 31 where {Ω2 µ} are the harmonic frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the poles of the harmonic propagator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The scattering matrix is ˆS(τ) = ˆS(τ, 0) = Tτe− � τ 0 dτ ′ ˆ Hanh(τ ′), (J3) here ˆHanh(τ) is the anharmonic part of the BO energy surface in the interacting picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The Matsubara transform is GAB(iΩn) = 1 2 � +β −β dτeiΩnτGAB(τ) (J4) with Ωn = 2πn β with n integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' First, we define the harmonic (non-interacting) Green’s function for position and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the harmonic polarization basis we have δ ˆ�R(τ)µ = ˆ�R(τ)µ − �Rµ = � ℏ 2Ωµ � ˆa(τ)µ + ˆa†(τ)µ � ˆ�P(τ)µ = −i � ℏΩµ 2 � ˆa(τ)µ − ˆa†(τ)µ � (J5) The Green’s functions in Matsubara frequencies are G(0)RR µν (iΩn) = δµν ℏ2 (iΩn)2 − (ℏΩµ)2 (J6a) G(0)P P µν (iΩn) = δµνΩ2 µG(0)RR µν (iΩn) (J6b) G(0)P R µν (iΩn) = iδµνℏ iΩn (iΩn)2 − (ℏΩµ)2 = i ℏ(iΩn)G(0)RR µν (iΩn) (J6c) G(0)RP µν (iΩn) = G(0)P R µν (−iΩn) = − i ℏ(iΩn)G(0)RR µν (iΩn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (J6d) Note that the analytical continuation of G(0)RR µν (iΩn) gives G(0)RR µν (iΩn → ℏω + i0+) = δµν (ω + i0+)2 − Ω2µ (J7) which coincides with our definition of harmonic free propagators, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The interacting momentum Green’s function is GP P µν (τ) = − � Tτ � ˆS(β, 0) �P(τ)µ �Pν(0) �� 0 = G(0)P P µν (τ) − � Tτ � ˆSanh(β, 0) �P(τ)µ �Pν(0) �� 0 (J8) where ˆSanh(β, 0) = ˆS(β, 0) − ˆ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The anharmonic correction is proportional to terms like � β 0 dτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. � β 0 dτm � Tτ �ˆ�P(τ)µ ˆ�R(τ1)α1 ˆ�R(τ1)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='ˆ�R(τn)αm ˆ�P ν(0) �� 0 (J9) where all the indices, except for µν, will be contracted with anharmonic vertices contained in the full BO energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (J9) is computed using the Wick theorem and contains terms that have the following form � β 0 dτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. � β 0 dτm � Tτ �ˆ�P(τ)µ ˆ�R(τ1)α1 �� 0 � Tτ �ˆ�R(τ1)α2 ˆ�R(τ2)α3 �� 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' � Tτ �ˆ�R(τn)αm ˆ�P(0)µ �� 0 = � β 0 dτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='. � β 0 dτmG(0)P R µα1(τ − τ1)G(0)RR α2α3(τ1 − τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='G(0)RP αmν(τm) (J10) When doing the contraction of the momentum variables we use G(0)P R µν (τ) = G(0)RP µν (−τ) and we take into account the multiplicity of the diagrams which cancels the n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' coming from the scattering matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Knowing that the Matsubara frequencies are conserved in all the diagrams and the relation between G(0)RP/P R µν (iΩn) and G(0)RR µν (iΩn) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (J6)), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (J10) becomes simply proportional to the anharmonic correction of the one-phonon Green’s function G(0)P R µα1(iΩn)π(iΩn)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.φm−1G(0)RP φmν(iΩn) =(iΩn)2G(0)RR µα1(iΩn)π(iΩn)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.αm−1G(0)RR αmν(iΩn) =(iΩn)2 � GRR µν (iΩn) − G(0)RR µν (iΩn) � (J11) 32 where π(iΩn)α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='.αm−1 is the Matsubara transform of the terms that contain only products of G(0)RR αiαj(τi − τj) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (J10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' In the end, using Eqs (J6c) (J6d), we get the following result for the interacting momentum Green’s function GP P µν (iΩn) = G(0)P P µν (iΩn) + (iΩn)2 ℏ2 � GRR µν (iΩn) − G(0)RR µν (iΩn) � = ω2 µG(0)RR µν (iΩn) + (iΩn)2 ℏ2 � GRR µν (iΩn) − G(0)RR µν (iΩn) � = −δµν + (iΩn)2 ℏ2 GRR µν (iΩn) (J12) Performing the analytical continuation iΩn → ℏω + i0+ and using the TDSCHA one-phonon Green’s function, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (74), we prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Appendix K: Prepare IR/Raman spectra calculation To compute IR spectra we need Eqs (90) (91) in polarization basis Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The first component of the re- sponse/perturbation vector is the one phonon vertex and contains equilibrium averages of the effective charges Zµ,aα = �∂pµ(R) ∂Raα � (0) = ⟨Z∗(R)µ,aα⟩(0) , (K1) where µ indicates the direction of the electric field, Ra,α is the position of atom a along the α coordinate and Z∗(R) is the effective charges tensor for a given configuration R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The second and third components of the response/perturbation vector contain the two-phonon vertex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' first derivatives of the effective charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Integration by parts leads to Zµ,aα,bβ = � ∂2pµ(R) ∂Raα∂Rbβ � (0) = N � c=1 3 � γ=1 α(0) bβ,cγ � δR(0) cγ � Z∗(R)µ,aα − Z∗(R(0))µ,aα �� (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K2) We subtract the equilibrium effective charges to reduce the noise in the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To compute Raman spectra we need Eqs (93) (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The first component of the response/perturbation vector contains equilibrium averages of the Raman tensor which give one-phonon processes Ξµ,ν,aα = �∂α(R)µν ∂Raα � (0) = ⟨Ξ(R)µν,aα⟩(0) , (K3) where µ ν indicates the photon polarization and Ξ(R) is the Raman tensor for a configurations R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The two phonon channel depends on the Raman tensor first derivatives and using integration by parts we have Ξµ,ν,aα,bβ = � ∂2α(R)µν ∂Raα∂Rbβ � (0) = N � c=1 3 � γ=1 α(0) bβ,cγ � δR(0) cγ � Ξ(R)µ,ν,aα − Ξ(R(0))µ,ν,aα �� (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K4) So to prepare the response and perturbation vector r and p we can use a stochastic approach as in [45] since all the averages have to be done on the equilibrium ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' We can enforce symmetries both for effective charges/Raman tensors and for their second-older counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' To symmetrize Z we note that the dipole p is related to the effective charge pµ = N � a=1 3 � α=1 Zµ,aαuaα (K5) where uaα is a displacement of atom a in the direction α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' If we apply a symmetry on u (defined in the supercell), the dipole will change according to the symmetry σ (3 × 3 unitary matrix) 3 � β=1 σµνpν = N � ab=1 3 � αβ=1 Zµ,aαSσ aα,bβubβ (K6) 33 where Sσ (3N × 3N matrix) is the symmetry operation associated with σ in the supercell Sσ aα,bβ = σαβδaσ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K7) j = σ(i) indicates that the symmetry σ maps i into j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' So using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K6) we get Zµ′,aα′ = 1 Ns Ns � σ=1 3 � µ,β=1 � σ† µ′µZµ,σ(a)ασαα′ � (K8) where Ns is the number of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' The symmetries for the second-order dipole moment are extracted noting that Pµ = N � ab=1 3 � αβ=1 Zµ,aα,bβuaαubβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K9) Since we know how the effective charges transform under a symmetry operation we can symmetrize Z Zν′,aα′,bβ′ = 1 Ns Ns � σ=1 3 � ν=1 3 � αβ=1 � σ† ν′νZν,σ(a)α,σ(b)βσαα′σββ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K10) We do the same for the Raman tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Similarly to what we do before, the polarizability α is related to the Raman tensors αµ,ν = N � a=1 3 � α=1 Ξµ,ν,aαua,α, αµ,ν = N � ab=1 3 � αβ=1 Ξµ,ν,aα,bβua,αub,β (K11) and we end up with the rules to symmetrize the averages of Raman-tensors Ξχ,φ,mµ′ = 1 Ns Ns � σ=1 � � 3 � αβ=1 3 � ν′=1 σχασφβΞα,β,σ(m),ν′σν′µ′ � � , (K12a) Ξχ,φ,pπ,rρ = 1 Ns Ns � σ=1 � � 3 � αβ=1 3 � µν=1 σχασφβΞα,β,σ(p),µ,σ(r),νσνρσµπ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' (K12b) [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Time-dependent self- consistent harmonic approximation: Anharmonic nuclear quantum dynamics and time correlation functions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 103, 104305 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lihm and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Park, Gaussian time-dependent variational principle for the finite-temperature anhar- monic lattice dynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Research 3, L032017 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rasetti, Raman spectra of crystals, Nature 127, 626 (1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fermi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rasetti, ¨Uber den ramaneffekt des stein- salzes, Zeitschrift f¨ur Physik 71, 689 (1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Goncharov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gregoryanz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hem- ley, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' kwang Mao, Spectroscopic studies of the vibrational and electronic properties of solid hydrogen to 285 gpa, Proceedings of the National Academy of Sciences 98, 14234 (2001), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='201528198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hautier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Jain, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Moore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ong, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ceder, Novel mixed polyanions lithium-ion battery cathode materials predicted by high-throughput ab initio computations, Journal of Materials Chemistry 21, 17147 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lilia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hennig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hirschfeld, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Profeta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sanna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Zurek, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pickett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Amsler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Dias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Eremets, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Heil, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hemley, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pier- leoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kolmogorov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rybin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Novoselov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Anisimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Oganov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pickard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Arita, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pellegrini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Requist, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gross, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Margine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Quan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hire, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fanfarillo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Stewart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hamlin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Stanev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gonnelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Piatti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Romanin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Daghero, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Valenti, The 2021 room-temperature superconductiv- ity roadmap, Journal of Physics: Condensed Matter 34, 183002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mounet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gibertini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Schwaller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Campi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Merkys, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Marrazzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sohier, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Castelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ce- pellotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pizzi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Marzari, Two-dimensional ma- 34 terials from high-throughput computational exfoliation of experimentally known compounds, Nature Nanotech- nology 13, 246 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pickard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nelson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Needs, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ma, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Quantum hydrogen-bond symmetrization in the super- conducting hydrogen sulfide system, Nature 532, 81 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [10] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Belli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sanna, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Koretsune, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tadano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Arita, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Flores-Livas, Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride, Nature 578, 66 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cherubini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, The mi- croscopic origin of the anomalous isotopic properties of ice relies on the strong quantum anharmonic regime of atomic vibration, The Journal of Chemical Physics 155, 184502 (2021), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='0062689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Pres- sure and stress tensor of complex anharmonic crystals within the stochastic self-consistent harmonic approxi- mation, Physical Review B 98, 024106 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Casula, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nakano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sorella, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Quantum phase diagram of high-pressure hy- drogen (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Drummond, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monserrat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lloyd-Williams, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R´ıos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pickard, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Needs, Quantum monte carlo study of the phase diagram of solid molecular hydrogen at extreme pressures, Nature Communications 6, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1038/ncomms8794 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, Anharmonicity and doping melt the charge density wave in single-layer TiSe2, Nano Letters 20, 4809 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Leroux, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Le Tacon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Souliou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gar- barino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cario, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bosak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rodi`ere, Strong anharmonicity induces quantum melt- ing of charge density wave in 2h − nbse2 under pressure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 92, 140303 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Quantum enhancement of charge density wave in NbS2 in the two-dimensional limit, Nano Letters 19, 3098 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Diego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Said, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mahatha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mona- celli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rossnagel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Blanco-Canosa, van der waals driven anharmonic melt- ing of the 3d charge density wave in VSe2, Nature Com- munications 12, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1038/s41467-020-20829-2 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Black metal hydrogen above 360 gpa driven by proton quantum fluctuations, Nature Physics 17, 63 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Loubeyre, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Occelli, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Dumas, Synchrotron in- frared spectroscopic evidence of the probable transition to metal hydrogen, Nature 577, 631 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bernasconi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Silvestrelli, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Parrinello, Ab initio infrared absorption study of the hydrogen-bond symmetrization in ice, Physical Review Letters 81, 1235 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Capitani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Langerome, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Brubach, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Roy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Drozdov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Eremets, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nicol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Carbotte, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Timusk, Spectroscopic evidence of a new energy scale for superconductivity in h3s, Nature Physics 13, 859 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ranalli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Verdi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kresse, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Franchini, Temperature-dependent an- harmonic phonons in quantum paraelectric ktao 3 by first principles and machine-learned force fields, arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='12036 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Verdi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ranalli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Franchini, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kresse, Quantum paraelectricity and structural phase transitions in strontium titanate beyond density-functional theory, arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='09616 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Juraschek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fechner, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Spaldin, Ultrafast structure switching through nonlinear phononics, Physi- cal Review Letters 118, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1103/physrevlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='054101 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Subedi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cavalleri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Georges, Theory of non- linear phononics for coherent light control of solids, Phys- ical Review B 89, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='220301 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tobey, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Dean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Itatani, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tomioka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tokura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Schoenlein, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cavalleri, Control of the electronic phase of a manganite by mode-selective vibrational excitation, Nature 449, 72 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Johnson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Knighton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Johnson, Distinguishing nonlinear terahertz excitation pathways with two-dimensional spectroscopy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 122, 073901 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cao and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Voth, The formulation of quantum sta- tistical mechanics based on the feynman path centroid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' algorithms for centroid molecular dynam- ics, The Journal of Chemical Physics 101, 6168 (1994), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='468399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Poulsen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nyman, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rossky, Practical evaluation of condensed phase quantum correlation func- tions: A feynman–kleinert variational linearized path in- tegral method, The Journal of Chemical Physics 119, 12179 (2003), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1626631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [31] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hele, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Willatt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Muolo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Althorpe, Boltzmann-conserving classical dynamics in quantum time-correlation functions: “matsubara dy- namics”, The Journal of Chemical Physics 142, 134103 (2015), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4916311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ceotto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Di Liberto, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Conte, Semiclassical “divide-and-conquer” method for spectroscopic calcula- tions of high dimensional molecular systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 119, 010401 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [33] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pl´e, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Huppert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Finocchi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Depondt, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonella, Anharmonic spectral features via trajectory- based quantum dynamics: A perturbative analysis of the interplay between dynamics and sampling, The Journal of Chemical Physics 155, 104108 (2021), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='0056824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Beutier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Borgis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Vuilleumier, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonella, Computing thermal wigner densities with the phase in- tegration method, The Journal of Chemical Physics 141, 084102 (2014), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4892597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pl´e, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Huppert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Finocchi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Depondt, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonella, Sampling the thermal wigner den- sity via a generalized langevin dynamics, The Journal of Chemical Physics 151, 114114 (2019), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='5099246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Poulsen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nyman, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rossky, Practical evaluation of condensed phase quantum correlation func- tions: A feynman–kleinert variational linearized path in- tegral method, The Journal of Chemical Physics 119, 12179 (2003), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1626631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [37] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Shi and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Geva, Semiclassical theory of vibra- tional energy relaxation in the condensed phase, The 35 Journal of Physical Chemistry A 107, 9059 (2003), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1021/jp030497+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [38] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Wigner, On the quantum correction for thermody- namic equilibrium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 40, 749 (1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tadano and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tsuneyuki, First-principles lattice dy- namics method for strongly anharmonic crystals, Jour- nal of the Physical Society of Japan 87, 041015 (2018), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='7566/JPSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='041015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tadano and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tsuneyuki, Self-consistent phonon cal- culations of lattice dynamical properties in cubic srtio3 with first-principles anharmonic force constants, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 92, 054301 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Imre, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ¨Ozizmir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rosenbaum, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Zweifel, Wigner method in quantum statistical mechanics, Jour- nal of Mathematical Physics 8, 1097 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Brogaard, Wigner function formalism in quantum me- chanics, Signature (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Novaes, Wigner and husimi functions in the double- well potential, Journal of Optics B: Quantum and Semi- classical Optics 5, S342 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Poulsen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Svensson, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nyman, Dy- namics of gaussian wigner functions derived from a time- dependent variational principle, AIP Advances 7, 115018 (2017), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='5004757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [45] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monacelli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cherubini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, The stochastic self-consistent har- monic approximation: calculating vibrational properties of materials with full quantum and anharmonic effects, Journal of Physics: Condensed Matter 33, 363001 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [46] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Georgescu and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mandelshtam, Self-consistent phonons revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' the role of thermal versus quantum fluctuations on structural transitions in large lennard- jones clusters, The Journal of Chemical Physics 137, 144106 (2012), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4754819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Brown, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Georgescu, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mandelsh- tam, Self-consistent phonons revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' a general and efficient method for computing free energies and vibrational spectra of molecules and clusters, The Journal of Chemical Physics 138, 044317 (2013), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='4788977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [48] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Monteferrante, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonella, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cic- cotti, Linearized symmetrized quantum time correlation functions calculation via phase pre- averaging, Molecular Physics 109, 3015 (2011), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1080/00268976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='619506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bianco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Errea, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Paulatto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Second-order structural phase transitions, free energy curvature, and temperature-dependent anhar- monic phonons in the self-consistent harmonic approx- imation: Theory and stochastic implementation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 96, 014111 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Maradudin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fein, Scattering of neutrons by an anharmonic crystal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 128, 2589 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [51] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mahan, Many-Particle Physics (Springer US, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [52] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tadano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gohda, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tsuneyuki, Anharmonic force constants extracted from first-principles molecu- lar dynamics: applications to heat transfer simulations, Journal of Physics: Condensed Matter 26, 225402 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lazzeri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Anharmonic phonon frequency shift in mgb 2, Physical Review B 68, 220509 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [54] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' G¨otze and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Michel, Elastic constants of nonionic anharmonic crystals, Zeitschrift f¨ur Physik A Hadrons and nuclei 217, 170 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [55] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Macheda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Barone, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Electron-phonon interaction and longitudinal-transverse phonon splitting in doped semiconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 129, 185902 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [56] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Macheda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sohier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Barone, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, Electron-phonon interaction and phonons in 2d doped semiconductors (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [57] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Oba, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tadano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Akashi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Tsuneyuki, First- principles study of phonon anharmonicity and negative thermal expansion in scf3, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Materials 3, 033601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [58] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Goldman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Horton, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Klein, An im- proved self-consistent phonon approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' 21, 1527 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [59] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Werthamer, Self-consistent phonon formulation of anharmonic lattice dynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 1, 572 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [60] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Juraschek and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Maehrlein, Sum-frequency ionic raman scattering, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 97, 174302 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Basini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Udina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pancaldi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Unikandanunni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonetti, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' benfatto, Terahertz ionic kerr effect (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [62] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' von Hoegen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mankowsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fechner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' F¨orst, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cavalleri, Probing the interatomic potential of solids with strong-field nonlinear phononics, Nature 555, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Deinzer and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Strauch, Two-phonon infrared absorp- tion spectra of germanium and silicon calculated from first principles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 69, 045205 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [64] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Silvestrelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bernasconi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Parrinello, Ab initio infrared spectrum of liquid water, Chemical Physics Letters 277, 478 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [65] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Windl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Karch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pavone, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sch¨utt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Strauch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Weber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hass, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rimai, Second-order raman spectra of sic: Experimental and theoretical results from ab initio phonon calculations, Physical Review B 49, 8764 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [66] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Windl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pavone, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Karch, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sch¨utt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Strauch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Giannozzi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Baroni, Second-order raman spectra of diamond from ab initio phonon calculations, Physical Review B 48, 3164 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [67] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Menahem, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Benshalom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Asher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Aharon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Korobko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Safran, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hellman, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Yaffe, The disorder origin of raman scattering in perovskites single crystals (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [68] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Miehlich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Savin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Stoll, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Preuss, Results obtained with the correlation energy density functionals of becke and lee, yang and parr, Chemical Physics Letters 157, 200 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [69] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Giannozzi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Baroni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bonini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Car, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cavazzoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ceresoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Chiarotti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cococ- cioni, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Dabo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Corso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' de Gironcoli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fabris, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Fratesi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gebauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gerstmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gougoussis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kokalj, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lazzeri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Martin-Samos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Marzari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mazzarello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Paolini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Pasquarello, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Paulatto, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sbraccia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Scandolo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sclauzero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Seitsonen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Smogunov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Umari, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Wentz- covitch, QUANTUM ESPRESSO: a modular and open- source software project for quantum simulations of mate- rials, Journal of Physics: Condensed Matter 21, 395502 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [70] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Giannozzi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Andreussi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Brumme, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Bunau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nardelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Calandra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Car, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cavazzoni, 36 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ceresoli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Cococcioni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Colonna, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Carnimeo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Corso, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' de Gironcoli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Delugas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' DiStasio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ferretti, A.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Jia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kawa- mura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Kokalj, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' K¨u¸c¨ukbenli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Lazzeri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Marsili, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Marzari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Mauri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='- V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Nguyen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' de-la Roza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Paulatto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Ponc´e, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rocca, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Sabatini, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Santra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Schlipf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Seitso- nen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Smogunov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Timrov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Thonhauser, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Umari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Vast, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Wu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Baroni, Advanced capabilities for materials modelling with quantum ESPRESSO, Journal of Physics: Condensed Matter 29, 465901 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [71] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Hamann, Optimized norm-conserving vanderbilt pseudopotentials, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' B 88, 085117 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' [72] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Rocca, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Gebauer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Saad, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content=' Baroni, Turbo charging time-dependent density-functional theory with lanczos chains, The Journal of Chemical Physics 128, 154105 (2008), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} +page_content='2899649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFAT4oBgHgl3EQfnR2x/content/2301.08628v1.pdf'} diff --git 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Denisenko1,a) and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Pandey2,b) 1Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna-141980, Moscow Region, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 2Graduate Engineer Trainee, Larsen & Toubro Limited, Faridabad, Haryana, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' a)iden@jinr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='ru b)rishav160999@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='com Abstract The space-time picture of hadron formation in high-energy collisions with nuclear targets is still poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The tests of hadron formation was suggested for the first stage of SPD running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' They will require measuring charged pion and proton spectra with the precision better than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' A research has been carried out to check feasibility of such studies at SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' In this work, 12C − 12C and 40Ca − 40Ca heavy ion collisions at center of mass energy of 11 GeV/nucleon were simulated using the SMASH event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Firstly, the generator-level events were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The distribution of track multiplicities and momentum distributions of different types of charged particles were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Secondly, the generated events passed through the full reconstruction using the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' At this stage particles were identified using dE/dx measurement and time-of-flight information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It allowed us to estimate charge track multiplicities in the tracking system and purities of charge particles spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The results on multiplicity are important to estimate occupancies in the tracking system, while the results on the pion and proton momentum spectra show that particle identification should be acceptable for validation of hadron formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This is the first study of moderate ion collisions for the SPD Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Keywords: Hadron formation effects, Heavy ion collision, SMASH, NICA-SPD, Rapidity, Charged track multiplicity, Particle physics event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='00997v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='ins-det] 3 Jan 2023 1 INTRODUCTION The SPD detector is primarily optimized to study spin dependent gluon structure of proton and deuteron using open charm production, charmonia production and prompt photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' At the same time, its physics program includes studies of various aspects of QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The work is devoted to studies of hadron formation in nuclear collisions proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Hadrons produced in hadron collisions emerge in the form of prehadrons, which interact with nucleons with reduced strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This suppression is poorly known and is described in model de- pendent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This suppression results in different spectra of final particles as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1 for rapidity distributions (in a similar way it affects the pT spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Naturally, these spectra can be used to study hadron formation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The required precision of such measurements is 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The aim of this work is to evaluate feasibility of such measurements with MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Here, ion collisions of 12C − 12C and 40Ca − 40Ca at √s = 11AGeV were generated using the SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) event generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Afterwards, the the full simulation and reconstruction was performed using the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 1: Rapidity spectra of protons and charged pions in 12C − 12C and 40Ca − 40Ca collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 2 12C + 12C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' s= 11 GeV 40Ca + 40Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' sN = 11 GeV 105 106 Protons Protons w/oformation w/oformation default default QDM QDM do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' mb 104 peut = 2 GeV/c - Peut = 2 GeV/c - Peut = 1 GeV/c Peut = 1 GeV/c 103 104 102 103 4 -3 -2 -1 0 1 2 3 4 4 3 -2 -1 0 1 2 3 4 y y 12C + 12C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' sN= 11 GeV 40Ca + 40Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' sNR = 11 GeV 104 105 do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' mb do/dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' mb 103 104 w/oformation w/oformation default default QDM QDM Peut = 2 GeV/c Peut = 2 GeV/c Pcut = 1 GeV/c pcut = 1 GeV/c 102 103 4 -3 -2 1 0 1 2 3 4 4 -3 -2 -1 0 1 2 3 4 y y2 NICA FACILITY The NICA (Nuclotron based Ion Collider fAcility) collider at Joint Institute for Nuclear Research in Dubna is is being built to provide beams for two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The first experiment, MPD (Multi Purpose Detector), will study properties of dense baryonic matter (matter present at extreme high density in QCD phase diagram) like Quark Gluon Plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The second experiment, SPD (Spin Physics Detector), is devoted to study of spin related phonomena and QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Once the NICA collider will be operational, scientists will be able to create a special state of matter in laboratory which existed for very short interval of time (˜20µ sec) just after the big bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This special state is called as QGP (Quark Gluon Plasma) and it filled the entire universe shortly after the big bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The main parts of NICA facility consists of two independent injector complex (injector for light ions, and injector for heavy ions-KRION 6T), Light Ion Linear Accelerator (LU20) for accelerating light ions like protons (H+), deutrons, and α-particles upto 5 MeV of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='E, then Heavy Ion Linear Accelerator (HILAC) to accelerate heavy ions upto Au to a maximum K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='E of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 MeV/n, then a Super Conducting (SC) Booster Synchrotron to create ultra high vacuum and to provide complete stripping of heavy ions, then a SC Heavy Ion Synchrotron Nuclotron to accelerate both light and heavy ions to required beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The accelerated beams will collide at two different locations where MPD detector and SPD detector are being built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The schematic view of NICA complex is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 2: Schematic view of NICA complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 3 SPD DETECTOR The Spin Physics Detector [2,3] is a 4π universal detector optimized to study spin-related phenomena via open charm, charmonia and promopt photons in the collisions of polarized p-p or d-d beams with √sNN up to 27 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' However, at first stage of NICA-SPD, the expected collision energy will be from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 up to 10 GeV, and later on after first upgrade, it is expected to reach upto 27 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The general layout depicting isometric projection of SPD setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The main parts involved in advanced tracking and particle identification capabilities have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (i) The beam pipe passes through the center of the detector, carries the accelerated beams of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (ii) The MicroMegas detector is to improves the momentum resolution and tracking efficiency of the tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (iii) The Straw Tracker (ST) detector is for the reconstruction of the primary and 3 BM@N Detector SPD Transport Channel HILAC Collider LU20 Booster MPD Nuclotronsecondary particle tracks and for determination of their momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (iv) The Time Of Flight (TOF) detector, is a part of Particle Identification (PID) system, and is used for identification of particles like π, k, and p with long trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (v) The magnet system shown by red color provides 1T of magnetic field along the beam axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This setup is limited to first stage of SPD operation, and will be considered only for the identification of stable charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Neutral particles, like n0, photons will be detected at later stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The main parts of SPD first stage have been explained in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' There is a possibility to have TOF system for the first stage studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 3: Layout of the SPD setup proposed for first stage at NICA-SPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1 CENTRAL TRACKER The innermost detector of SPD consists of a MicroMegas-based Central Tracker (MCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Its purpose is to identify the primary vertex coordinate and to improve momentum resolution and tracking efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It is based on MicroMegas (Micro Mesh Gaseous Structure) technology and detects charged particle by amplifying the charges produced due to ionization of the gas molecules present in detector volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' When an ionizing particle track passes through detector volume, it ionizes the gas molecules and creates few hundreds of e−-ion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Electrons are accelerated opposite to the direction of applied electric field of 600 V/cm in ionization gap, while ions are attracted towards cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' When the e− crosses micromesh, it faces intense electric field (> 30 KV/cm) and gains enough energy to ionize other gas molecules in its path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' During this process an avalanche of e−-ion pair is produced (1e− produces 104 e−-ion pairs) which is significant to create an electronic signal which is read out by readout electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 STRAW TRACKER ST is mainly for the reconstruction of primary and secondary particle tracks and measuring their momenta, but also participates in identification of π, K, and p on via energy deposit (dE/dx) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It consists of two major parts - barrel (covers radius from 270 to 850 mm) and two end-caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The barrel is divided into 8 modules enclosed in a carbon fiber capsule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Each module has 30 double layers of straw tubes (dia 1cm) which runs parallel (long straw tubes) and perpendicular 4 Straw tracker Magnet Range system MicroMegas Endcap RangesystemEndcap MicroMegas Beam-beamcounter Beam pipe Strawtracker Endcap zoomx4 Zero degree calorimeter(short straw tubes) to the beam axis and contains 1500 and 6000 parallel and perpendicular straw tubes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Straw tubes are made of polyethylene terephthalate and outer surface is coated with very thin layer of Cu and Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Carbon capsule is meant to protect the outer surface of these tubes from humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' One side and two opposite ends of capsule are provided with small holes where end plugs are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' FEE are connected to these end plugs to read the detector signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Any particle which passes through the long straws will send detector signal to both opposite ends while a particle passing through short straw will send detector signal to any one side of capsule where FEE is attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Thus, long straws will be read from two opposite ends while short straws will be read from one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The end-caps of ST are divided into 3 modules and each module has 4 hexadecimal cameras (U, V, X, Y) to record the four coordinates of any physical quantity like four-momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The FEE to be used can be similar to the one used at NA64 experiment (for the search of dark matter), or DUNE experiment (to detect and study properties of neutrino).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='3 TIME OF FLIGHT DETECTOR TOF detector is the part of PID system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Similar to ST, the TOF provides identification of π, k, and p by measuring their flight time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The energy loss data registered by ST can be used together with the data from TOF for correct identification of particle tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The TOF distinguishes charged particles (mainly π and k) in the momentum range up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The major parts of TOF comprises of a barrel and two end-caps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' For the first stage of NICA-SPD, two different designs of TOF has been suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' First one is TOF based on multigap timing Resistive Plate Chambers (mRPC), which will consist 220 rectangular plate chambers (160 for the barrel and 30 each for end-caps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Second one is based on Plastic Scintillator Tiles and will comprise 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1K small scintillator tiles (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4K for barrel and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4K for each end-caps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Scintillator has a property of emitting light in visible region when an ionizing radiation passes through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' So, in this design when a particle passes through TOF, scintillated photons are produced which are detected by four Si Photo Multipliers (SiPMs) present at each sensor board attached at two extreme ends of scintillator tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 4 EVENT GENERATION 12C −12C and 40Ca−40Ca heavy ion collisions at √s = 11 AGeV with maximum impact parameter set to 8 fm for C-C and 11 fm for Ca-Ca were simulated using SMASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The fermi motion was assumed to be “frozen” and 100K events were generated for each heavy ion collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The SMASH input file for C-C collision is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' *********** SMASH INPUT ************ config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='yaml file for C-C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Logging: default: INFO General: Modus: Collider Time_Step_Mode: Fixed Delta_Time: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1 End_Time: 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='0 Randomseed: 1 Nevents: 100000 5 Output: Output_Interval: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='0 Particles: Format: ["Oscar2013"] Modi: Collider: Projectile: Particles: {2212: 6, 2112: 6} #C-12 Target: Particles: {2212: 6, 2112: 6} #C-12 Sqrtsnn: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='0 Impact: Sample: "quadratic" Range: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='0, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='0] Fermi_Motion: "frozen" ************************************ Multiplicity of generated charged particles for C − C and Ca − Ca collisions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The peaks at 12 for 12C+12C collisions and at 40 for 40Ca+40Ca collisions correspond to events where no interaction occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The rapidity distributions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The spectra obtained from SMASH output show qualitative agreement with the ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Peaks for protons correspond to particles moving close to the initial beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Moreover, fractions of different particle types can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It can be seen that for |y| < 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' within the acceptance of the detector) charge particles are dominated by pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Apart from p±, π±, & K±, marginal numbers of sigmas, cascades, and omegas were also generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The PID efficiency depends on the particle momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The momentum spectra for protons, pions and kaons are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6 in the midrapidity region (|y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 for which theoretical predictions has been given) Most of the pions have momentum below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 GeV and protons - below 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It means that types of these particles should be well resolved by dE/dx measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' When studying pion or proton spectra, there is high probability of kaon/pion misidentification, but fraction of such events is strongly suppressed by small initial kaon numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6 (a) (b) Figure 4: Generator-level multiplicity of charged particles for 12C − 12C collision (a) and 40Ca − 40Ca collisions (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (a) (b) Figure 5: Rapidity distribution of charged particles in 12C − 12C (a) and 40Ca − 40Ca (b) collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 7 Total Multiplicity of Charged Particles, C-12 + C-12 104 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of events 103 102 10 0 10 20 30 40 50 60 70 80 90 100 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of charged particlesTotal Multiplicity of Charged Particles, Ca-40 + Ca-40 104 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of events 103 0 10 20 30 40 50 60 70 80 90 100 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of charged particlesRapidity distribution of charged particles, C-12 + C-12 protons 105 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='. pions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' kaons 104 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of charged particles 103 102 10 3 2 3 5 Rapidity of charaed particles (yRapidity distribution of charged particles, Ca-40 + Ca-40 106 protons pions 105 kaons No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' of charged particles 104 103 102 10 5 3 2 Y Rapidity of charged particles (y)(a) p distribution of p± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (b) p distribution of p± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (c) p distribution of π± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (d) p distribution of π± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (e) p distribution of k± in 12C − 12C collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (f) p distribution of k± in 40Ca − 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 6: Total momentum distribution of protons, pions, and kaons at generator level in 12C − 12C and 40Ca− 40Ca collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 8 Total momentum distribution of pions, C-12 + C-12 16000 14000 12000 pions 10000 8000 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6000 4000 2000 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 Total momentum of pions (p)Total momentum distribution of pions, Ca-40 + Ca-40 60000 50000 40000 ON 30000 20000 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 Total momentum of pions (p)Total momentum distribution of kaons, C-12 + C-12 1000 800 of kaons 600 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 400 200 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 Total momentum of kaons (p)Total momentum distribution of kaons, Ca-40 + Ca-40 4000 3500 3000 kaons 2500 2000 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 1500 1000 500 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 Total momentum of kaons (p)Total momentum distribution of protons, C-12 + C-12 1600 1400 1200 protons 1000 800 ON 600 400 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 Total momentum of protons (p)Total momentum distribution of protons, Ca-40 + Ca-40 6000 5000 of protons 4000 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 3000 2000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 Total momentum of protons (p)5 DETECTOR SIMULATION AND EVENT RECONSTRUCTION The detector simulation and reconstruction was performed with the SpdRoot framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' To read SMASH generated events the SpdRoot code was modified and additional C++ class was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' During the simulation stage the particles were transported through the detector geometrical model using Geant4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' At the reconstruction stage, Geant4 tracks and vertices were reconstructed and particle identification with dE/dx and time of flight measurements was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' For the PID three hypotheses were considered: pion, kaon and proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The reconstructed ionization energy losses and “measured” time of flight were used to construct conditional probabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' p(t|pid), where t is the measured time and pid is a particle type hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Out of 100K events generated by SMASH, first 1K events were considered for detector simulation due to slow data processing in SpdRoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6 ANALYSIS A physical analysis was performed using C++ codes and ROOT library based on SpdRoot output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' All tracks reconstructed in the detector with measured momentum were accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' For the particle type the one that gives the largest conditional probability is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Multiplicity, as well as kinematic distributions for pions, kaons and protons are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' For particle momentum spectra there are no notable differences between C − C and for Ca − Ca collisions, so only the first ones will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1 CHARGED TRACK MULTIPLICITY The SPD detector set-up is optimized for p − p and d − d collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Thus knowing charged track multiplicities for ion collisions is important to estimate CT and ST occupancies and feasibility of such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 7 shows the total multiplicity of charged particles reconstructed by the tracking system in 12C − 12C and 40Ca − 40Ca collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The numbers of reconstructed tracks are much lower compared to generator-level studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It is because the geometry of the tracking system is such that, tracks with polar angle, θ < 10◦ or > 170◦ do not hit the tracker and passes along the beam pipe itself, so such tracks are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Also, there were events without nuclei interactions which resulted in no track reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' So, to avoid a large peak at zero due to mentioned reasons, the X-axis count starts from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 PION MOMENTUM SPECTRUM (12C − 12C) The spectra of particles identified as pions separately by ionization losses and by TOF are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 8 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The spectra show resemblance with the generator plot of pion momentum distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Based pn MC-truth information backround from misidentification other charged particles (K±, p±, e±, & µ±) is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The obtained distribution for “pions identified as pions” only slightly deviates from distribution of all selected pion candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The estimated relative contamination of the pion spectra is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It can seen that purity above 90% can be obtained up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 GeV using either dE/dx or TOF measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 9 Figure 7: Charged track multiplicity reconstructed by in 12C − 12C (left), 40Ca − 40Ca (right) collisions (shown by red) and number of particles for which TOF information is available (shown by blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (a) Total momentum distribution of reconstructed charged particles identified as π± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (b) Total momentum distribution of reconstructed charged particles identified as π± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 8: Total momentum distribution of reconstructed π± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 9: Purity of the selected pion candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 10 Total multiplicity of charged particles passing through tracking system, C12-C12 60 TOF 50 ST 40 events 30 NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 20 10 一 10 20 30 40 50 60 70 80 90 10090 TOF 80 ST 70 events 60 50 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100Charged Particles ldentified as Pions by ST, C12-C12 350 Pions identified as pions Kaons identified as pions Protons identified as pions 300 Electrons identified as pions Muons identified as pions 250 Chargedparticlesidentifiedaspions 200 150 100 50 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 p(GeV/c)Charged Particles ldentified as Pions by TOF, C12-C12 Pions identified as pions 300 Kaons identified as pions Protons identified as pions 250 Electrons identified as pions Muons identified as pions Charged particles identified as pions 200 Counts 150 100 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 p(GeV/c)Pion spectra precision, C12-C12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 Counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 Precision recordedbyTOF Precision recorded by ST 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 p(GeV/c)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='3 KAON MOMENTUM SPECTRUM (12C − 12C) The kaon momentum spectrum was explicitly mentioned among observables to study hadron for- mation effects in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Nevertheless, kaon production may be interesting for the reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The obtained spectra of kaon candidates is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 10 separately for ionization losses and TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' First of all, the shown data lack statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Secondly, it can bee seen that there is a huge contamina- tion from misidentified pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' This is explained by very small fraction of generated kaons and the fact that probability to select misidentified particle is proportional to their number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The relative fraction of correctly identified kaons in shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (a) Total momentum distribution of reconstructed charged particles identified as K± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (b) Total momentum distribution of reconstructed charged particles identified as K± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 10: Total momentum distribution of reconstructed K± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 11: Purity of the selected kaon candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 11 Charged Particles ldentified as Kaons by ST, C12-C12 Kaons identified as kaons 60 Pions identified as kaons Protons identified as kaons Electrons identified as kaons 50 Muons identified as kaons Charged particles identified askaons 40 Counts 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 p(GeV/c)Charged Particles ldentified as Kaons by TOF, C12-C12 25 Kaons identified as kaons Pions identified as kaons Protons identifiedaskaons 20 Electrons identified as kaons Muons identified as kaons Charged particles identified as kaons 15 Counts 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 0 2 p(GeV/c)Kaon spectra precision, C12-C12 Precision recorded by TOF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='9 Precision recorded by ST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 unts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 Col 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 2 p(GeV/c)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 PROTON MOMENTUM SPECTRUM (12C − 12C) Finally, proton momentum spectra have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' In this study protons and antiprotons were considered together, but the fraction of produced antiprotons is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The proton candidate distributions and the contributions from misidentification are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The purity of the selected samples is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' It can be seen dE/dx measurements alone will not allow precise determination of proton spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The reasonably good results can be expected only in case of combined identification by ionization losses and TOF system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (a) Total momentum distribution of reconstructed charged particles identified as p± by ionization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' (b) Total momentum distribution of reconstructed charged particles identified as p± by TOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 12: Total momentum distribution of reconstructed p± candidates in 12C − 12C collision (Detector level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Figure 13: Purity of the selected proton candidates as a function of their momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 12 Charged Particles ldentified as Protons by ST, C12-C12 90 Protons identified as protons Kaons identified as protons 80 Pions identified as protons Electrons identified as protons Muons identified as protons 70 Charged particles identified as protons 60 Counts 50 40 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 0 5 p(GeV/c)Charged Particles ldentified as Protons by TOF, C12-C12 90 Protons identified as protons Kaons identified as protons 80 Pions identified as protons Electrons identified asprotons 70 Muons identified as protons Charged particles identified as protons 60 Counts 50 40 30 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 0 5 p(GeV/c)Proton spectra precision, C12-C12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='8 Counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='2 Precision recorded by TOF Precision recorded by ST 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='5 5 p(GeV/c)7 SUMMARY The goal of this work was to check the feasibility of hadron formation effects studies at the first stage of SPD operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' For this purpose an analysis of 12C − 12C and 40Ca − 40Ca collisions were performed at the generator level and then the full event reconstruction was done at detector level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The multiplicity distributions indicate that occupancies of tracking detectors should be checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Part of the events with high number of charged tracks may not be fully reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Particle identification with ionization losses and TOF was considered separately (for future dE/dx only or their combination can be expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' The purity of the measured charged pion distribution for both types of ion collisions using dE/dx only is rather good and meets mentioned before requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' In case of combination of information from ionization losses and time of flight system purity of proton distribution may be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Abramov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Aleshko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Baskov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Boos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Bunichev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Dalkarov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' El-Kholy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Galoyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Guskov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Kim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 52 (2021) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='6, 1044-1119 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='1134/S1063779621060022 [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='08477 [hep-ph]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' Abazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' [SPD proto], [arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content='00442 [hep-ex]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' [3] SPD TDR [unpublished].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfEvoZ/content/2301.00997v1.pdf'} diff --git a/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/2301.02413v1.pdf.txt b/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/2301.02413v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..656cff81b645f175cfa1dc9f2f9540fe889f343f --- /dev/null +++ b/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/2301.02413v1.pdf.txt @@ -0,0 +1,3062 @@ +Benchmarking Gaussian Basis Sets in +Quantum-Chemical Calculations of +Photoabsorption Spectra of Light Atomic +Clusters +Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,† +†Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, +India +‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, +Helmholtzstr. 20, 01069 Dresden, Germany +E-mail: mahamiyavikram@gmail.com; pritambhattacharyya01@gmail.com; shukla@iitb.ac.in +Abstract +The choice of Gaussian basis functions for computing the ground-state properties of +molecules, and clusters, employing wave-function-based electron-correlated approaches, +is a well-studied subject. +However, the same cannot be said when it comes to the +excited-state properties of such systems, in general, and optical properties, in particular. +The aim of the present study is to understand how the choice of basis functions affects +the calculations of linear optical absorption in clusters, qualitatively, and quantitatively. +For this purpose, we have calculated linear optical absorption spectra of several small +charged and neutral clusters, namely, Li2, Li3, Li4, B+ +2 , B+ +3 , Be+ +2 , and Be+ +3 , using a +variety of Gaussian basis sets. The calculations were performed within the frozen-core +approximation, and a rigorous account of electron correlation effects in the valence +1 +arXiv:2301.02413v1 [physics.chem-ph] 6 Jan 2023 + +sector was taken by employing various levels of configuration interaction (CI) approach +both for the ground and excited states. +Our results on the peak locations in the +absorption spectra of Li3 and Li4 are in very good agreement with the experiments. +Our general recommendation is that for excited-state calculations, it is very important +to utilize those basis sets which contain augmented functions. Relatively smaller aug- +cc-pVDZ basis sets also yield high-quality results for photoabsorption spectra, and are +recommended for such calculations if the computational resources are limited. +Introduction +Gaussian basis functions (GBFs) were initially proposed by Boys for use in computational +atomic and molecular quantum mechanics,1 and over the years have become the preferred +basis functions in quantum chemistry.1 The reason behind the popularity of GBFs is the +so-called Gaussian product theorem1,2 which allows for analytical results for the expressions +of multi-center integrals involving various physical quantities. Nevertheless, one has to be +always careful about various convergence related issues when using GBFs, because, unlike +Slater basis functions, they do not exhibit correct asymptotic behavior far away from the +nuclei. +This generally leads to the requirement that a large number of GBFs should be +used to achieve convergence, leading to huge memory and CPU-time requirements because +the required number of integrals scale as ≈ N 4, where N is the total number of basis +functions. Keeping this in mind, several groups have studied the convergence properties +of GBFs over the years, and have come up with schemes to balance accuracy with the +computational effort (see, e.g., Refs.3,4 for comprehensive reviews). Huzinaga was one of +the earliest researchers to optimize GBFs for Hartree-Fock (HF) calculations on atoms.5 +Ruedenberg and coworkers devised the so-called even-tempered basis set,6,7 while Huzinaga +and coworkers developed well-tempered basis functions.8 Huzinaga and coworkers further +developed several contracted basis sets,9,10 discussed in detail in Ref.4 Pople and coworkers +developed a large number of basis sets3,4 which enjoy continued popularity even in present +2 + +times. One of the most popular minimal basis sets introduced by Pople and coworkers is STO- +3G contracted basis set,11 whose purpose was to emulate Slater-type orbitals, using GTFs. +Split-valence basis sets are among the most popular extended basis sets introduced by Pople +et al.,12–15 in which for inner shells contracted minimal basis functions are used, but for the +valence shells a split set of basis functions is employed, which consist of both contracted and +primitive GTFs. Depending upon the contraction schemes, these basis sets were given names +such as 3-21G,13 4-31G,12 6-21G,14 and 6-311G15, etc. Pople and coworkers also proposed +further enlarged basis sets containing polarization and diffuse functions of higher angular +momenta, which have since become popular choices in quantum chemistry.15–20 Dunning +and coworkers introduced a series of extended basis sets, called “correlation-consistent” (CC) +basis sets, which are of varying sizes, containing both polarization and diffuse functions.21–23 +The basic idea behind these CC basis sets is that they recover a significant amount of electron +correlation energy in post-Hartree-Fock treatments of corresponding atoms. In addition to +the basis sets mentioned here, numerous other sets of basis functions have been developed +over the years, for which we refer the reader to review articles by Davidson and Feller,3 and +Huzinaga.4 +Even though so many basis sets have been developed by numerous groups, in most of the +reports the criteria for their selection appears to be driven by a good description of the ground +state energies of the atoms involved either at the Hartree-Fock level, or in electron-correlated +calculations.3,4 Previously, Balakina et al.24 have explored the basis set dependence of the +linear and non-linear optical properties of conjugated organic molecule p-nitroaniline. They +reported that the [4s3p2d/3s] basis set also provides similar results as aug-cc-pVDZ basis +set for the calculations of (hyper)polarizability. Parsons et al.25 have explored the basis +set dependence of optical rotation calculations of various types of gauges. They found that +the origin-invariant length gauge (LG-OI) gauge with aug-cc-pVTZ basis set provides a +balance of cost and accuracy for DFT method. +Reis and Papadopoulos26 reported that +the inclusion of f-functions in the Dunning’s basis sets does not have a large effect on the +3 + +electric properties of B4 cluster. Lauderdale and Coolidge27 have explored the effect of basis +sets on the non-linear optical properties (hyperpolarizabilities) of linear diacetylenes using +time-dependent Hartree-Fock theory. They found that the inclusion of a diffuse ‘d’ function +to a standard double-zeta plus polarization basis can significantly improve the frequency- +dependent hyperpolarizability. Jabłonski and Palusiak28 have explored the influence of basis +sets in Hartree-Fock (HF) and DFT/B3LYP calculations for the values atoms in molecules +(AIM) parameters. They found that smaller Dunning’s basis sets, including cc-pVDZ and +aug-cc-pVDZ provide poor results as compared to medium-sized Pople-type basis sets. We +are not aware of a systematic study in which the basis sets have been examined from the +perspective of their performance in excited state calculations. Furthermore, we have also +not come across a study which examines the basis sets from the point of view their ability to +compute optical properties of atoms and molecules, which involves calculations of transition +dipole moments, in addition to excited state energies, and wave functions. +In order to +fill this void, we decided to undertake a systematic investigation of the influence of basis +sets on the qualitative and quantitative description of optical absorption spectra of atomic +clusters. In this paper, we have performed calculations of linear optical absorption spectra +of several small neutral and cationic clusters, e.g., Li2, Li3, Li4, B+ +2 , B+ +3 , Be+ +2 , and Be+ +3 , +using the configuration-interaction (CI) approach. For this purpose, a number of basis sets, +namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug- +cc-pVTZ, were employed, and their influence on the convergence of excited state energies, +wave functions, and transition dipole moments has been systematically examined. In this +study, the reason behind our choice of smaller sized atomic clusters and their ions, as against +larger ones, is that it is possible to perform highly accurate CI calculations on smaller systems +so that the difference between results obtained with different basis sets will be due the nature +of basis sets, and not due to the CI approach employed. Based upon our calculations, the +main conclusion is that it is very important to include diffuse basis functions in the basis set +in order to obtain a good description of the photoabsorption spectra. +4 + +Theoretical Approach and Computational Details +General Methodology +All the calculations were performed using the first-principles wave-function-based electron- +correlated approaches, using the standard Hamiltonian within the Born-Oppenheimer ap- +proximation. The molecular orbitals are expressed in terms of the linear combination of +Cartesian-Gaussian type basis functions, also called atomic orbitals (AOs). Although, for +such calculations, a number of program packages are available, we employed GAUSSIAN1629 +and MELD30 for our calculations. The geometries of all the clusters considered in this work +were optimized using GAUSSIAN16 package29 at the coupled-clusters singles-double (CCSD) +level of theory, employing a large augmented correlation-consistent polarized valence triple- +zeta (aug-cc-pVTZ) basis set. +We perform excited-states calculations for various clusters employing their ground-state +optimized geometries, using the configuration-interaction (CI) methodology at various levels +of approximation, as implemented in the program package MELD30. The CI calculations +yield the vertical excitation energies, the ground and excited state wave functions, and the +transition dipole matrix elements connecting the ground and the excited states, which, in +turn, are used to compute the optical absorption spectra of various clusters. +The level +of CI employed in the calculations depends on the size of cluster, the number of valence +electrons in cluster, and the number of active orbitals. The linear optical absorption spectra +of Li2, Li3, Li4, Be+ +2 , B+ +2 clusters were computed at the full CI (FCI) level, while for Be+ +3 +and B+ +3 calculations were performed at the quadruple CI (QCI), and the multi-reference +singles-doubles CI (MRSDCI) levels, respectively. +We start the calculations on a given cluster by first performing restricted Hartree-Fock +(RHF) calculations on it, and obtain the molecular orbitals (MOs), expressed as linear com- +binations of the chosen AOs. In order to perform CI calculations, the one- and two-electron +Hamiltonian matrix elements are transformed from the AO representation to the MO rep- +5 + +resentation. For the FCI calculations, all possible configurations obtained by placing all the +valence electrons of the cluster in the given set of MOs, in all possible ways, consistent with +the Pauli exclusion principle. In the QCI approach, we first choose a reference configuration, +and then generate configurations which are singly-, doubly-, triply-, and quadruply-excited +with respect to it. For the ground-state calculations, the reference configuration is normally +taken to be the RHF configuration, while for the excited-state calculations one chooses an ex- +cited configuration which is closest to the excited state one is trying to calculate. However, +both the FCI and the QCI approaches can lead to a very large number of configurations +if the number of electrons and the MO basis is large, thus, making the calculations in- +tractable. Therefore, for the larger clusters, we employed the multi-reference singles-doubles +configuration-interaction (MRSDCI) approach, as implemented in the MELD package. In +this approach, the singly- and doubly-excited configurations are generated from a list of +configurations called the reference configurations, chosen by the user. We performed the +MRSDCI calculations in an incremental manner, by starting out with a small set of refer- +ence configurations that are close to the states (ground or excited) we are targeting. Then +we analyze the optical absorption spectra of the cluster calculated from that MRSDCI cal- +culation, and identify a new set of configurations which need to be included in the list of +reference configurations based upon their contributions in the wave functions of the targeted +states. The procedure is iterated until the calculated optical absorption spectrum converges +to within a user-defined threshold. In all the CI calculations, the configurations are actu- +ally configuration-state functions (CSFs) which are eigenstates of the point-group symmetry +operators, and the total spin operators S2 and Sz.31–42 +The linear optical absorption spectrum of a given cluster is calculated under the electric- +dipole approximation, using the formula +σ(ω) = 4πα +� +i +ωio|⟨i|ˆe.r|0⟩|2γ2 +(ωi0 − ω)2 + γ2 +(1) +6 + +Above: (i) σ(ω) represents the optical absorption cross section, (ii) ω is the frequency of +incident light, (iii)ˆe denotes the polarization direction of the incident light, (iv) r is the posi- +tion operator, (v) α is the fine structure constant, (vi) ℏωi0 is the energy difference between +ground state (0) and the ith excited state (i), and (vii) γ is the uniform line width associated +with each excited state. The line width γ is taken to be 0.1 eV in all our calculations. The +sum over index i denotes the sum over all possible excited states. We have restricted this +sum in our calculations up to the states corresponding to excitation energies of 10 eV, or +less. Additionally, the oscillator strength fn corresponding to an optical transition from the +ground state to the n-th excited state is computed using the standard formula +fn = 2me +3ℏ2 ∆En +� +j=x,y,z +� +α +|⟨nα|Oj|0⟩|2 +(2) +above me is the electron mass, |0⟩ and |nα⟩ are, respectively the CI wave functions of the +ground state and the excited state in question, with α being a degeneracy label, Oj denotes +j-th Cartesian component of the electric-dipole operator, while ∆En = En − E0 is the +excitation energy of the excited state. +Computational Parameters +In this section, we will discuss the convergence of the results with respect two parameters, +related to the basis-set-size: (a) number of active orbitals in the CI calculations, and (b) +number of CSFs included in the calculations. +Active molecular orbitals +It is well-known that the computational cost at configuration interaction (CI) level of theory +increases as N 6 +act, where Nact is the total number of active molecular orbitals used in the +CI calculations. +Therefore, the time needed to perform a CI calculation will proliferate +rapidly with the increasing values of Nact. We have adopted two approaches to reduce the +7 + +size of the active MO set: (a) we adopt the frozen-core approximation to eliminate the core +orbitals of each atom of the cluster, and (b) for certain cases involving large CI matrices, +we delete all those virtual (unoccupied) orbitals from our calculations whose single-particle +energies are larger than 1 Hartree. The frozen-core approximation is a standard approach +which also has the added advantage of considerably reducing the number of active electrons +(nelec) in the calculation. The “1 Hartree cutoff” also doesn’t reduce the accuracy of the +calculations because we are interested in low-lying optical excitations below 10 eV, while +our cutoff eliminates only those orbitals from the calculations whose energy is larger than +27.21 eV. Both these approximations have been investigated rigorously in our group in earlier +calculations.36,41–43. +To be specific, in the present set of calculations, we have considered all the virtual orbitals +for Li2 and Be+ +2 clusters, while for Li3, Li4, Be+ +3 , B+ +2 and B+ +3 clusters we have imposed the 1 +Hartree cutoff. +Size of CI expansion +Another important parameter that controls the quality of calculations is the total number +of CSFs, Ntotal, included in the CI expansion of the many-particle wave functions of the +clusters concerned, both for their ground and the excited states. As mentioned earlier, for +a given set of active electrons and MOs, the best possible CI expansion corresponds to the +FCI expansion, which becomes intractable for systems with large values of nelec and Nact. +However, whenever FCI is not possible, we employ one of the restricted CI approaches such +as the QCI or the MRSDCI methods. Of the two, it is crucial to examine the convergence of +the MRSDCI approach which is based upon singles and doubles excitations from a number +of reference configurations (Nref) leading to the final CI expansion with Ntotal CSFs. We +examined the convergence of the optical absorption spectrum for the B+ +3 cluster calculated +using the MRSDCI method, with respect to Nref, and Ntotal, as presented in Fig. 1. +8 + +Figure 1: Convergence of the optical absorption spectrum of the B+ +3 computed using the +MRSDCI method, with the increasing numbers of reference configurations (Nref). For cal- +culations labeled MRSDCI1, MRSDCI2, and MRSDCI3, values of Nref were 58, 101, and +144, respectively. +In the figure, we plot the absorption spectra of B+ +3 obtained from three MRSDCI calcula- +tions of increasing sizes labeled as MRSDCI1, MRSDCI2, and MRSDCI3. In these calcula- +tions, the values of parameters Nref and Ntotal were Nref= 58, Ntotal= 4007873, Nref= 101, +Ntotal= 5781436, and Nref= 144, Ntotal= 8422193, respectively. From Fig. 1., it is obvious +that the spectra obtained using MRSDCI2 and MRSDCI3 calculations are very close to each +other, signaling convergence with respect to the size of the MRSDCI expansion. +Results and Discussion +Before discussing the results of our calculations of the optical absorption spectra of various +clusters, we first summarize their ground state geometries in Table 1, optimized at the CCSD +9 + +300 +MRSDCI1 +250 +MRSDCI2 +MRSDCI3 +Intensity(arb. units) +200 +150 +100 +50 +0 +0 +8 +10 +Energy(eV)level of theory, employing GAUSSIAN16 suite of programs29, and large aug-cc-pVTZ basis +sets. +Table 1: The nature of the structure, along with the point group symmetry utilized, during +the coupled-cluster singles-doubles (CCSD) geometry optimization calculations are presented +below. Additionally, for each cluster, the symmetry of the ground-state wave function, total +Hartree-Fock (HF) energy in Hartree (Ha), total CCSD energy (Ha), and the correlation +energy (eV) are also presented. During the calculations, aug-cc-pVTZ basis set was employed +for each cluster. +Cluster +Structure +Point group +Symmetry of the +HF energy +CCSD energy +Correlation energy +GS wave function +(Ha) +(Ha) +(eV) +Li2 +Linear +D2h +1Ag +-14.8715509 +-14.9033549 +0.87 +Li3 +Linear +D2h +2Ag +-22.3088776 +-22.3454346 +0.99 +Li3 +Isosceles triangle +C2v +2A1 +-22.3170594 +-22.3557287 +1.05 +Li4 +Rhombus +D2h +1Ag +-29.7619144 +-29.8354840 +2.00 +Be+ +2 +Linear +D2h +2Ag +-28.9205835 +-28.9672583 +1.27 +Be+ +3 +Linear +D2h +2Ag +-43.5410215 +-43.6332325 +2.51 +B+ +2 +Linear +D2h +2Ag +-48.8344277 +-48.9626672 +3.49 +B+ +3 +Equilateral triangle +D3h +1A +′ +1 +-73.4445744 +-73.7127527 +7.27 +For each cluster, the table lists the nature of its ground-state structure, point group +employed in the calculations, symmetry of the ground state wave function, total energy of the +ground state, and the correlation energy. In Table 2, the details related to our CI calculations +performed for computing the optical absorption spectra of various clusters are provided. For +various clusters, the table lists: (a) the type of CI calculation, (b) the point-group symmetry +employed in the calculations, (c) irreducible representations considered for each point group, +and (d) for each irreducible representation, the size of the CI expansion (Ntotal) for each +cluster are depicted. From Table 2 it is obvious that most of the CI calculations were of the +FCI type, which are exact for the chosen set of active MOs. Furthermore, in the calculations +in which approaches such as QCI or MRSDCI were used, the size of the CI expansion is +quite large. This means that the CI calculations performed in this work are fairly large +scale, indicating that the computed optical absorption spectra are numerically accurate. +Next, for these clusters, we discuss in detail the calculated ground state geometries, followed +by their optical absorption spectra. +10 + +Table 2: For each cluster, the type of CI approach used for the calculations of the optical +properties, point group symmetry employed during the CI calculations, and the total number +of configurations (Ntotal) in the calculation are listed below. The value of Ntotal corresponds +to aug-cc-pVTZ basis set based CI calculations. +Cluster +Structure +Method +Point group +Symmetry +Ntotal +used +Li2 +Linear +FCI +C1 +1A +5886 +Li3 +Linear +FCI +C1 +2A +575960 +Li3 +Isosceles triangle +FCI +C2v +2A1 +137956 +2B1 +129520 +2B2 +137396 +Li4 +Rhombus +FCI +D2h +1Ag +1853578 +1B1u +1846246 +1B2u +1844485 +1B3u +1802190 +Be+ +2 +Linear +FCI +C1 +2A +419868 +Be+ +3 +Linear +QCI +D2h +2Ag +7393226 +2B1u +7393210 +2B2u +7286869 +2B3u +7286869 +B+ +2 +Linear +FCI +D2h +2Ag +6365216 +2B1u +6365216 +2B2u +6323328 +2B3u +6323328 +B+ +3 +Equilateral triangle +MRSDCI +C1 +1A +8422193 +11 + +Geometry +The simplest cluster of lithium is lithium dimer with the D∞h point group symmetry. We +obtained the optimized bond length of Li2 cluster to be 2.70 Å, as shown in Fig. 2(a). This +result is in excellent agreement with the bond length 2.68 Å reported by Wheeler et al.44, +who performed the calculations at the CCSD/CCSD(T) level of theory using the Dunning +correlation-consistent polarized core-valence triple/quadruple-zeta cc-pwCVXZ basis sets. +Florez et al.45 performed density functional theory (DFT) calculations using the B3LYP +and BLYP functionals, and reported the bond lengths to be 2.70 Å, and 2.71 Å, respectively, +again in excellent agreement with our result. Furthermore, our calculated bond length is +also in a very good agreement with the experimentally measured value 2.67 Å, reported by +Huber46. +As far as Li3 cluster is concerned, two isomers namely linear and isosceles triangle were +found to be stable. The equilateral triangular structure of Li3 cluster is not stable, and +undergoes Jahn-Teller distortion to acquire the isosceles triangular structure. The linear +structure has the D∞h point-group symmetry, with the optimized equal bond lengths of 2.90 +Å (see Fig. 2(b)), in excellent agreement with the value 2.89 Å, reported by Jones et al.47. +The lowest-energy geometry of the Li3 cluster is an isosceles triangle with the C2v point- +group symmetry. The CCSD-level optimized bond lengths for this structure are found to be +2.68 and 3.07 Å, with the bond angles 51.73◦ and 64.13◦(see Fig. 2(c)). We note that by +performing DFT calculations, Jones et al.47 obtained the bond lengths of 2.82 and 3.37 Å, +that are significantly different as compared to our results. +12 + +Figure 2: Optimized geometry of (a) Li2, (b) Li3 linear, (c) Li3 isosceles triangular, (d) Li4, +(e) Be+ +2 , (f) Be+ +3 linear, (g) B+ +2 , and (h) B+ +3 equilateral triangular clusters considered in +this work. The geometry optimization has been performed using the CCSD method, and +aug-cc-pVTZ basis sets. All the listed bond lengths are in Å units. +The lowest-energy structure for the Li4 cluster has a rhombus shape, with D2h point +group44,47, as shown in Fig. +2(d). +Our optimized bond lengths of the side and minor +diagonal of the rhombus structure are 3.02 and 2.65 Å, respectively, which are in excellent +agreement with the values 3.04 and 2.62 Å reported by Jones et al.47. +The optimized bond length of the Be+ +2 cluster with the D∞h point group is found to be +2.25 Å (see Fig. 2(e)), in good agreement with the reported bond length 2.21 Å, obtained +from DFT calculations by Srinivas et al.48. Our lowest-energy optimized structure of Be+ +3 +cluster also has a linear geometry, with two equal bond lengths 2.22 Å, as shown in Fig. 2(f). +This value of the bond length is in very good agreement with the value 2.19 Å, computed +by Srinivas et al.48 using DFT. +As far as B+ +2 cluster is concerned, we computed its minimum-energy bond length to be +2.18 Å (see Fig. 2(g)), which is 0.18 Å larger than the value 2 Å reported by Hanley et al.49. +13 + +(a) +(b) +2.70 +2.90 +Liz +Lis chain +(c) +(d) +3.07 +2.65 +2.68 +3.02 +Li3 +(f) +Li4 +(e) +2.25 +2.22 +Be2* +Be3+ +(g) +(μ) +2.18 +1.58 +B, +B3We attribute this difference to two factors, namely, smaller basis set (6-31G∗), coupled with a +lower-level CI methodology used by the authors.49. Our optimized structure of B+ +3 cluster is +an equilateral triangle of sides 1.58 Å, with the D3h point-group symmetry, as shown in Fig. +2(h). Hanley et al.49 using a CI approach, along with the 6-31G∗ basis set, also obtained the +optimized structure to be an equilateral triangle for the B+ +3 , but with a bond length of 1.53 +Å, which is 0.05 Å smaller than our result. We again attribute the differences to the choice +of a smaller basis set, coupled with a lower-level correlation methodology as compared to +the CCSD approach used by us. +Peak locations +Li2 dimer, with just two active electrons within the frozen-core approximation, is the small- +est many-electron cluster considered in this work. Therefore, very high-quality correlated- +electron calculations using large basis sets are possible for this system, not just for its ground +states, but also for the excited states. As a result, this case can provide us deep insights into +the influence of the choice of basis functions on the calculated excited state properties and the +photoabsorption spectra. For the calculations, we employed the frozen-core FCI method us- +ing six basis sets of varying sizes, namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ, +cc-pVTZ, aug-cc-pVDZ and aug-cc-pVTZ, and the computed spectra are presented in Fig. +3(a). All the virtual molecular orbitals generated during the RHF calculations were used in +the CI calculations, i.e., no unoccupied orbitals were discarded. As a result, the frozen-core +FCI results presented here are the best ones possible for the chosen basis sets. +For the Li2 dimer, the peak locations in the computed spectra are presented in Table S1 +of the Supporting information (SI), from which it is obvious that for the first two peaks the +excitation energies calculated using different basis sets are in very good agreement with each +other. This is encouraging because from Fig. 3(a) it is obvious that most of the oscillator +strength of the absorption spectrum is confined to these two peaks. However, starting from +the third peak onward, we start seeing differences in the excitation energies predicted by +14 + +different basis sets. For the third peak, the predicted peak locations can be classified in two +groups: (a) those predicted by correlation-consistent basis sets cc-pVDZ and cc-pVTZ, and +(b) the ones predicted by 6-311G++ and augmented correlation consistent (aug-cc-) class of +basis sets. We note that the peak locations predicted by the former class of basis functions +have values significantly larger than those predicted by the latter class. Another noteworthy +point is that there is very good agreement among the peak locations predicted by the second +class of basis sets. As far as the location of the fourth peak is concerned, there is good +agreement among the predictions by 6-311++G(3df,3pd) and aug-cc class of basis functions, +while the remaining three basis functions predict very different values. The case of the fifth +peak is somewhat anomalous in that the agreement among the predictions by any of the +basis sets is not good. However, for higher peaks we note that the results from the aug-cc +class of basis functions are in good agreement with each other, while other basis functions +predict widely differing results. Hong et al.50 also performed first-principles calculations of +the photoabsorption spectra of several Lin clusters employing the time-dependent density- +functional theory (TDDFT) methodology, and for Li2 their predicted locations of the first +two peaks are 1.92 eV and 2.53 eV50. On comparing these with our best values of 1.83 eV +and 2.57 eV, respectively, we note: (a) our excitation energy for peak I is about 0.09 eV +smaller than theirs, while (b) our location for peak II is about 0.04 eV larger than theirs. +We attribute these differences to different computational methodologies adopted in the two +sets of calculations, and it will be interesting to compare the computational results with the +experimental ones, whenever they are available. +The peak locations of the photoabsorption spectra of Li3 chain are presented in Table S2 +of SI, from which it is clear that the locations of the first two peaks converge completely for +all the basis sets, similar to the case of dimer. The third peak is the most intense peak of +the computed spectra as shown in Fig. 3(b), whose location is in good agreement for all the +basis sets except for cc-pVDZ, which predicts higher excitation energy as compared to the +rest. From the fourth peak onward, the peak locations can be classified in two similar group +15 + +as discussed previously for the case of dimer: the peak locations predicted from correlation- +consistent basis sets cc-pVDZ and cc-pVTZ are towards the higher energy side as compared +to all other basis sets. It can also be seen that the peak positions corresponding to the two +classes of basis sets are in good agreement within the class. +Next, we examine the peak locations in the photoabsorption spectra of Li3 triangular +cluster computed using various basis sets. We note that the peak locations corresponding +to the first five peaks are in very good agreement with each other for different basis sets +as is obvious from Table S3 of SI. This result is very encouraging because peak IV is the +most intense (MI) peak of the computed spectra as presented in Fig. 3(c), and it is crucial +for a basis set to be able to accurately describe the MI peaks. The location of this peak is +2.43 eV computed using the aug-cc-pVTZ basis set, which is in a decent agreement with the +experimentally detected peak at 2.58 eV by Blanc et al.51. From the sixth peak onward it +was observed that the peak locations calculated using correlation consistent basis sets (cc- +pVDZ and cc-pVTZ) do not match with the other classes of basis sets. However, the peak +locations computed using the 6-31G class and the aug-cc-pVTZ continue to be in very good +agreement with each other till peak VIII, located near 3.8 eV. The locations of higher-energy +peaks beyond peak VIII computed using these basis sets are presented in Table S4 for the +SI. +16 + +Figure 3: Optical absorption spectra of (a) Li2, (b) Li3 linear, (c) Li3 triangular, and (d) Li4 +clusters computed using various basis sets and the frozen core FCI method. The uniform +line-width 0.1 eV is used to plot the spectrum. +The peak positions of the photoabsorption spectra of Li4 cluster computed using various +basis sets are presented in Table S5 of SI, while the spectra are plotted in Fig. 3(d). We note +that for this cluster, the excitation energies of the first five peaks computed using different +basis sets are in very good agreement with each other. The first three peaks are much more +intense as compared to the higher energy peaks, and in peak III there are slight differences +(≈0.1 eV) in the peak locations predicted by different basis sets. The two largest basis sets +(6-311++G(3df,3pd), and aug-cc-pVTZ) predict the location of peak III at 2.93 eV, while +the predictions by the rest of the basis sets are in the range 3.03–3.08 eV. From peak VI +17 + +SO +(a) +60K +6-311++G(2d,2p) +6-311++G(3df,3pd) +(b) +III +6-311++G(2d,2p) +400 +cc-pVDZ +500 +6-311++G(3df,3pd) +ZLAd-0 +cc-pVDZ +aug-cc-pVDZ +cC-pVTZ +awg-cc-pVTZ +aug-cc-pVDZ +(s)) +400 +aug-cc-pVTZ. +300 +200 +100 +100 +I +VVI +VII +6 +Energy (eV) +300 +400 +(c) +6-311++G(2d,2p) +(d) +6-311++G(2d,2p) +250 +6-311++G(3df,3pd) +cc-pVDZ +6-311++G(3df,3pd) +ZLAd-33 +II +cc-pVDZ +aug-cc-pVDZ +300 +cc-pVTZ +200 +ZLAd-30-8ne +aug-cc-pVDZ +aug-cc-pVTZ +Intensity +150 +Intensity +100 +100 +50 +VIII +X +2 +6 +Energy (eV) +Energy (eV)onward we begin to observe differences among the locations predicted by different basis sets, +with a tendency towards clustering into different classes. However, the noteworthy point +is that the intensity corresponding to these higher energy peaks is very low. As far as the +comparison with the experiments is concerned, the first three photoabsorption peaks of the +Li4 cluster located at 1.87 eV, 2.65 eV, and 2.93 eV for aug-cc-pVTZ basis set are in excellent +agreement with the experimental measurements of Blanc et al.51 who detected these peaks +at 1.83 eV, 2.65 eV, and 2.93 eV, respectively. +For Be+ +2 cluster, we present the spectra computed by different basis sets in Fig. 4(a), +while the corresponding peak locations are presented in Table S6 of SI. We note excellent +convergence of the excitation energies up to the sixth peak, beyond which results obtained +by different basis sets do not agree much with each other. We further note that Peak V +located near 6.30 eV is the most intense peak, and, for that, the predictions of the different +basis sets are in a fairly narrow energy range 6.30-6.37 eV. +Figure 4: Optical absorption spectra of (a) Be+ +2 and (b) Be+ +3 clusters computed using various +basis sets and frozen core FCI and QCI methods, respectively. The uniform line-width of +0.1 eV is used to plot the spectrum. +The excited-states peak locations of Be+ +3 cluster for different basis sets are presented in +Table S7 of SI. We notice excellent agreement of the excited-states peak locations up to the +18 + +600 +(a) +6-311++G(2d,2p) +VII +6-311++G(2d,2p) +300 +6-311++G(3df.3pd) +500 +6-311++G(3df.3pd) +cc-pVDZ +cc-pVDZ +cc-pVTZ +cc-pVTZ +aug-cc-pVDZ +aug-cc-pVDZ +aug-cc-pVTZ +400 +aug-cc-pVTZ +(sirun +200 +Intensity (arb. +300 +Intensity +200 +100 +100 +2 +3 +4 +5 +6 +9 +[0 +3 +5 +Energy (eV) +Energy (eV)seventh peak which is also the most intense peak of the spectra located near 6.5 eV, as shown +in Fig. 4(b). Although the peak location of the sixth peak computed using cc-pVDZ basis +set is slightly towards the higher energy region as compared to all other basis sets, but the +difference is small. Noteworthy point is that these basis sets are able to achieve convergence +in the peak positions in Be+ +2 and Be+ +3 photoabsorption spectra up to much higher excitation +energies, as compared to the Li clusters. +The excited-states peak positions corresponding to the photoabsorption spectra of B+ +2 +cluster are presented in Table S8 of SI. We notice excellent agreement of the peak energies +corresponding to first three peaks for all the basis sets. The fourth peak is the most intense +peak of the spectra, as shown in Fig. +5(a) for whose location excellent agreement has +been achieved for 6-311++G (2d, 2p), cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ basis sets, +indicating complete convergence. However, the excitation energies for peak IV computed +using the 6-311++G (3df, 3pd) and cc-pVDZ basis sets are about 0.1 eV higher, as compared +to other basis sets. As far as peak V is concerned, which is a very weak shoulder of peak IV, +we again observe excellent convergence for all the basis sets, except cc-pVDZ which fails to +predict the peak. From the sixth peak onward, as discussed previously, the predicted peak +locations can be classified into two groups: (a) larger basis sets of 6-311++G and aug-cc- +type, and (b) smaller basis sets cc-pVDZ and cc-pVTZ, with the peak locations predicted +by individual classes being in very good agreement with each other. +19 + +Figure 5: Optical absorption spectra of (a) B+ +2 and (b) B+ +3 cluster computed using various +basis sets and frozen core FCI and MRSDCI methods, respectively. The uniform line-width +0.1 eV is used to plot the spectrum. +The peak locations corresponding to the excited-states of the photoabsorption spectra +of B+ +3 cluster are presented in Table S9 of SI, while the calculated spectra are plotted in +Fig. 5(b) . For this cluster, we get eight well-separated peaks in the explored energy range, +with peak VIII located near 8.9 eV being the most intense. We note that the peak energies +corresponding to all the basis sets converge excellently up to the peak VIII, except those +predicted by the cc-pVDZ basis set, which are consistently higher. +We have noticed in +Fig.4 and Fig.5, only the deep valence excitation energies are dependent on the choice of +basis sets. This behavior can be a consequence of the frozen-core approximation, which we +have employed in the calculations. To verify this, we have computed the optical absorption +spectra of Li2 and Be+ +2 clusters by also including the core excitations within the large-scale +QCI method. We found that the optical spectra of these clusters computed by including core +excitations agrees completely with the absorption spectra computed after employing frozen- +core approximation, as shown in Fig.S1 and Fig.S2 of the SI. Therefore, the frozen-core +approximation does not alter the absorption spectra of small clusters. +Based on the peak positions of the individual clusters discussed above, we observe the +20 + +500 +250 +(a) +(b) +6-311++G(2d,2p) +6-311++G(2d,2p) +6-311++G(3df,3pd) +VII +6-311++G(3df,3pd) +400 +cc-pVDZ +IV +200 +cc-pVDZ +cc-pVTZ +VII +cc-pVTZ +aug-cc-pVDZ +aug-cc-pVDZ +aug-cc-pVTZ +ZLAd-03-8ne +150 +Intensity (arb. +200 +VI +100 +100 +50 +III +II +1 +2 +3 +4 +6 +8 +9 +10 +2 +3 +1 +6 +7 +8 +10 +Energy (eV) +Energy (eV)following general trends: (a) peak locations for all the clusters used in this study are in very +good agreement for all the basis sets up to the most intense peak of the spectra, except +for cc-pVDZ basis set. (b) the excited-states peak locations beyond the most intense peak +can be classified in two groups, in which the peak locations calculated using correlation- +consistent basis sets do not match with the peak locations computed using all other basis +sets, and (c) for the cc-pVDZ basis sets peaks are located at higher energies as compared +to the rest of the basis functions. As the basis-set dependence of the optical properties is +different for the density functional theory compared to the wave function-based large-scale +configuration-interaction method, it will be interesting to explore the optical properties of +clusters using time-dependent density functional theory (TD-DFT) and compare it with our +results. We found that the first peak of the optical absorption spectra of B+ +3 cluster is located +at 0.84 eV when computed using large-scale MRSDCI calculations along with a large aug- +cc-pVTZ basis set. However, when the calculations are performed using TD-DFT method +with B3LYP functional and the same basis set, it is obtained at 0.95 eV. The most intense +peak of the optical absorption spectra is located at 8.85 eV using the MRSDCI approach, +which is found to be at 9.35 eV by employing the TD-DFT method. The calulated optical +absorption spectra and excited-states peak locations of B+ +3 cluster corresponding to TD- +DFT calculations are provided in the Fig.S3 and Table S10 of the SI. We also report that +the variations in the peak locations of the photoabsorption spectra of B+ +3 computed using +various basis sets and TD-DFT method are lesser than the wave function-based CI method. +Oscillator strength +In addition to the excitation energy, the next important quantity determining the profile of +the absorption spectrum is the oscillator strength (f) corresponding to various optical tran- +sitions, connecting the ground state to the excited state in question. The oscillator strength +calculated using Eq. 2 is determined by the excitation energy of the state involved, and the +corresponding transition dipole moment (TDM). The TDM being a matrix element, is, in +21 + +turn, determined by the many-particle wave functions of the ground and the excited state +that it connects. Another important quantity is the polarization of the photon involved in a +given optical transition (peak), which can be measured in oriented samples. The polarization +is a consequence of the point-group symmetry of the concerned molecule, and hence should +be independent of the basis set employed. In this section, we discuss the convergence of the +oscillator strengths and photon polarizations associated with various peaks of the calculated +spectra. In Table 3, we present the oscillator strengths corresponding to the first peak, and +the most intense peak (peak II) of the spectra of Li2 computed using different basis func- +tions. Additionally, the table also contains the dominant configurations contributing to the +many-particle wave functions of the excited states involved. +Table 3: Comparison of oscillator strengths and the dominant configurations contributing to +the many-particle wave functions for peaks I and II of Li2 cluster calculated using different +basis sets. In the “Polarization” column, ∥ indicates photon polarization along the direction +of the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to +the molecular axis (transverse polarization). Note that the transversely polarized states are +doubly degenerate, therefore, the oscillator strength corresponding to those is the sum of +both the contributions. ’H’ and ’L’ stand for HOMO and LUMO orbitals. +Basis Set +Peak I +Peak II +Polarization +f +Configurations +Polarization +f +Configurations +6-311++G(2d,2p) +∥ +0.460 +|H → L⟩ +⊥ +0.971 +|H → L + 2⟩ +|H → L + 3⟩ +|H → L + 7⟩ +6-311++G(3df,3pd) +∥ +0.456 +|H → L⟩ +⊥ +0.966 +|H → L + 2⟩ +|H → L + 3⟩ +|H → L + 7⟩ +cc-PVDZ +∥ +0.463 +|H → L⟩ +⊥ +0.970 +|H → L + 1⟩ +|H → L; H → L + 2⟩ +|H → L + 1; H → L + 5⟩ +cc-pVTZ +∥ +0.455 +|H → L⟩ +⊥ +0.969 +|H → L + 1⟩ +|H → L + 4⟩ +|H → L + 6⟩ +aug-cc-pVDZ +∥ +0.462 +|H → L⟩ +⊥ +0.972 +|H → L + 1⟩ +|H → L + 3⟩ +|H → L + 6⟩ +aug-cc-pVTZ +∥ +0.454 +|H → L⟩ +⊥ +0.966 +|H → L + 2⟩ +|H → L + 3⟩ +|H → L + 7⟩ +From Table 3 it is obvious that the oscillator strengths computed using various basis +functions for both the peaks are in very good agreement with each other. We also note that +the direction of the polarization of the photons involved in a given optical transitions are of +22 + +the excited-states corresponding to the first and second peak of the spectra of Li2 cluster are +parallel and perpendicular to molecular axis, respectively, irrespective of the basis set. +The oscillator strengths corresponding to the first and most intense peaks of the spectra +of Li3 chain and triangular clusters are presented in Table S11 and Table S12, respectively. +We note that the oscillator strengths of the first peaks both of Li3 chain, and the triangular +cluster, computed using the different basis sets are in excellent agreement with each other. +The oscillator strengths corresponding to the most intense peaks of Li3, i.e. peak III of +the chain and peak IV for the triangular cluster, calculated using various basis sets can +be classified into two groups: (a) those calculated using correlation-consistent (cc-pVTZ +and cc-pVDZ) class of basis sets, and (b) those computed using 6-33++G- and augmented +correlation-consistent (aug-cc-) class of basis sets. The oscillator strength calculated by the +first class of basis sets is comparatively higher than the second class of basis sets. But, the +relative maximum difference between oscillator strengths of different classes is close to 6%, +which is fairly acceptable. +The oscillator strengths corresponding to the first and the most intense peak (peak II) +in the photoabsorption spectra of Li4 cluster are presented in Table S13. +We note that +the oscillator strengths of peak I are in good agreement with each other for all the basis +sets except for cc-pVDZ and aug-cc-pVTZ. For these basis sets the oscillator strength is +comparatively larger. +For peak II, we note that the difference in the oscillator strength +computed by cc-pVDZ and 6-311++G (3df, 3pd) basis sets is about 5%, which is again +quite small. +We present the oscillator strengths corresponding to the first and the most intense peaks +of the cationic beryllium clusters Be+ +2 and Be+ +3 in tables S14, and S15, respectively. We +note very good agreement on the oscillator strengths of both the peaks of the Be+ +2 and +Be+ +3 clusters for all the basis sets. The maximum relative disagreement we find among the +oscillator strengths for a given peak is around 6%. +Finally, we discuss the oscillator strengths of the first and the most intense peaks of the +23 + +B+ +2 and B+ +3 clusters presented in tables S16, and S17, respectively. We note that both for +B+ +2 and B+ +3 clusters, the oscillator strengths of the first peaks are two orders of magnitude +smaller than those of their most intense peaks, indicating that the first peaks for both the +clusters are relatively feeble. Nevertheless, the oscillator strengths of the first peaks of the +photoabsorption spectra of the two clusters calculated using various basis sets are in very +good agreement with each other. As far as the most intense peaks are concerned, both for B+ +2 +and B+ +3 we see the following pattern: oscillator strengths computed using 6-311++G- and +aug-cc-pVTZ basis sets are in very good agreement with each other, while those computed +using other basis sets differ from them somewhat. +Wave function analysis +Next, we examine the dominant configurations contributing to the CI wave functions of the +excited states contributing to various peaks. The dominant configurations corresponding +to the excited-states CI wave functions of peak I and peak II of Li2 are presented in Table +3. We note that for peak I, the main contribution to the corresponding excited state wave +function is from the singly excited configuration |H → L⟩ for all the basis sets. However, +the next important configuration to the same wave function depends on the class of basis set +employed: (a) it is |H → L+3⟩ single excitation when calculations are performed using larger +basis sets of the type 6-311++ and aug-cc, but (b) for smaller basis sets, this configuration is +found to be |H → L + 4⟩ for cc-PVTZ basis, and |H → L; H → L + 2⟩ for the cc-PVDZ set. +Peak II is due to two degenerate excited states to which the dominant contributions are from +configurations |H → L+2⟩ and |H → L+7⟩, for the calculations performed using 6-31G++ +and aug-cc-PVTZ type basis sets. But, for the calculations performed with smaller basis +sets, the dominant configurations is |H → L + 1⟩, while the next important configuration +can be |H → L + 6⟩ or |H → L + 1; H → L + 5⟩, depending on the basis set. Thus, we +can draw the following general conclusion regarding this: (a) for large basis set calculations, +for a given peak, the configurations are in perfect agreement with each other, and (b) the +24 + +configurations predicted by calculations performed using smaller basis sets such as cc-PVDZ +are found to be different as compared to those obtained in larger basis set calculations. +The dominant configurations for the wave functions corresponding to peak I and the +most intense peak (peak III) of Li3 chain, computed using various basis sets are presented +in Table S11. We find that for the first peak the dominant configuration is |H − 1 → H⟩ for +all the basis sets except for aug-cc-pVTZ. For the aug-cc-pVTZ the dominant configurations +contributing to the excited state wave function are different compared to other basis sets, +because of the reversal of ground and excited states. However, because the peak energies and +oscillator strength for the state are in excellent agreement with all other basis sets implies +that we have obtained correct quantitative description of the excited states even with this +basis set. For peak III the main contribution to the excited state wave function is from +|H − 1 → L + 2⟩ for the larger 6-311++ and aug-cc class of basis sets, while it is from +|H − 1 → L + 1⟩ for the smaller cc-pVTZ and cc-pVDZ basis sets. +The main configurations contributing to the excited states wave functions of peak I and +the most intense peak (peak IV) of Li3 triangular cluster, computed using various basis sets +are presented in Table S12. We note that for the first peak, the main contribution to the +wave function is from configurations |H → L+14⟩ or |H → L+13⟩ for the larger 6-311++G +and aug-cc class of basis sets. For the correlation-consistent basis sets (cc-pVDZ and cc- +pVTZ) the main contribution is due to the configuration |H → L + 2⟩. For the fourth peak, +the dominant configuration is |H − 1 → L⟩, irrespective of the type of basis set used for the +calculation. +The dominant configurations corresponding to the excited states wave functions of peak +I and the most intense peak of the spectra (peak II) of the Li4 cluster are presented in Table +S13. For the first peak, the main contribution is from the configuration |H → L + 1⟩ for all +the basis sets, while for peak II it is |H−1 → L⟩ for all the basis sets. Thus, we have excellent +agreement among all the basis sets when it comes to the most important configuration for +both the peaks of the Li4 cluster. +25 + +The important configurations corresponding to the excited states wave function of peak +I and the most intense peak (peak V) of the photoabsorption spectra of Be+ +2 cluster are +presented in Table S14. The dominant configuration contributing to peak I is |H → L⟩ for +the 6-311++G class of basis sets, and |H → L + 2⟩ for the rest. For peak V, the main +configuration contributing to the CI wave function is |H − 1 → L + 1⟩ for 6-311++G class +of basis sets, and |H − 1 → L⟩ for the rest of the sets. +The configurations dominating the excited state CI wave functions of peak I and the +most intense peak (peak VII) of Be+ +3 cluster are listed in Table S15. +We note that the +most important configurations contributing to peak I can be classified in two groups: (a) +for larger 6-311++G(3df,3pd) and aug-cc class of basis sets the dominant configuration is +|H → L + 1⟩, (b) while for smaller basis sets dominant configuration is highly basis set +dependent. +For peak VII, the doubly-excited configurations |H − 2 → L; H → L + 2⟩ +and |H − 1 → L; H → L + 4⟩ dominate the excited-state wave functions for the larger +6-311++G(3df,3pd) and aug-cc class of basis sets, but vary significantly for the rest. +Most important configurations contributing to the wave functions for peak I and the +most intense peak (peak IV) of B+ +2 cluster are presented in Table S16. It is obvious that the +double-excitation |H − 1 → L; H → L⟩ contributes the most to peak I for all the basis sets. +The dominant configurations contributing to the wave functions of peak IV are |H → L+5⟩ +and |H → L + 11⟩ for all the basis sets except the cc-pVDZ/cc-pVTZ, for which instead of +|H → L + 11⟩, the double excitation |H − 1 → L; H → L⟩ contributes. +Finally, we present the dominant configurations in the CI wave functions corresponding +to peak I, and the most intense peak (peak VIII), of B+ +3 cluster in Table S17. The configura- +tion with maximum contribution to the excited state wave functions for peak I is |H → L⟩, +irrespective of the basis set. The next dominant configuration is basis-set dependent, how- +ever, it is a double excitation in all the cases. The dominant configuration corresponding to +the CI wave function of peak VIII is the double excitation |H − 1 → L; H → L + 3⟩ for all +the basis sets. +26 + +The detailed wave function analysis for all the peaks of the optical absorption spectra of +clusters considered in this work using the largest aug-cc-pVTZ basis set is provided in Table +S18-S25 of the SI. +Conclusion +In this work, we presented electron-correlated calculations of the optical absorption spectra +of small neutral and ionic clusters using various basis sets. First, the stable geometries of +various clusters were determined at the CCSD level of theory, using the aug-cc-PVTZ basis +set. For the ground and the excited state wave functions calculations needed to compute +the absorption spectra, we used the FCI, QCI, and MRSDCI approaches depending upon +the size of the clusters. The CI calculations were performed using six different basis sets, +namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ , cc-pVTZ, aug-cc-pVDZ, and +aug-cc-pVTZ. +We observed that the optical absorption spectra of all these clusters exhibit excellent +convergence for all the basis sets in the lower energy range. However, usually after the first +two peaks, the shift in peak locations for cc-pVDZ and cc-pVTZ basis set are noted in all +likelihood because of the lack of diffuse basis functions in these sets. If we use augmented +basis sets, the absorption spectra show good agreement with the results computed using +other similar basis sets. Although aug-cc-pVDZ basis set has a relatively smaller number of +basis functions as compared to aug-cc-pVTZ basis set, the agreement between the spectra +computed using the two basis sets is very good. Because the number of two-electron integrals +increases as N 4 where N is the number of basis functions in basis set, we can reduce the +computational cost significantly by using aug-cc-pVDZ basis set instead of larger Pople’s +basis sets, and aug-cc-pVTZ basis set. Thus, our general recommendation is that for optical +absorption calculations one should use a basis set containing diffuse functions, i.e., of the +aug-cc- type. However, whether one should use aug-cc-pVDZ, or a larger set, should be +27 + +decided by the available computational resources. +We believe that the CI calculations presented in this work are quite accurate, as is obvious +from the fact that our obtained results are in very good agreement with the experiments for +Li3 and Li4 clusters. Therefore, it will be of interest to compare our results on other clusters +also with the experiments, as and when they are performed. +Associated Content +Supporting Information +In the supporting information file, we have provided the peak locations, oscillator strengths, +and dominant excited state configurations corresponding to the optical absorption spectra +of all the clusters for all the basis sets considered in this work. The SI file also contains the +details of the many-particle wave functions of excited states contributing to the peaks in the +optical absorption spectrum of clusters for aug-cc-pVTZ basis set. +Author Information +Corresponding Author +Alok Shukla: Department of Physics, Indian Institute of Technology Bombay, Powai, Mum- +bai 400076, India; *E-mail: shukla@phy.iitb.ac.in +Acknowledgment +This work was supported by senior research fellowship (DST-Inspire) provided by department +of science and technology, India. +28 + +Authors +Vikram Mahamiya: Department of Physics, Indian Institute of Technology Bombay, Powai, +Mumbai 400076, India; E-mail: mahamiyavikram@gmail.com +Pritam Bhattacharyya: Institute for Theoretical Solid State Physics, Leibniz IFW Dres- +den, Helmholtzstr. 20, 01069 Dresden, Germany; E-mail: pritambhattacharyya01@gmail.com +Notes +The authors declare no competing financial interests. +29 + +References +(1) Boys, S. F. Electronic Wave Functions. I. A General Method of Calculation for the +Stationary States of Any Molecular System. Proceedings of the Royal Society of London +Series A 1950, 200, 542–554. +(2) McMurchie, L. E.; Davidson, E. R. One- and two-electron integrals over cartesian gaus- +sian functions. Journal of Computational Physics 1978, 26, 218 – 231. +(3) Davidson, E. R.; Feller, D. Basis set selection for molecular calculations. Chemical +Reviews 1986, 86, 681–696. +(4) Huzinaga, S. Basis sets for molecular calculations. Computer Physics Reports 1985, 2, +281 – 339. +(5) Huzinaga, S. Gaussian-Type Functions for Polyatomic Systems. I. The Journal of +Chemical Physics 1965, 42, 1293–1302. +(6) Ruedenberg, K.; Raffenetti, R. C.; Bardo, R. D. Structure and Reactivity, Proceedings +of the 1972 Boulder Conference; Wiley: New York, 1973. +(7) Bardo, R. D.; Ruedenberg, K. Even-tempered atomic orbitals. VI. Optimal orbital +exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and +oxygen in molecules. The Journal of Chemical Physics 1974, 60, 918–931. +(8) Huzinaga, S.; Klobukowski, M.; Tatewaki, H. The well-tempered GTF basis sets and +their applications in the SCF calculations on N2, CO, Na2, and P2. Canadian Journal +of Chemistry 1985, 63, 1812–1828. +(9) Tatewaki, H.; Huzinaga, S. A systematic preparation of new contracted Gaussian type +orbital set. I. Transition metal atoms from Sc to Zn. The Journal of Chemical Physics +1979, 71, 4339–4348. +30 + +(10) Tatewaki, H.; Huzinaga, S. A systematic preparation of new contracted Gaussian-type +orbital sets. III. Second-row atoms from Li through ne. Journal of Computational Chem- +istry 1980, 1, 205–228. +(11) Hehre, W. J.; Stewart, R. F.; Pople, J. A. Self-Consistent Molecular-Orbital Methods. I. +Use of Gaussian Expansions of Slater-Type Atomic Orbitals. The Journal of Chemical +Physics 1969, 51, 2657–2664. +(12) Ditchfield, R.; Hehre, W. J.; Pople, J. A. Self-Consistent Molecular-Orbital Meth- +ods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic +Molecules. The Journal of Chemical Physics 1971, 54, 724–728. +(13) Binkley, J. S.; Pople, J. A.; Hehre, W. J. Self-consistent molecular orbital methods. 21. +Small split-valence basis sets for first-row elements. Journal of the American Chemical +Society 1980, 102, 939–947. +(14) Binkley, J. S.; Pople, J. A. Self-consistent molecular orbital methods. XIX. Split-valence +Gaussian-type basis sets for beryllium. The Journal of Chemical Physics 1977, 66, 879– +880. +(15) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. Self-consistent molecular or- +bital methods. XX. A basis set for correlated wave functions. The Journal of Chemical +Physics 1980, 72, 650–654. +(16) Hariharan, P. C.; Pople, J. A. The influence of polarization functions on molecular +orbital hydrogenation energies. Theoretica chimica acta 1973, 28, 213–222. +(17) Collins, J. B.; von R. Schleyer, P.; Binkley, J. S.; Pople, J. A. Self-consistent molecular +orbital methods. XVII. Geometries and binding energies of second-row molecules. A +comparison of three basis sets. The Journal of Chemical Physics 1976, 64, 5142–5151. +31 + +(18) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; DeFrees, D. J.; +Pople, J. A. Self-consistent molecular orbital methods. XXIII. A polarization-type basis +set for second-row elements. The Journal of Chemical Physics 1982, 77, 3654–3665. +(19) Pietro, W. J.; Francl, M. M.; Hehre, W. J.; DeFrees, D. J.; Pople, J. A.; Binkley, J. S. +Self-consistent molecular orbital methods. 24. Supplemented small split-valence basis +sets for second-row elements. Journal of the American Chemical Society 1982, 104, +5039–5048. +(20) Frisch, M. J.; Pople, J. A.; Binkley, J. S. Self-consistent molecular orbital methods +25. Supplementary functions for Gaussian basis sets. The Journal of Chemical Physics +1984, 80, 3265–3269. +(21) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations.I The +atoms boron through neon and hydrogen. The Journal of Chemical Physics 1989, 90, +1007–1023. +(22) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms +revisited.Systematic basis sets and wave functions. The Journal of Chemical Physics +1992, 96, 6796–6806. +(23) Woon, D. E.; Dunning, T. H. J. The Pronounced Effect of Microsolvation on Diatomic +Alkali Halides: Ab Initio Modeling of MX(H2O)n (M = Li, Na; X=F, Cl; n = 1-3). +Journal of the American Chemical Society 1995, 117, 1090–1097. +(24) Balakina, M.; Nefediev, S. The choice of basis set for calculations of linear and nonlinear +optical properties of conjugated organic molecules in gas and in dielectric medium by the +example of p-nitroaniline. Computational Materials Science 2007, 38, 467–472, Selected +papers from the International Conference on Computational Methods in Sciences and +Engineering 2004. +32 + +(25) Parsons, T.; Balduf, T.; Cheeseman, J. R.; Caricato, M. Basis Set Dependence of +Optical Rotation Calculations with Different Choices of Gauge. The Journal of Physical +Chemistry A 2022, 126, 1861–1870, PMID: 35271772. +(26) Reis, H.; Papadopoulos, M. G. Nonlinear optical properties of the rhombic B4-cluster. +Journal of Computational Chemistry 1999, 20, 679–687. +(27) Lauderdale, W. J.; Coolidge, M. B. Basis set effects on the nonlinear optical properties +of selected linear diacetylenes. The Journal of Physical Chemistry 1995, 99, 9368–9373. +(28) Jabłoński, M.; Palusiak, M. Basis Set and Method Dependence in Quantum Theory of +Atoms in Molecules Calculations for Covalent Bonds. The Journal of Physical Chem- +istry A 2010, 114, 12498–12505, PMID: 21049895. +(29) Frisch, M. J. et al. Gaussian 16 Revision C.01. 2016. +(30) McMurchie, L. E.; Elbert, S. T.; Langhoff, S. R.; Davidson, E. R. MELD package from +Indiana University. It has been modified by us to handle bigger systems. +(31) Shinde, R.; Shukla, A. Large-scale first principles configuration interaction calculations +of optical absorption in aluminum clusters. Phys. Chem. Chem. Phys. 2014, 16, 20714– +20723. +(32) Rai, D. K.; Chakraborty, H.; Shukla, A. Tunable Optoelectronic Properties of Triply +Bonded Carbon Molecules with Linear and Graphyne Substructures. The Journal of +Physical Chemistry C 2018, 122, 1309–1317. +(33) Chakraborty, H.; Shukla, A. Pariser - Parr - Pople Model Based Investigation of Ground +and Low - Lying Excited States of Long Acenes. The Journal of Physical Chemistry A +2013, 117, 14220–14229. +(34) Aryanpour, K.; Shukla, A.; Mazumdar, S. Electron correlations and two-photon states +33 + +in polycyclic aromatic hydrocarbon molecules: A peculiar role of geometry. The Journal +of Chemical Physics 2014, 140, 104301. +(35) Chakraborty, H.; Shukla, A. Theory of triplet optical absorption in oligoacenes: From +naphthalene to heptacene. The Journal of Chemical Physics 2014, 141, 164301. +(36) SHINDE, R.; SHUKLA, A. LARGE-SCALE FIRST PRINCIPLES CONFIGURA- +TION INTERACTION CALCULATIONS OF OPTICAL ABSORPTION IN BORON +CLUSTERS. Nano LIFE 2012, 02, 1240004. +(37) Shukla, A. Correlated theory of triplet photoinduced absorption in phenylene-vinylene +chains. Phys. Rev. B 2002, 65, 125204. +(38) Shukla, A. Theory of nonlinear optical properties of phenyl-substituted polyacetylenes. +Phys. Rev. B 2004, 69, 165218. +(39) Sony, P.; Shukla, A. Large-scale correlated calculations of linear optical absorption and +low-lying excited states of polyacenes: Pariser-Parr-Pople Hamiltonian. Phys. Rev. B +2007, 75, 155208. +(40) Basak, T.; Chakraborty, H.; Shukla, A. Theory of linear optical absorption in diamond- +shaped graphene quantum dots. Phys. Rev. B 2015, 92, 205404. +(41) Priya, P. K.; Rai, D. K.; Shukla, A. Photoabsorption in sodium clusters: first principles +configuration interaction calculations. The European Physical Journal D 2017, 71, 116. +(42) Shinde, R.; Shukla, A. First principles electron-correlated calculations of optical ab- +sorption in magnesium clusters. The European Physical Journal D 2017, 71, 301. +(43) Bhattacharyya, P.; Rai, D. K.; Shukla, A. Systematic First-Principles Configuration- +Interaction Calculations of Linear Optical Absorption Spectra in Silicon Hydrides: +Si2H2n (n = 1-3). The Journal of Physical Chemistry A 2019, 123, 8619–8631. +34 + +(44) Wheeler, S. E.; Sattelmeyer, K. W.; Schleyer, P. v. R.; Schaefer, H. F. Binding energies +of small lithium clusters (Lin) and hydrogenated lithium clusters (LinH). The Journal +of Chemical Physics 2004, 120, 4683–4689. +(45) Florez, E.; Fuentealba, P. A theoretical study of alkali metal atomic clusters: From Lin +to Csn (n = 2-8). International Journal of Quantum Chemistry 2009, 109, 1080–1093. +(46) HUBER, K. P. Molecular Structure Constants of Diatomic molecules. Molecular Spectra +and molecular Structure Constants of Diatomic molecules 1979, +(47) Jones, R. O.; Lichtenstein, A. I.; Hutter, J. Density functional study of structure and +bonding in lithium clusters Lin and their oxides LinO. The Journal of Chemical Physics +1997, 106, 4566–4574. +(48) Srinivas, S.; Jellinek, J. Structural and electronic properties of small beryllium clusters: +A theoretical study. The Journal of Chemical Physics 2004, 121, 7243–7252. +(49) Hanley, L.; Whitten, J. L.; Anderson, S. L. Collision-induced dissociation and ab initio +studies of boron cluster ions: determination of structures and stabilities. The Journal +of Physical Chemistry 1988, 92, 5803–5812. +(50) Hong, X.; Wang, F. TDDFT calculation for photoabsorption spectra of Lin (n=2-11,20) +clusters. Physics Letters A 2011, 375, 1883 – 1888. +(51) Blanc, J.; Broyer, M.; Chevaleyre, J.; Dugourd, P.; Kühling, H.; Labastie, P.; Ul- +bricht, M.; Wolf, J. P.; Wöste, L. High resolution spectroscopy of small metal clusters. +Zeitschrift für Physik D Atoms, Molecules and Clusters 1991, 19, 7–12. +35 + +For Table of Contents Only +36 + +400 +Optical absorption spectra of Li4 +cluster using various basis sets +6-311++G(2d,2p) +6-311++G(3df,3pd) +III +cC-pVDZ +300 +cc-pVTZ +aug-cc-pVDZ +(arb. units) +aug-cc-pVTZ +200 +Intensity +100 +VI +VII +0 +0 +2 +4 +Energy (eV)Supporting Information For: Benchmarking +Gaussian Basis Sets in Quantum-Chemical +Calculations of Photoabsorption Spectra of +Light Atomic Clusters +Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,† +†Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, +India +‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, +Helmholtzstr. 20, 01069 Dresden, Germany +E-mail: mahamiyavikram@gmail.com; pritambhattacharyya01@gmail.com; shukla@phy.iitb.ac.in +Table S1: Comparison of the peak locations of the optical absorption spectra of Li2 cluster +computed using various basis sets. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +Peak VIII +Peak IX +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +1.82 +2.61 +3.85 +4.87 +5.88 +6.50 +7.06 +- +- +6-311++G(3df,3pd) +1.83 +2.57 +3.85 +4.61 +4.84 +5.64 +5.88 +6.08 +7.00 +cc-pVDZ +1.82 +2.65 +4.43 +5.95 +7.12 +- +- +- +- +cc-pVTZ +1.83 +2.58 +4.26 +5.43 +5.98 +6.64 +7.03 +- +- +aug-cc-pVDZ +1.82 +2.60 +3.84 +4.58 +5.06 +5.94 +6.67 +6.96 +- +aug-cc-pVTZ +1.83 +2.57 +3.87 +4.59 +5.43 +5.93 +6.66 +7.03 +- +S1 +arXiv:2301.02413v1 [physics.chem-ph] 6 Jan 2023 + +Table S2: Comparison of the peak locations of the optical absorption spectra of Li3 chain +computed using various basis sets. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +0.72 +1.27 +2.58 +3.39 +3.91 +4.28 +6-311++G(3df,3pd) +0.72 +1.27 +2.54 +3.38 +3.90 +4.14 +cc-pVDZ +0.72 +1.26 +2.65 +3.47 +4.37 +4.92 +cc-pVTZ +0.72 +1.28 +2.57 +3.45 +4.39 +4.58 +aug-cc-pVDZ +0.72 +1.26 +2.56 +3.38 +3.88 +- +aug-cc-pVTZ +0.72 +1.27 +2.53 +3.37 +3.89 +- +Table S3: The peak locations of the optical absorption spectra of Li3 isosceles triangular +cluster computed using various basis sets are compared. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +1.09 +1.41 +2.14 +2.41 +2.66 +2.97 +3.21 +6-311++G(3df,3pd) +1.07 +1.42 +2.12 +2.39 +2.65 +2.96 +3.19 +cc-pVDZ +1.11 +1.40 +2.18 +2.43 +2.61 +2.81 +3.13 +cc-pVTZ +1.08 +1.42 +2.15 +2.40 +2.70 +3.02 +4.27 +aug-cc-pVDZ +1.09 +1.41 +2.13 +2.40 +2.65 +2.96 +3.20 +aug-cc-pVTZ +1.07 +1.42 +2.11 +2.43 +2.65 +2.95 +3.20 +Table S4: High energy peak locations of the optical absorption spectra of Li3 Isosceles +triangular cluster computed using various basis sets are compared. +Li3 Cluster +Peak VIII +Peak IX +Peak X +Peak XI +Peak XII +Basis Set +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +3.77 +4.25 +5.29 +5.99 +- +6-311++G(3df,3pd) +3.75 +4.17 +4.91 +5.25 +5.79 +cc-pVDZ +3.57 +4.24 +4.94 +5.33 +6.05 +cc-pVTZ +4.69 +5.64 +- +- +- +aug-cc-pVDZ +4.16 +4.44 +5.35 +5.59 +5.94 +aug-cc-pVTZ +3.77 +4.11 +5.39 +5.60 +- +S2 + +Table S5: The peak locations of the optical absorption spectra of Li4 rhombus cluster com- +puted using various basis sets. The higher energy peak locations are presented in the Table +I of the Supporting Information. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +1.84 +2.65 +3.05 +3.64 +4.24 +4.52 +5.15 +6-311++G(3df,3pd) +1.83 +2.64 +2.93 +3.63 +4.21 +4.51 +5.11 +cc-pVDZ +1.87 +2.67 +3.08 +3.65 +4.27 +4.86 +5.42 +cc-pVTZ +1.84 +2.65 +3.03 +3.64 +4.22 +4.54 +5.30 +aug-cc-pVDZ +1.85 +2.65 +3.05 +3.61 +4.23 +4.62 +- +aug-cc-pVTZ +1.87 +2.65 +2.93 +3.66 +4.22 +4.61 +5.30 +Table S6: The peak locations of the optical absorption spectra of Be+ +2 cluster computed +using various basis sets are compared. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +Peak VIII +Peak IX +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +1.75 +3.66 +4.18 +6.04 +6.32 +8.32 +8.90 +9.33 +9.65 +6-311++G(3df,3pd) +1.74 +3.68 +4.18 +6.04 +6.30 +8.33 +8.84 +9.31 +9.63 +cc-pVDZ +1.76 +3.69 +4.20 +6.12 +6.37 +8.28 +9.61 +- +- +cc-pVTZ +1.74 +3.68 +4.19 +6.05 +6.31 +8.44 +9.50 +- +- +aug-cc-pVDZ +1.77 +3.69 +4.18 +6.08 +6.36 +8.38 +9.09 +9.47 +- +aug-cc-pVTZ +1.74 +3.68 +4.18 +6.03 +6.30 +8.34 +8.92 +9.38 +- +Table S7: The peak locations of the optical absorption spectra of Be+ +3 cluster computed +using various basis sets are compared. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +1.05 +3.16 +3.66 +4.90 +5.39 +5.86 +6.53 +6-311++G(3df,3pd) +1.02 +3.17 +3.63 +4.84 +5.40 +5.83 +6.47 +cc-pVDZ +1.04 +3.16 +3.67 +4.91 +5.39 +5.89 +6.52 +cc-pVTZ +1.01 +3.14 +3.60 +4.87 +5.37 +5.83 +6.50 +aug-cc-pVDZ +1.02 +3.16 +3.66 +4.87 +5.41 +5.87 +6.51 +aug-cc-pVTZ +1.02 +3.16 +3.66 +4.87 +5.40 +5.85 +6.52 +S3 + +Table S8: The peak locations of the optical absorption spectra of B+ +2 cluster computed using +various basis sets are compared. +Basis Set +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +Peak VIII +Peak IX +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +3.65 +4.86 +6.01 +7.00 +7.63 +9.68 +10.26 +11.42 +12.71 +6-311++G(3df,3pd) +3.65 +4.77 +5.98 +7.14 +7.62 +9.65 +10.22 +11.40 +12.61 +cc-pVDZ +3.64 +4.79 +6.03 +7.16 +- +9.88 +11.29 +12.74 +cc-pVTZ +3.66 +4.80 +5.99 +7.05 +7.63 +9.87 +11.12 +12.77 +- +aug-cc-pVDZ +3.63 +4.78 +5.98 +7.03 +7.60 +9.66 +10.27 +11.31 +12.52 +aug-cc-pVTZ +3.66 +4.80 +5.99 +7.04 +7.65 +9.65 +10.21 +11.41 +12.20 +Table S9: The peak locations of the optical absorption spectra of B+ +3 cluster computed using +various basis sets are compared. +B+ +3 Cluster +Peak I +Peak II +Peak III +Peak IV +Peak V +Peak VI +Peak VII +Peak VIII +Basis Set +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +0.82 +3.24 +5.00 +5.38 +6.02 +7.12 +8.09 +8.85 +6-311++G(3df,3pd) +0.83 +3.21 +4.99 +5.44 +6.06 +7.17 +8.07 +8.87 +cc-pVDZ +0.96 +3.28 +5.17 +5.59 +6.11 +7.39 +8.21 +8.95 +cc-pVTZ +0.82 +3.23 +4.98 +5.35 +6.03 +7.01 +8.06 +8.79 +aug-cc-pVDZ +0.84 +3.22 +5.00 +5.41 +6.02 +7.10 +8.03 +8.80 +aug-cc-pVTZ +0.84 +3.22 +5.01 +5.45 +6.03 +7.16 +8.08 +8.85 +Table S10: The peak locations of the optical absorption spectra of B+ +3 cluster employing TD- +DFT method with B3LYP functional and computed using various basis sets are compared. +B+ +3 Cluster +Peak I +Peak II +Peak III +Peak IV +Basis Set +(eV) +(eV) +(eV) +(eV) +6-311++G(2d,2p) +0.94 +2.95 +5.70 +9.35 +6-311++G(3df,3pd) +0.95 +2.94 +5.70 +9.35 +cc-pVDZ +0.96 +3.00 +5.73 +9.52 +cc-pVTZ +0.96 +2.95 +5.70 +9.40 +aug-cc-pVDZ +0.96 +2.99 +5.73 +9.40 +aug-cc-pVTZ +0.95 +2.94 +5.70 +9.36 +S4 + +Table S11: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak III) +of Li3 chain calculated using different basis sets. In the “Polarization” column, ∥ indicates +photon polarization along the direction of the molecule (longitudinal polarization), while ⊥ +indicates polarization perpendicular to the molecular axis (transverse polarization). ’H’ and +’L’ stand for HOMO and LUMO orbitals. +Basis Set +Peak I +Peak III +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.106 +|H − 1 → H⟩ +⊥ +0.591 +|H − 1 → L + 2⟩ +|H → L + 6⟩ +6-311++G(3df,3pd) +∥ +0.106 +|H − 1 → H⟩ +⊥ +0.595 +|H − 1 → L + 2⟩ +|H → L + 5⟩ +|H → L + 11⟩ +cc-PVDZ +∥ +0.097 +|H − 1 → H⟩ +⊥ +0.631 +|H − 1 → L + 1⟩ +|H → L⟩ +|H → L + 3⟩ +cc-pVTZ +∥ +0.105 +|H − 1 → H⟩ +⊥ +0.618 +|H − 1 → L + 1⟩ +|H → L⟩ +|H → L + 3⟩ +aug-cc-pVDZ +∥ +0.101 +|H − 1 → H⟩ +⊥ +0.603 +|H − 1 → L + 2⟩ +|H → L + 7⟩ +|H → L + 10⟩ +aug-cc-pVTZ +∥ +0.106 +|HF⟩ +⊥ +0.594 +|H − 1 → L + 2⟩ +|H − 1 → H; +|H − 1 → H; +H − 1 → L + 5⟩ +H − 1 → L + 11⟩ +Table S12: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV) +of Li3 triangular calculated using different basis sets. The rest of the information is same as +in the caption of Table S11. +Basis Set +Peak I +Peak IV +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.127 +|H → L + 14⟩ +∥ +0.465 +|H − 1 → L⟩ +|H → L + 1⟩ +|H → L + 9⟩ +6-311++G(3df,3pd) +∥ +0.131 +|H → L + 14⟩ +∥ +0.459 +|H − 1 → L⟩ +|H → L + 1⟩ +|H → L + 5⟩ +cc-PVDZ +∥ +0.121 +|H → L + 2⟩ +∥ +0.488 +|H − 1 → L⟩ +|H − 1 → L + 2⟩ +|H → L + 4⟩ +cc-pVTZ +∥ +0.129 +|H → L + 2⟩ +∥ +0.478 +|H − 1 → L⟩ +|H − 1 → L + 2⟩ +|H → L + 4⟩ +aug-cc-pVDZ +∥ +0.126 +|H → L + 14⟩ +∥ +0.469 +|H − 1 → L⟩ +|H → L + 10⟩ +|H → L + 16⟩ +aug-cc-pVTZ +∥ +0.129 +|H → L + 13⟩ +∥ +0.462 +|H − 1 → L⟩ +|H → L + 1⟩ +|H → L + 5⟩ +S5 + +Table S13: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak II) +of Li4 cluster calculated using different basis sets. The rest of the information is same as in +the caption of Table S11. +Basis Set +Peak I +Peak II +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.628 +|H → L + 1⟩ +∥ +0.648 +|H − 1 → L⟩ +|H → L + 8⟩ +|H − 1 → L + 5⟩ +6-311++G(3df,3pd) +∥ +0.615 +|H → L + 1⟩ +∥ +0.683 +|H − 1 → L⟩ +|H → L + 9⟩ +|H − 1 → L + 6⟩ +cc-PVDZ +∥ +0.659 +|H → L + 1⟩ +∥ +0.649 +|H − 1 → L⟩ +|H → L + 4⟩ +|H → L + 3⟩ +cc-pVTZ +∥ +0.624 +|H → L + 1⟩ +∥ +0.678 +|H − 1 → L⟩ +|H → L + 5⟩ +|H → L + 4⟩ +aug-cc-pVDZ +∥ +0.634 +|H → L + 1⟩ +∥ +0.654 +|H − 1 → L⟩ +|H → L + 9⟩ +|H − 1 → L + 5⟩ +aug-cc-pVTZ +∥ +0.656 +|H → L + 1⟩ +∥ +0.660 +|H − 1 → L⟩ +|H → L + 9⟩ +|H − 1 → L + 5⟩ +Table S14: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak V) +of Be+ +2 cluster calculated using different basis sets. The rest of the information is same as in +the caption of Table S11 +Basis Set +Peak I +Peak V +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.113 +|H → L⟩ +⊥ +0.745 +|H − 1 → L + 1⟩ +|H − 1 → H⟩ +|H − 1 → L; H → L + 2⟩ +6-311++G(3df,3pd) +∥ +0.119 +|H → L⟩ +⊥ +0.749 +|H − 1 → L + 1⟩ +|H − 1 → H⟩ +|H − 1 → L; H → L + 2⟩ +cc-PVDZ +∥ +0.114 +|H → L + 2⟩ +⊥ +0.736 +|H − 1 → L⟩ +|H → L; H − 1 → L⟩ +|H − 1 → L + 2; H → L + 3⟩ +cc-pVTZ +∥ +0.120 +|H → L + 2⟩ +⊥ +0.749 +|H − 1 → L⟩ +|H → L; H − 1 → L⟩ +|H − 1 → L + 2; H → L + 3⟩ +aug-cc-pVDZ +∥ +0.114 +|H → L + 2⟩ +⊥ +0.768 +|H − 1 → L⟩ +|H → L; H − 1 → L⟩ +|H − 1 → L + 2; H → L + 3⟩ +aug-cc-pVTZ +∥ +0.120 +|H → L + 2⟩ +⊥ +0.757 +|H − 1 → L⟩ +|H → L; H − 1 → L⟩ +|H − 1 → L + 2; H → L + 3⟩ +S6 + +Table S15: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak VII) +of Be+ +3 cluster calculated using different basis sets. The rest of the information is same as in +the caption of Table S11 +Basis Set +Peak I +Peak VII +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.172 +|HF⟩ +⊥ +0.655 +|H − 2 → L + 1⟩ +|H → L⟩ +|H − 1 → L + 2⟩ +6-311++G(3df,3pd) +∥ +0.184 +|H → L + 1⟩ +⊥ +0.647 +|H − 2 → L; H → L + 2⟩ +|H − 1 → L; H → L⟩ +|H − 1 → L; H → L + 4⟩ +cc-PVDZ +∥ +0.178 +|H → L⟩ +⊥ +0.640 +|H − 2 → L + 1⟩ +|H − 1 → H⟩ +|H − 1 → L + 2⟩ +cc-pVTZ +∥ +0.169 +|HF⟩ +⊥ +0.635 +|H − 2 → L; H → L + 1⟩ +|H − 1 → L; H → L⟩ +|H − 1 → L; H → L + 2⟩ +aug-cc-pVDZ +∥ +0.180 +|H → L + 1⟩ +⊥ +0.646 +|H − 2 → L; H → L + 2⟩ +|H − 1 → L; H → L⟩ +|H − 1 → L; H → L + 4⟩ +aug-cc-pVTZ +∥ +0.179 +|H → L + 1⟩ +⊥ +0.657 +|H − 2 → L; H → L + 2⟩ +|H − 1 → L; H → L⟩ +|H − 1 → L; H → L + 4⟩ +Table S16: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV) +of B+ +2 cluster calculated using different basis sets. The rest of the information is same as in +the caption of Table S11 +Basis Set +Peak I +Peak IV +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +∥ +0.026 +|H − 1 → L; H → L⟩ +∥ +0.910 +|H → L + 5⟩ +|H → L + 11⟩ +|H → L + 11⟩ +6-311++G(3df,3pd) +∥ +0.025 +|H − 1 → L; H → L⟩ +∥ +0.915 +|H → L + 5⟩ +|H → L + 11⟩ +|H → L + 11⟩ +cc-PVDZ +∥ +0.025 +|H − 1 → L; H → L⟩ +∥ +0.965 +|H → L + 5⟩ +|H → L + 5⟩ +|H − 1 → L; H → L⟩ +cc-pVTZ +∥ +0.024 +|H − 1 → L; H → L⟩ +∥ +0.942 +|H → L + 5⟩ +|H → L + 5⟩ +|H − 1 → L; H → L⟩ +aug-cc-pVDZ +∥ +0.028 +|H − 1 → L; H → L⟩ +∥ +0.895 +|H → L + 11⟩ +|H → L + 11⟩ +|H → L + 5⟩ +aug-cc-pVTZ +∥ +0.026 +|H − 1 → L; H → L⟩ +∥ +0.918 +|H → L + 11⟩ +|H → L + 11⟩ +|H → L + 5⟩ +S7 + +Table S17: Comparison of oscillator strengths and the dominant configurations contributing +to the many-particle wave functions for peaks I and the maximum intensity peak (peak VIII) +of B+ +3 cluster calculated using different basis sets. The rest of the information is same as in +the caption of Table S11 +Basis Set +Peak I +Peak VIII +Polarization +f +Wave-function +Polarization +f +Wave-function +6-311++G(2d,2p) +⊥ +0.005 +|H → L⟩ +∥ +0.508 +|H − 1 → L; H → L + 3⟩ +|H − 2 → L; H → L + 1⟩ +|H − 1 → L; H → L + 2⟩ +6-311++G(3df,3pd) +⊥ +0.005 +|H → L⟩ +∥ +0.514 +|H − 1 → L; H → L + 3⟩ +|H − 1 → L; H → L + 1⟩ +|H − 1 → L; H → L + 2⟩ +cc-PVDZ +⊥ +0.007 +|H → L⟩ +∥ +0.480 +|H − 1 → L; H → L + 3⟩ +|H − 1 → L; H → L + 1⟩ +|H − 1 → L; H → L + 3⟩ +cc-pVTZ +⊥ +0.005 +|H → L⟩ +∥ +0.491 +|H − 1 → L; H → L + 3⟩ +|H − 1 → L; H → L + 1⟩ +|H − 1 → L; H → L + 2⟩ +aug-cc-pVDZ +⊥ +0.005 +|H → L⟩ +∥ +0.467 +|H − 1 → L; H → L + 3⟩ +|H − 1 → L; H → L + 2⟩ +|H − 1 → L; H → L + 3⟩ +aug-cc-pVTZ +⊥ +0.006 +|H → L⟩ +∥ +0.519 +|H − 1 → L; H → L + 3⟩ +|H − 1 → L; H → L + 2⟩ +|H − 1 → L; H → L + 3⟩ +S8 + +Table S18: Many-particle wave functions of excited states contributing to the peaks in +the optical absorption spectrum of Li2 cluster for aug-cc-pVTZ basis set. ’E’ corresponds +to excitation energy (in eV) of an excited state, f denotes the oscillator strength for a +particular electric dipole transition. In the “|TDM|” column,we present the magnitudes of the +transition dipole moments (TDMs) to understand the extent of coupling between the relevant +excited state and the ground state. ∥ indicates photon polarization along the direction of +the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to the +molecular axis (transverse polarization). ’H’ and ’L’ stand for HOMO and LUMO orbitals. +In the “Wave function” column, each number inside the parentheses denotes the coefficient +of the corresponding configuration in the CI wave function. GS indicates the ground states +wave function of the cluster. +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|HF⟩(0.9520) +|H → L + 7; H → L + 15⟩(0.0879) +I +1.83 +0.454 +3.184 +∥ +|H → L⟩(0.7523) +|H → L + 3⟩(0.3959) +II +2.57 +0.966 +2.770 +⊥ +|H → L + 2⟩(0.7046) +|H → L + 7⟩(0.5914) +III +3.87 +0.049 +0.509 +⊥ +|H → L + 7⟩(0.6291) +|H → L + 2⟩(0.5439) +V +5.42 +0.024 +0.303 +⊥ +|H → L + 16⟩(0.8509) +|H → L + 8; H → L + 16⟩(0.1702) +VI +5.93 +0.026 +0.421 +∥ +|H → L + 12⟩(0.2911) +|H → L; H → L + 8⟩(0.2523) +S9 + +Table S19: Many-particle wave functions of excited states contributing to the peaks in the +optical absorption spectrum of Li3 linear cluster for aug-cc-pVTZ basis set. The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|H − 1 → H⟩(0.9246) +|H − 1 → L + 13⟩(0.0919) +I +0.72 +0.106 +2.453 +∥ +|HF⟩(0.8814) +|H − 1 → H; H − 1 → L + 5⟩(0.1436) +II +1.27 +0.503 +4.023 +∥ +|H − 1 → H; H − 1 → L⟩(0.5569) +|H − 1 → H; H − 1 → L + 5⟩(0.5102) +III +2.53 +0.594 +3.096 +⊥ +|H − 1 → L + 2⟩(0.6262) +|H − 1 → H; H − 1 → L + 11⟩(0.3628) +IV +3.37 +0.215 +1.141 +⊥ +|H − 1 → L + 7⟩(0.4255) +|H − 1 → L + 14⟩(0.4237) +V +3.89 +0.011 +0.343 +∥ +|H − 1 → L + 8⟩(0.4214) +|H − 1 → H; H − 1 → L + 13⟩(0.3520) +S10 + +Table S20: Many-particle wave functions of excited states contributing to the peaks in the +optical absorption spectrum of Li3 isosceles triangular cluster for aug-cc-pVTZ basis set. +The rest of the information is same as in the caption of TableS18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|HF⟩(0.9102) +|H − 1 → H⟩(0.0933) +I +1.07 +0.129 +2.222 +∥ +|H → L + 13⟩(0.4241) +|H → L + 1⟩(0.4114) +II +1.42 +0.020 +0.767 +∥ +|H → L + 16⟩(0.5005) +|H − 1 → L⟩(0.4115) +III +2.11 +0.352 +2.611 +∥ +|H − 1 → H⟩(0.5806) +|H − 1 → L + 15⟩(0.2483) +IV +2.43 +0.462 +2.885 +∥ +|H − 1 → L⟩(0.5537) +|H → L + 5⟩(0.3869) +V +2.65 +0.088 +1.163 +⊥ +|H → L + 2⟩(0.5541) +|H → L + 12⟩(0.3375) +VI +2.95 +0.267 +1.922 +⊥ +|H − 1 → L + 2⟩(0.4766) +|H − 1 → L + 12⟩(0.4269) +VII +3.20 +0.117 +1.223 +⊥ +|H − 1 → L + 2⟩(0.3983) +|H → L + 12⟩(0.3579) +VIII +3.77 +0.016 +0.417 +∥ +|H → L + 10⟩(0.3841) +|H → L + 19⟩(0.3349) +IX +4.11 +0.029 +0.533 +∥ +|H − 1 → L + 14⟩(0.4375) +|H − 1 → L⟩(0.3910) +X +5.39 +0.004 +0.171 +∥ +|H → L + 35⟩(0.2813) +|H → L + 37⟩(0.2744) +S11 + +Table S21: Many-particle wave functions of excited states contributing to the peaks in +the optical absorption spectrum of Li4 cluster for aug-cc-pVTZ basis set. The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|HF⟩(0.8913) +|(H − 1) → L; H → L + 18⟩(0.0791) +I +1.87 +0.656 +3.785 +∥ +|H → L + 1⟩(0.5922) +|H → L + 9⟩(0.4041) +II +2.65 +0.660 +3.188 +∥ +|H − 1 → L⟩(0.6193) +|H − 1 → L + 5⟩(0.2910) +III +2.93 +0.316 +2.098 +⊥ +|H → L + 7⟩(0.4577) +|H → L + 20⟩(0.4174) +IV +3.43 +0.045 +0.736 +⊥ +|H − 1 → L + 3⟩(0.3080) +|H → L + 7⟩(0.2299) +V +3.66 +0.103 +1.070 +∥ +|H → L + 6⟩(0.5653) +|H → L + 18⟩(0.3856) +VI +4.22 +0.077 +0.862 +⊥ +|H − 1 → L + 3⟩(0.2380) +|H → L + 20; H → L + 21⟩(0.2227) +VII +4.61 +0.063 +0.746 +⊥ +|H − 1 → L + 3⟩(0.4044) +|H − 1 → L + 12⟩(0.4026) +VIII +5.30 +0.012 +0.300 +∥ +|H → L + 33⟩(0.3681) +|H → L; H → L + 6⟩(0.2294) +Table S22: Many-particle wave functions of excited states contributing to the peaks in the +optical absorption spectrum of Be+ +2 cluster for aug-cc-pVTZ basis set. +The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|H → L⟩(0.9351) +|H − 1 → L; H → L + 2⟩(0.1684) +I +1.74 +0.120 +1.680 +∥ +|H → L + 2⟩(0.8403) +|H − 1 → L; H → L⟩(0.3820) +II +3.67 +0.121 +0.821 +⊥ +|H → L + 3⟩(0.6705) +|H → L + 4⟩(0.6375) +III +4.19 +0.386 +1.940 +∥ +|H − 1 → L; H → L⟩(0.7268) +|H → L + 2⟩(0.3897) +IV +6.01 +0.201 +0.827 +⊥ +|H − 1 → L⟩(0.6418) +|H − 1 → L; H → L + 1⟩(0.6116) +V +6.30 +0.757 +1.566 +⊥ +|H − 1 → L⟩(0.8762) +|H − 1 → L + 2; H → L + 3⟩(0.8573) +S12 + +Table S23: Many-particle wave functions of excited states contributing to the peaks in the +optical absorption spectrum of Be+ +3 cluster for aug-cc-pVTZ basis set. +The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|H → L⟩(.8776) +|H − 1 → L + 1; H → L⟩(.1913) +I +1.02 +0.179 +2.678702 +∥ +|H → L + 1⟩(0.8200) +|H − 1 → L; H → L⟩(0.3257) +II +3.16 +0.433 +2.365586 +∥ +|H − 1 → L; H → L⟩(0.6152) +|H − 1 → L + 1; H → L + 1⟩(0.4714) +III +3.66 +0.028 +0.563211 +⊥ +|H − 2 → L; H → L + 2⟩(0.5209) +|H → L + 17⟩(0.4125) +IV +4.87 +0.020 +0.407823 +⊥ +|H − 1 → L + 1; H → L + 2⟩(0.5211) +|H → L + 17⟩(0.3217) +V +5.40 +0.204 +1.242734 +∥ +|H − 2 → L; H → L + 1⟩(0.6337) +|H − 1 → L; H → L⟩(0.3104) +VI +5.85 +0.054 +0.612656 +⊥ +|H − 1 → L; H → L + 4⟩(0.4315) +|H − 1 → L; H − 1 → L + 2; H → L⟩(0.2864) +VII +6.52 +0.657 +2.028291 +⊥ +|H − 2 → L; H → L + 2⟩(0.4960) +|H − 1 → L; H → L + 4⟩(0.4653) +Table S24: Many-particle wave functions of excited states contributing to the peaks in +the optical absorption spectrum of B+ +2 cluster for aug-cc-pVTZ basis set. The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|H → L⟩(.9033) +|H − 2 → L; H − 1 → L + 2⟩(.1271) +I +3.66 +0.026 +0.537 +∥ +|H − 1 → L; H → L⟩(0.7250) +|H → L + 11⟩(0.2622) +II +4.80 +0.029 +0.496 +∥ +|H − 1 → L + 1; H → L + 1⟩(0.5092) +|H − 1 → H⟩(0.4944) +III +5.99 +0.019 +0.364 +⊥ +|H − 1 → L; H → L + 2⟩(0.6138) +|H − 2 → L⟩(0.4445) +IV +7.04 +0.918 +2.307 +∥ +|H → L + 11⟩(0.4808) +|H → L + 5⟩(0.4713) +V +7.65 +0.009 +0.216 +⊥ +|H − 2 → L⟩(0.5322) +|H − 1 → L; H → L + 2⟩(0.4844) +S13 + +Table S25: Many-particle wave functions of excited states contributing to the peaks in +the optical absorption spectrum of B+ +3 cluster for aug-cc-pVTZ basis set. The rest of the +information is same as in the caption of Table S18 +Peak +E (eV) +f +|TDM| +Polarization +Wave function +GS +|HF⟩(0.8535) +|H − 1 → L + 1⟩(0.1393) +I +0.84 +0.006 +0.524 +⊥ +|H → L⟩(0.8572) +|H − 1 → L; H → L + 2⟩(0.1319) +II +3.22 +0.083 +0.726 +∥ +|H − 1 → L⟩(0.8246) +|H − 1 → L; H − 1 → L + 1⟩(0.1642) +III +5.01 +0.053 +0.463 +∥ +|H − 1 → L; H − 1 → L⟩(0.5482) +|H → L; H → L + 1⟩(0.3184) +IV +5.45 +0.013 +0.217 +∥ +|H → L; H → L + 1⟩(0.5721) +|H → L; H → L + 2⟩(0.5675) +V +6.03 +0.026 +0.295 +∥ +|H → L + 3⟩(0.3937) +|H − 1 → L + 2⟩(0.3723) +VI +7.16 +0.026 +0.382 +∥ +|H → L + 1; H → L + 2⟩(0.5866) +|H → L; H → L + 1⟩(0.4308) +VII +8.08 +0.059 +0.380 +∥ +|H − 1 → L; H − 1 → L + 1⟩(0.4529) +|H − 1 → L; H − 1 → L + 2⟩(0.3162) +VIII +8.85 +0.519 +1.091 +∥ +|H → L + 3; H − 1 → L⟩(0.3762) +|H → L + 3; H − 1 → L⟩(0.3748) +Figure S1: Validation of frozen core approximation for the optical absorption spectra of Li2 +cluster employing QCI method. +S14 + +400 +With core-electrons +Frozen core approximated +300 +(arb. units) +200 +Intensity +100 +0 +0 +2 +6 +8 +10 +Energy (eV)Figure S2: Validation of frozen core approximation for the optical absorption spectra of Be+ +2 +cluster employing QCI method. +Figure S3: Optical absorption spectra of B+ +3 cluster computed using various basis sets em- +ploying B3LYP functional and TD-DFT method. +S15 + +400 +With core-electrons +Frozen core approximated +300 +(arb. units) +200 +Intensity +100 +0 +0 +2 +4 +6 +8 +10 +Energy (eV)250 +IV +6-311++G(2d,2p) +6-311++G(3df,3pd) +200 +cc-pVDZ +cc-pVTZ +alug-cc-pVDZ +Intensity (arb. units) +aug-cc-pVTZ +150 +100 +50 +II +III +II +0 +0 +1 +2 +3 +4 +5 +6 +8 +9 +10 +Energy (eV) \ No newline at end of file diff --git a/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/load_file.txt b/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..806a1b24081cb96e1fa50c77464a1ad85cfa07c9 --- /dev/null +++ b/CNE0T4oBgHgl3EQfgAHQ/content/tmp_files/load_file.txt @@ -0,0 +1,1908 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf,len=1907 +page_content='Benchmarking Gaussian Basis Sets in Quantum-Chemical Calculations of Photoabsorption Spectra of Light Atomic Clusters Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,† †Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India ‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, Helmholtzstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 20, 01069 Dresden, Germany E-mail: mahamiyavikram@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' pritambhattacharyya01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' shukla@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='in Abstract The choice of Gaussian basis functions for computing the ground-state properties of molecules, and clusters, employing wave-function-based electron-correlated approaches, is a well-studied subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, the same cannot be said when it comes to the excited-state properties of such systems, in general, and optical properties, in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The aim of the present study is to understand how the choice of basis functions affects the calculations of linear optical absorption in clusters, qualitatively, and quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For this purpose, we have calculated linear optical absorption spectra of several small charged and neutral clusters, namely, Li2, Li3, Li4, B+ 2 , B+ 3 , Be+ 2 , and Be+ 3 , using a variety of Gaussian basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The calculations were performed within the frozen-core approximation, and a rigorous account of electron correlation effects in the valence 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02413v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='chem-ph] 6 Jan 2023 sector was taken by employing various levels of configuration interaction (CI) approach both for the ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Our results on the peak locations in the absorption spectra of Li3 and Li4 are in very good agreement with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Our general recommendation is that for excited-state calculations, it is very important to utilize those basis sets which contain augmented functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Relatively smaller aug- cc-pVDZ basis sets also yield high-quality results for photoabsorption spectra, and are recommended for such calculations if the computational resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Introduction Gaussian basis functions (GBFs) were initially proposed by Boys for use in computational atomic and molecular quantum mechanics,1 and over the years have become the preferred basis functions in quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 The reason behind the popularity of GBFs is the so-called Gaussian product theorem1,2 which allows for analytical results for the expressions of multi-center integrals involving various physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Nevertheless, one has to be always careful about various convergence related issues when using GBFs, because, unlike Slater basis functions, they do not exhibit correct asymptotic behavior far away from the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This generally leads to the requirement that a large number of GBFs should be used to achieve convergence, leading to huge memory and CPU-time requirements because the required number of integrals scale as ≈ N 4, where N is the total number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Keeping this in mind, several groups have studied the convergence properties of GBFs over the years, and have come up with schemes to balance accuracy with the computational effort (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3,4 for comprehensive reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Huzinaga was one of the earliest researchers to optimize GBFs for Hartree-Fock (HF) calculations on atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5 Ruedenberg and coworkers devised the so-called even-tempered basis set,6,7 while Huzinaga and coworkers developed well-tempered basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8 Huzinaga and coworkers further developed several contracted basis sets,9,10 discussed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4 Pople and coworkers developed a large number of basis sets3,4 which enjoy continued popularity even in present 2 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' One of the most popular minimal basis sets introduced by Pople and coworkers is STO- 3G contracted basis set,11 whose purpose was to emulate Slater-type orbitals, using GTFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Split-valence basis sets are among the most popular extended basis sets introduced by Pople et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=',12–15 in which for inner shells contracted minimal basis functions are used, but for the valence shells a split set of basis functions is employed, which consist of both contracted and primitive GTFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Depending upon the contraction schemes, these basis sets were given names such as 3-21G,13 4-31G,12 6-21G,14 and 6-311G15, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople and coworkers also proposed further enlarged basis sets containing polarization and diffuse functions of higher angular momenta, which have since become popular choices in quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='15–20 Dunning and coworkers introduced a series of extended basis sets, called “correlation-consistent” (CC) basis sets, which are of varying sizes, containing both polarization and diffuse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='21–23 The basic idea behind these CC basis sets is that they recover a significant amount of electron correlation energy in post-Hartree-Fock treatments of corresponding atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In addition to the basis sets mentioned here, numerous other sets of basis functions have been developed over the years, for which we refer the reader to review articles by Davidson and Feller,3 and Huzinaga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4 Even though so many basis sets have been developed by numerous groups, in most of the reports the criteria for their selection appears to be driven by a good description of the ground state energies of the atoms involved either at the Hartree-Fock level, or in electron-correlated calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3,4 Previously, Balakina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='24 have explored the basis set dependence of the linear and non-linear optical properties of conjugated organic molecule p-nitroaniline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' They reported that the [4s3p2d/3s] basis set also provides similar results as aug-cc-pVDZ basis set for the calculations of (hyper)polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='25 have explored the basis set dependence of optical rotation calculations of various types of gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' They found that the origin-invariant length gauge (LG-OI) gauge with aug-cc-pVTZ basis set provides a balance of cost and accuracy for DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Reis and Papadopoulos26 reported that the inclusion of f-functions in the Dunning’s basis sets does not have a large effect on the 3 electric properties of B4 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Lauderdale and Coolidge27 have explored the effect of basis sets on the non-linear optical properties (hyperpolarizabilities) of linear diacetylenes using time-dependent Hartree-Fock theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' They found that the inclusion of a diffuse ‘d’ function to a standard double-zeta plus polarization basis can significantly improve the frequency- dependent hyperpolarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Jabłonski and Palusiak28 have explored the influence of basis sets in Hartree-Fock (HF) and DFT/B3LYP calculations for the values atoms in molecules (AIM) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' They found that smaller Dunning’s basis sets, including cc-pVDZ and aug-cc-pVDZ provide poor results as compared to medium-sized Pople-type basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We are not aware of a systematic study in which the basis sets have been examined from the perspective of their performance in excited state calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Furthermore, we have also not come across a study which examines the basis sets from the point of view their ability to compute optical properties of atoms and molecules, which involves calculations of transition dipole moments, in addition to excited state energies, and wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In order to fill this void, we decided to undertake a systematic investigation of the influence of basis sets on the qualitative and quantitative description of optical absorption spectra of atomic clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In this paper, we have performed calculations of linear optical absorption spectra of several small neutral and cationic clusters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=', Li2, Li3, Li4, B+ 2 , B+ 3 , Be+ 2 , and Be+ 3 , using the configuration-interaction (CI) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For this purpose, a number of basis sets, namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug- cc-pVTZ, were employed, and their influence on the convergence of excited state energies, wave functions, and transition dipole moments has been systematically examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In this study, the reason behind our choice of smaller sized atomic clusters and their ions, as against larger ones, is that it is possible to perform highly accurate CI calculations on smaller systems so that the difference between results obtained with different basis sets will be due the nature of basis sets, and not due to the CI approach employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Based upon our calculations, the main conclusion is that it is very important to include diffuse basis functions in the basis set in order to obtain a good description of the photoabsorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 4 Theoretical Approach and Computational Details General Methodology All the calculations were performed using the first-principles wave-function-based electron- correlated approaches, using the standard Hamiltonian within the Born-Oppenheimer ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The molecular orbitals are expressed in terms of the linear combination of Cartesian-Gaussian type basis functions, also called atomic orbitals (AOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Although, for such calculations, a number of program packages are available, we employed GAUSSIAN1629 and MELD30 for our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The geometries of all the clusters considered in this work were optimized using GAUSSIAN16 package29 at the coupled-clusters singles-double (CCSD) level of theory, employing a large augmented correlation-consistent polarized valence triple- zeta (aug-cc-pVTZ) basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We perform excited-states calculations for various clusters employing their ground-state optimized geometries, using the configuration-interaction (CI) methodology at various levels of approximation, as implemented in the program package MELD30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The CI calculations yield the vertical excitation energies, the ground and excited state wave functions, and the transition dipole matrix elements connecting the ground and the excited states, which, in turn, are used to compute the optical absorption spectra of various clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The level of CI employed in the calculations depends on the size of cluster, the number of valence electrons in cluster, and the number of active orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The linear optical absorption spectra of Li2, Li3, Li4, Be+ 2 , B+ 2 clusters were computed at the full CI (FCI) level, while for Be+ 3 and B+ 3 calculations were performed at the quadruple CI (QCI), and the multi-reference singles-doubles CI (MRSDCI) levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We start the calculations on a given cluster by first performing restricted Hartree-Fock (RHF) calculations on it, and obtain the molecular orbitals (MOs), expressed as linear com- binations of the chosen AOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In order to perform CI calculations, the one- and two-electron Hamiltonian matrix elements are transformed from the AO representation to the MO rep- 5 resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the FCI calculations, all possible configurations obtained by placing all the valence electrons of the cluster in the given set of MOs, in all possible ways, consistent with the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the QCI approach, we first choose a reference configuration, and then generate configurations which are singly-, doubly-, triply-, and quadruply-excited with respect to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the ground-state calculations, the reference configuration is normally taken to be the RHF configuration, while for the excited-state calculations one chooses an ex- cited configuration which is closest to the excited state one is trying to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, both the FCI and the QCI approaches can lead to a very large number of configurations if the number of electrons and the MO basis is large, thus, making the calculations in- tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Therefore, for the larger clusters, we employed the multi-reference singles-doubles configuration-interaction (MRSDCI) approach, as implemented in the MELD package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In this approach, the singly- and doubly-excited configurations are generated from a list of configurations called the reference configurations, chosen by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We performed the MRSDCI calculations in an incremental manner, by starting out with a small set of refer- ence configurations that are close to the states (ground or excited) we are targeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Then we analyze the optical absorption spectra of the cluster calculated from that MRSDCI cal- culation, and identify a new set of configurations which need to be included in the list of reference configurations based upon their contributions in the wave functions of the targeted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The procedure is iterated until the calculated optical absorption spectrum converges to within a user-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In all the CI calculations, the configurations are actu- ally configuration-state functions (CSFs) which are eigenstates of the point-group symmetry operators, and the total spin operators S2 and Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='31–42 The linear optical absorption spectrum of a given cluster is calculated under the electric- dipole approximation, using the formula σ(ω) = 4πα � i ωio|⟨i|ˆe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='r|0⟩|2γ2 (ωi0 − ω)2 + γ2 (1) 6 Above: (i) σ(ω) represents the optical absorption cross section, (ii) ω is the frequency of incident light, (iii)ˆe denotes the polarization direction of the incident light, (iv) r is the posi- tion operator, (v) α is the fine structure constant, (vi) ℏωi0 is the energy difference between ground state (0) and the ith excited state (i), and (vii) γ is the uniform line width associated with each excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The line width γ is taken to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV in all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The sum over index i denotes the sum over all possible excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We have restricted this sum in our calculations up to the states corresponding to excitation energies of 10 eV, or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' the oscillator strength fn corresponding to an optical transition from the ground state to the n-th excited state is computed using the standard formula fn = 2me 3ℏ2 ∆En � j=x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='z � α |⟨nα|Oj|0⟩|2 (2) above me is the electron mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' |0⟩ and |nα⟩ are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' respectively the CI wave functions of the ground state and the excited state in question,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' with α being a degeneracy label,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Oj denotes j-th Cartesian component of the electric-dipole operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' while ∆En = En − E0 is the excitation energy of the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Computational Parameters In this section, we will discuss the convergence of the results with respect two parameters, related to the basis-set-size: (a) number of active orbitals in the CI calculations, and (b) number of CSFs included in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Active molecular orbitals It is well-known that the computational cost at configuration interaction (CI) level of theory increases as N 6 act, where Nact is the total number of active molecular orbitals used in the CI calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Therefore, the time needed to perform a CI calculation will proliferate rapidly with the increasing values of Nact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We have adopted two approaches to reduce the 7 size of the active MO set: (a) we adopt the frozen-core approximation to eliminate the core orbitals of each atom of the cluster, and (b) for certain cases involving large CI matrices, we delete all those virtual (unoccupied) orbitals from our calculations whose single-particle energies are larger than 1 Hartree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The frozen-core approximation is a standard approach which also has the added advantage of considerably reducing the number of active electrons (nelec) in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The “1 Hartree cutoff” also doesn’t reduce the accuracy of the calculations because we are interested in low-lying optical excitations below 10 eV, while our cutoff eliminates only those orbitals from the calculations whose energy is larger than 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='21 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Both these approximations have been investigated rigorously in our group in earlier calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='36,41–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' To be specific, in the present set of calculations, we have considered all the virtual orbitals for Li2 and Be+ 2 clusters, while for Li3, Li4, Be+ 3 , B+ 2 and B+ 3 clusters we have imposed the 1 Hartree cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Size of CI expansion Another important parameter that controls the quality of calculations is the total number of CSFs, Ntotal, included in the CI expansion of the many-particle wave functions of the clusters concerned, both for their ground and the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As mentioned earlier, for a given set of active electrons and MOs, the best possible CI expansion corresponds to the FCI expansion, which becomes intractable for systems with large values of nelec and Nact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, whenever FCI is not possible, we employ one of the restricted CI approaches such as the QCI or the MRSDCI methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Of the two, it is crucial to examine the convergence of the MRSDCI approach which is based upon singles and doubles excitations from a number of reference configurations (Nref) leading to the final CI expansion with Ntotal CSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We examined the convergence of the optical absorption spectrum for the B+ 3 cluster calculated using the MRSDCI method, with respect to Nref, and Ntotal, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 8 Figure 1: Convergence of the optical absorption spectrum of the B+ 3 computed using the MRSDCI method, with the increasing numbers of reference configurations (Nref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For cal- culations labeled MRSDCI1, MRSDCI2, and MRSDCI3, values of Nref were 58, 101, and 144, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the figure, we plot the absorption spectra of B+ 3 obtained from three MRSDCI calcula- tions of increasing sizes labeled as MRSDCI1, MRSDCI2, and MRSDCI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In these calcula- tions, the values of parameters Nref and Ntotal were Nref= 58, Ntotal= 4007873, Nref= 101, Ntotal= 5781436, and Nref= 144, Ntotal= 8422193, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=', it is obvious that the spectra obtained using MRSDCI2 and MRSDCI3 calculations are very close to each other, signaling convergence with respect to the size of the MRSDCI expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Results and Discussion Before discussing the results of our calculations of the optical absorption spectra of various clusters, we first summarize their ground state geometries in Table 1, optimized at the CCSD 9 300 MRSDCI1 250 MRSDCI2 MRSDCI3 Intensity(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' units) 200 150 100 50 0 0 8 10 Energy(eV)level of theory, employing GAUSSIAN16 suite of programs29, and large aug-cc-pVTZ basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Table 1: The nature of the structure, along with the point group symmetry utilized, during the coupled-cluster singles-doubles (CCSD) geometry optimization calculations are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Additionally, for each cluster, the symmetry of the ground-state wave function, total Hartree-Fock (HF) energy in Hartree (Ha), total CCSD energy (Ha), and the correlation energy (eV) are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' During the calculations, aug-cc-pVTZ basis set was employed for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Cluster Structure Point group Symmetry of the HF energy CCSD energy Correlation energy GS wave function (Ha) (Ha) (eV) Li2 Linear D2h 1Ag 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8715509 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9033549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 Li3 Linear D2h 2Ag 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3088776 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3454346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='99 Li3 Isosceles triangle C2v 2A1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3170594 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3557287 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='05 Li4 Rhombus D2h 1Ag 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7619144 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8354840 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='00 Be+ 2 Linear D2h 2Ag 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9205835 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9672583 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 Be+ 3 Linear D2h 2Ag 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5410215 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6332325 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='51 B+ 2 Linear D2h 2Ag 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8344277 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9626672 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='49 B+ 3 Equilateral triangle D3h 1A ′ 1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4445744 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7127527 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 For each cluster, the table lists the nature of its ground-state structure, point group employed in the calculations, symmetry of the ground state wave function, total energy of the ground state, and the correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In Table 2, the details related to our CI calculations performed for computing the optical absorption spectra of various clusters are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For various clusters, the table lists: (a) the type of CI calculation, (b) the point-group symmetry employed in the calculations, (c) irreducible representations considered for each point group, and (d) for each irreducible representation, the size of the CI expansion (Ntotal) for each cluster are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From Table 2 it is obvious that most of the CI calculations were of the FCI type, which are exact for the chosen set of active MOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Furthermore, in the calculations in which approaches such as QCI or MRSDCI were used, the size of the CI expansion is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This means that the CI calculations performed in this work are fairly large scale, indicating that the computed optical absorption spectra are numerically accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Next, for these clusters, we discuss in detail the calculated ground state geometries, followed by their optical absorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 10 Table 2: For each cluster, the type of CI approach used for the calculations of the optical properties, point group symmetry employed during the CI calculations, and the total number of configurations (Ntotal) in the calculation are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The value of Ntotal corresponds to aug-cc-pVTZ basis set based CI calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Cluster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Point group ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='B+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Equilateral triangle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='MRSDCI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8422193 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Geometry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='The simplest cluster of lithium is lithium dimer with the D∞h point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We obtained the optimized bond length of Li2 cluster to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 Å, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This result is in excellent agreement with the bond length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='68 Å reported by Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='44, who performed the calculations at the CCSD/CCSD(T) level of theory using the Dunning correlation-consistent polarized core-valence triple/quadruple-zeta cc-pwCVXZ basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Florez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='45 performed density functional theory (DFT) calculations using the B3LYP and BLYP functionals, and reported the bond lengths to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 Å, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='71 Å, respectively, again in excellent agreement with our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Furthermore, our calculated bond length is also in a very good agreement with the experimentally measured value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='67 Å, reported by Huber46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as Li3 cluster is concerned, two isomers namely linear and isosceles triangle were found to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The equilateral triangular structure of Li3 cluster is not stable, and undergoes Jahn-Teller distortion to acquire the isosceles triangular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The linear structure has the D∞h point-group symmetry, with the optimized equal bond lengths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='90 Å (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(b)), in excellent agreement with the value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='89 Å, reported by Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The lowest-energy geometry of the Li3 cluster is an isosceles triangle with the C2v point- group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The CCSD-level optimized bond lengths for this structure are found to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='68 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='07 Å, with the bond angles 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='73◦ and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='13◦(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that by performing DFT calculations, Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='47 obtained the bond lengths of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='82 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='37 Å, that are significantly different as compared to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 12 Figure 2: Optimized geometry of (a) Li2, (b) Li3 linear, (c) Li3 isosceles triangular, (d) Li4, (e) Be+ 2 , (f) Be+ 3 linear, (g) B+ 2 , and (h) B+ 3 equilateral triangular clusters considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The geometry optimization has been performed using the CCSD method, and aug-cc-pVTZ basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' All the listed bond lengths are in Å units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The lowest-energy structure for the Li4 cluster has a rhombus shape, with D2h point group44,47, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Our optimized bond lengths of the side and minor diagonal of the rhombus structure are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 Å, respectively, which are in excellent agreement with the values 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='04 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='62 Å reported by Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The optimized bond length of the Be+ 2 cluster with the D∞h point group is found to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='25 Å (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(e)), in good agreement with the reported bond length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='21 Å, obtained from DFT calculations by Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Our lowest-energy optimized structure of Be+ 3 cluster also has a linear geometry, with two equal bond lengths 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 Å, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This value of the bond length is in very good agreement with the value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='19 Å, computed by Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='48 using DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as B+ 2 cluster is concerned, we computed its minimum-energy bond length to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='18 Å (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(g)), which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='18 Å larger than the value 2 Å reported by Hanley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 13 (a) (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='90 Liz Lis chain (c) (d) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02 Li3 (f) Li4 (e) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 Be2* Be3+ (g) (μ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='58 B, B3We attribute this difference to two factors, namely, smaller basis set (6-31G∗), coupled with a lower-level CI methodology used by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Our optimized structure of B+ 3 cluster is an equilateral triangle of sides 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='58 Å, with the D3h point-group symmetry, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hanley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='49 using a CI approach, along with the 6-31G∗ basis set, also obtained the optimized structure to be an equilateral triangle for the B+ 3 , but with a bond length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='53 Å, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='05 Å smaller than our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We again attribute the differences to the choice of a smaller basis set, coupled with a lower-level correlation methodology as compared to the CCSD approach used by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Peak locations Li2 dimer, with just two active electrons within the frozen-core approximation, is the small- est many-electron cluster considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Therefore, very high-quality correlated- electron calculations using large basis sets are possible for this system, not just for its ground states, but also for the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As a result, this case can provide us deep insights into the influence of the choice of basis functions on the calculated excited state properties and the photoabsorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the calculations, we employed the frozen-core FCI method us- ing six basis sets of varying sizes, namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ, cc-pVTZ, aug-cc-pVDZ and aug-cc-pVTZ, and the computed spectra are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' All the virtual molecular orbitals generated during the RHF calculations were used in the CI calculations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=', no unoccupied orbitals were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As a result, the frozen-core FCI results presented here are the best ones possible for the chosen basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the Li2 dimer, the peak locations in the computed spectra are presented in Table S1 of the Supporting information (SI), from which it is obvious that for the first two peaks the excitation energies calculated using different basis sets are in very good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This is encouraging because from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 3(a) it is obvious that most of the oscillator strength of the absorption spectrum is confined to these two peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, starting from the third peak onward, we start seeing differences in the excitation energies predicted by 14 different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the third peak, the predicted peak locations can be classified in two groups: (a) those predicted by correlation-consistent basis sets cc-pVDZ and cc-pVTZ, and (b) the ones predicted by 6-311G++ and augmented correlation consistent (aug-cc-) class of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the peak locations predicted by the former class of basis functions have values significantly larger than those predicted by the latter class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Another noteworthy point is that there is very good agreement among the peak locations predicted by the second class of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as the location of the fourth peak is concerned, there is good agreement among the predictions by 6-311++G(3df,3pd) and aug-cc class of basis functions, while the remaining three basis functions predict very different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The case of the fifth peak is somewhat anomalous in that the agreement among the predictions by any of the basis sets is not good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, for higher peaks we note that the results from the aug-cc class of basis functions are in good agreement with each other, while other basis functions predict widely differing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='50 also performed first-principles calculations of the photoabsorption spectra of several Lin clusters employing the time-dependent density- functional theory (TDDFT) methodology, and for Li2 their predicted locations of the first two peaks are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='92 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='53 eV50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' On comparing these with our best values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='83 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='57 eV, respectively, we note: (a) our excitation energy for peak I is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='09 eV smaller than theirs, while (b) our location for peak II is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='04 eV larger than theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We attribute these differences to different computational methodologies adopted in the two sets of calculations, and it will be interesting to compare the computational results with the experimental ones, whenever they are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The peak locations of the photoabsorption spectra of Li3 chain are presented in Table S2 of SI, from which it is clear that the locations of the first two peaks converge completely for all the basis sets, similar to the case of dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The third peak is the most intense peak of the computed spectra as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 3(b), whose location is in good agreement for all the basis sets except for cc-pVDZ, which predicts higher excitation energy as compared to the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From the fourth peak onward, the peak locations can be classified in two similar group 15 as discussed previously for the case of dimer: the peak locations predicted from correlation- consistent basis sets cc-pVDZ and cc-pVTZ are towards the higher energy side as compared to all other basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' It can also be seen that the peak positions corresponding to the two classes of basis sets are in good agreement within the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Next, we examine the peak locations in the photoabsorption spectra of Li3 triangular cluster computed using various basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the peak locations corresponding to the first five peaks are in very good agreement with each other for different basis sets as is obvious from Table S3 of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This result is very encouraging because peak IV is the most intense (MI) peak of the computed spectra as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 3(c), and it is crucial for a basis set to be able to accurately describe the MI peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The location of this peak is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='43 eV computed using the aug-cc-pVTZ basis set, which is in a decent agreement with the experimentally detected peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='58 eV by Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From the sixth peak onward it was observed that the peak locations calculated using correlation consistent basis sets (cc- pVDZ and cc-pVTZ) do not match with the other classes of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, the peak locations computed using the 6-31G class and the aug-cc-pVTZ continue to be in very good agreement with each other till peak VIII, located near 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The locations of higher-energy peaks beyond peak VIII computed using these basis sets are presented in Table S4 for the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 16 Figure 3: Optical absorption spectra of (a) Li2, (b) Li3 linear, (c) Li3 triangular, and (d) Li4 clusters computed using various basis sets and the frozen core FCI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The uniform line-width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV is used to plot the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The peak positions of the photoabsorption spectra of Li4 cluster computed using various basis sets are presented in Table S5 of SI, while the spectra are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that for this cluster, the excitation energies of the first five peaks computed using different basis sets are in very good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The first three peaks are much more intense as compared to the higher energy peaks, and in peak III there are slight differences (≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV) in the peak locations predicted by different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The two largest basis sets (6-311++G(3df,3pd), and aug-cc-pVTZ) predict the location of peak III at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 eV, while the predictions by the rest of the basis sets are in the range 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='03–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='08 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From peak VI 17 SO (a) 60K 6-311++G(2d,2p) 6-311++G(3df,3pd) (b) III 6-311++G(2d,2p) 400 cc-pVDZ 500 6-311++G(3df,3pd) ZLAd-0 cc-pVDZ aug-cc-pVDZ cC-pVTZ awg-cc-pVTZ aug-cc-pVDZ (s)) 400 aug-cc-pVTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 300 200 100 100 I VVI VII 6 Energy (eV) 300 400 (c) 6-311++G(2d,2p) (d) 6-311++G(2d,2p) 250 6-311++G(3df,3pd) cc-pVDZ 6-311++G(3df,3pd) ZLAd-33 II cc-pVDZ aug-cc-pVDZ 300 cc-pVTZ 200 ZLAd-30-8ne aug-cc-pVDZ aug-cc-pVTZ Intensity 150 Intensity 100 100 50 VIII X 2 6 Energy (eV) Energy (eV)onward we begin to observe differences among the locations predicted by different basis sets, with a tendency towards clustering into different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, the noteworthy point is that the intensity corresponding to these higher energy peaks is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as the comparison with the experiments is concerned, the first three photoabsorption peaks of the Li4 cluster located at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 eV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 eV, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 eV for aug-cc-pVTZ basis set are in excellent agreement with the experimental measurements of Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='51 who detected these peaks at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='83 eV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 eV, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For Be+ 2 cluster, we present the spectra computed by different basis sets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 4(a), while the corresponding peak locations are presented in Table S6 of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note excellent convergence of the excitation energies up to the sixth peak, beyond which results obtained by different basis sets do not agree much with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We further note that Peak V located near 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30 eV is the most intense peak, and, for that, the predictions of the different basis sets are in a fairly narrow energy range 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='37 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Figure 4: Optical absorption spectra of (a) Be+ 2 and (b) Be+ 3 clusters computed using various basis sets and frozen core FCI and QCI methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The uniform line-width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV is used to plot the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The excited-states peak locations of Be+ 3 cluster for different basis sets are presented in Table S7 of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We notice excellent agreement of the excited-states peak locations up to the 18 600 (a) 6-311++G(2d,2p) VII 6-311++G(2d,2p) 300 6-311++G(3df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3pd) 500 6-311++G(3df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3pd) cc-pVDZ cc-pVDZ cc-pVTZ cc-pVTZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVTZ 400 aug-cc-pVTZ (sirun 200 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 300 Intensity 200 100 100 2 3 4 5 6 9 [0 3 5 Energy (eV) Energy (eV)seventh peak which is also the most intense peak of the spectra located near 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5 eV, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Although the peak location of the sixth peak computed using cc-pVDZ basis set is slightly towards the higher energy region as compared to all other basis sets, but the difference is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Noteworthy point is that these basis sets are able to achieve convergence in the peak positions in Be+ 2 and Be+ 3 photoabsorption spectra up to much higher excitation energies, as compared to the Li clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The excited-states peak positions corresponding to the photoabsorption spectra of B+ 2 cluster are presented in Table S8 of SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We notice excellent agreement of the peak energies corresponding to first three peaks for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The fourth peak is the most intense peak of the spectra, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 5(a) for whose location excellent agreement has been achieved for 6-311++G (2d, 2p), cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ basis sets, indicating complete convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, the excitation energies for peak IV computed using the 6-311++G (3df, 3pd) and cc-pVDZ basis sets are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV higher, as compared to other basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as peak V is concerned, which is a very weak shoulder of peak IV, we again observe excellent convergence for all the basis sets, except cc-pVDZ which fails to predict the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' From the sixth peak onward, as discussed previously, the predicted peak locations can be classified into two groups: (a) larger basis sets of 6-311++G and aug-cc- type, and (b) smaller basis sets cc-pVDZ and cc-pVTZ, with the peak locations predicted by individual classes being in very good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 19 Figure 5: Optical absorption spectra of (a) B+ 2 and (b) B+ 3 cluster computed using various basis sets and frozen core FCI and MRSDCI methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The uniform line-width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1 eV is used to plot the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The peak locations corresponding to the excited-states of the photoabsorption spectra of B+ 3 cluster are presented in Table S9 of SI, while the calculated spectra are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 5(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For this cluster, we get eight well-separated peaks in the explored energy range, with peak VIII located near 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9 eV being the most intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the peak energies corresponding to all the basis sets converge excellently up to the peak VIII, except those predicted by the cc-pVDZ basis set, which are consistently higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We have noticed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5, only the deep valence excitation energies are dependent on the choice of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' This behavior can be a consequence of the frozen-core approximation, which we have employed in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' To verify this, we have computed the optical absorption spectra of Li2 and Be+ 2 clusters by also including the core excitations within the large-scale QCI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We found that the optical spectra of these clusters computed by including core excitations agrees completely with the absorption spectra computed after employing frozen- core approximation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='S1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='S2 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Therefore, the frozen-core approximation does not alter the absorption spectra of small clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Based on the peak positions of the individual clusters discussed above, we observe the 20 500 250 (a) (b) 6-311++G(2d,2p) 6-311++G(2d,2p) 6-311++G(3df,3pd) VII 6-311++G(3df,3pd) 400 cc-pVDZ IV 200 cc-pVDZ cc-pVTZ VII cc-pVTZ aug-cc-pVDZ aug-cc-pVDZ aug-cc-pVTZ ZLAd-03-8ne 150 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 200 VI 100 100 50 III II 1 2 3 4 6 8 9 10 2 3 1 6 7 8 10 Energy (eV) Energy (eV)following general trends: (a) peak locations for all the clusters used in this study are in very good agreement for all the basis sets up to the most intense peak of the spectra, except for cc-pVDZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (b) the excited-states peak locations beyond the most intense peak can be classified in two groups, in which the peak locations calculated using correlation- consistent basis sets do not match with the peak locations computed using all other basis sets, and (c) for the cc-pVDZ basis sets peaks are located at higher energies as compared to the rest of the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As the basis-set dependence of the optical properties is different for the density functional theory compared to the wave function-based large-scale configuration-interaction method, it will be interesting to explore the optical properties of clusters using time-dependent density functional theory (TD-DFT) and compare it with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We found that the first peak of the optical absorption spectra of B+ 3 cluster is located at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='84 eV when computed using large-scale MRSDCI calculations along with a large aug- cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, when the calculations are performed using TD-DFT method with B3LYP functional and the same basis set, it is obtained at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The most intense peak of the optical absorption spectra is located at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 eV using the MRSDCI approach, which is found to be at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='35 eV by employing the TD-DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The calulated optical absorption spectra and excited-states peak locations of B+ 3 cluster corresponding to TD- DFT calculations are provided in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='S3 and Table S10 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We also report that the variations in the peak locations of the photoabsorption spectra of B+ 3 computed using various basis sets and TD-DFT method are lesser than the wave function-based CI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Oscillator strength In addition to the excitation energy, the next important quantity determining the profile of the absorption spectrum is the oscillator strength (f) corresponding to various optical tran- sitions, connecting the ground state to the excited state in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The oscillator strength calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2 is determined by the excitation energy of the state involved, and the corresponding transition dipole moment (TDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The TDM being a matrix element, is, in 21 turn, determined by the many-particle wave functions of the ground and the excited state that it connects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Another important quantity is the polarization of the photon involved in a given optical transition (peak), which can be measured in oriented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The polarization is a consequence of the point-group symmetry of the concerned molecule, and hence should be independent of the basis set employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In this section, we discuss the convergence of the oscillator strengths and photon polarizations associated with various peaks of the calculated spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In Table 3, we present the oscillator strengths corresponding to the first peak, and the most intense peak (peak II) of the spectra of Li2 computed using different basis func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Additionally, the table also contains the dominant configurations contributing to the many-particle wave functions of the excited states involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Table 3: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and II of Li2 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the “Polarization” column, ∥ indicates photon polarization along the direction of the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to the molecular axis (transverse polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Note that the transversely polarized states are doubly degenerate, therefore, the oscillator strength corresponding to those is the sum of both the contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ’H’ and ’L’ stand for HOMO and LUMO orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Polarization f Configurations Polarization f Configurations 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='460 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='971 |H → L + 2⟩ |H → L + 3⟩ |H → L + 7⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='456 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='966 |H → L + 2⟩ |H → L + 3⟩ |H → L + 7⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='463 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='970 |H → L + 1⟩ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 5⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='455 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='969 |H → L + 1⟩ |H → L + 4⟩ |H → L + 6⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='462 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='972 |H → L + 1⟩ |H → L + 3⟩ |H → L + 6⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='454 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='966 |H → L + 2⟩ |H → L + 3⟩ |H → L + 7⟩ From Table 3 it is obvious that the oscillator strengths computed using various basis functions for both the peaks are in very good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We also note that the direction of the polarization of the photons involved in a given optical transitions are of 22 the excited-states corresponding to the first and second peak of the spectra of Li2 cluster are parallel and perpendicular to molecular axis, respectively, irrespective of the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The oscillator strengths corresponding to the first and most intense peaks of the spectra of Li3 chain and triangular clusters are presented in Table S11 and Table S12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the oscillator strengths of the first peaks both of Li3 chain, and the triangular cluster, computed using the different basis sets are in excellent agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The oscillator strengths corresponding to the most intense peaks of Li3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' peak III of the chain and peak IV for the triangular cluster, calculated using various basis sets can be classified into two groups: (a) those calculated using correlation-consistent (cc-pVTZ and cc-pVDZ) class of basis sets, and (b) those computed using 6-33++G- and augmented correlation-consistent (aug-cc-) class of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The oscillator strength calculated by the first class of basis sets is comparatively higher than the second class of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' But, the relative maximum difference between oscillator strengths of different classes is close to 6%, which is fairly acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The oscillator strengths corresponding to the first and the most intense peak (peak II) in the photoabsorption spectra of Li4 cluster are presented in Table S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the oscillator strengths of peak I are in good agreement with each other for all the basis sets except for cc-pVDZ and aug-cc-pVTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For these basis sets the oscillator strength is comparatively larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For peak II, we note that the difference in the oscillator strength computed by cc-pVDZ and 6-311++G (3df, 3pd) basis sets is about 5%, which is again quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We present the oscillator strengths corresponding to the first and the most intense peaks of the cationic beryllium clusters Be+ 2 and Be+ 3 in tables S14, and S15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note very good agreement on the oscillator strengths of both the peaks of the Be+ 2 and Be+ 3 clusters for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The maximum relative disagreement we find among the oscillator strengths for a given peak is around 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Finally, we discuss the oscillator strengths of the first and the most intense peaks of the 23 B+ 2 and B+ 3 clusters presented in tables S16, and S17, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that both for B+ 2 and B+ 3 clusters, the oscillator strengths of the first peaks are two orders of magnitude smaller than those of their most intense peaks, indicating that the first peaks for both the clusters are relatively feeble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Nevertheless, the oscillator strengths of the first peaks of the photoabsorption spectra of the two clusters calculated using various basis sets are in very good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' As far as the most intense peaks are concerned, both for B+ 2 and B+ 3 we see the following pattern: oscillator strengths computed using 6-311++G- and aug-cc-pVTZ basis sets are in very good agreement with each other, while those computed using other basis sets differ from them somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Wave function analysis Next, we examine the dominant configurations contributing to the CI wave functions of the excited states contributing to various peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configurations corresponding to the excited-states CI wave functions of peak I and peak II of Li2 are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that for peak I, the main contribution to the corresponding excited state wave function is from the singly excited configuration |H → L⟩ for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, the next important configuration to the same wave function depends on the class of basis set employed: (a) it is |H → L+3⟩ single excitation when calculations are performed using larger basis sets of the type 6-311++ and aug-cc, but (b) for smaller basis sets, this configuration is found to be |H → L + 4⟩ for cc-PVTZ basis, and |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ for the cc-PVDZ set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Peak II is due to two degenerate excited states to which the dominant contributions are from configurations |H → L+2⟩ and |H → L+7⟩, for the calculations performed using 6-31G++ and aug-cc-PVTZ type basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' But, for the calculations performed with smaller basis sets, the dominant configurations is |H → L + 1⟩, while the next important configuration can be |H → L + 6⟩ or |H → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 5⟩, depending on the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Thus, we can draw the following general conclusion regarding this: (a) for large basis set calculations, for a given peak, the configurations are in perfect agreement with each other, and (b) the 24 configurations predicted by calculations performed using smaller basis sets such as cc-PVDZ are found to be different as compared to those obtained in larger basis set calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configurations for the wave functions corresponding to peak I and the most intense peak (peak III) of Li3 chain, computed using various basis sets are presented in Table S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We find that for the first peak the dominant configuration is |H − 1 → H⟩ for all the basis sets except for aug-cc-pVTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the aug-cc-pVTZ the dominant configurations contributing to the excited state wave function are different compared to other basis sets, because of the reversal of ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, because the peak energies and oscillator strength for the state are in excellent agreement with all other basis sets implies that we have obtained correct quantitative description of the excited states even with this basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For peak III the main contribution to the excited state wave function is from |H − 1 → L + 2⟩ for the larger 6-311++ and aug-cc class of basis sets, while it is from |H − 1 → L + 1⟩ for the smaller cc-pVTZ and cc-pVDZ basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The main configurations contributing to the excited states wave functions of peak I and the most intense peak (peak IV) of Li3 triangular cluster, computed using various basis sets are presented in Table S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that for the first peak, the main contribution to the wave function is from configurations |H → L+14⟩ or |H → L+13⟩ for the larger 6-311++G and aug-cc class of basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the correlation-consistent basis sets (cc-pVDZ and cc- pVTZ) the main contribution is due to the configuration |H → L + 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the fourth peak, the dominant configuration is |H − 1 → L⟩, irrespective of the type of basis set used for the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configurations corresponding to the excited states wave functions of peak I and the most intense peak of the spectra (peak II) of the Li4 cluster are presented in Table S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the first peak, the main contribution is from the configuration |H → L + 1⟩ for all the basis sets, while for peak II it is |H−1 → L⟩ for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Thus, we have excellent agreement among all the basis sets when it comes to the most important configuration for both the peaks of the Li4 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 25 The important configurations corresponding to the excited states wave function of peak I and the most intense peak (peak V) of the photoabsorption spectra of Be+ 2 cluster are presented in Table S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configuration contributing to peak I is |H → L⟩ for the 6-311++G class of basis sets, and |H → L + 2⟩ for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For peak V, the main configuration contributing to the CI wave function is |H − 1 → L + 1⟩ for 6-311++G class of basis sets, and |H − 1 → L⟩ for the rest of the sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The configurations dominating the excited state CI wave functions of peak I and the most intense peak (peak VII) of Be+ 3 cluster are listed in Table S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We note that the most important configurations contributing to peak I can be classified in two groups: (a) for larger 6-311++G(3df,3pd) and aug-cc class of basis sets the dominant configuration is |H → L + 1⟩, (b) while for smaller basis sets dominant configuration is highly basis set dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For peak VII, the doubly-excited configurations |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ and |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩ dominate the excited-state wave functions for the larger 6-311++G(3df,3pd) and aug-cc class of basis sets, but vary significantly for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Most important configurations contributing to the wave functions for peak I and the most intense peak (peak IV) of B+ 2 cluster are presented in Table S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' It is obvious that the double-excitation |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ contributes the most to peak I for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configurations contributing to the wave functions of peak IV are |H → L+5⟩ and |H → L + 11⟩ for all the basis sets except the cc-pVDZ/cc-pVTZ, for which instead of |H → L + 11⟩, the double excitation |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Finally, we present the dominant configurations in the CI wave functions corresponding to peak I, and the most intense peak (peak VIII), of B+ 3 cluster in Table S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The configura- tion with maximum contribution to the excited state wave functions for peak I is |H → L⟩, irrespective of the basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The next dominant configuration is basis-set dependent, how- ever, it is a double excitation in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The dominant configuration corresponding to the CI wave function of peak VIII is the double excitation |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ for all the basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 26 The detailed wave function analysis for all the peaks of the optical absorption spectra of clusters considered in this work using the largest aug-cc-pVTZ basis set is provided in Table S18-S25 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Conclusion In this work, we presented electron-correlated calculations of the optical absorption spectra of small neutral and ionic clusters using various basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' First, the stable geometries of various clusters were determined at the CCSD level of theory, using the aug-cc-PVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' For the ground and the excited state wave functions calculations needed to compute the absorption spectra, we used the FCI, QCI, and MRSDCI approaches depending upon the size of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The CI calculations were performed using six different basis sets, namely, 6-311++G(2d,2p), 6-311++G(3df,3pd), cc-pVDZ , cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We observed that the optical absorption spectra of all these clusters exhibit excellent convergence for all the basis sets in the lower energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, usually after the first two peaks, the shift in peak locations for cc-pVDZ and cc-pVTZ basis set are noted in all likelihood because of the lack of diffuse basis functions in these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' If we use augmented basis sets, the absorption spectra show good agreement with the results computed using other similar basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Although aug-cc-pVDZ basis set has a relatively smaller number of basis functions as compared to aug-cc-pVTZ basis set, the agreement between the spectra computed using the two basis sets is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Because the number of two-electron integrals increases as N 4 where N is the number of basis functions in basis set, we can reduce the computational cost significantly by using aug-cc-pVDZ basis set instead of larger Pople’s basis sets, and aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Thus, our general recommendation is that for optical absorption calculations one should use a basis set containing diffuse functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=', of the aug-cc- type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' However, whether one should use aug-cc-pVDZ, or a larger set, should be 27 decided by the available computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' We believe that the CI calculations presented in this work are quite accurate, as is obvious from the fact that our obtained results are in very good agreement with the experiments for Li3 and Li4 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Therefore, it will be of interest to compare our results on other clusters also with the experiments, as and when they are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Associated Content Supporting Information In the supporting information file, we have provided the peak locations, oscillator strengths, and dominant excited state configurations corresponding to the optical absorption spectra of all the clusters for all the basis sets considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The SI file also contains the details of the many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of clusters for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Author Information Corresponding Author Alok Shukla: Department of Physics, Indian Institute of Technology Bombay, Powai, Mum- bai 400076, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' *E-mail: shukla@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='in Acknowledgment This work was supported by senior research fellowship (DST-Inspire) provided by department of science and technology, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 28 Authors Vikram Mahamiya: Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E-mail: mahamiyavikram@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com Pritam Bhattacharyya: Institute for Theoretical Solid State Physics, Leibniz IFW Dres- den, Helmholtzstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 20, 01069 Dresden, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E-mail: pritambhattacharyya01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com Notes The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 29 References (1) Boys, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Electronic Wave Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A General Method of Calculation for the Stationary States of Any Molecular System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Proceedings of the Royal Society of London Series A 1950, 200, 542–554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (2) McMurchie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Davidson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' One- and two-electron integrals over cartesian gaus- sian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of Computational Physics 1978, 26, 218 – 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (3) Davidson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Feller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis set selection for molecular calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chemical Reviews 1986, 86, 681–696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (4) Huzinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis sets for molecular calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Computer Physics Reports 1985, 2, 281 – 339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (5) Huzinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Gaussian-Type Functions for Polyatomic Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1965, 42, 1293–1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (6) Ruedenberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Raffenetti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Bardo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Structure and Reactivity, Proceedings of the 1972 Boulder Conference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Wiley: New York, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (7) Bardo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Ruedenberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Even-tempered atomic orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Optimal orbital exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and oxygen in molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1974, 60, 918–931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (8) Huzinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Klobukowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Tatewaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The well-tempered GTF basis sets and their applications in the SCF calculations on N2, CO, Na2, and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Canadian Journal of Chemistry 1985, 63, 1812–1828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (9) Tatewaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Huzinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A systematic preparation of new contracted Gaussian type orbital set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Transition metal atoms from Sc to Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1979, 71, 4339–4348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 30 (10) Tatewaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Huzinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A systematic preparation of new contracted Gaussian-type orbital sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Second-row atoms from Li through ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of Computational Chem- istry 1980, 1, 205–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (11) Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Stewart, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-Consistent Molecular-Orbital Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Use of Gaussian Expansions of Slater-Type Atomic Orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1969, 51, 2657–2664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (12) Ditchfield, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-Consistent Molecular-Orbital Meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1971, 54, 724–728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (13) Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Small split-valence basis sets for first-row elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of the American Chemical Society 1980, 102, 939–947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (14) Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' XIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Split-valence Gaussian-type basis sets for beryllium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1977, 66, 879– 880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (15) Krishnan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Seeger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular or- bital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A basis set for correlated wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1980, 72, 650–654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (16) Hariharan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The influence of polarization functions on molecular orbital hydrogenation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Theoretica chimica acta 1973, 28, 213–222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (17) Collins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' von R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Schleyer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' XVII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Geometries and binding energies of second-row molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A comparison of three basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1976, 64, 5142–5151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 31 (18) Francl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pietro, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' DeFrees, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' XXIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A polarization-type basis set for second-row elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1982, 77, 3654–3665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (19) Pietro, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Francl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' DeFrees, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Supplemented small split-valence basis sets for second-row elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of the American Chemical Society 1982, 104, 5039–5048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (20) Frisch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Self-consistent molecular orbital methods 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Supplementary functions for Gaussian basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1984, 80, 3265–3269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (21) Dunning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Gaussian basis sets for use in correlated molecular calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='I The atoms boron through neon and hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1989, 90, 1007–1023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (22) Kendall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Dunning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Harrison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Electron affinities of the first-row atoms revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='Systematic basis sets and wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1992, 96, 6796–6806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (23) Woon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Dunning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Pronounced Effect of Microsolvation on Diatomic Alkali Halides: Ab Initio Modeling of MX(H2O)n (M = Li, Na;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' X=F, Cl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' n = 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of the American Chemical Society 1995, 117, 1090–1097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (24) Balakina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Nefediev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The choice of basis set for calculations of linear and nonlinear optical properties of conjugated organic molecules in gas and in dielectric medium by the example of p-nitroaniline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Computational Materials Science 2007, 38, 467–472, Selected papers from the International Conference on Computational Methods in Sciences and Engineering 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 32 (25) Parsons, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Balduf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Cheeseman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Caricato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Dependence of Optical Rotation Calculations with Different Choices of Gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry A 2022, 126, 1861–1870, PMID: 35271772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (26) Reis, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Papadopoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Nonlinear optical properties of the rhombic B4-cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Journal of Computational Chemistry 1999, 20, 679–687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (27) Lauderdale, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Coolidge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis set effects on the nonlinear optical properties of selected linear diacetylenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry 1995, 99, 9368–9373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (28) Jabłoński, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Palusiak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set and Method Dependence in Quantum Theory of Atoms in Molecules Calculations for Covalent Bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chem- istry A 2010, 114, 12498–12505, PMID: 21049895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (29) Frisch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Gaussian 16 Revision C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (30) McMurchie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Elbert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Langhoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Davidson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' MELD package from Indiana University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' It has been modified by us to handle bigger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (31) Shinde, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Large-scale first principles configuration interaction calculations of optical absorption in aluminum clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 2014, 16, 20714– 20723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (32) Rai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chakraborty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Tunable Optoelectronic Properties of Triply Bonded Carbon Molecules with Linear and Graphyne Substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry C 2018, 122, 1309–1317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (33) Chakraborty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Pariser - Parr - Pople Model Based Investigation of Ground and Low - Lying Excited States of Long Acenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry A 2013, 117, 14220–14229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (34) Aryanpour, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Mazumdar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Electron correlations and two-photon states 33 in polycyclic aromatic hydrocarbon molecules: A peculiar role of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 2014, 140, 104301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (35) Chakraborty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Theory of triplet optical absorption in oligoacenes: From naphthalene to heptacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 2014, 141, 164301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (36) SHINDE, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' SHUKLA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' LARGE-SCALE FIRST PRINCIPLES CONFIGURA- TION INTERACTION CALCULATIONS OF OPTICAL ABSORPTION IN BORON CLUSTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Nano LIFE 2012, 02, 1240004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (37) Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Correlated theory of triplet photoinduced absorption in phenylene-vinylene chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B 2002, 65, 125204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (38) Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Theory of nonlinear optical properties of phenyl-substituted polyacetylenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B 2004, 69, 165218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (39) Sony, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Large-scale correlated calculations of linear optical absorption and low-lying excited states of polyacenes: Pariser-Parr-Pople Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B 2007, 75, 155208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (40) Basak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chakraborty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Theory of linear optical absorption in diamond- shaped graphene quantum dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B 2015, 92, 205404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (41) Priya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Photoabsorption in sodium clusters: first principles configuration interaction calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The European Physical Journal D 2017, 71, 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (42) Shinde, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' First principles electron-correlated calculations of optical ab- sorption in magnesium clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The European Physical Journal D 2017, 71, 301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (43) Bhattacharyya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Rai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Shukla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Systematic First-Principles Configuration- Interaction Calculations of Linear Optical Absorption Spectra in Silicon Hydrides: Si2H2n (n = 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry A 2019, 123, 8619–8631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 34 (44) Wheeler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Sattelmeyer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Schleyer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Schaefer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Binding energies of small lithium clusters (Lin) and hydrogenated lithium clusters (LinH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 2004, 120, 4683–4689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (45) Florez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Fuentealba, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' A theoretical study of alkali metal atomic clusters: From Lin to Csn (n = 2-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' International Journal of Quantum Chemistry 2009, 109, 1080–1093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (46) HUBER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Molecular Structure Constants of Diatomic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Molecular Spectra and molecular Structure Constants of Diatomic molecules 1979, (47) Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Lichtenstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Hutter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Density functional study of structure and bonding in lithium clusters Lin and their oxides LinO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 1997, 106, 4566–4574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (48) Srinivas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Jellinek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Structural and electronic properties of small beryllium clusters: A theoretical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Chemical Physics 2004, 121, 7243–7252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (49) Hanley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Whitten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Anderson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Collision-induced dissociation and ab initio studies of boron cluster ions: determination of structures and stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The Journal of Physical Chemistry 1988, 92, 5803–5812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (50) Hong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' TDDFT calculation for photoabsorption spectra of Lin (n=2-11,20) clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Physics Letters A 2011, 375, 1883 – 1888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' (51) Blanc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Broyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Chevaleyre, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Dugourd, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Kühling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Labastie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Ul- bricht, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Wolf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Wöste, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' High resolution spectroscopy of small metal clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Zeitschrift für Physik D Atoms, Molecules and Clusters 1991, 19, 7–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 35 For Table of Contents Only 36 400 Optical absorption spectra of Li4 cluster using various basis sets 6-311++G(2d,2p) 6-311++G(3df,3pd) III cC-pVDZ 300 cc-pVTZ aug-cc-pVDZ (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' units) aug-cc-pVTZ 200 Intensity 100 VI VII 0 0 2 4 Energy (eV)Supporting Information For: Benchmarking Gaussian Basis Sets in Quantum-Chemical Calculations of Photoabsorption Spectra of Light Atomic Clusters Vikram Mahamiya,∗,† Pritam Bhattacharyya,∗,†,‡ and Alok Shukla∗,† †Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India ‡Present Address: Institute for Theoretical Solid State Physics, Leibniz IFW Dresden, Helmholtzstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' 20, 01069 Dresden, Germany E-mail: mahamiyavikram@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' pritambhattacharyya01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' shukla@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='in Table S1: Comparison of the peak locations of the optical absorption spectra of Li2 cluster computed using various basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII Peak VIII Peak IX (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='88 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='06 6-311++G(3df,3pd) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='88 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='00 cc-pVDZ 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02413v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='chem-ph] 6 Jan 2023 Table S2: Comparison of the peak locations of the optical absorption spectra of Li3 chain computed using various basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='28 6-311++G(3df,3pd) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='54 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='58 aug-cc-pVDZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='88 aug-cc-pVTZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='89 Table S3: The peak locations of the optical absorption spectra of Li3 isosceles triangular cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='21 6-311++G(3df,3pd) 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='20 Table S4: High energy peak locations of the optical absorption spectra of Li3 Isosceles triangular cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Li3 Cluster Peak VIII Peak IX Peak X Peak XI Peak XII Basis Set (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='99 6-311++G(3df,3pd) 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='59 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='94 aug-cc-pVTZ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='60 S2 Table S5: The peak locations of the optical absorption spectra of Li4 rhombus cluster com- puted using various basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The higher energy peak locations are presented in the Table I of the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='15 6-311++G(3df,3pd) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 3.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='62 aug-cc-pVTZ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30 Table S6: The peak locations of the optical absorption spectra of Be+ 2 cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII Peak VIII Peak IX (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='75 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='34 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='92 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='38 Table S7: The peak locations of the optical absorption spectra of Be+ 3 cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='86 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='52 S3 Table S8: The peak locations of the optical absorption spectra of B+ 2 cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII Peak VIII Peak IX (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='86 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='01 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='21 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='41 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='20 Table S9: The peak locations of the optical absorption spectra of B+ 3 cluster computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B+ 3 Cluster Peak I Peak II Peak III Peak IV Peak V Peak VI Peak VII Peak VIII Basis Set (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='08 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 Table S10: The peak locations of the optical absorption spectra of B+ 3 cluster employing TD- DFT method with B3LYP functional and computed using various basis sets are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' B+ 3 Cluster Peak I Peak II Peak III Peak IV Basis Set (eV) (eV) (eV) (eV) 6-311++G(2d,2p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='35 6-311++G(3df,3pd) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='35 cc-pVDZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='73 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='52 cc-pVTZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='40 aug-cc-pVDZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='73 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='40 aug-cc-pVTZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='36 S4 Table S11: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak III) of Li3 chain calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the “Polarization” column, ∥ indicates photon polarization along the direction of the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to the molecular axis (transverse polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ’H’ and ’L’ stand for HOMO and LUMO orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak III Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='106 |H − 1 → H⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='591 |H − 1 → L + 2⟩ |H → L + 6⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='106 |H − 1 → H⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='595 |H − 1 → L + 2⟩ |H → L + 5⟩ |H → L + 11⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='097 |H − 1 → H⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='631 |H − 1 → L + 1⟩ |H → L⟩ |H → L + 3⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='105 |H − 1 → H⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='618 |H − 1 → L + 1⟩ |H → L⟩ |H → L + 3⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='101 |H − 1 → H⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='603 |H − 1 → L + 2⟩ |H → L + 7⟩ |H → L + 10⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='106 |HF⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='594 |H − 1 → L + 2⟩ |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 5⟩ H − 1 → L + 11⟩ Table S12: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV) of Li3 triangular calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak IV Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='127 |H → L + 14⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='465 |H − 1 → L⟩ |H → L + 1⟩ |H → L + 9⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='131 |H → L + 14⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='459 |H − 1 → L⟩ |H → L + 1⟩ |H → L + 5⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='121 |H → L + 2⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='488 |H − 1 → L⟩ |H − 1 → L + 2⟩ |H → L + 4⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='129 |H → L + 2⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='478 |H − 1 → L⟩ |H − 1 → L + 2⟩ |H → L + 4⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='126 |H → L + 14⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='469 |H − 1 → L⟩ |H → L + 10⟩ |H → L + 16⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='129 |H → L + 13⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='462 |H − 1 → L⟩ |H → L + 1⟩ |H → L + 5⟩ S5 Table S13: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak II) of Li4 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Basis Set Peak I Peak II Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='628 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='648 |H − 1 → L⟩ |H → L + 8⟩ |H − 1 → L + 5⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='615 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='683 |H − 1 → L⟩ |H → L + 9⟩ |H − 1 → L + 6⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='659 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='649 |H − 1 → L⟩ |H → L + 4⟩ |H → L + 3⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='624 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='678 |H − 1 → L⟩ |H → L + 5⟩ |H → L + 4⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='634 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='654 |H − 1 → L⟩ |H → L + 9⟩ |H − 1 → L + 5⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='656 |H → L + 1⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='660 |H − 1 → L⟩ |H → L + 9⟩ |H − 1 → L + 5⟩ Table S14: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak V) of Be+ 2 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11 Basis Set Peak I Peak V Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='113 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='745 |H − 1 → L + 1⟩ |H − 1 → H⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='119 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='749 |H − 1 → L + 1⟩ |H − 1 → H⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='114 |H → L + 2⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='736 |H − 1 → L⟩ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩ |H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='120 |H → L + 2⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='749 |H − 1 → L⟩ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩ |H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='114 |H → L + 2⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='768 |H − 1 → L⟩ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩ |H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='120 |H → L + 2⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='757 |H − 1 → L⟩ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩ |H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ S6 Table S15: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak VII) of Be+ 3 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11 Basis Set Peak I Peak VII Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='172 |HF⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='655 |H − 2 → L + 1⟩ |H → L⟩ |H − 1 → L + 2⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='184 |H → L + 1⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='647 |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='178 |H → L⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='640 |H − 2 → L + 1⟩ |H − 1 → H⟩ |H − 1 → L + 2⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='169 |HF⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='635 |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='180 |H → L + 1⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='646 |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='179 |H → L + 1⟩ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='657 |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩ Table S16: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak IV) of B+ 2 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11 Basis Set Peak I Peak IV Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='910 |H → L + 5⟩ |H → L + 11⟩ |H → L + 11⟩ 6-311++G(3df,3pd) ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='025 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='915 |H → L + 5⟩ |H → L + 11⟩ |H → L + 11⟩ cc-PVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='025 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='965 |H → L + 5⟩ |H → L + 5⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='024 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='942 |H → L + 5⟩ |H → L + 5⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ aug-cc-pVDZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='028 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='895 |H → L + 11⟩ |H → L + 11⟩ |H → L + 5⟩ aug-cc-pVTZ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='918 |H → L + 11⟩ |H → L + 11⟩ |H → L + 5⟩ S7 Table S17: Comparison of oscillator strengths and the dominant configurations contributing to the many-particle wave functions for peaks I and the maximum intensity peak (peak VIII) of B+ 3 cluster calculated using different basis sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S11 Basis Set Peak I Peak VIII Polarization f Wave-function Polarization f Wave-function 6-311++G(2d,2p) ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='005 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='508 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ 6-311++G(3df,3pd) ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='005 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='514 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ cc-PVDZ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='007 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='480 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ cc-pVTZ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='005 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='491 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ aug-cc-pVDZ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='005 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='467 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ aug-cc-pVTZ ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='006 |H → L⟩ ∥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='519 |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩ S8 Table S18: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Li2 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ’E’ corresponds to excitation energy (in eV) of an excited state, f denotes the oscillator strength for a particular electric dipole transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the “|TDM|” column,we present the magnitudes of the transition dipole moments (TDMs) to understand the extent of coupling between the relevant excited state and the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ∥ indicates photon polarization along the direction of the molecule (longitudinal polarization), while ⊥ indicates polarization perpendicular to the molecular axis (transverse polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' ’H’ and ’L’ stand for HOMO and LUMO orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' In the “Wave function” column, each number inside the parentheses denotes the coefficient of the corresponding configuration in the CI wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' GS indicates the ground states wave function of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Peak E (eV) f |TDM| Polarization Wave function GS |HF⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9520) |H → L + 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 15⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='0879) I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='454 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='184 ∥ |H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7523) |H → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3959) II 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='966 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='770 ⊥ |H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7046) |H → L + 7⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5914) III 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='509 ⊥ |H → L + 7⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6291) |H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5439) V 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='303 ⊥ |H → L + 16⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8509) |H → L + 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 16⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1702) VI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='421 ∥ |H → L + 12⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2911) |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 8⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2523) S9 Table S19: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Li3 linear cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |H − 1 → H⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9246) |H − 1 → L + 13⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='0919) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='453 ∥ |HF⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8814) |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 5⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1436) II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='503 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='023 ∥ |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5569) |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 5⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5102) III 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='594 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='096 ⊥ |H − 1 → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6262) |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 11⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3628) IV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='215 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='141 ⊥ |H − 1 → L + 7⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4255) |H − 1 → L + 14⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4237) V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='343 ∥ |H − 1 → L + 8⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4214) |H − 1 → H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 13⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3520) S10 Table S20: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Li3 isosceles triangular cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of TableS18 Peak E (eV) f |TDM| Polarization Wave function GS |HF⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9102) |H − 1 → H⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='0933) I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='129 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='222 ∥ |H → L + 13⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4241) |H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4114) II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='767 ∥ |H → L + 16⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5005) |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4115) III 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='352 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='611 ∥ |H − 1 → H⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5806) |H − 1 → L + 15⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2483) IV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='462 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='885 ∥ |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5537) |H → L + 5⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3869) V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='088 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='163 ⊥ |H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5541) |H → L + 12⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3375) VI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='267 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='922 ⊥ |H − 1 → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4766) |H − 1 → L + 12⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4269) VII 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='117 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='223 ⊥ |H − 1 → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3983) |H → L + 12⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3579) VIII 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='417 ∥ |H → L + 10⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3841) |H → L + 19⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3349) IX 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='533 ∥ |H − 1 → L + 14⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4375) |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3910) X 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='171 ∥ |H → L + 35⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2813) |H → L + 37⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2744) S11 Table S21: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Li4 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |HF⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8913) |(H − 1) → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 18⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='0791) I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='656 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='785 ∥ |H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5922) |H → L + 9⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4041) II 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='660 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='188 ∥ |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6193) |H − 1 → L + 5⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2910) III 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='316 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='098 ⊥ |H → L + 7⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4577) |H → L + 20⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4174) IV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='736 ⊥ |H − 1 → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3080) |H → L + 7⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2299) V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='070 ∥ |H → L + 6⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5653) |H → L + 18⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3856) VI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='862 ⊥ |H − 1 → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2380) |H → L + 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 21⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2227) VII 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='746 ⊥ |H − 1 → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4044) |H − 1 → L + 12⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4026) VIII 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='300 ∥ |H → L + 33⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3681) |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 6⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2294) Table S22: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Be+ 2 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9351) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1684) I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='680 ∥ |H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8403) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3820) II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='821 ⊥ |H → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6705) |H → L + 4⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6375) III 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='386 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='940 ∥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7268) |H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3897) IV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='827 ⊥ |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6418) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6116) V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='757 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='566 ⊥ |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8762) |H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8573) S12 Table S23: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of Be+ 3 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |H → L⟩(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8776) |H − 1 → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1913) I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='179 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='678702 ∥ |H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8200) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3257) II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='433 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='365586 ∥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6152) |H − 1 → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4714) III 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='563211 ⊥ |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5209) |H → L + 17⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4125) IV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='407823 ⊥ |H − 1 → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5211) |H → L + 17⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3217) V 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='242734 ∥ |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6337) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3104) VI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='612656 ⊥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4315) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2864) VII 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='657 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='028291 ⊥ |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4960) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 4⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4653) Table S24: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of B+ 2 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |H → L⟩(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='9033) |H − 2 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 2⟩(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1271) I 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='537 ∥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='7250) |H → L + 11⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='2622) II 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='496 ∥ |H − 1 → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5092) |H − 1 → H⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4944) III 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='364 ⊥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='6138) |H − 2 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4445) IV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='918 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='307 ∥ |H → L + 11⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4808) |H → L + 5⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4713) V 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='216 ⊥ |H − 2 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5322) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4844) S13 Table S25: Many-particle wave functions of excited states contributing to the peaks in the optical absorption spectrum of B+ 3 cluster for aug-cc-pVTZ basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' The rest of the information is same as in the caption of Table S18 Peak E (eV) f |TDM| Polarization Wave function GS |HF⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8535) |H − 1 → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1393) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='524 ⊥ |H → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8572) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1319) II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='726 ∥ |H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='8246) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='1642) III 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='463 ∥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5482) |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3184) IV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='217 ∥ |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5721) |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5675) V 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='295 ∥ |H → L + 3⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3937) |H − 1 → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3723) VI 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='382 ∥ |H → L + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='5866) |H → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4308) VII 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='380 ∥ |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 1⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='4529) |H − 1 → L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L + 2⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3162) VIII 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='519 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='091 ∥ |H → L + 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3762) |H → L + 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' H − 1 → L⟩(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content='3748) Figure S1: Validation of frozen core approximation for the optical absorption spectra of Li2 cluster employing QCI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S14 400 With core-electrons Frozen core approximated 300 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' units) 200 Intensity 100 0 0 2 6 8 10 Energy (eV)Figure S2: Validation of frozen core approximation for the optical absorption spectra of Be+ 2 cluster employing QCI method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' Figure S3: Optical absorption spectra of B+ 3 cluster computed using various basis sets em- ploying B3LYP functional and TD-DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' S15 400 With core-electrons Frozen core approximated 300 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' units) 200 Intensity 100 0 0 2 4 6 8 10 Energy (eV)250 IV 6-311++G(2d,2p) 6-311++G(3df,3pd) 200 cc-pVDZ cc-pVTZ alug-cc-pVDZ Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} +page_content=' units) aug-cc-pVTZ 150 100 50 II III II 0 0 1 2 3 4 5 6 8 9 10 Energy (eV)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE0T4oBgHgl3EQfgAHQ/content/2301.02413v1.pdf'} diff --git a/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/2301.00589v1.pdf.txt b/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/2301.00589v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..00ba96b3a404f678a86aba5801ff25579ce12836 --- /dev/null +++ b/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/2301.00589v1.pdf.txt @@ -0,0 +1,702 @@ +Tailoring the escape rate of a Brownian particle by combining a vortex +flow with a magnetic field +I. Abdoli,1, a) H. Löwen,2 J.-U. Sommer,1, a) and A. Sharma1, a) +1)Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, +Germany +2)Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, 40225, +Germany +(*Electronic mail: abdoli@ipfdd.de.) +The probability per unit time for a thermally activated Brownian particle to escape over a potential well is in general +well-described by Kramers theory. Kramers showed that the escape time decreases exponentially with increasing +barrier height. The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an +external magnetic field. This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers +escape rate without any impact on the barrier-height-dependent exponent. Here we show that the barrier height can +be effectively changed when the charged particle is subjected to an external vortex flow. While the external vortex +alone does not affect the mean escape time of the particle, when combined with a magnetic field it effectively pushes +the fluctuating particle either radially outside or inside depending on its sign relative to that of the magnetic field. In +particular, the effective potential over which the particle escapes can be changed to a flat, a stable, and an unstable +potential by tuning the signs and magnitudes of the external vortex and the applied magnetic field. Notably, the last +case corresponds to enhanced escape dynamics. +I. +INTRODUCTION +A Brownian particle undergoes erratic motion as a result +of its collisions with the solvent molecules. If the particle is +being initially put at the bottom of a potential well, the ther- +mal activation of the particle may cause an escape from the +potential well over an energetic barrier. Using the flux-over- +population method1, Kramers first derived the escape rate of a +Brownian particle over an energy barrier moving in a bistable +potential, regardless of what happens after this escape2. He +showed that the probability per unit time for the particle to es- +cape the potential well exponentially decays with the height +of the energy barrier. Kramers derived limiting expressions +for weak friction and strong damping and realized a global +maximum at some intermediate value of the damping, which +is known as Kramers turnover3–5. The problem has been gen- +eralized to include memory friction6–8 and athermal fluctua- +tions9–13 and was extended to quantum field theory14,15. +While Kramers’ framework and its extensions have been +thoroughly studied with the relevant deterministic potential +force fields16–21, much less is known when the deterministic +force is nonconservative, namely when it is not of potential +type22–24. Recently, by taking into account a nonconservative +force in the form of Lorentz force, we have studied the escape +dynamics of a two-dimensional Brownian system with broken +spatial symmetry via two noises with different strengths25. We +have shown that while the escape process becomes anisotropic +(i.e. particles tend to escape the potential well more along the +axis with larger noise strength) due to two different noises, +when subjected to an external magnetic field, the spatial sym- +metry can be restored25. However, to our knowledge, it is +a)Also at Technische Universität Dresden, Institut für Theoretische Physik, +01069 Dresden, Germany +expected that the escape process is reduced (or unaffected in +the direction of the applied magnetic field) by external con- +stant magnetic fields24,25 which is evident via a rescaling of +the diffusion coefficient. It has been shown that the combined +influence of a nonconservative force and a magnetic field may +cause an instability in the system26. Here, taking advantage +of such an instability, we show that the Lorentz force due to a +constant magnetic field can result in enhanced escape dynam- +ics. +In this work, we study the escape dynamics of a Brown- +ian particle from a harmonic trap which is cut-off at a certain +distance in the presence of an external vortex and the Lorentz +force due to an external constant magnetic field. Taking ad- +vantage of the spatial isotropy in the system we derive an ex- +act expression for the mean first passage time. While the ex- +ternal vortex alone does not affect the escape dynamics, we +observe a nontrivial result when an external magnetic field is +present: the mean first passage time can be reduced or en- +hanced. This is attributed to the shape change of the effec- +tive potential well. By tuning the external magnetic field or +alternatively the strength of the external vortex the effective +potential can change shape to a flat, a stable, or an unstable +potential. This means that by tuning either parameters the +barrier energy over which the particle may escape can be ef- +fectively altered to a smaller or larger one whose origin can +be understood as follows: the combination of the vortex flow +and the magnetic field effectively pushes the fluctuating parti- +cle either radially outside or inside depending on their signs. +In other words, the combination of the two fields, which indi- +vidually induces no radial force, gives rise to a radial force.In +what follows, we first introduce the model. Next we calculate +the mean first passage time, which can be written in terms of +an effective potential. We then study the trends of the escape +time with respect to the magnetic field strength and the vortex +flow and finally we discuss several experimental realizations +of the set-up considered here. +arXiv:2301.00589v1 [cond-mat.stat-mech] 2 Jan 2023 + +2 +FIG. 1. A single charged particle diffusing in a two-dimensional har- +monic potential U(x,y) = k(x2 + y2)/2, shown by concentric con- +tours, with k being its stiffness. The particle is subjected to an ex- +ternal magnetic field B in the −ˆz direction and a nonconservative +force Fnc = ε(−y,x) with ε being its strength. The nonconserva- +tive force is shown for ε > 0. The particle can escape the trap when +reaches the boundary, truncated at r = a, shown by dashed circle, +where r = +� +x2 +y2 is the distance from the origin. +II. +MODEL +We consider an overdamped charged Brownian particle +with the charge q subjected to an external magnetic field B +in the −ˆz direction. Since the Lorentz force due to the field +does not affect the motion of the particle in the z direction, we +effectively reduce the system to a two-dimensional one and +study the motion of the particle in the xy plane. The parti- +cle is trapped in an isotropic potential U(x,y) = k(x2 +y2)/2 +and undergoes a vortex flow due to the nonconservative force +Fnc = ε(−y,x)⊤. Here k and ε are the stiffness of the poten- +tial and the strength of the nonconservative force, respectively. +A schematic of the system is shown in Fig. 1. It is experi- +mentally and theoretically known that even statically optically +trapped Brownian particles in the overdamped limit represent +nonequilibrium behavior characterized by Brownian vortices. +This is due to the nonconservative forces generated by opti- +cal scattering forces27–30. Moreover, by applying a prescribed +external vortex flow field such as a rotating bucket to an un- +derdamped Brownian particle one can induce similar terms to +the nonconservative force, i.e. −εy and εx31. +It has been shown that the overdamped dynamics of the par- +ticle derived by simply setting the inertia term to zero can +yield an incorrect description in the presence of a magnetic +field32. In this case, the overdamped Langevin equation de- +scribing dynamics of the system can be derived using the low- +mass approach25,33,34, which can be written as +˙x = +1 +γ(1+κ2) [−kx−εy+kκy−εκx]+ξx(t), +(1) +˙y = +1 +γ(1+κ2) [−ky+εx−kκx−εκy]+ξy(t), +(2) +FIG. 2. The effective potential from Eq. (4) for different values of +the stiffness ke f f = k + εκ. By varying the parameter κ (or ε) the +effective potential can change shape to a stable one if ke f f > 0, a flat +one if ke f f = 0, or to an unstable one if ke f f < 0. +where γ is the friction coefficient and κ = qB/γ is the diffu- +sive Hall parameter quantifying the strength of the Lorentz +force relative to the frictional force. +We note that κ can +be positive or negative depending on the sign of the applied +magnetic field. Here ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite +noise with zero mean and time correlation ⟨ξ(t)ξ⊤(t′)⟩ = +TG−1δ+(t − t′) + T(G−1)⊤δ−(t − t′) where T is the tem- +perature, G = γ +� 1 +κ +−κ 1 +� +, and the notations δ±(s = t − t′) +are the modified Dirac delta functions which are zero for +s ̸= 0 while +� ∞ +0 dsδ+(s) = +� 0 +−∞ dsδ−(s) = 1 and +� ∞ +0 dsδ−(s) = +� 0 +−∞ dsδ+(s) = 0. Throughout this work we set the Boltzmann +constant kB to unity. Length and time are measured in units of +� +T/k and γ/k, respectively. +We use Itô calculus to reduce the Langevin equations in +Eq. (1) and Eq. (2) to a one-dimensional problem for the vari- +able r = +� +x2 +y2, which is given as35 +dr = +1 +1+κ2 +� +−k +εκ +γ +r + D +r +� +dt + +� +2D +1+κ2 η(t)dt, +(3) +where D = T/γ is the coefficient of a freely diffusing particle +and η(t) is Gaussian white noise with zero mean and the Dirac +delta time correlation ⟨η(t)η(t′)⟩ = δ(t −t′). The terms in the +square brackets on the right hand side of Eq.(3) describe the +force on the the particle due to an effective potential, given as +Uef f (r) = +kef f +2γ(1+κ2)r2 − +D +1+κ2 log(r), +(4) +where kef f = k +εκ is the stiffness of the effective potential. +As it is evident from the effective stiffness, in the absence of +the vortex flow the potential simply gets rescaled by the factor +1/(1 + κ2), as we have shown in the supplemental informa- +tion of Ref.25. Moreover, in the absence of the magnetic field +there is no effect of the vortex flow on the effective potential +and Eq. (4) reduces to the well known results in Ref.35 for a +rotationally symmetric Ornstein-Uhlenbeck process in two di- +mensions. The second term on the right and side comes from + +1.0 +a. +y 0.0 +-1.0 +-1.0 +0.0 +1.0keff= - 0.6 +keff = 0.6 +30 + keff= 0.0 +keff= k +keff= 2.0 +20 +10 ++ +0 +-10 +0 +2 +3 +4 +5 +1 +63 +FIG. 3. The mean escape time as a function of the diffusive Hall pa- +rameter κ from Eq. (5) and Eq. (7) for different values of the scaled +barrier height β∆E with β = 1.0, γ = 1.0 and ε = 0.2. Obviously +the mean escape time increases with increasing the barrier height. It +can increase or decrease by tuning the parameter κ: the presence of +an external vortex field can work together with the applied magnetic +field to effectively push the fluctuating particle either radially out- +side, if κ < −k/ε, or inside, if κ > −k/ε. The former corresponds +to the case in which the combination helps the particle to escape. +The point κ = 0 corresponds to unaffected escape time by the exter- +nal vortex (i.e. ke f f = k). In the inset, we show the mean escape time +which is scaled by the mean escape time in the absence of the exter- +nal vortex ⟨t0⟩ where the subscript 0 indicates zero strength length +of the vortex flow. It implies that the mean escape time can decrease +with increasing κ as compared to the mean escape time without the +vortex flow. +the transformation to r and corresponds to an extremely re- +pulsive potential at the origin due to reduced number of states +on the circle of radius r. This term influences the motion of +the particle only near the origin and is negligible for larger +distances as compared to the first term. +Figure. 2 represents the scaled effective potential from +Eq. (4) for different values of the parameter ke f f without the +logarithmic term. By tuning the diffusive Hall parameter or al- +ternatively the strength of the nonconservative force, the effec- +tive potential changes shape: the potential is stable if kef f > 0, +flat if kef f = 0, and unstable if ke f f < 0. It becomes simple +quadratic potential in the absence of ε or/and κ. +III. +MEAN ESCAPE TIME +We consider a particle which is trapped in an isotropic po- +tential U(x,y) which taking advantage of the spatial symmetry +whose distance from the origin, r = |r|, can be described by +Eq. (3). We are interested in the mean time at which the par- +ticle reaches the boundary, truncated at r = a, as shown in +Fig. 1. As we show in the Appendix A, the mean escape time +can be exactly calculated from Eq. (3) which reads +⟨t⟩ = γ(1+κ2) +2kef f +� +Ei +� +β∆Ee f f +� +−log +� +β∆Ee f f +� +−γEM +� +, (5) +FIG. 4. +The mean escape time with respect to the strength of the +conservative force ε from Eq. (5) and Eq. (7) for different values of +the scaled barrier heights with β = 1.0 and γ = 1.0. The lines with +circles and squares correspond to the results with κ = 2.0 and κ = +−2.0, respectively. The mean escape time can increase or decrease +with increasing the strength of the vortex flow, which depends on +its sign relative to that of κ and their magnitude compared to the +stiffness of the potential k. +if kef f > 0 corresponding to the effective stable potential and +⟨t⟩ = γ(1+κ2) +2kef f +� +−Ei +� +−β|∆Eef f | +� ++log +� +β|∆Eef f | +� ++γEM +� +, +(6) +if kef f < 0 corresponding to the unstable effective potential +where β is the inverse of the temperature, γEM is the Euler- +Mascheroni constant, and Ei(x) is the exponential integral. +Here ∆Eef f = ∆E + εκa2/2 is the effective barrier energy +which is the real barrier height ∆E = ka2/2 augmented by the +coupling between the magnitude of the applied magnetic field +and the strength of the external vortex. Using the series ex- +pansion of the exponential integral at kef f = 0 for Eq. (5) and +Eq. (6), the mean escape time for the effective flat potential +reads +⟨t⟩ ∼ (1+κ2) +4D +a2, +(7) +which is the mean escape time for a freely diffusing parti- +cle scaled by 1 + κ2. +In the limit of large barrier heights +the exponential integral in Eq. (5) can be expanded and as +a consequence the mean escape time reduces to ⟨t⟩ ∼ γ(1 + +κ2)exp(β∆Eef f )/(2kef f β∆Eef f ). In the absence of the exter- +nal vortex, which corresponds to ε = 0, the result reduces to +the Kramers result rescaled by 1 + κ2 arising from the trivial +rescaling of the diffusion coefficient. The expression becomes +the same as the Kramers one when the magnetic field is absent +κ = 0. This confirms that the external vortex field alone does +not affect the mean escape time. The intuitive reason for that +is that, for κ = 0, the presence of a vortex field only changes +the azimuthal motion but not the radial one which leaves the +redial particle escape unaffected. +Figure 3 shows the mean escape time with respect to the +diffusive Hall parameter κ. Obviously it takes the particle +longer time to escape over larger barrier heights as is evident +in the figure. The magnetic field together with the vortex flow + +103 +102 +106 +101 +105 +100 +10-1 +104 +10-2 +(t) +103 +-8 +9- +-4 +-2 +0 +2 +4 +102 +101 +β△E = 0.5 +β△E = 2.0 +100 +β△E = 4.5 +β△E = 8.0 +-8 +-6 +0 +f2 +4 +K106 +β△E = 0.5 +β△E = 2.0 +0 +K=2.0 +口 +K= - 2.0 +βE = 4.5 +105 +β△E = 8.0 +104 +0 +103 +(t) +102 +Q +Q +N +2 +101 +100 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.54 +creates additional fluctuations in radial direction which can +be directed either outwards or inwards depending on its sign. +The former corresponds to the case in which the combination +of the vortex flow and the magnetic field helps the particle to +escape. The inset shows the mean escape time scaled by the +mean escape time in the absence of the external vortex, which +is indicated by the subscript 0. The mean escape time can +decrease with increasing magnetic field as compared to the +mean escape time without the vortex flow and remains almost +constant for small barrier height. +In Fig.4, we show that tuning the strength of the vortex flow +is an alternative way to vary the mean escape time which is +evident in Eq.(5) and Eq.(6) via the production of the two pa- +rameters, i.e. εκ. Therefore the similar trends are expected. +The figure represents the mean escape time with respect to the +parameter ε for a system with κ = 2.0, denoted by lines with +circles, and a system with κ = −2.0, denoted by lines with +squares. Our results imply that the mean escape time can be +decreased or increased by tuning the vortex flow strength de- +pending on its sign relative to that of the magnetic field and +their magnitude compared to the stiffness of the potential k. +IV. +DISCUSSION +In this work, we studied the effect of a vortex flow on the es- +cape dynamics of a Brownian magneto-system made of single +charged Brownian particle subjected to an external magnetic +field. We expressed the potential in an effective form which +can change shape to a stable, a flat, or an unstable potential +depending on the stiffness of the effective potential. Taking +advantage of the spatial isotropy in the system we obtained an +exact expression for the mean escape time. In the absence of +the external vortex, exerted by the nonconservative force, the +Lorentz force due to the external magnetic field slows down +the dynamics of the system without any qualitative change, +which is evident via the trivial rescaling of the diffusion coef- +ficient. We showed that while the external vortex alone does +not affect the mean escape time, when coupled to the mag- +netic field it can enhance or reduce the escape time: this is +intuitive as the magnetic field together with the vortex flow +creates additional fluctuations in radial direction which can be +directed either outwards or inwards depending on its sign. In +other words, the combination of the two fields, which individ- +ually induces no radial force, gives rise to a radial force. We +showed that the barrier over which the particle escapes can be +effectively changed to a larger or smaller one depending on +the relative signs of the strength of the vortex flow and the ap- +plied magnetic field as well as their magnitude compared to +the stiffness of the potential in which the particle is trapped. +Moreover, the trap can be effectively switched-off by an ap- +propriate sign and value of the magnetic field. +A possible experimental realisation is to trap the particle us- +ing optical tweezers either in a radio-frequency plasma sheath +with a vertical magnetic field37,38 or in a rotating frame of +reference. By rotating the reference frame a Coriolis force +can be induced which acts the same as the Lorentz force due +to an external magnetic field39–41. +As it has been shown +that even statically optically trapped Brownian particles un- +dergoe a nonconservative force induced by optical scattering +forces27–30,36, we expect that the study of the enhanced es- +cape dynamics does not require an additional external vortex. +Another possibility is to apply a rotating bucket to an under- +damped Brownian particle which induces similar terms to the +nonconservative force in the overdamped limit31. +From a future perspective, it could be interesting to study +the escape dynamics of an opposite charged dimer42. In the +limit of low persistence length, an active chiral particle fol- +lows curved trajectories, similar to the Brownian motion of a +charged particle43,44 under a magnetic field. Therefore, an- +other study of interest would be the escape dynamics of a +chiral active Brownian particle in the presence of an external +vortex. Finally, it could be interesting to study how an exter- +nal magnetic field can affect an active turnover for an active +particle in a bistable potential45–an optimal correlation time +where the transition rate is maximized– and how an external +vortex influences new turnovers observed in the presence of a +fluctuating magnetic field22,23. +ACKNOWLEDGMENTS +A. Sharma and H. Löwen acknowledge the support by the +Deutsche Forschungsgemeinschaft (DFG) within the projects +SH 1275/3-1 (A.S.) and LO 418/25-1 (H.L.). +DATA AVAILABILITY STATEMENT +The data that support the findings of this study are available +from the corresponding author upon reasonable request. +AUTHOR DECLARATIONS +The authors have no conflicts to disclose. +Appendix A: Derivation of the mean escape time +The main purpose of this section is to derive the mean +escape time in Eq. (5) to Eq. (7). +We start with the un- +derdamped Langevin equation describing the dynamics of a +charged Brownian particle with mass m and charge q sub- +jected to a magnetic field B in the −ˆz direction. The veloc- +ity Langevin equation for the position r = (x,y)⊤ and the +velocity v = (vx,vy)⊤ of the particle under the effect of the +linear nonconservative force Fnc = ε(−y,x)⊤ and the con- +servative force Fc = −k(x,y)⊤ due to the isotropic potential +U(x,y) = k(x2 +y2)/2, can be written as +m ˙v = −Kr −Gv(t)+ +� +2γTη(t), +(A1) +where η(t) = (ηx(t),ηy(t))⊤ is the Gaussian white noise with +zero mean and Dirac delta correlation ⟨η(t)η⊤(t′)⟩ = δ(t −t′) + +5 +with γ being the friction coefficient and T the temperature. +The matrices G and K are defined as +G = γ +� +1 +κ +−κ 1 +� +, K = +� +k +ε +−ε k +� +, +(A2) +with κ = qB/γ being the diffusive Hall parameter which quan- +tifies the strength of the Lorentz force relative to the frictional +force. Using the low-mass approach, the corresponding over- +damped Langevin equation can be written as25,33,34 +˙r = Ar +ξ(t), +(A3) +where A = G−1K and ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite +noise with +⟨ξ(t)⟩ = 0, +(A4) +⟨ξ(t)ξ⊤(t′)⟩ = TG−1δ+(t −t′)+T(G−1)⊤δ−(t −t′), (A5) +where δ±(s = t − t′) are the modified Dirac delta functions +which are zero for s ̸= 0 while +� ∞ +0 dsδ+(s) = +� 0 +−∞ dsδ−(s) = 1 +and +� ∞ +0 dsδ−(s) = +� 0 +−∞ dsδ+(s) = 0. +Equation (A3) can be rewritten as Eq. (1) and Eq. (2) in +the Cartesian coordinates and thereafter using Itô calculus can +be reduced to a one-dimensional equation for the variable r, +which is the distance from the origin and is given by Eq. (3). +The mean time for the particle to escape the trap, truncated at +r = a, can be obtained by the following equation +⟨t⟩ = 1+κ2 +D +� a +0 y−1 exp +�ke f f +2γDy2 +� +dy +� y +0 zexp +� +−kef f +2γDz2 +� +dz, +(A6) +where D = T/γ is the diffusion coefficient for a freely moving +particle. This equation can be exactly solved: using a change +of variables the second integral on the right hand side gives +(γD/kef f ) +� +1−exp +� +−ke f f y2/2γD +�� +. By substitution of this +solution into Eq. (A6), the resulting integral can be exactly +solved which gives Eq. (5) and Eq. (6). +REFERENCES +1L. Farkas, “Keimbildungsgeschwindigkeit in übersättigten Dämpfen,” +Zeitschrift für physikalische Chemie 125, 236–242 (1927). +2H. A. Kramers, “Brownian motion in a field of force and the diffusion +model of chemical reactions,” Physica 7, 284–304 (1940). +3H. Grabert and S. Linkwitz, “Effect of time-delayed friction on the escape +from a metastable well,” Physical Review A 37, 963 (1988). +4L. I. McCann, M. Dykman, and B. Golding, “Thermally activated tran- +sitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 +(1999). +5L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, and L. Novotny, +“Direct measurement of Kramers turnover with a levitated nanoparticle,” +Nature Nanotechnology 12, 1130–1133 (2017). +6R. F. Grote and J. T. Hynes, “The stable states picture of chemical reactions. +ii. rate constants for condensed and gas phase reaction models,” The Journal +of Chemical Physics 73, 2715–2732 (1980). +7B. Carmeli and A. Nitzan, “Non-Markoffian theory of activated rate pro- +cesses,” Physical Review Letters 49, 423 (1982). +8R. Ianconescu and E. Pollak, “A study of Kramers’ turnover theory in the +presence of exponential memory friction,” The Journal of Chemical Physics +143, 104104 (2015). +9P. Hänggi, F. Marchesoni, +and P. Grigolini, “Bistable flow driven by +coloured Gaussian noise: a critical study,” Zeitschrift für Physik B Con- +densed Matter 56, 333–339 (1984). +10P. Jung and P. Hänggi, “Bistability and colored noise in nonequilibrium +systems: theory versus precise numerics,” Physical Review Letters 61, 11 +(1988). +11A. Sharma, R. Wittmann, and J. M. Brader, “Escape rate of active parti- +cles in the effective equilibrium approach,” Physical Review E 95, 012115 +(2017). +12A. Scacchi, J. M. Brader, and A. Sharma, “Escape rate of transiently ac- +tive Brownian particle in one dimension,” Physical Review E 100, 012601 +(2019). +13L. Caprini, U. Marini Bettolo Marconi, A. Puglisi, and A. Vulpiani, “Ac- +tive escape dynamics: The effect of persistence on barrier crossing,” The +Journal of Chemical Physics 150, 024902 (2019). +14A. Berera, J. Mabillard, B. W. Mintz, and R. O. Ramos, “Formulating the +Kramers problem in field theory,” Physical Review D 100, 076005 (2019). +15L. Darmé, J. Jaeckel, and M. Lewicki, “Generalized escape paths for dy- +namical tunneling in qft,” Physical Review D 100, 096012 (2019). +16G. R. Fleming and P. Hänggi, Activated Barrier Crossing: applications in +physics, chemistry and biology, Vol. 4 (World Scientific, 1993). +17P. Talkner and P. Hänggi, New trends in Kramers’ reaction rate theory, +Vol. 11 (Springer Science & Business Media, 1995). +18P. Hänggi and F. Mojtabai, “Thermally activated escape rate in presence of +long-time memory,” Physical Review A 26, 1168 (1982). +19E. Pollak, “Theory of activated rate processes: +A new derivation of +Kramers’ expression,” The Journal of Chemical Physics 85, 865–867 +(1986). +20E. Pollak, H. Grabert, and P. Hänggi, “Theory of activated rate processes +for arbitrary frequency dependent friction: Solution of the turnover prob- +lem,” The Journal of Chemical Physics 91, 4073–4087 (1989). +21E. Pollak and J. Ankerhold, “Improvements to Kramers turnover theory,” +The Journal of Chemical Physics 138, 164116 (2013). +22A. Baura, S. Ray, and B. C. Bag, “Tuning of barrier crossing time of a par- +ticle by time dependent magnetic field,” The Journal of Chemical Physics +138, 244110 (2013). +23S. Mondal, A. Baura, S. Das, and B. C. Bag, “A generic signature of a fluc- +tuating magnetic field: An additional turnover prior to the Kramers’ one,” +Physica A: Statistical Mechanics and its Applications 502, 58–76 (2018). +24R. Filliger and P. Reimann, “Kramers escape rate for a charged particle in a +magnetic field,” EPL (Europhysics Letters) 77, 30008 (2007). +25I. Abdoli, J.-U. Sommer, H. Löwen, and A. Sharma, “Escape dynamics +in an anisotropically driven Brownian magneto-system,” EPL (Europhysics +Letters) 139, 21003 (2022). +26S. Lee and C. Kwon, “Nonequilibrium driven by an external torque in the +presence of a magnetic field,” Physical Review E 99, 052142 (2019). +27B. Sun, D. G. Grier, and A. Y. Grosberg, “Minimal model for Brownian +vortexes,” Physical Review E 82, 021123 (2010). +28B. Sun, J. Lin, E. Darby, A. Y. Grosberg, +and D. G. Grier, “Brownian +vortexes,” Physical Review E 80, 010401 (2009). +29Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of non- +conservative optical forces on the dynamics of optically trapped colloidal +spheres: the fountain of probability,” Physical Review Letters 101, 128301 +(2008). +30H. W. Moyses, R. O. Bauer, A. Y. Grosberg, and D. G. Grier, “Perturbative +theory for Brownian vortexes,” Physical Review E 91, 062144 (2015). +31B. Liebchen and H. Löwen, “Optimal navigation strategies for active parti- +cles,” EPL (Europhysics Letters) 127, 34003 (2019). +32H. D. Vuijk, J. M. Brader, and A. Sharma, “Anomalous fluxes in over- +damped Brownian dynamics with Lorentz force,” Journal of Statistical Me- +chanics: Theory and Experiment 2019, 063203 (2019). +33H.-M. Chun, X. Durang, and J. D. Noh, “Emergence of nonwhite noise in +Langevin dynamics with magnetic Lorentz force,” Physical Review E 97, +032117 (2018). +34I. Abdoli, R. Wittmann, J. M. Brader, J.-U. Sommer, H. Löwen, +and +A. Sharma, “Tunable Brownian magneto heat pump,” Scientific Reports +12, 1–10 (2022). + +6 +35C. Gardiner, Stochastic methods, Vol. 4 (Springer Berlin, 2009). +36M. Mangeat, Y. Amarouchene, Y. Louyer, T. Guérin, and D. S. Dean, “Role +of nonconservative scattering forces and damping on Brownian particles in +optical traps,” Physical Review E 99, 052107 (2019). +37J. Carstensen, F. Greiner, L.-J. Hou, H. Maurer, and A. Piel, “Effect of +neutral gas motion on the rotation of dust clusters in an axial magnetic +field,” Physics of Plasmas 16, 013702 (2009). +38A. Piel, Plasma physics: an introduction to laboratory, space, and fusion +plasmas (Springer, 2017). +39H. Kählert, J. Carstensen, M. Bonitz, H. Löwen, F. Greiner, and A. Piel, +“Magnetizing a complex plasma without a magnetic field,” Physical Review +Letters 109, 155003 (2012). +40P. Hartmann, Z. Donkó, T. Ott, H. Kählert, +and M. Bonitz, “Magneto- +plasmons in rotating dusty plasmas,” Physical Review Letters 111, 155002 +(2013). +41P. Hartmann, J. C. Reyes, E. G. Kostadinova, L. S. Matthews, T. W. Hyde, +R. U. Masheyeva, K. N. Dzhumagulova, T. S. Ramazanov, T. Ott, H. Käh- +lert, et al., “Self-diffusion in two-dimensional quasimagnetized rotating +dusty plasmas,” Physical Review E 99, 013203 (2019). +42R. Shinde, J. U. Sommer, H. Löwen, and A. Sharma, “Strongly enhanced +dynamics of a charged rouse dimer by an external magnetic field,” PNAS +Nexus 1, pgac119 (2022). +43S. Van Teeffelen and H. Löwen, “Dynamics of a Brownian circle swimmer,” +Physical Review E 78, 020101 (2008). +44C. Scholz, A. Ldov, T. Pöschel, M. Engel, and H. Löwen, “Surfactants and +rotelles in active chiral fluids,” Science Advances 7, eabf8998 (2021). +45A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjohanns, L. Novotny, +and C. Dellago, “Escape dynamics of active particles in multistable poten- +tials,” Nature Communications 12, 1–6 (2021). + diff --git a/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/load_file.txt b/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cd6b6aee949140c1be6825fd0b856d4cfec4219 --- /dev/null +++ b/NdAyT4oBgHgl3EQfs_n5/content/tmp_files/load_file.txt @@ -0,0 +1,465 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf,len=464 +page_content='Tailoring the escape rate of a Brownian particle by combining a vortex flow with a magnetic field I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Abdoli,1, a) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sommer,1, a) and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma1, a) 1)Leibniz-Institut für Polymerforschung Dresden, Institut Theorie der Polymere, 01069 Dresden, Germany 2)Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, 40225, Germany (*Electronic mail: abdoli@ipfdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=') The probability per unit time for a thermally activated Brownian particle to escape over a potential well is in general well-described by Kramers theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kramers showed that the escape time decreases exponentially with increasing barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The dynamics slow down when the particle is charged and subjected to a Lorentz force due to an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This is evident via a rescaling of the diffusion coefficient entering as a prefactor in the Kramers escape rate without any impact on the barrier-height-dependent exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Here we show that the barrier height can be effectively changed when the charged particle is subjected to an external vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' While the external vortex alone does not affect the mean escape time of the particle, when combined with a magnetic field it effectively pushes the fluctuating particle either radially outside or inside depending on its sign relative to that of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In particular, the effective potential over which the particle escapes can be changed to a flat, a stable, and an unstable potential by tuning the signs and magnitudes of the external vortex and the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Notably, the last case corresponds to enhanced escape dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' INTRODUCTION A Brownian particle undergoes erratic motion as a result of its collisions with the solvent molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' If the particle is being initially put at the bottom of a potential well, the ther- mal activation of the particle may cause an escape from the potential well over an energetic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Using the flux-over- population method1, Kramers first derived the escape rate of a Brownian particle over an energy barrier moving in a bistable potential, regardless of what happens after this escape2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' He showed that the probability per unit time for the particle to es- cape the potential well exponentially decays with the height of the energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kramers derived limiting expressions for weak friction and strong damping and realized a global maximum at some intermediate value of the damping, which is known as Kramers turnover3–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The problem has been gen- eralized to include memory friction6–8 and athermal fluctua- tions9–13 and was extended to quantum field theory14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' While Kramers’ framework and its extensions have been thoroughly studied with the relevant deterministic potential force fields16–21, much less is known when the deterministic force is nonconservative, namely when it is not of potential type22–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Recently, by taking into account a nonconservative force in the form of Lorentz force, we have studied the escape dynamics of a two-dimensional Brownian system with broken spatial symmetry via two noises with different strengths25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We have shown that while the escape process becomes anisotropic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' particles tend to escape the potential well more along the axis with larger noise strength) due to two different noises, when subjected to an external magnetic field, the spatial sym- metry can be restored25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' However, to our knowledge, it is a)Also at Technische Universität Dresden, Institut für Theoretische Physik, 01069 Dresden, Germany expected that the escape process is reduced (or unaffected in the direction of the applied magnetic field) by external con- stant magnetic fields24,25 which is evident via a rescaling of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It has been shown that the combined influence of a nonconservative force and a magnetic field may cause an instability in the system26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Here, taking advantage of such an instability, we show that the Lorentz force due to a constant magnetic field can result in enhanced escape dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In this work, we study the escape dynamics of a Brown- ian particle from a harmonic trap which is cut-off at a certain distance in the presence of an external vortex and the Lorentz force due to an external constant magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Taking ad- vantage of the spatial isotropy in the system we derive an ex- act expression for the mean first passage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' While the ex- ternal vortex alone does not affect the escape dynamics, we observe a nontrivial result when an external magnetic field is present: the mean first passage time can be reduced or en- hanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This is attributed to the shape change of the effec- tive potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' By tuning the external magnetic field or alternatively the strength of the external vortex the effective potential can change shape to a flat, a stable, or an unstable potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This means that by tuning either parameters the barrier energy over which the particle may escape can be ef- fectively altered to a smaller or larger one whose origin can be understood as follows: the combination of the vortex flow and the magnetic field effectively pushes the fluctuating parti- cle either radially outside or inside depending on their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In other words, the combination of the two fields, which indi- vidually induces no radial force, gives rise to a radial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='In what follows, we first introduce the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Next we calculate the mean first passage time, which can be written in terms of an effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We then study the trends of the escape time with respect to the magnetic field strength and the vortex flow and finally we discuss several experimental realizations of the set-up considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='00589v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='stat-mech] 2 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' A single charged particle diffusing in a two-dimensional har- monic potential U(x,y) = k(x2 + y2)/2, shown by concentric con- tours, with k being its stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The particle is subjected to an ex- ternal magnetic field B in the −ˆz direction and a nonconservative force Fnc = ε(−y,x) with ε being its strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The nonconserva- tive force is shown for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The particle can escape the trap when reaches the boundary, truncated at r = a, shown by dashed circle, where r = � x2 +y2 is the distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' MODEL We consider an overdamped charged Brownian particle with the charge q subjected to an external magnetic field B in the −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Since the Lorentz force due to the field does not affect the motion of the particle in the z direction, we effectively reduce the system to a two-dimensional one and study the motion of the particle in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The parti- cle is trapped in an isotropic potential U(x,y) = k(x2 +y2)/2 and undergoes a vortex flow due to the nonconservative force Fnc = ε(−y,x)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Here k and ε are the stiffness of the poten- tial and the strength of the nonconservative force, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' A schematic of the system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It is experi- mentally and theoretically known that even statically optically trapped Brownian particles in the overdamped limit represent nonequilibrium behavior characterized by Brownian vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This is due to the nonconservative forces generated by opti- cal scattering forces27–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Moreover, by applying a prescribed external vortex flow field such as a rotating bucket to an un- derdamped Brownian particle one can induce similar terms to the nonconservative force, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' −εy and εx31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It has been shown that the overdamped dynamics of the par- ticle derived by simply setting the inertia term to zero can yield an incorrect description in the presence of a magnetic field32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In this case, the overdamped Langevin equation de- scribing dynamics of the system can be derived using the low- mass approach25,33,34, which can be written as ˙x = 1 γ(1+κ2) [−kx−εy+kκy−εκx]+ξx(t), (1) ˙y = 1 γ(1+κ2) [−ky+εx−kκx−εκy]+ξy(t), (2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The effective potential from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (4) for different values of the stiffness ke f f = k + εκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' By varying the parameter κ (or ε) the effective potential can change shape to a stable one if ke f f > 0, a flat one if ke f f = 0, or to an unstable one if ke f f < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' where γ is the friction coefficient and κ = qB/γ is the diffu- sive Hall parameter quantifying the strength of the Lorentz force relative to the frictional force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We note that κ can be positive or negative depending on the sign of the applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Here ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite noise with zero mean and time correlation ⟨ξ(t)ξ⊤(t′)⟩ = TG−1δ+(t − t′) + T(G−1)⊤δ−(t − t′) where T is the tem- perature, G = γ � 1 κ −κ 1 � , and the notations δ±(s = t − t′) are the modified Dirac delta functions which are zero for s ̸= 0 while � ∞ 0 dsδ+(s) = � 0 −∞ dsδ−(s) = 1 and � ∞ 0 dsδ−(s) = � 0 −∞ dsδ+(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Throughout this work we set the Boltzmann constant kB to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Length and time are measured in units of � T/k and γ/k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We use Itô calculus to reduce the Langevin equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (2) to a one-dimensional problem for the vari- able r = � x2 +y2, which is given as35 dr = 1 1+κ2 � −k +εκ γ r + D r � dt + � 2D 1+κ2 η(t)dt, (3) where D = T/γ is the coefficient of a freely diffusing particle and η(t) is Gaussian white noise with zero mean and the Dirac delta time correlation ⟨η(t)η(t′)⟩ = δ(t −t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The terms in the square brackets on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (3) describe the force on the the particle due to an effective potential, given as Uef f (r) = kef f 2γ(1+κ2)r2 − D 1+κ2 log(r), (4) where kef f = k +εκ is the stiffness of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' As it is evident from the effective stiffness, in the absence of the vortex flow the potential simply gets rescaled by the factor 1/(1 + κ2), as we have shown in the supplemental informa- tion of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Moreover, in the absence of the magnetic field there is no effect of the vortex flow on the effective potential and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (4) reduces to the well known results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='35 for a rotationally symmetric Ornstein-Uhlenbeck process in two di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The second term on the right and side comes from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0keff= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='6 keff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='6 30 keff= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 keff= k keff= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 20 10 + 0 10 0 2 3 4 5 1 63 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The mean escape time as a function of the diffusive Hall pa- rameter κ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (7) for different values of the scaled barrier height β∆E with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Obviously the mean escape time increases with increasing the barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It can increase or decrease by tuning the parameter κ: the presence of an external vortex field can work together with the applied magnetic field to effectively push the fluctuating particle either radially out- side, if κ < −k/ε, or inside, if κ > −k/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The former corresponds to the case in which the combination helps the particle to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The point κ = 0 corresponds to unaffected escape time by the exter- nal vortex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' ke f f = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In the inset, we show the mean escape time which is scaled by the mean escape time in the absence of the exter- nal vortex ⟨t0⟩ where the subscript 0 indicates zero strength length of the vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It implies that the mean escape time can decrease with increasing κ as compared to the mean escape time without the vortex flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' the transformation to r and corresponds to an extremely re- pulsive potential at the origin due to reduced number of states on the circle of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This term influences the motion of the particle only near the origin and is negligible for larger distances as compared to the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 2 represents the scaled effective potential from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (4) for different values of the parameter ke f f without the logarithmic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' By tuning the diffusive Hall parameter or al- ternatively the strength of the nonconservative force, the effec- tive potential changes shape: the potential is stable if kef f > 0, flat if kef f = 0, and unstable if ke f f < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' It becomes simple quadratic potential in the absence of ε or/and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' MEAN ESCAPE TIME We consider a particle which is trapped in an isotropic po- tential U(x,y) which taking advantage of the spatial symmetry whose distance from the origin, r = |r|, can be described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We are interested in the mean time at which the par- ticle reaches the boundary, truncated at r = a, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' As we show in the Appendix A, the mean escape time can be exactly calculated from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (3) which reads ⟨t⟩ = γ(1+κ2) 2kef f � Ei � β∆Ee f f � −log � β∆Ee f f � −γEM � , (5) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The mean escape time with respect to the strength of the conservative force ε from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (7) for different values of the scaled barrier heights with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The lines with circles and squares correspond to the results with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 and κ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The mean escape time can increase or decrease with increasing the strength of the vortex flow, which depends on its sign relative to that of κ and their magnitude compared to the stiffness of the potential k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' if kef f > 0 corresponding to the effective stable potential and ⟨t⟩ = γ(1+κ2) 2kef f � −Ei � −β|∆Eef f | � +log � β|∆Eef f | � +γEM � , (6) if kef f < 0 corresponding to the unstable effective potential where β is the inverse of the temperature, γEM is the Euler- Mascheroni constant, and Ei(x) is the exponential integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Here ∆Eef f = ∆E + εκa2/2 is the effective barrier energy which is the real barrier height ∆E = ka2/2 augmented by the coupling between the magnitude of the applied magnetic field and the strength of the external vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Using the series ex- pansion of the exponential integral at kef f = 0 for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (6), the mean escape time for the effective flat potential reads ⟨t⟩ ∼ (1+κ2) 4D a2, (7) which is the mean escape time for a freely diffusing parti- cle scaled by 1 + κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In the limit of large barrier heights the exponential integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) can be expanded and as a consequence the mean escape time reduces to ⟨t⟩ ∼ γ(1 + κ2)exp(β∆Eef f )/(2kef f β∆Eef f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In the absence of the exter- nal vortex, which corresponds to ε = 0, the result reduces to the Kramers result rescaled by 1 + κ2 arising from the trivial rescaling of the diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The expression becomes the same as the Kramers one when the magnetic field is absent κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This confirms that the external vortex field alone does not affect the mean escape time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The intuitive reason for that is that, for κ = 0, the presence of a vortex field only changes the azimuthal motion but not the radial one which leaves the redial particle escape unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Figure 3 shows the mean escape time with respect to the diffusive Hall parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Obviously it takes the particle longer time to escape over larger barrier heights as is evident in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The magnetic field together with the vortex flow 103 102 106 101 105 100 10-1 104 10-2 (t) 103 8 9- 4 2 0 2 4 102 101 β△E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 β△E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 100 β△E = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 β△E = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 8 6 0 f2 4 K106 β△E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 β△E = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 0 K=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 口 K= - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 βE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 105 β△E = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 104 0 103 (t) 102 Q Q N 2 101 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='54 creates additional fluctuations in radial direction which can be directed either outwards or inwards depending on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The former corresponds to the case in which the combination of the vortex flow and the magnetic field helps the particle to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The inset shows the mean escape time scaled by the mean escape time in the absence of the external vortex, which is indicated by the subscript 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The mean escape time can decrease with increasing magnetic field as compared to the mean escape time without the vortex flow and remains almost constant for small barrier height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='4, we show that tuning the strength of the vortex flow is an alternative way to vary the mean escape time which is evident in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (6) via the production of the two pa- rameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' εκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Therefore the similar trends are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The figure represents the mean escape time with respect to the parameter ε for a system with κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0, denoted by lines with circles, and a system with κ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='0, denoted by lines with squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Our results imply that the mean escape time can be decreased or increased by tuning the vortex flow strength de- pending on its sign relative to that of the magnetic field and their magnitude compared to the stiffness of the potential k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' DISCUSSION In this work, we studied the effect of a vortex flow on the es- cape dynamics of a Brownian magneto-system made of single charged Brownian particle subjected to an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We expressed the potential in an effective form which can change shape to a stable, a flat, or an unstable potential depending on the stiffness of the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Taking advantage of the spatial isotropy in the system we obtained an exact expression for the mean escape time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In the absence of the external vortex, exerted by the nonconservative force, the Lorentz force due to the external magnetic field slows down the dynamics of the system without any qualitative change, which is evident via the trivial rescaling of the diffusion coef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We showed that while the external vortex alone does not affect the mean escape time, when coupled to the mag- netic field it can enhance or reduce the escape time: this is intuitive as the magnetic field together with the vortex flow creates additional fluctuations in radial direction which can be directed either outwards or inwards depending on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In other words, the combination of the two fields, which individ- ually induces no radial force, gives rise to a radial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We showed that the barrier over which the particle escapes can be effectively changed to a larger or smaller one depending on the relative signs of the strength of the vortex flow and the ap- plied magnetic field as well as their magnitude compared to the stiffness of the potential in which the particle is trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Moreover, the trap can be effectively switched-off by an ap- propriate sign and value of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' A possible experimental realisation is to trap the particle us- ing optical tweezers either in a radio-frequency plasma sheath with a vertical magnetic field37,38 or in a rotating frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' By rotating the reference frame a Coriolis force can be induced which acts the same as the Lorentz force due to an external magnetic field39–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' As it has been shown that even statically optically trapped Brownian particles un- dergoe a nonconservative force induced by optical scattering forces27–30,36, we expect that the study of the enhanced es- cape dynamics does not require an additional external vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Another possibility is to apply a rotating bucket to an under- damped Brownian particle which induces similar terms to the nonconservative force in the overdamped limit31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' From a future perspective, it could be interesting to study the escape dynamics of an opposite charged dimer42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' In the limit of low persistence length, an active chiral particle fol- lows curved trajectories, similar to the Brownian motion of a charged particle43,44 under a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Therefore, an- other study of interest would be the escape dynamics of a chiral active Brownian particle in the presence of an external vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Finally, it could be interesting to study how an exter- nal magnetic field can affect an active turnover for an active particle in a bistable potential45–an optimal correlation time where the transition rate is maximized– and how an external vortex influences new turnovers observed in the presence of a fluctuating magnetic field22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' ACKNOWLEDGMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the projects SH 1275/3-1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=') and LO 418/25-1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' AUTHOR DECLARATIONS The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Appendix A: Derivation of the mean escape time The main purpose of this section is to derive the mean escape time in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' We start with the un- derdamped Langevin equation describing the dynamics of a charged Brownian particle with mass m and charge q sub- jected to a magnetic field B in the −ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The veloc- ity Langevin equation for the position r = (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='y)⊤ and the velocity v = (vx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='vy)⊤ of the particle under the effect of the linear nonconservative force Fnc = ε(−y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='x)⊤ and the con- servative force Fc = −k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='y)⊤ due to the isotropic potential U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='y) = k(x2 +y2)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' can be written as m ˙v = −Kr −Gv(t)+ � 2γTη(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (A1) where η(t) = (ηx(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='ηy(t))⊤ is the Gaussian white noise with zero mean and Dirac delta correlation ⟨η(t)η⊤(t′)⟩ = δ(t −t′) 5 with γ being the friction coefficient and T the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The matrices G and K are defined as G = γ � 1 κ −κ 1 � , K = � k ε −ε k � , (A2) with κ = qB/γ being the diffusive Hall parameter which quan- tifies the strength of the Lorentz force relative to the frictional force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Using the low-mass approach, the corresponding over- damped Langevin equation can be written as25,33,34 ˙r = Ar +ξ(t), (A3) where A = G−1K and ξ(t) = (ξx,ξy)⊤ is Gaussian nonwhite noise with ⟨ξ(t)⟩ = 0, (A4) ⟨ξ(t)ξ⊤(t′)⟩ = TG−1δ+(t −t′)+T(G−1)⊤δ−(t −t′), (A5) where δ±(s = t − t′) are the modified Dirac delta functions which are zero for s ̸= 0 while � ∞ 0 dsδ+(s) = � 0 −∞ dsδ−(s) = 1 and � ∞ 0 dsδ−(s) = � 0 −∞ dsδ+(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Equation (A3) can be rewritten as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (2) in the Cartesian coordinates and thereafter using Itô calculus can be reduced to a one-dimensional equation for the variable r, which is the distance from the origin and is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' The mean time for the particle to escape the trap, truncated at r = a, can be obtained by the following equation ⟨t⟩ = 1+κ2 D � a 0 y−1 exp �ke f f 2γDy2 � dy � y 0 zexp � −kef f 2γDz2 � dz, (A6) where D = T/γ is the diffusion coefficient for a freely moving particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' This equation can be exactly solved: using a change of variables the second integral on the right hand side gives (γD/kef f ) � 1−exp � −ke f f y2/2γD �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' By substitution of this solution into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (A6), the resulting integral can be exactly solved which gives Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' REFERENCES 1L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Farkas, “Keimbildungsgeschwindigkeit in übersättigten Dämpfen,” Zeitschrift für physikalische Chemie 125, 236–242 (1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica 7, 284–304 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grabert and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Linkwitz, “Effect of time-delayed friction on the escape from a metastable well,” Physical Review A 37, 963 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' McCann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Dykman, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Golding, “Thermally activated tran- sitions in a bistable three-dimensional optical trap,” Nature 402, 785–787 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 5L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Rondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Gieseler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ricci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Quidant, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Dellago, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Novotny, “Direct measurement of Kramers turnover with a levitated nanoparticle,” Nature Nanotechnology 12, 1130–1133 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 6R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grote and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hynes, “The stable states picture of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' rate constants for condensed and gas phase reaction models,” The Journal of Chemical Physics 73, 2715–2732 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 7B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Carmeli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Nitzan, “Non-Markoffian theory of activated rate pro- cesses,” Physical Review Letters 49, 423 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ianconescu and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Pollak, “A study of Kramers’ turnover theory in the presence of exponential memory friction,” The Journal of Chemical Physics 143, 104104 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 9P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Marchesoni, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grigolini, “Bistable flow driven by coloured Gaussian noise: a critical study,” Zeitschrift für Physik B Con- densed Matter 56, 333–339 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 10P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Jung and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi, “Bistability and colored noise in nonequilibrium systems: theory versus precise numerics,” Physical Review Letters 61, 11 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 11A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Wittmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Brader, “Escape rate of active parti- cles in the effective equilibrium approach,” Physical Review E 95, 012115 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 12A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Scacchi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Brader, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, “Escape rate of transiently ac- tive Brownian particle in one dimension,” Physical Review E 100, 012601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 13L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Caprini, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Marini Bettolo Marconi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Puglisi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Vulpiani, “Ac- tive escape dynamics: The effect of persistence on barrier crossing,” The Journal of Chemical Physics 150, 024902 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 14A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Berera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Mabillard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Mintz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ramos, “Formulating the Kramers problem in field theory,” Physical Review D 100, 076005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 15L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Darmé, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Jaeckel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Lewicki, “Generalized escape paths for dy- namical tunneling in qft,” Physical Review D 100, 096012 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 16G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Fleming and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi, Activated Barrier Crossing: applications in physics, chemistry and biology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 4 (World Scientific, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 17P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Talkner and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi, New trends in Kramers’ reaction rate theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 11 (Springer Science & Business Media, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 18P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Mojtabai, “Thermally activated escape rate in presence of long-time memory,” Physical Review A 26, 1168 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 19E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Pollak, “Theory of activated rate processes: A new derivation of Kramers’ expression,” The Journal of Chemical Physics 85, 865–867 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 20E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Pollak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grabert, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hänggi, “Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover prob- lem,” The Journal of Chemical Physics 91, 4073–4087 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 21E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Pollak and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ankerhold, “Improvements to Kramers turnover theory,” The Journal of Chemical Physics 138, 164116 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 22A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Baura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ray, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Bag, “Tuning of barrier crossing time of a par- ticle by time dependent magnetic field,” The Journal of Chemical Physics 138, 244110 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 23S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Mondal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Baura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Das, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Bag, “A generic signature of a fluc- tuating magnetic field: An additional turnover prior to the Kramers’ one,” Physica A: Statistical Mechanics and its Applications 502, 58–76 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 24R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Filliger and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Reimann, “Kramers escape rate for a charged particle in a magnetic field,” EPL (Europhysics Letters) 77, 30008 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 25I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Abdoli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, “Escape dynamics in an anisotropically driven Brownian magneto-system,” EPL (Europhysics Letters) 139, 21003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 26S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Lee and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kwon, “Nonequilibrium driven by an external torque in the presence of a magnetic field,” Physical Review E 99, 052142 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 27B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grosberg, “Minimal model for Brownian vortexes,” Physical Review E 82, 021123 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 28B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Lin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Darby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grosberg, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grier, “Brownian vortexes,” Physical Review E 80, 010401 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 29Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Roichman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Stolarski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grier, “Influence of non- conservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Physical Review Letters 101, 128301 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 30H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Moyses, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Bauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grosberg, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Grier, “Perturbative theory for Brownian vortexes,” Physical Review E 91, 062144 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 31B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Liebchen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, “Optimal navigation strategies for active parti- cles,” EPL (Europhysics Letters) 127, 34003 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 32H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Vuijk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Brader, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, “Anomalous fluxes in over- damped Brownian dynamics with Lorentz force,” Journal of Statistical Me- chanics: Theory and Experiment 2019, 063203 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 33H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Chun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Durang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Noh, “Emergence of nonwhite noise in Langevin dynamics with magnetic Lorentz force,” Physical Review E 97, 032117 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 34I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Abdoli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Wittmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Brader, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, “Tunable Brownian magneto heat pump,” Scientific Reports 12, 1–10 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 6 35C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Gardiner, Stochastic methods, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 4 (Springer Berlin, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 36M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Mangeat, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Amarouchene, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Louyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Guérin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Dean, “Role of nonconservative scattering forces and damping on Brownian particles in optical traps,” Physical Review E 99, 052107 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 37J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Carstensen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Greiner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Maurer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Piel, “Effect of neutral gas motion on the rotation of dust clusters in an axial magnetic field,” Physics of Plasmas 16, 013702 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 38A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Piel, Plasma physics: an introduction to laboratory, space, and fusion plasmas (Springer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 39H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kählert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Carstensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Bonitz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Greiner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Piel, “Magnetizing a complex plasma without a magnetic field,” Physical Review Letters 109, 155003 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 40P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hartmann, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Donkó, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ott, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kählert, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Bonitz, “Magneto- plasmons in rotating dusty plasmas,” Physical Review Letters 111, 155002 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 41P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hartmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Reyes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Kostadinova, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Matthews, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Hyde, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Masheyeva, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Dzhumagulova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ramazanov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ott, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Käh- lert, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=', “Self-diffusion in two-dimensional quasimagnetized rotating dusty plasmas,” Physical Review E 99, 013203 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 42R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Shinde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sommer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Sharma, “Strongly enhanced dynamics of a charged rouse dimer by an external magnetic field,” PNAS Nexus 1, pgac119 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 43S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Van Teeffelen and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, “Dynamics of a Brownian circle swimmer,” Physical Review E 78, 020101 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 44C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Scholz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Ldov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Pöschel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Engel, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Löwen, “Surfactants and rotelles in active chiral fluids,” Science Advances 7, eabf8998 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' 45A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Militaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Innerbichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Frimmer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Tebbenjohanns, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Novotny, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} +page_content=' Dellago, “Escape dynamics of active particles in multistable poten- tials,” Nature Communications 12, 1–6 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf'} diff --git a/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/2301.02470v1.pdf.txt b/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/2301.02470v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc30d640a289db73b3f470d47aee3f4083f054f1 --- /dev/null +++ b/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/2301.02470v1.pdf.txt @@ -0,0 +1,3454 @@ +Long-time behaviour of an advection-selection equation +Jules Guilberteau∗, Camille Pouchol† and Nastassia Pouradier Duteil‡ +Abstract +We study the long-time behaviour of the advection-selection equation +∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t)) n(t, x), +ρ(t) = +� +Rd n(t, x)dx +t ≥ 0, x ∈ Rd, +with an initial condition n(0, ·) = n0. In the field of adaptive dynamics, this equation typically describes +the evolution of a phenotype-structured population over time. In this case, x �→ n(t, x) represents the +density of the population characterised by a phenotypic trait x, the advection term ‘∇ · (f(x)n(t, x))’ a +cell differentiation phenomenon driving the individuals toward specific regions, and the selection term +‘(r(x) − ρ(t)) n(t, x)’ the growth of the population, which is of logistic type through the total population +size ρ(t) = +� +Rd n(t, x)dx. +In the one-dimensional case x ∈ R, we prove that the solution to this equation can either converge to +a weighted Dirac mass or to a function in L1. Depending on the parameters n0, f and r, we determine +which of these two regimes of convergence occurs, and we specify the weight and the point where the +Dirac mass is supported, or the expression of the L1-function which is reached. +1 +Introduction +1.1 +Advection-selection equation +We consider the asymptotic behaviour of the advection-selection equation +� +� +� +� +� +∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t)) n(t, x), +t ≥ 0, x ∈ Rd +ρ(t) = +� +Rd n(t, x)dx, +t ≥ 0 +n(0, x) = n0(x), +x ∈ Rd. +(1) +This type of model typically comes up in the field of adaptive dynamics. The aim is to understand how, +among heterogeneous populations of individuals structured by a so-called continuous trait or phenotype x, +the distribution of the density x �→ n(t, x) evolves over time, and which phenotypes prevail in large times +t → +∞. +In the model above (1), the partial differential equation (PDE) takes into account +• advection via the term ∇ · (f(x)n(t, x)), whereby individuals follow the flow associated with f, +• growth via the term (r(x) − ρ(t))n(t, x), which is of logistic type through the total population size +ρ(t) = +� +Rd n(t, x) dx. +The literature concerning so-called phenotype-structured partial differential equations for adaptive dy- +namics is abundant [1, 5, 3, 7, 6, 10, 12, 15, 28, 30, 34, 35]. +These models usually take into account +selection, which favors individuals with the most adapted traits in terms of growth, and mutations, which +∗Sorbonne Universit´e, CNRS, Universit´e Paris Cit´e, Inria, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France. +jules.guilberteau@sorbonne-universite.fr +†Universit´e Paris Cit´e, FP2M, CNRS FR 2036, MAP5 UMR 8145, F-75006 Paris, France. camille.pouchol@u-paris.fr +‡Sorbonne Universit´e, Inria, Universit´e Paris Cit´e, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France +nastassia.pouradier duteil@sorbonne-universite.fr +1 +arXiv:2301.02470v1 [math.AP] 6 Jan 2023 + +induce a slight phenotypic change upon reproduction. +Mutation is often assumed to be rare and small +compared to selection, [14, 19, 31]. Models with no mutation at all have also been the subject of several +studies [2, 13, 21, 25, 29, 36]. +One way to analyse how the population adapts is to study the long-time behaviour for solutions of +such PDE models. In particular, determining if the population becomes monomorphic (i.e. the solution +concentrates around a certain trait, called Evolutionary Stable Strategy (ESS) [23]), or if phenotypic diversity +is preserved is a fundamental question when studying such models. Broadly speaking, it has been shown +that selection leads to concentration (around a finite number of phenotypic traits), while mutations, on the +contrary, tend to regularise solutions, and, possibly, their limits [4, 21]. +However, less emphasis has been put on studying the effect of advection, except for the recent few +examples [10, 27, 11] where most results are of numerical nature, or assume a very specific form of the +functions r and f. +Yet, considering advection is relevant in various contexts. From the phenomenological point of view, +it may represent how the environment drives the individuals towards specific regions, as opposed to more +random mutations. It is also the rigorous way to model phenotype changes that are intrinsic to the individual, +mediated by an ordinary differential equation (ODE) of the form +˙x(t) = f(x(t)), +(2) +where x(t) ∈ Rd denotes the phenotypic trait of the individual at time t ≥ 0. +As is well known, the +PDE for the density of individuals corresponding to the sole model (2) is indeed the advection equation +∂tn(t, x) + ∇ · (f(x)n(t, x)) = 0. Our original motivation is that of cell differentiation, for which very refined +ODE models have been developed in systems biology (see for instance [20, 37, 38, 40]). +The goal of the present article is to investigate the combined effect of selection and advection, assuming +that mutations are absent or sufficiently small to be neglected. We hence study the long-time behaviour of +the PDE (1), where n0 is the initial population distribution, and ρ(t) is the size of the population at time +t ≥ 0. The equation incorporates advection with the flow f of the corresponding ODE, and selection (or +growth) through the non-linear and non-local term (r(x) − ρ(t))n(t, x). Here, r(x) − ρ(t) can be interpreted +as the fitness of individuals with trait x inside the environment created by the total population, where +the individuals are in a blind competition with all the other ones, regardless of their phenotype. We note +that such models can rigorously be derived from stochastic individual based-models, in the limit of large +populations [8, 9]. +In the absence of differentiation (f ≡ 0), the long-time behaviour of this model has been studied in +detail by Benoˆıt Perthame [34], Tommaso Lorenzi and Camille Pouchol [29], and it has been proved that, in +general, solutions typically concentrate onto a single trait. This result is rather intuitive, since this model +does not take mutations into account. Solutions of the advection equation alone are also known to converge +to weighted Dirac masses located at the roots of f which are asymptotically stable for the ODE (2) [16]. On +the contrary, when considering both selection and advection as in equation (1), the long-time behaviour is +not known, to the best of our knowledge. Intuitively, two antagonistic effects will compete: +• advection will push the solution towards the asymptotically stable equilibria of ODE (1). +• growth will push the solution towards regions where r is maximised. +When coupling these two phenomena, our aim is to uncover whether the solution of (1) converges to a +weighted Dirac mass, or if it converges to a smooth function. We show that both phenomena can occur, +depending on the parameters n0, f and r. Perhaps surprisingly, the model (1) features convergence to smooth +functions even in the absence of terms modelling mutations. +Determining which parameters lead to convergence to a continuous function seems rather intricate in full +generality. In particular, this problem cannot be addressed with traditional entropy methods as developed +in [32], since in the absence of mutations, there is no decrease of entropy. +1.2 +Main results +In this paper, we thus develop a different strategy allowing to reduce this problem to the study of parameter- +dependent integrals, which is mainly applied to the one-dimensional case (x ∈ R). In this case, we elucidate +2 + +the asymptotic behaviour for a large class of parameter values, and we show that there exist many different +subcases depending on the number of zeros of the function f. A general statement encompassing all our +results is hence rather convoluted. In order to illustrate our main results, we here focus on a few example +cases which highlight the main two parameter regimes encountered for the asymptotic behaviour of (1). +Proposition 1. Let us assume that the parameter functions f, n0 and r are smooth enough, that f has +a unique root (that we denote xs), and that f ′(xs) < 0 (which means that xs is asymptotically stable for +ODE (2)). Then, ρ converges to r(xs), and n converges to a weighted Dirac mass at xs, when t goes to +∞. +Hence, in the presence of a single asymptotically stable equilibrium point for ODE (2), the solution of +PDE (1) converges to a Dirac mass at this point. In other words, the selection term is dominated by the +advection term, which determines the point in which the solution concentrates. As soon as f has at least +two roots, the situation is much more complex and solutions may converge to L1 functions, as illustrated in +Figure 1 and exposed in the following proposition: +Proposition 2. Let us assume that the functions f, n0 and r are smooth enough, that f has exactly two roots +(that we denote xu and xs, with xu < xs), such that f ′(xu) > 0 and f ′(xs) < 0, which means that the points +xu and xs are respectively asymptotically unstable and asymptotically stable for the ODE (2). Moreover, let +us assume that n0 has its support in [xu, xs], and that n0(xu) > 0. Then, the following alternative holds: +• If r(xs) > r(xu) − f ′(xu), n converges to a weighted Dirac mass at xs, and ρ converges to r(xs). +• If r(xs) < r(xu) − f ′(xu), n converges to a function in L1(xu, xs), and ρ converges to r(xu) − f ′(xu). +This proposition can be interpreted as follows: since f is positive on (xu, xs), the advection term drives the +solution towards xs. On the other hand, since xu is an equilibrium, albeit unstable, it acts as a counterweight +by controlling the speed of the transition towards xs in the neighbourhood of xu. Hence, in the case where +r(xu) − f ′(xu) is large enough (r(xu) − f ′(xu) > r(xs)), the growth rate around xu is large enough to +compensate for the advection term, leading to the convergence of n to a continuous function. In the other +case, the advection term is dominant, and n converges to a weighted Dirac mass at xs. If n0(xu) = 0, the +toggle value between the two regimes (i.e. the convergence to a smooth function or to a Dirac mass) changes, +depending on how n0 vanishes at xu, and other limit functions can be reached: the complete result is detailed +in Proposition 9. The method of analysis proposed in this article allows in fact to solve this problem for any +function f with a finite number of roots, as detailed in Proposition 10. The case where f is equal to zero on +a whole interval can also be studied with our method, as highlighted by Proposition 11. +1.3 +Discussion +Open problems. +Some limit cases of the problem remain unclear: we do not deal with the case of non- +hyperbolic equilibria, i.e. x ∈ R which satisfy f(x) = f ′(x) = 0, and we are not able to determine what +happens in the case where several carrying capacities, as defined in Section 3, converge to the same maximum +limit. This last case might lead to other asymptotic behaviours, such as convergence to a sum of weighted +Dirac masses, or a sum of weighted Dirac masses and L1-functions. Lastly, we did not manage to elucidate +the equality cases (of the form r(xs) = r(xu) − f ′(xu)). +Furthermore, even if the framework introduced in Section 3 could theoretically be applied in any dimen- +sion, computing the limits of the carrying capacities seems out of reach in the multidimensional case. As +shown by the semi-explicit expression introduced in Subsection 3.1, the behaviour of n is closely linked to +that of the solutions of ODE ˙x = f(x), which suggests that other asymptotic behaviours, such as convergence +to a limit cycle, or chaotic behaviours (if the dimension is greater than or equal to 3) might occur. +These behaviours may be excluded by making specific assumptions regarding the function f, for example +by requiring in the 2D case that ODE ˙x = f(x) be competitive or cooperative. Additionally if the roots of +f are hyperbolic and none of them is a repellor, then n cannot converge to a L1-function (Proposition 13). +Nevertheless, the question of the asymptotic limit of n in this case remains open, and might be, in the +presence of a saddle point, a singular measure which is not a sum of weighted Dirac masses. This situation +is commonplace for some applications, since toggle switches used to model cell differentiation phenomena +are usually competitive or cooperative ODE models. +3 + +Figure 1: The two possible regimes of convergence stated in Proposition 2. In both cases, we have chosen +f(x) = x(1 − x), n0 ≡ 6, and we work on the segment (0, 1) (hence xu = 0, xs = 1). The three figures +above (in red) show the time evolution of the solution in the case where r(x) = 6 − 0.5x (and thus 5.5 = +r(1) > r(0)−f ′(0) = 5), which implies, according to Proposition 2, that the solution converges to a weighted +Dirac mass at 1. The three figures below (in blue) show the time evolution of the solution in the case where +r(x) = 6 − 4x, (and thus 2 = r(1) < r(0) − f ′(0) = 5), which implies that the solution converges to a +continuous function in L1. The black dashed curve represents this limit function, which can explicitly be +computed (see Proposition 9). +Perspectives. +A natural generalisation for the model would be to model mutations, either by means of +a Laplacian term or an integral term. Because of their smoothing effect, convergence to Dirac masses will +typically be lost. The method developed in this paper does not seem to handle such cases well. However, it +is an interesting perspective to tackle the asymptotic behaviour with entropy methods when mutations are +added [32]. +From the numerical point of view, we have proved that the solution of this equation could be approximated +with a particle method, with which we obtained the plots of Figure 1. The details of the scheme, and the +proof of its convergence will be published in a forthcoming article [17]. +Outline of the paper. +This paper is organised as follows: Section 2 introduces the measure-theoretic +framework in which convergence is considered, and includes several important reminders regarding ODE +theory which will be used throughout the article. +Section 3 details the method used to determine the +asymptotic behaviour of (1), and Section 4 corresponds to a direct application of this method to several +examples in the one-dimensional case. Lastly, section 5 presents two results in higher dimension which allow +to determine, in some specific cases, if some initial solution can lead to a convergence to a smooth function +or not. +4 + +T=1 +16 +14 +12 +10 - +8 +9 +4 +2 ++0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0T=5 +16 +14 +12 +10 - +8 +9 +4 +2 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0T=10 +16 +14 +12 +10- +8 +9 +4 +2 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0T=1 +16 +14 +12 +10 +8 +6 +4 +2 ++0 +0.4 +0.8 +0.0 +0.2 +0.6 +1.0T=2 +16 +14 +12 +10 +8 +6 +4 +2 +-0 +0.8 +0.0 +0.2 +0.4 +0.6 +1.0T=4 +16 +14 +12 +10 +8 +6 +4 +2 ++0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.02 +Framework and reminders +We consider the asymptotic behaviour of the integro-differential PDE +� +� +� +� +� +∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t))n(t, x), +t ≥ 0, x ∈ Rd +ρ(t) = +� +Rd n(t, x)dx, +t ≥ 0 +n(0, x) = n0(x), +x ∈ Rd. +(1) +All along the article, we make the following regularity hypotheses +• f is Lipschitz-continuous, and is in C2(Rd). +• r is positive, is in L1(Rd) ∩ C1(Rd), and goes to zero when ∥x∥ goes to +∞. Let us note that these +hypotheses imply that r is bounded. +• n0 is in C1 +c(Rd) (the space of C1 functions with a compact support), is non-negative and is not the zero +function. +Whenever possible, we will indicate whether these hypotheses can be weakened for a given specific result. If +not specified, it will be assumed that these three hypotheses hold. +From the modelling point of view, they can be justified as follows: n0 denoting the initial density, it is +reasonable to consider that a bounded range of phenotypic traits is initially represented; the hypothesis on +r at +∞ is made in order to prevent an unlikely proliferation of individuals with more and more extreme +(∥x∥ → +∞) phenotypic traits. +Under the above hypotheses, we can prove that there exists a unique solution n ∈ C +� +R+, L1(R) +� +for this +Cauchy problem by coupling the well-known method of characteristics for the advection equation [16] with +the method applied in [34] for the case f ≡ 0. We do not elaborate further here on the issue of existence +and uniqueness, that will be addressed in a more general framework in an upcoming article [17]. +Since we are concerned with the long-time behaviour of the PDE (1) and we expect to obtain convergence +either to Dirac masses or to regular functions, the space of Radon measures is a natural setting. We start +with a few usual reminders. +2.1 +The space of Radon measures +We recall that the space of finite Radon measures can be identified with the topological dual space of +Cc(Rd), i.e. the space of continuous functions on Rd with a compact support. Thus, we say that a sequence +of finite Radon measures (µk)k∈N weakly converges to a finite Radon measure µ (denoted uk ⇀ µ) if +∀ϕ ∈ Cc(Rd), +� +Rd ϕ(x)dµk(x) −→ +k→+∞ +� +Rd ϕ(x)dµ(x). +In this article, we will be confronted mainly with convergence to Dirac masses or to L1 functions. It is +clear that the convergence in L1 to a certain function implies the weak convergence to this function. The +following standard lemma provides a sufficient condition to prove the weak convergence to a single Dirac +mass. For completeness, we provide a proof. +Lemma 1. Let u : R+ × Rd → R be a non-negative mapping such that u(t, ·) ∈ L1(Rd) for all t ≥ 0, +and u(t, ·) is compactly supported, uniformly in t ≥ 0. We assume that there exists x ∈ Rd, such that for +all compact set Kx which does not contain x, +� +Kx u(t, x)dx +−→ +t→+∞ 0, and that there exists Vx a compact +neighbourhood of x and C ∈ R such that +� +Vx u(t, x)dx −→ +t→+∞ C. Then, u(t, ·) +⇀ +t→+∞ Cδx. +Proof. Let ϕ ∈ Cc(Rd), and let K be a compact set such that, for all t ≥ 0, supp(u(t, ·)) ∪ Vx ⊂ K. Then, +���� +� +Rd ϕ(x)u(t, x)dx − Cϕ(x) +���� = +���� +� +K +ϕ(x)u(t, x)dx − +� +K +ϕ(x)u(t, x)dx + +� +K +ϕ(x)u(t, x)dx − Cϕ(x) +���� +≤ +� +K +|ϕ(x) − ϕ(x)|u(t, x)dx + |ϕ(x)| +���� +� +K +u(t, x)dx − C +����. +5 + +The second term tends to 0 since K contains Vx. It remains to prove that t �→ +� +Rd |ϕ(x) − ϕ(x)|u(t, x)dx +converges to zero. Let ε > 0 be given. Since ϕ is continuous, there exists Bx a neighbourhood of x, which +can be chosen as a subset of Vx, such that |ϕ(x) − ϕ(x)| ≤ ε, for all x ∈ Bx. Thus, for all t ≥ 0, +� +K +|ϕ(x) − ϕ(x)|u(t, x)dx = +� +K\Bx +|ϕ(x) − ϕ(x)|u(t, x)dx + +� +Bx +|ϕ(x) − ϕ(x)|u(t, x)dx +≤ 2∥ϕ∥∞ +� +K\Bx +u(t, x)dx + ε +� +Bx +u(t, x)dx. +This concludes the proof, since t �→ +� +K\Vx u(t, x)dx converges to zero and for any t large enough, +� +Bx u(t, x)dx ≤ +� +Vx u(t, x)dx ≤ C + ε. +2.2 +General statement regarding the characteristics curves +We are led to consider the characteristics curves associated with the advection term. In this section, we +introduce some notations and state some classical results from ODE theory, that will prove to be useful later +on. +Since f is assumed to be Lipschitz-continuous, the global Cauchy-Lipschitz theorem ensures the global +existence on R+ and the uniqueness of the characteristic curves related to f defined for all y ∈ Rd as the +solution to the ODE +� ˙X(t, y) = f(X(t, y)) +t ≥ 0 +X(0, y) = y +. +(3) +It is well-known that for all t ≥ 0, y �→ X(t, y) is a C1-diffeomorphism between Rd and itself [16], and that +the inverse function of X(t, ·), that we denote x �→ Y (t, x), is the unique solution of +� ˙Y (t, x) = −f(Y (t, x)) +t ≥ 0 +Y (0, x) = x +. +(4) +Moreover, Liouville’s formula states that for all t ≥ 0 and y ∈ Rd, +det (JacyX(t, y)) = e +� t +0 ∇·f(X(s,y))ds. +(5) +It follows from the uniqueness of solutions to (3) that for all 0 ≤ s ≤ t, +X(s, Y (t, x)) = Y (t − s, x). +(6) +Specific results in R. Let us note that the behaviour of the characteristic curves is particularly simple +in R. Indeed, an elementary ODE analysis shows that for all x, y ∈ R, t �→ X(t, y) and t �→ Y (t, x) are +monotonic functions. This implies that these characteristic curves either converge to a root of f, or go to +±∞ as t → +∞. More precisely, if f has a finite number of roots, then for all y ∈ R such that f(y) > 0, +t �→ X(t, y) converges to the closest root of f which is greater than y, or to +∞ if y is greater than the +greatest root of f. Similarly, for all y ∈ R such that f(y) < 0, t �→ X(t, y) converges to the closest root of f +which is lesser y, and to −∞ if y is lesser the smallest root of f. +Moreover, if each of these roots are hyperbolic equilibrium points for the ODE ˙x = f(x), i.e. if f ′(x) ̸= 0 +for all x root of f, then a given root of f is either asymptotically unstable (i.e. f ′(x) > 0), which implies +that its basin of attraction is limited to itself, or asymptotically stable (i.e. f ′(x) < 0), which implies that +its basin of attraction in an open interval containing x. +Lastly, let us recall that under these hypotheses, the convergence to an asymptotically stable point +happens with an exponential speed, which means that for all y ∈ R, x root of f, +X(t, y) −→ +t→+∞ x +⇒ +∃δy > 0 : X(t, y) − x = +O +t→+∞(e−δy t) +Since the reverse characteristic curves satisfy (4), the same results hold for Y (t, x), provided that we replace +f by −f. +In brief, the asymptotically stable equilibria become unstable for the reverse ODE, and vice +versa, and if t �→ X(t, y) is increasing (respectively decreasing), then t �→ Y (t, x) is decreasing (respectively +increasing). +6 + +3 +Resolution method +The method of resolution to determine the asymptotic behaviour of n that we propose here is based on the +following two propositions, which are developed in the following two subsections, respectively: +1. For all t ≥ 0, x ∈ Rd, we can express n(t, x) as a function which only depends on t, x, on the functions +n0, f and r, on the inverse characteristic curves Y (t, x), and on the population size ρ. Therefore, +knowing the limit of Y (t, x) and ρ(t) as t goes to +∞ is enough to understand the long-time behaviour +of n. +2. The population size ρ is the solution of a non-autonomous ODE, and its long-time behaviour may be +inferred from the limit of some parameter-dependent integrals. +Combining these two propositions allows us to reduce the study of the asymptotic behaviour of n to that +of parameter-dependent integrals. +3.1 +Semi-explicit expression of the solution +According to the definition of the characteristic curves (3), for all t ≥ 0 and all y ∈ Rd, +d +dtn(t, X(t, y)) = +� +r(X(t, y)) − ∇ · f(X(t, y)) − ρ(t) +� +n(t, X(t, y)), +i.e. +n(t, X(t, y)) = e +� t +0 (r(X(s,y))−∇·f(X(s,y))−ρ(s))dsn0(y). +Replacing y by Y (t, x) in this last expression, we get a semi-explicit expression for n, which is expressed +as a function of t, x and ρ: +n(t, x) = n0(Y (t, x))e +� t +0 ((r−∇·f)(X(s,Y (t,x)))−ρ(s))ds += n0(Y (t, x))e +� t +0 ((r−∇·f)(Y (s,x))−ρ(s))ds, +(7) +The second equality holds according to equality (6) and the change of variable s′ = t − s. +Beyond the non-negativity of n, this semi-explicit expression shows that determining the asymptotic +behaviour of ρ and Y is enough to uncover that of n. In the following section, we show that ρ is the solution +of a non-autonomous ODE, and that its asymptotic behaviour is related to that of parameter-dependent +integrals. +This expression also provides exhaustive information about the support of of n(t, ·): indeed, it ensures +that for all t ≥ 0, +supp (n(t, ·)) = supp +� +n0 ◦ Y (t, ·) +� += X +� +t, supp +� +n0�� +. +(8) +Since n0 is assumed to have a compact support, then so does n(t, ·) for any t ≥ 0. +We recall that a set E ⊂ Rd is said to be positively invariant for the ODE ˙x = f(u) if for all t ≥ 0, +X(t, E) ⊂ E. +With this definition in mind, it becomes clear, according to (8), that if supp +� +n0� +is positively invariant +for the ODE ˙x = f(x), then supp (n(t, ·)) ⊂ supp +� +n0� +, for all t ≥ 0, and, more generally, that if there exists +E ⊂ Rd a set which is positively invariant for this ODE such that supp +� +n0� +⊂ E, then supp (n(t, ·)) ⊂ E, for +all t ≥ 0. Hence, even if PDE (1) is defined for all x ∈ Rd, if the support of n0 is included in a compact +subset of Rd which is positively invariant, then everything happens as if we were working in this compact +set. In particular, the functions f and r do not need to be defined outside this set. +7 + +3.2 +ODE satisfied by the population size +Let us start with a basic lemma which ensures that the population size ρ does not blow up as t tends to +∞. +Lemma 2 (Bounds on ρ). Let ρ be defined as in (1). Then for all t ≥ 0, ρ(t) ≤ max (∥r∥∞, ρ(0)). +Proof. According to (8), since, n0 is assumed to have a compact support, n(t, ·) has a compact support for +all t ≥ 0. Hence, when integrating the fist line of (1), the advection term vanishes, and we get +˙ρ(t) = +� +Rd +� +r(x) − n(t, x) +� +n(t, x)dx ≤ (∥r∥∞ − ρ(t)) ρ(t). +In other words, ρ is a sub-solution of the logistic ODE ˙u = (∥r∥∞ − u) u, which proves the result. +In the remainder of this section, we show that ρ is in fact the solution to a non-autonomous logistic +equation, which can be written in different forms. In order to lighten the future expressions, we now denote +˜r := r − ∇ · f. +Let E ⊂ Rd be any measurable subset of Rd, and let us denote +ρE(t) := +� +ε +n(t, x)dx, +which is well-defined and bounded, according to Lemma 3.2. By integrating the semi-explicit expression (7) +of n over E, we obtain the equality +ρE(t) = SE(t)e− +� t +0 ρ(s)ds, +(9) +where +SE(t) := +� +E +n0(Y (t, x))e +� t +0 ˜r(Y (s,x))dsdx +is a function which only depends on the parameters f, r and n0. This function is well-defined, and differen- +tiable, thanks to our regularity assumptions, and since for all t ≥ 0 n0(Y (t, ·)) has compact support. Thus, +under the hypothesis that for all t ≥ 0, SE(t) > 0, we obtain +ln (ρE(t)) = ln (SE(t)) − +� t +0 +ρ(s)ds, +and finally, by differentiating and multiplying by ρE on both sides, +˙ρE(t) = +� ˙SE(t) +SE(t) − ρ(t) +� +ρE(t). +(10) +At this stage, one might be tempted to choose E = Rd to obtain, denoting S := SRd(t), +˙ρ(t) = +� ˙S(t) +S(t) − ρ(t) +� +ρ(t). +(11) +This proves that ρ is the solution to a non-autonomous logistic equation, and the study of such equations +[22] proves that if the time-dependant carrying capacity t �→ +˙S(t) +S(t) converges, then ρ converges to the same +limit. Unfortunately, computing the limit of t �→ +˙S(t) +S(t) is intricate (except in very specific cases). This brings +us to introducing a more general framework, which involves simpler functions whose limit can be computed +(at least in the case x ∈ R). The idea is to partition the space Rd into several well-chosen subsets, and to +consider the size of the population on each of these sets. As seen above, to obtain equations of the type (10), +we must be cautious when choosing these subsets in order for the corresponding functions SE to be positive. +All this leads us the following proposition: +8 + +Proposition 3. Let U ⊂ Rd be a set such that +X(R+ × supp(n0)) ⊂ U +(12) +and let (Oi)i∈{1,...,N} be a finite family of open subsets of U such that +(i) ∀i ̸= j, +Oi ∩ Oj = ∅. +(ii) ν +� +U\ +N� +i=1 +Oi +� += 0, where ν denotes the Lebesgue measure. +(iii) ∀i ∈ {1, ...N}, +∀t ≥ 0, +X +� +t, supp +� +n0� � +∩ Oi ̸= ∅. +Then, by denoting for all i ∈ {1, ..., N} +ρi(t) := +� +Oi +n(t, x)dx, +(13) +Si(t) := +� +Oi +n0(Y (t, x))e +� t +0 ˜r(Y (s,x))dsdx, +(14) +Ri(t) := +˙Si(t) +Si(t), +(15) +the following equation holds: +� +� +� +� +� +� +� +˙ρi(t) = (Ri(t) − ρ(t)) ρi(t) +∀t ≥ 0, +∀i ∈ {1, ..., N} +ρ(t) = +N +� +i=0 +ρi(t) +∀t ≥ 0 +ρi(0) > 0 +∀i ∈ {1, ..., N} +. +(16) +Remark. Note that a sufficient condition for the third condition (iii) to hold is the following: for any +i ∈ {1, ..., N}, there exists xi in the closure of Oi such that f(xi) = 0 and n0(xi) > 0. +Proof. As a consequence of the discussion at the beginning of this section, it is enough to prove that +1. For all i ∈ {1, ..., N} and all t ≥ 0, +Si(t) > 0 +2. For all t ≥ 0, +ρ(t) = +N +� +i=1 +ρi(t). +First, notice that hypothesis (iii) is equivalent to supp +� +n0� +∩Y (t, Oi) ̸= ∅ for all i ∈ {1, ..., N} and all t ≥ +0. Moreover, Oi is an open set, which ensures, thanks to the continuity of n0, that {x ∈ Oi : n0(Y (t, x)) > 0} +has a positive measure for all t ≥ 0. This proves the first point by definition of Si. Since ρi(0) = Si(0), we +also infer ρi(0) > 0. +The second point is due to hypothesis (12): Indeed, for any t ≥ 0, according to the semi-explicit expression +of n provided by (7), n(t, x) = 0 if Y (t, x) /∈ supp(n0) i.e. if x /∈ X(t, supp(n0)), which ensures that +ρ(t) = +� +Rd n(t, x)dx = +� +U +n(t, x)dx. +The first two hypotheses satisfied by the sets Oi ensure that ρ(t) = +N +� +i=1 +ρi(t). +Proof of the remark: Let xi be a root of f. A classical ODE result ensures that for all t ≥ 0, x ∈ Rd, +∥Y (t, x) − xi∥ ≤ eLt∥x − xi∥, with L > 0 the Lipschitz constant of f. Since n0(xi) > 0 and n0 is continuous, +there exists ε > 0 such that B(xi, ε) ⊂ supp(n0). Let t ≥ 0, x ∈ Oi ∩B(xi, εe−Lt/2) (such a point does exist, +by definition of the closure). Then, Y (t, x) ∈ B(xi, ε) ⊂ supp(n0), which ensures that x ∈ X +� +t, supp +� +n0� � +, +and thus concludes the proof. +9 + +In the one-dimensional case, assuming that f has a finite number of roots, an efficient choice for the sets +Oi is to take the segments between the roots of f which interseect the support of n0, as the following result +shows. +Lemma 3. Let x ∈ R and assume that f : R → R has a finite number of roots, that we denote x1 < x2 < +... < xN. Let us denote +O0 := (−∞, x1), +Oi := (xi, xi+1), i ∈ {1, ..., N − 1}, +ON := (xN, +∞), +and, among these segments, let us consider Oi1, ...OiM those which have an non-empty intersection with +supp +� +n0� +. +Then, the set U := +� +1≤j≤M +Oij and the family of sets +� +Oij +� +1≤j≤M satisfy the hypotheses of +Proposition 3. +Proof. By applying the results stated at the end of Section 2.2, we note that for all i ∈ {1, ..., N}, Oi +is positively invariant for the ODE ˙x = f(x). +Thus, for all y ∈ supp(n0) ⊂ U, t ≥ 0, X(t, y) ∈ U, +which ensures that X(R+ × supp(n0)) ⊂ U. Moreover, the same results show that for all j ∈ {1, ..., M}, +X(t, supp(n0) ∩ Oij) ⊂ Oij, and thus that X(t, supp(n0)) ∩ Oij ̸= ∅. The other two points are automatically +satisfied, thanks to the definition of U and the sets Oi. +Proposition 3 shows us that ρ satisfies ODE (16). Our next result shows that the long-time behaviour +of this ODE depends on the long-time behaviour of the functions Ri. In particular, it states that if all +the functions Ri converge, then ρ converges to the maximum of their limit. Before stating the result, we +introduce some notations. +Notation. For any function g : R+ → R, we denote: +g := lim inf +t→+∞ g(t) +and +g := lim sup +t→+∞ g(t), +and we say that g converges to l ∈ R with an exponential speed if there exist δ > 0 such that +g(t) − l = +O +t→+∞ +� +e−δt� +. +Proposition 4. The coupled system of ODEs (16) has the following properties: +(i) For all i ∈ {1, ..., N} and all t ≥ 0, ρi(t) > 0. +(ii) ρ ≥ min +1≤i≤N +� +Ri +� +and +ρ ≤ max +1≤i≤N +� +Ri +� +. +(iii) Let j ∈ {1, ..., N}. If there exists i ∈ {1, ..., N} such that Rj < Ri, then ρj(t) −→ +t→+∞ 0. +(iv) Let us assume that there exists l ∈ R+ ∪ {+∞}, and a non empty set I ⊂ {1, ..., N} (where potentially +I = {1, ..., N}) such that for all i ∈ I, Ri(t) −→ +t→+∞ l, and Rj < l for all j /∈ I. Then, ρ(t) −→ +t→+∞ l. +(v) Under the hypotheses of (iv), if moreover 0 < l < +∞ and for all i ∈ I the function Ri converges to l +with an exponential speed, then ρ converges to l with an exponential speed. +Proof. +(i) According to the first line of ODE (16), ρi(t) = e +� t +0 Ri(s)−ρ(s)dsρi(0), which is positive according +to the third line. +(ii) If +min +1≤i≤N +� +Ri +� += 0, there is nothing to prove: we assume +min +1≤i≤N +� +Ri +� +> 0 and let m < +min +1≤i≤N +� +Ri +� +. +There exists Tm ≥ 0 such that for all t ≥ Tm, and all i ∈ {1, ..., N}, Ri(t) ≥ m. Thus +˙ρ(t) = +N +� +i=1 +˙ρi(t) = +N +� +i=1 +(Ri(t) − ρ(t)) ρi(t) ≥ (m − ρ(t)) ρ(t), +which means that ρ is a super-solution of a logistic equation which converges to m, and thus that +ρ ≥ m. Since this inequality holds for any m < +min +1≤i≤N +� +Ri +� +it proves that ρ ≥ +min +1≤i≤N +� +Ri +� +. By +proceeding in the same way with the limit superior, we get the second inequality. +10 + +(iii) Let i, j ∈ {1, ..., N} such that Rj < Ri. The latter inequality is written with the convention that if +Ri = +∞, then Rj ∈ R. Using the first point, ρj, ρi > 0 on R+. We can compute +d +dt ln +� ρi(t) +ρj(t) +� += Ri(t) − Rj(t) > ε, +for a certain ε > 0 and t large enough. Thus, ρ(t) ≥ ρi(t) ≥ Ceεtρj(t), for a certain constant C > 0, +which yields +˙ρj(t) ≤ +� +sup +t>0 +Rj(t) − Ceεtρj(t) +� +ρj(t), +with sup +t>0 +Rj(t) < +∞ by hypothesis, and thus ρj goes to zero as t goes to +∞. +(iv) Let us denote ρJ := � +j /∈I +ρj. (This first step is not necessary in the case I = {1, ..., N}). According to +the previous property, ρJ converges to zero. By denoting ˜Ri := Ri − ρJ, we can thus rewrite system +(16) as: +� +� +� +� +� +� +� +˙ρi(t) = +� +˜Ri(t) − ρI(t) +� +ρi(t) +∀t ≥ 0, +∀i ∈ I +ρI(t) = � +i∈I +ρi(t) +∀t ≥ 0 +ρi(0) > 0 +∀i ∈ {1, ..., N} +. +Applying Property (ii) to this new system proves the desired result, since +min +i∈I +� +˜Ri +� += max +i∈I +� +˜Ri +� += l. +(v) Let l ∈ (0, +∞). According to the previous point, ρ is bounded by two positive constants (and so is ρI), +that we denote ρm < ρM. Using the same argument as in the proof of the third point, one proves that +for all j /∈ I, there exists ε > 0 such that ρJ(t) ≤ Ce−εtρM, and thus that ρJ converges to 0 with +an exponential speed. Thus, it remains to prove that the convergence of ρI to l also occurs with an +exponential speed. By hypothesis, there exists C, δ > 0 such that for all t ≥ 0, � +i∈I +| ˜Ri(t) − l| ≤ Ce−δt. +Thus, by denoting C′ := C∥ρI(·) − l∥∞ ρM, we find +d +dt +1 +2 +� +ρI(t) − l +�2 = (ρI(t) − l) +� +i∈I +(( ˜Ri(t) − l) − (ρI(t) − l))ρi(t) +≤ C′e−δt − ρm (ρI(t) − l)2 , +which concludes the proof, according to Gr¨onwall’s lemma. +4 +Results in the one-dimensional case +4.1 +Asymptotic behaviour of the carrying capacities +As evidenced by the previous section and in particular by Proposition 4, the long-time behaviour of ρ is +completely determined by that of the functions Ri, which we call carrying capacities by analogy with the +logistic equation. As their definition suggests, computing the limit of these functions is a delicate issue: this +section is dedicated to these computations. The multidimensional case seems out of reach with this method, +because, as we shall see, we use a change of variable that requires to be working in 1D. +11 + +In order to simplify the notations, we will now denote R instead of RE or Ri, when there is no ambiguity +as to which sets we are working with. We are thus interested in the asymptotic behaviour of the function +R(t) = +˙S(t) +S(t), +with +S(t) = +� +E +n0(Y (t, x))e +� t +0 ˜r(Y (s,x))dsdx, +(17) +where E ⊂ Rd is an open set which satisfies supp(n0) ∩ Y (t, E) ̸= ∅ for all t ≥ 0. +First, let us note that for all l ∈ R, +R(t) − l = +d +dt +� +S(t)e−lt� +S(t)e−lt +. +(18) +Thus, in order to prove that R converges to l ∈ R with an exponential speed, it in enough to prove that: +(a) lim inf +t→+∞ S(t)e−lt > 0. +(b) t �→ eδt d +dt +� +S(t)e−lt� +is bounded for a certain δ > 0. +Indeed, we immediately deduce from (18), and the fact that S is positive, according to its definition (14), +that these two hypotheses imply that for any δ′ ∈ (0, δ), +R(t) − l = +O +t→+∞ +� +e−δ′t� +. +4.1.1 +Integral formulae for the carrying capacities +This section aims at listing several alternative formulae of S. In the following section, we will use one or the +other, depending on the studied case. +We recall that S is defined as +S(t) = +� +E +n0(Y (t, x))e +� t +0 ˜r(Y (s,x))dsdx, +(19) +with ˜r := r − ∇ · f. +As seen in the first section, for any t ≥ 0, x �→ Y (t, x) is a C1-diffeomorphism from E to Y (t, E). +Thus, the change of variable y = Y (t, x), and Liouville’s formula which ensures that |det(Jac(Y (t, x)))| = +e +� t +0 −∇· f(Y (s,x))ds provide a second expression for S, namely +S(t) = +� +Y (t,E) +n0(y)e +� t +0 r(X(s,y))dsdy. +(20) +Moreover, in the one-dimensional case x ∈ R, if E is an interval on which f ̸= 0, then for all y ∈ E, +t �→ X(t, y) is also a C1-diffeomorphism from (0, t) to (y, X(t, y)) or (X(t, y), y). This allows us to make the +change of variable s′ = Y (s, x) and s′ = X(s, y) in the two expressions for S, thereby obtaining two new +formulations +S(t) = +� +E +n0(Y (t, x))e +� x +Y (t,x) +˜r(s) +f(s) dsdx = +� +Y (t,E) +n0(y)e +� X(t,y) +y +r(s) +f(s) dsdy, +(21) +and, in the same way, for all l ∈ R, +S(t)e−lt = +� +E +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−l +f(s) dsdx = +� +Y (t,E) +n0(y)e +� X(t,y) +y +r(s)−l +f(s) dsdy. +(22) +Likewise, by differentiating expressions (19) and (20), we are led to several formulae for d +dt +� +S(t)e−lt� +, namely +d +dt +� +S(t)e−lt� += +� +E +m(Y (t, x))e +� t +0 ˜r(Y (s,x))−l dsdx = +� +E +m(y)e +� t +0 r(X(s,y))−l dsdx, +(23) +12 + +with +m(y) := n0(y) (˜r(y) − l ) − f(y)n0′(y). +(24) +In the one-dimensional case x ∈ R, assuming that E is an interval in which f ̸= 0, we get the additional +expressions +d +dt +� +S(t)e−lt� += +� +E +m(Y (t, x))e +� x +Y (t,x) +˜r(s)−l +f(s) +dsdx = +� +E +m(y)e +� X(t,y) +y +r(s)−l +f(s) +dsdy. +(25) +Lastly, in the particular one-dimensional case where E is an interval such that Y (t, E) = E for all t ≥ 0, +and f ̸= 0 on E, (which is the case if E is an interval delimited by two consecutive roots of f) one can +differentiate (20) to get +d +dt +� +S(t)e−lt� += +� +E +n0(y) (r(X(t, y)) − l) e +� t +0 r(X(s,y))−l dsdy +(26) +and the second expression of (22) to get +d +dt +� +S(t)e−lt� += +� +E +n0(y) (r(X(t, y)) − l) e +� X(t,y) +y +r(s)−l +f(s) +dsdy = +� +E +n0(Y (t, x))(r(x) − l)e +� x +Y (t,x) +˜r(s)−l +f(s) +dsdx. +(27) +4.1.2 +An important estimate +The lemma stated in this section will be crucial in computing limits of the relevant parameter-dependent +integrals in the next section. +Notation. Let x0 ∈ R ∪ {±∞}, and h and g be two functions defined in the neighbourhood of x0. If there +exist C1, C2 > 0 such that +C1|g(x)| ≤ |h(x)| ≤ C2|g(x)| +for any x close enough to x0, we write +h(x) = +Θ +x→x0(g(x)). +Remark. According to the definition of Θ, is is clear that for any x0 ∈ R ∪ {±∞}, g, h defined in the +neighbourhood of x0, f such that h(x) = +Θ +x→x0(g(x)), h is integrable near x0 if and only if g is integrable +near x0. +Lemma 4. Let x0, y ∈ R, with x0 ̸= y, and let β ∈ C2([x0, y]) such that β(y) = 0, β′(y) ̸= 0 and β ̸= 0 on +[x0, y), and α ∈ C1([x0, y]). Then, +e +� x +x0 +α(s) +β(s) ds = Θ +x→y +� +|y − x| +α(y) +β′(y) +� +. +Proof. According to the regularity of α and β, for all s ∈ (x0, y), +α(s) = α(y) + O(s − y), +and +β(s) = (s − y)β′(y) + O((s − y)2) +Thus, +α(s) +β(s) − +α(y) +(s − y)β′(y) = α(s)(s − y)β′(y) − α(y)β(s) +β(s)(s − y)β′(y) += +O(s − y)2 +β′(y)2(s − y)2 + O((s − y)3) = O(1). +Hence, +e +� x +x0 +α(s) +β(s) ds = e +� x +x0 +α(y) +β′(y) +1 +s−y +O(1)ds = eO(1)|y − x| +α(y) +β′(y) , +which proves the result of this lemma. +13 + +4.1.3 +Asymptotic behaviour of the carrying capacity in one dimension +We here focus on the one-dimensional case. We recall that we assume that n0 ∈ Cc(R), f ∈ C2(R) ∩ Lip(R), +r ∈ C1(R) ∩ L1(R), and that r(x) goes to 0 as x goes to ±∞ In this section, we further assume that +f ∈ BV(R), i.e f ′ ∈ L1(R), and that f converges to a non-zero limit at ±∞. +In order to apply Proposition 4 (as explained in Lemma 3), the most insightful division is to consider +each segment between the roots of f. Hence, we must first compute the limit of the function R when the +chosen set E is such a segment. +To be more precise, we must therefore distinguish between several cases, depending on whether the +considered interval is bounded (delimited by two consecutive roots of f) or not (delimited by the smallest +or the greatest root of f), and the sign of the derivative at these boundary roots. +In fact, when n0 vanishes at a given root a, the limit may depend on how fast n0 vanishes, i.e. on the +value α > 0 such that n0(y) vanishes like (y − a)α. For our method of proof to accommodate this case, we +will need to make a slightly stronger assumption involving the derivative of n0. +We will see in the next section that a slight change in the limit of R may have a drastic impact on the +long-time behaviour of n. We also deal with cases where f does not have any root (which ensures, as one +might expect, that R converges to 0), and the case where f is zero on a whole interval. Hence, this result +can be seen as a generalisation of the one stated in [29]. +Proposition 5. In each case, we assume that E ∩ supp(n0) ̸= ∅. +(i) If E = (a, +∞), f < 0 on E, f(a) = 0 and f ′(a) < 0, then R converges to r(a). +(ii) If E = (−∞, a), f > 0 on E, f(a) = 0 and f ′(a) < 0, then R converges to r(a). +(iii) If E = (a, +∞), f > 0 on E, f(a) = 0, f ′(a) > 0, then +• If n0(a) > 0, then +– If r(a) − f ′(a) > 0, then R converges to r(a) − f ′(a). +– If r(a) − f ′(a) < 0, then R converges to 0. +• If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + +O +y→a+((y − a)α), then +– If r(a) − (1 + α)f ′(a) > 0, then R converges to r(a) − (1 + α)f ′(a). +– If r(a) − (1 + α)f ′(a) < 0, then R converges to 0. +• If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a, a + ε], then R converges to 0. +(iv) If E = (−∞, a), f < 0 on E, f(a) = 0, f ′(a) > 0, then +• If n0(a) > 0, then +– If r(a) − f ′(a) > 0, then R converges to r(a) − f ′(a). +– If r(a) − f ′(a) < 0, then R converges to 0. +• If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = −Cα(a−y)α−1 + +O +y→a−((a−y)α), then +– If r(a) − (1 + α)f ′(a) > 0, then R converges to r(a) − (1 + α)f ′(a). +– If r(a) − (1 + α)f ′(a) < 0, then R converges to 0. +• If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a − ε, a], then R converges to 0. +(v) If E = (a, b), f > 0 on (a, b), f(a) = f(b) = 0, f ′(a) > 0, f ′(b) < 0, then +• If n0(a) > 0, then +– If r(b) > r(a) − f ′(a), then R converges to r(b). +– If r(b) < r(a) − f ′(a), then R converges to r(a) − f ′(a). +• If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + +O +y→a+((y − a)α), then +14 + +– If r(b) > r(a) − (α + 1)f ′(a), then R converges to r(b). +– If r(b) < r(a) − (α + 1)f ′(a), then R converges to r(a) − (α + 1)f ′(a). +• If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a, a + ε], then R converges to r(b). +(vi) If E = (a, b), f < 0 on (a, b), f(a) = f(b) = 0, f ′(a) < 0, f ′(b) > 0, then +• If n0(b) > 0, then +– If r(a) > r(b) − f ′(b), then R converges to r(a). +– If r(a) < r(b) − f ′(b), then R converges to r(b) − f ′(b). +• If n0(b) = 0, and if there exist C, α > 0 such that n0′(y) = −Cα(b − y)α−1 + +O +y→b−((b − y)α), then +– If r(a) > r(b) − (α + 1)f ′(b), then R converges to r(a). +– If r(a) < r(b) − (α + 1)f ′(b), then R converges to r(b) − (α + 1)f ′(b). +• If n0(b) = 0, and if there exists ε > 0 such that n0(·) = 0 on [b − ε, b], then R converges to r(a). +(vii) If E = R, and f > 0 on R, then R converges to 0. +(viii) If E = R, and f < 0 on R, then R converges to 0. +(ix) If E is a interval in which f ≡ 0, and n0 > 0, and arg max +E +r = {x1, ..., xp} ⊂ E, with +r′(xi) = 0, +r′′(xi) < 0 +for all i ∈ {1, ..., p}, +then, R converges to r := max +x∈E r(x). +Moreover, except in this last case, R converges with an exponential speed whenever it does not converge to 0. +Proof. As explained at the beginning of this section, whenever we show that R converges with an exponential +speed, we must prove successively that +(a) lim inf +t→+∞ S(t)e−lt > 0 +(b) t �→ eδt d +dt +� +S(t)e−lt� +is bounded for a certain δ > 0, +where l is the expected limit. By Fatou’s lemma, the point (a) can be proven by showing that the integrand +involved in the expression of S (which depends on the chosen formula) converges pointwise to a non-negative +function which is positive on a set of positive Lebesgue measure. Depending on the case, we will use different +expressions for S and S′ among those determined in Section 4.1.1. In order to lighten the proof, we assume +without loss of generality that a = 0 and b = 1, and we denote +˜r = r − f ′ +and +˜rα := ˜r − αf ′ = r − (α + 1)f ′ +for α ∈ R. +Moreover since the cases (ii), (iv), (vi) and (viii) are symmetric to the cases (i), (iii), (v) and (vii) respec- +tively, we omit their proof. +(i) Note that, according to the hypotheses satisfied by f, for all y ∈ (0, +∞), t �→ X(t, y) converges to 0. +(a) According to (20), +S(t)e−r(0)t = +� +∞ +0 +n0(y)e +� t +0 r(X(s,y))−r(0)dsdy = +� M +0 +n0(y)e +� t +0 r(X(s,y))−r(0)dsdy, +for a certain M > 0, since n0 has a compact support. Since f ′(0) < 0, there exist C, δ > 0 such that +X(t, y) ≤ Ce−δt for all y ∈ [0, M], t ≥ 0. This proves that for all y ∈ [0, M], s �→ r(X(s, y)) − r(0) +is integrable on (0, +∞), and thus that y �→ n0(y)e +� +∞ +0 +r(X(s,y))−r(0)ds is well-defined on [0, M]. +Since this function is positive on a sub-interval of [0, M], its integral on this segment is positive. +Moreover, t �→ n0(y)e +� t +0 r(X(s,y))−r(0)ds converges pointwise to this function. +15 + +(b) As seen in the first point, there exist C, δ > 0 such that for all y ∈ [0, M] and all t ≥ 0, +0 ≤ X(t, y) ≤ Ce−δt. Thus, using expression (26), and the mean value theorem, +����eδt d +dt +� +S(t)e−r(0)t� ���� = eδt +���� +� M +0 +n0(y) +� +r(X(t, y)) − r(0) +� +e +� t +0 r(X(t,y))−r(0)dsdy +���� +≤ 2∥n0∥∞∥r∥L∞(0,M)C +� M +0 +e +� t +0 |r(X(s,y))−r(0)|dsdy +≤ ∥n0∥∞∥r′∥L∞(0,M)CMe +� t +0 C∥r′∥L∞(0,M)e−δsds +which is bounded. +(iii) Note that, according to the hypothesis on f, for all x, y ∈ (0, +∞), t �→ X(t, y) is increasing and goes +to +∞, and t �→ Y (t, x) is decreasing and converges to 0. +• Let us assume that n0(0) > 0. We distinguish two cases: +– Case r(0) − f ′(0) > 0: +(a) According to (22), +S(t)e−˜r(0)t = +� +E +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +dsdx. +For all x ∈ (0, +∞), n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +ds +−→ +t→+∞ n0(0)e +� x +0 +˜r(s)−˜r(0) +f(s) +ds, which is well +defined since s �→ ˜r(s)−˜r(0) +f(s) +is continuous on [0, x), thanks to the regularity of r and f, +and positive, since n0(0) > 0 by hypothesis. +(b) Let δ ∈ +� +0, min(˜r(0), f ′(0)) +� +. Since δ − ˜r(0) < 0, r goes to 0 at +∞ and f is positive, we +can find M ≥ 0 such that r(s)−˜r(0)+δ +f(s) +≤ 0 for all s ∈ [M, +∞), and supp +� +n0� +∩E ⊂ [0, M]. +Thus, for all t ≥ 0, and all y ∈ (0, M), +� X(t,y) +y +r(s) − ˜r(0) + δ +f(s) +ds ≤ +� M +y +���� +r(s) − ˜r(0) + δ +f(s) +����ds. +According to (25), +eδt d +dt +� +S(t)e−˜r(0)t� += +� +∞ +0 +m(y)e +� X(t,y) +y +r(s)−˜r(0)+δ +f(s) +dsdy. +Thus, since supp(m) ∩ E = supp(n0) ∩ E ⊂ [0, M], and by the previous inequality, +����eδt d +dt +� +S(t)e−˜r(0)t� ���� ≤ +� M +0 +|m(y)|e +� M +y +|r(s)−˜r(0)+δ| +f(s) +dsdy. +Since m(y) = n0(y)(˜r(y) − ˜r(0)) − f(y)n0′(y), |m(y)| = O +y→0(y). Moreover, since |r(0) − +˜r(0) + δ| = f ′(0) + δ, Lemme 4 yields e +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds = O +y→0(y−1−δ/f ′(0)). Therefore, +|m(y)|e +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds = O +y→0(y−δ/f ′(0)), +and is thus integrable since δ < f ′(0). +16 + +– Case r(0) − f ′(0) < 0: in this case, we do not show that convergence occurs with an expo- +nential speed. Thus, we do not prove the two points as before, but simply that +lim sup +t→+∞ S(t) > 0 +and +lim +t→+∞S′(t) = 0, +which will imply, by definition of R (17), that R converges to 0. According to (22), +S(t) = +� +∞ +0 +n0(y)e +� X(t,y) +y +r(s) +f(s) dsdy. +By hypothesis, f converges to a positive limit. Thus, for all y > 0, there exist εy > 0 such +that f(s) > εy, for all s ≥ y. Thus, for all y > 0, +� +∞ +y +r(s) +f(s)ds ≤ +1 +εy ∥r∥L1 < +∞. This implies +that y �→ n0(y)e +� +∞ +y +r(s) +f(s) ds is well defined on R+. Moreover, this function is positive at any y +such that n0(y) > 0, hence its integral is positive. Finally, t �→ n0(y)e +� X(t,y) +y +r(s) +f(s) ds converges +to this function pointwise,. +Owing to (26) (with l = 0), +S′(t) = +� +supp(n0) +n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) dsdy. +By hypothesis, there exist ε, M > 0 such that f(s) ≥ ε for all s ≥ M. Thus, for all y > 0, +� X(t,y) +y +r(s) +f(s)ds ≤ +� +∞ +y +r(s) +f(s)ds ≤ +� +∞ +M +r(s) +f(s)ds +� +�� +� +≤ +∥r∥L1 +ε ++ +� M +y +r(s) +f(s)ds 1(0,M)(y). +Since r ∈ L1(R+), by hypothesis, this proves that there exists a constant K > 0 such that for +all t ≥ 0, y > 0, +����n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) ds +���� ≤ ∥n0∥∞∥r∥∞eKe +� M +y +r(s) +f(s) ds 1(0,M)(y). +By virtue of Lemma 4, this last quantity in integrable, since +e +� M +y +r(s) +f(s) ds = O +y→0 +� +y− r(0) +f′(0) +� +, +with r(0) < f ′(0), by hypothesis. Moreover, since t �→ r(X(t, y)) converges to 0 as t goes to ++∞ for any y > 0, n0(y) r(X(t, y)) e +� X(t,y) +y +r(s) +f(s) dsdy converges to 0 pointwise. According to +the dominated convergence theorem, S′ thus converges to 0. +• Let us assume that n0(a) = 0, and that the hypothesis of the theorem regarding n0′ holds. We +follow exactly the same steps and use the same formulae as in the case ‘n0(0) > 0’, by adapting +the computations. We distinguish again two cases. +– Case ˜rα(0) > 0: +(a) According to (22), +S(t)e−˜rα(0)t = +� +E +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +dsdx. +For all x ∈ (0, +∞), +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜rα(0) +f(s) +ds = n0(Y (t, x)) +Y (t, x)α e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +ds Y (t, x)αe +� x +Y (t,x) +αf′(0) +f(s) ds. +17 + +Let x > 0. On the one hand, +n0(Y (t, x)) +Y (t, x)α e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +ds −→ +t→+∞ Ce +� x +0 +˜r(s)−˜r(0) +f(s) +ds +which is well defined since s �→ ˜r(s)−˜r(0) +f(s) +is continuous on [0, x), according to the regularity +assumptions on r and f, and positive. On the other hand, by rewriting +Y (t, x)αe +� x +Y (t,x) +−αf′(0) +f(s) +ds = eα(ln(Y (t,x))−ln(x))xαe +� x +Y (t,x) +αf′(0) +f(s) ds += xαe +� x +Y (t,x) +αf′(0) +f(s) − α +s ds, +and by noting that s �→ αf ′(0) +f(s) − α +s is continuous at 0, since +αf ′(0) +f(s) +− α +s = αf ′(0)s − αf(s) +sf(s) += αf ′(0)s − αf ′(0)s + f ′′(0)/2s2 + o(s2) +f ′(0)s2 + o(s2) +−→ +s→0 −αf ′′(0) +2f ′(0) , +we show that +Y (t, x)αe +� x +Y (t,x) +−αf′(0) +f(s) +ds −→ +t→+∞ xαe +� x +0 +αf′(0) +f(s) − α +s ds, +which is also well defined, and positive. +(b) Let δ ∈ +� +0, min(˜rα(0), f ′(0)) +� +. Since δ−˜rα(0) < 0, r goes to 0 at +∞ and f is positive, we +can find M ≥ 0 such that r(s)−˜rα(0)+δ +f(s) +≤ 0 for all s ∈ [M, +∞), and supp +� +n0� +⊂ [0, M]. +Thus, for all t ≥ 0, and all y ∈ (0, M), +� X(t,y) +y +r(s) − ˜rα(0) + δ +f(s) +ds ≤ +� M +y +���� +r(s) − ˜rα(0) + δ +f(s) +����ds. +According to (25), +eδt d +dt +� +S(t)e−˜rα(0)t� += +� +∞ +0 +m(y)e +� X(t,y) +y +r(s)−˜rα(0)+δ +f(s) +dsdy. +Thus, since supp(m) ∩ E = supp(n0) ∩ E ⊂ [0, M], and thanks to the previous inequality, +����eδt d +dt +� +S(t)e−˜rα(0)t� ���� ≤ +� M +0 +|m(y)|e +� M +y +|r(s)−˜rα(0)+δ| +f(s) +dsdy. +Let us prove that this integral is bounded. First, let us note that +m(y) = n0(y)(˜r(y) − ˜rα(0)) − f(y)n0′(y) = +O +y→0+(yα+1) +Indeed, since n0(y) = Cyα + +O +y→0+(yα+1) and n0′(y) = Cαyα−1 + +O +y→0+(yα), +|m(y)| +yα+1 ≤ n0(y) +yα +|˜r(y) − ˜r(0)| +y ++ |αf ′(0)n0(y) − f(y)n0′(y)| +yα+1 +≤ n0(y) +yα ∥˜r′∥∞ + |Cαf ′(0)yα − Cαf ′(0)yα + O(yα+1)| +yα+1 += +O +y→0+(1). +18 + +Moreover, according to Lemma 4, since |r(0) − ˜rα(0) + δ| = (α + 1)f ′(0) + δ, +e +� M +y +|r(s)−˜rα(0)+δ| +f(s) +ds = +O +y→0+(y−α−1−δ/f ′(0)). +Therefore, +|m(y)|e +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds = +O +y→0+(y−δ/f ′(0)), +and is thus integrable since δ < f ′(0). +– Case ˜rα(0) < 0: again, we just prove that +lim sup +t→+∞ S(t) > 0 +and +lim +t→+∞S′(t) = 0. +According to (22), +S(t) = +� +∞ +0 +n0(y)e +� X(t,y) +y +r(s) +f(s) dsdy. +By hypothesis, f converges to a positive limit. Thus, for all y > 0, there exist εy > 0 such +that f(s) > εy, for all s ≥ y. Hence, for all y > 0, +� +∞ +y +r(s) +f(s)ds ≤ +1 +εy ∥r∥L1 < +∞. This ensures +that y �→ n0(y)e +� +∞ +y +r(s) +f(s) ds is well defined on R+. Moreover, this function is positive for every +y such that n0(y) > 0, which ensures that its integral is positive, and t �→ n0(y)e +� X(t,y) +y +r(s) +f(s) ds +converges to this function pointwise. According to (26), (with l = 0), +S′(t) = +� +supp(n0) +n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) dsdy. +By hypothesis, there exist ε, M > 0 such that f(s) ≥ ε for all s ≥ M. Thus, for all y > 0, +� X(t,y) +y +r(s) +f(s)ds ≤ +� +∞ +y +r(s) +f(s)ds ≤ +� +∞ +M +r(s) +f(s)ds +� +�� +� +≤ +∥r∥L1 +ε ++ +� M +y +r(s) +f(s)ds 1(0,M)(y). +Since r ∈ L1(R+), this proves that there exist a constant K > 0 such that for all t ≥ 0, y > 0, +����n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) ds +���� ≤ ∥r∥∞eKn0(y)e +� M +y +r(s) +f(s) ds 1(0,M)(y). +By hypothesis, and according to Lemma 4, +n0(y) = +O +y→0+(yα) +and +e +� M +y +r(s) +f(s) ds = +O +y→0+ +� +y− r(0) +f′(0) +� +. +Thus, +n0(y)e +� M +y +r(s) +f(s) ds = +O +y→0+ +� +yα− r(0) +f′(0) +� +, +with α− r(0) +f ′(0) > −1. Moreover, since t �→ r(X(t, y)) converges to 0 as t goes to +∞ for any y > +0, n0(y) r(X(t, y)) e +� X(t,y) +y +r(s) +f(s) dsdy converges to 0 pointwise. By the dominated convergence +theorem, S′ thus converges to 0. +• We can prove this point exactly as we treat the case f > 0 on R. We therefore leave it to the +reader and refer to the proof of (vii). +19 + +(v) Let us note that, for any x, y ∈ (0, 1), t �→ X(t, y) is increasing and converges to 1, and t �→ Y (t, x) is +decreasing and converges to 0. +• Let us assume that n0(a) > 0. We distinguish again between two cases: +– Case r(1) > ˜r(0): +(a) Let us use the second expression (22) for S, i.e. +S(t)e−r(1)t = +� 1 +0 +e +� X(t,y) +y +r(s)−r(1) +f(s) +dsdy. +For all y ∈ (0, 1), n0(y)e +� X(t,y) +y +r(s)−r(1) +f(s) +ds +−→ +t→+∞ n0(y)e +� 1 +y +r(s)−r(1) +f(s) +ds, which is well-defined +for all y ∈ (0, 1), since s �→ +r(s)−r(1) +f(s) +is continuous on (0, 1], and positive on a set of +non-zero measure, since it is positive where n0 is positive. +(b) Let δ ∈ +� +0, min (r(1) − ˜r(0), −f ′(1)) +� +. Since ˜r(0) − r(1) + δ < 0, there exists m ∈ (0, 1) +such that ˜r(s) − r(1) + δ for all s ∈ (0, m]. Thus, for all x ∈ (0, 1), t ≥ 0, +� x +Y (t,x) +˜r(s) − r(1) + δ +f(s) +ds ≤ +� x +m +|˜r(s) − r(1) + δ| +f(s) +ds 1(m,1)(x). +Thus, using expression (27), +eδt d +dt +� +S(t)e−r(1)t� += +� 1 +0 +n0 (Y (t, x)) (r(x) − r(1)) e +� x +Y (t,x) +˜r(s)−r(1)+δ +f(s) +dsdx +≤ ∥n0∥ +� 1 +0 +|r(x) − r(1)|e +� x +m +|˜r(s)−r(1)+δ| +f(s) +ds 1(m,1)(x)dx < +∞. +This last integral is finite since |˜r(1) − r(1) + δ| = −f ′(1) + δ, and thus e +� x +a +|˜r(s)−˜r(0)| +f(s) +ds = +O +x→1 +� +|x − 1| +δ +f′(1) −1� +, by Lemma 4) |r(x)−r(1)| = O +x→1|x−1|, and +δ +f ′(1) > −1 by hypothesis. +– Case ˜r(0) > r(1): +(a) Using (22), we find +S(t)e−˜r(0)t = +� 1 +0 +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +dsdx. +For all x ∈ (0, 1), n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(0) +f(s) +ds +−→ +t→+∞ n0(0)e +� x +0 +˜r(s)−˜r(0) +f(s) +ds, which is well- +defined since s �→ ˜r(s)−˜r(0) +f(s) +is continuous on [0, 1), and positive by hypothesis on n0. +(b) Let δ ∈ +� +0, min(˜r(0) − r(1), f ′(0)) +� +. Since r(1) − ˜r(0) + δ < 0, there exists M ∈ (0, 1) such +that r(s) − ˜r(0) + δ < 0 for all s ≥ M. Thus, for all y ∈ (0, 1), t ≥ 0, +� X(t,y) +y +r(s)−˜r(0)+δ +f(s) +≤ +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds 1(0,M)(y). Thus, +����eδt d +dt +� +S(t)e−˜r(0)t� ���� = +���� +� 1 +0 +m(y)e +� X(t,y) +y +r(s)−˜r(0)+δ +f(s) +ds +���� ≤ +� 1 +0 +|m(y)|e +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds 1(0,M)(y)dy, +which is a finite integral, since |r(0) − ˜r(0) + δ| = f ′(0) + δ, and thus e +� b +y +|r(s)−˜r(0)+δ| +f(s) +ds = +O +y→0 +� +y− +δ +f′(0) −1� +(by Lemma 4), m(y) = n0(y) (˜r(y) − ˜r(0)) − f(y)n0′(y) = O +y→0 (y), and +δ +f ′(0) < 1 thanks to our choice for δ. +20 + +• Let us assume that n0(a) = 0, and that the hypothesis on n0′ of the theorem holds. As usual, we +distinguish two cases. +– Case r(1) > ˜rα(0): +(a) This first point is exactly the same as in the case n0 > 0. Let us use the second expres- +sion (22) for S, i.e. +S(t)e−r(1)t = +� 1 +0 +e +� X(t,y) +y +r(s)−r(1) +f(s) +dsdy. +For all y ∈ (0, 1), n0(y)e +� X(t,y) +y +r(s)−r(1) +f(s) +ds +−→ +t→+∞ n0(y)e +� 1 +y +r(s)−r(1) +f(s) +ds, which is well-defined +for all y ∈ (0, 1), since s �→ +r(s)−r(1) +f(s) +is continuous on (0, 1], and positive on a set of +measure non-zero, since it is positive where n0 is positive. +(b) Let δ ∈ +� +0, min (r(1) − ˜rα(0), −f ′(1)) +� +. First, let us note that we can rewrite +Y (t, x)α = eln(Y (t,x))−ln(x)xα = xαe +� x +Y (t,x) − α +s ds. +Thus, by using expression (27), we get +eδt d +dt +� +S(t)e−r(1)t� += +� 1 +0 +n0 (Y (t, x)) (r(x) − r(1)) e +� x +Y (t,x) +˜r(s)−r(1)+δ +f(s) +dsdx += +� 1 +0 +n0(Y (t, x)) +Y (t, x)α xα(r(x) − r(1))e +� x +Y (t,x) +ϕ(s) +f(s) dsdx, +with +ϕ(s) := ˜r(s) − r(1) + δ − αf(s) +s +. +By hypothesis on n0, f and r, ˜n0 : y �→ +n0(y) +yα +and ϕ are both continuous on [0, 1]. +Moreover, since ϕ(0) = ˜rα(0) − r(1) + δ < 0, there exists ε ∈ (0, 1) such that ϕ(s) < 0 for +all s ∈ [0, ε]. Thus, +����eδt d +dt +� +S(t)e−r(1)t� ���� ≤ ∥ ˜n0∥∞ +� 1 +0 +|r(x) − r(1)|xαe +� x +ε +|ϕ(s)| +f(s) ds1(ε,1)(x)dx. +since |ϕ(1)| = δ − f ′(1), Lemma 4 yields +e +� x +ε +|ϕ(s)| +f(s) ds = O +x→1(|x − 1| +δ +f′(1) −1). +Since |r(x) − r(1)| = O +x→1(x) and +δ +f ′(1) > −1 (by hypothesis on δ), this proves that this +last integral is bounded. +– Case ˜rα(0) > r(1): +(a) According to (22), +S(t)e−˜rα(0)t = +� 1 +0 +n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜rα(0) +f(s) +dsdx += +� 1 +0 +n0(Y (t, x)) +Y (t, x)α Y (t, x)αe +� x +Y (t,x) +˜r(s)−˜rα(0) +f(s) +dsdx. +By rewriting Y (t, x)α = xαe− +� x +Y (t,x) +α +s ds, we get +S(t)e−˜rα(0)t = +� 1 +0 +n0(Y (t, x)) +Y (t, x)α xαe +� x +Y (t,x) +˜r(s)−˜rα(0) +f(s) +− α +s dsdx. +Since n0(Y (t,x)) +Y (t,x)α xαe +� x +Y (t,x) +˜r(s)−˜rα(0) +f(s) +− α +s ds converges pointwise to C xα e +� x +0 +˜r(s)−˜rα(0) +f(s) +− α +s ds, which +is well-defined, since s �→ ˜r(s)−˜rα(0) +f(s) +− α +s is continuous at 0 and positive, we are done. +21 + +(b) Let δ ∈ +� +0, min(˜rα(0) − r(1), f ′(0)) +� +. Since r(1) − ˜rα(0) + δ < 0, there exists M ∈ (0, 1) +such that r(s) − ˜rα(0) + δ < 0 for all s ≥ M. Thus, for all y ∈ (0, 1), t ≥ 0, +� X(t,y) +y +r(s) − ˜rα(0) + δ +f(s) +≤ +� M +y +|r(s) − ˜rα(0) + δ| +f(s) +ds 1(0,M)(y). +Hence, using expression (25), we get +����eδt d +dt +� +S(t)e−˜r(0)t� ���� = +���� +� 1 +0 +m(y)e +� X(t,y) +y +r(s)−˜r(0)+δ +f(s) +ds +���� ≤ +� 1 +0 +|m(y)|e +� M +y +|r(s)−˜r(0)+δ| +f(s) +ds 1(0,M)(y)dy, +which is a finite integral, since |r(0) − ˜rα(0) + δ| = (1 + α)f ′(0) + δ. Lemma 4 leads to +e +� b +y +|r(s)−˜r(0)+δ| +f(s) +ds = O +y→0 +� +y− +δ +f′(0) −α−1� +. +The integrability follows from m(y) = n0(y) (˜r(y) − ˜rα(0)) − f(y)n0′(y) = O +y→0 +� +yα+1� +(as +seen previously), and +δ +f ′(0) < 1 thanks to our choice for δ. +• We prove this case with exactly the same arguments that for the case of a unique root which is +asymptotically unstable. We therefore apply the proof of (i). +(vii) In this case, since f > 0, X(t, y) −→ +t→+∞ +∞, for all y ∈ R. Let us prove that +lim inf +t→+∞ S(t) > 0 +and +lim +t→+∞ S(t) = 0. +According to (21), +S(t) = +� +supp(n0) +n0(y)e +� X(t,y) +y +r(s) +f(s) dsdy. +The integrand n0(y)e +� X(t,y) +y +r(s) +f(s) ds converges pointwise to n0(y)e +� +∞ +y +r(s) +f(s) ds, which is well defined (with +values in [0, +∞]), and positive for all y ∈ supp(n0), since r +f is positive. According to (27), +S′(t) = +� +supp(n0) +n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) dsdy. +Since f is continuous, positive, and converges to positive constants at ±∞, ε := min +s∈R f(s) > 0. Thus, +for all y ∈ R, t ≥ 0, +����n0(y)r(X(t, y))e +� X(t,y) +y +r(s) +f(s) ds +���� ≤ ∥n0∥∞∥r∥∞e +∥r∥L1 +ε +< +∞. +Combined with the fact that r(X(t, y)) converges to 0 as t goes to +∞ pointwise, we deduce that S′ +converges to 0 by the dominated convergence theorem. +(ix) Since f ≡ 0 on E, Y (t, x) = x for all (t, x) ∈ R+ × E. Thus, according to formula (20), +S(t) = +� +E +n0(x)er(x)tdx +and +S′(t) = +� +E +n0(x)r(x)er(x)tdx. +By Laplace’s formula (see [39]), +S(t) +∼ +t→+∞ +√ +2π +� p +� +i=1 +n0(xi) +� +|r′′(xi)| +� +ert +√ +t +22 + +and +S′(t) +∼ +t→+∞ +√ +2π +� p +� +i=1 +n0(xi)r(xi) +� +|r′′(xi)| +� +ert +√ +t = +√ +2π r +� p +� +i=1 +n0(xi) +� +|r′′(xi)| +� +ert +√ +t +∼ +t→+∞ r S(t). +Thus, R(t) = S′(t) +S(t) +−→ +t→+∞ r. +4.2 +Applications +Summary of the method. The method that we propose in order to study the asymptotic behaviour of +PDE (1) can be summarised by the following three steps: +1. Choose an appropriate family of set (Oi) which satisfies the assumptions of Proposition 12, and such +that we can compute the asymptotic behaviour of the functions Ri: a good choice when f has a finite +number of roots is to take the interval between the roots, as suggested in Lemma 3. +2. Use Proposition 4 in order to determine the limit of ρ, and its speed of convergence when possible. +3. Use the semi-explicit expression of n provided by equation (7), and eventually Proposition 1 to deduce +the asymptotic behaviour of n. +In each of the following subsections, we apply the three points detailed in this summary to study the +asymptotic behaviour of n in different cases. +Remark regarding the regularity of parameter functions. As in subsection 4.1.3, we make the +further assumptions that f ∈ BV(R), and that f converges to a non-zero limit at ±∞. Moreover, we easily +check that all the results of this previous section remain true if we assume that n0 is C1 on each interval +between the roots of f, and not necessarily on the whole of R. As far as f is concerned, it is enough to assume +that it is globally Lipschitz, and C2 only on a neighbourhood of its roots. It will sometimes be advisable to +make these two additional assumptions: we will indicate this at the beginning of each statement whenever +this is the case. +4.2.1 +Case of a unique stable equilibrium +We start by assuming that f has a unique root (denoted a), which is asymptotically stable for the ODE +˙u = f(u). In this case, solutions converge to a weighted Dirac mass at a, regardless of the functions r and n0. +The weight in front of the Dirac mass is determined by the value of r at a. Note that this result can be +generalised to higher dimensions, see Proposition 12. +Proposition 6. Let us assume that f has a unique root (denoted a), and that f ′(a) < 0. Then, ρ converges +to r(a) and n(t, ·) +⇀ +t→+∞ r(a)δa. +Proof. We apply the three points detailed in the summary: +1. Let us denote O1 := (−∞, a), O2 := (a, +∞), which satisfy the assumptions of Proposition 3, by +Lemma 3. By proposition 5, R1 and R2 both converge to r(a) (with an exponential speed). +2. By Proposition 4, ρ converges to r(a) with an exponential speed. +3. According to to the semi-explicit expression (7), n(t, x) = n0(Y (t, x))e +� t +0 ˜r(Y (s,x))−ρ(s)ds. Let δ > 0. +Since ∥Y (t, x)∥ +−→ +t→+∞ +∞ for all x ∈ Rd\{a}, and n0 has a compact support, there exists T0 such +that n(t, x) = 0 for all t ≥ T0, x ∈ Rd\[a − δ, a + δ]. Since ρ(t) = +� +supp(n0) n(t, x)dx converges to r(a), +Propositions 1 allows us to conclude that n(t, ·) +⇀ +t→+∞ r(a)δa. +23 + +4.2.2 +Case of a unique unstable equilibrium +We now assume that f has a unique root (denoted a) which is asymptotically unstable for the ODE ˙u = f(u). +Under theses hypotheses, the growth term can counterbalance the advection term: there exist two regimes +of convergence, depending on how r(a) and f ′(a) compare. +Proposition 7. Let us assume that f has a unique root (denoted a), and that f ′(a) > 0, Then: +• If r(a) < f ′(a), then ρ(t) −→ +t→+∞ 0 and n(t, ·) −→ +t→+∞ 0 in L1(R). +• If r(a) > f ′(a), and n0(a) > 0, then ρ(t) −→ +t→+∞ r(a) − f ′(a), and n(t, ·) −→ +t→+∞ n in L1(R), where +n(x) := Ce +� x +a +˜r(s)−˜r(a) +f(s) +ds, +with ˜r = r − f ′ and C such that +� +R n(x)dx = r(a) − f ′(a). +Proof. We apply the three points detailed in the summary: +• Let us assume that r(a) < f ′(a): +1. Let us denote O1 := (−∞, a), O2 := (a, +∞), which satisfy the assumptions of Proposition 3, by +Lemma 3. Proposition 5 shows that R1 and R2 both converge to 0. +2. By Proposition 4, ρ converges to 0. +3. We immediately deduce from the previous point that n(t, ·) −→ +t→+∞ 0 in L1(R), by definition of ρ. +• Let us assume that r(a) > f ′(a): +1. With the same choice for O1 and O2, Proposition 5 shows that R1 and R2 both converge to +r(a) − f ′(a). +2. By Proposition 4, ρ converges to ˜r(a) with an exponential speed. +3. By the semi-explicit expression (7), +n(t, x) = n0(Y (t, x))e +� t +0 ˜r(Y (s,x))−ρ(s)ds = n0(Y (t, x))e +� t +0 ˜r(Y (s,x))−˜r(a)dse +� t +0 ˜r(a)−ρ(s)ds += n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(a) +f(s) +dse +� t +0 ˜r(a)−ρ(s)ds +(we use the change of variable s′ = Y (s, x) in the first integral to get this last expression). Thus, +n(t, ·) converges pointwise to +x �→ n0(a)e +� x +a +˜r(s)−˜r(a) +f(s) +dse +� +∞ +0 +˜r(a)−ρ(s)ds, +which is well-defined, since ρ converges to ˜r(a) with an exponential speed, f > 0 on (a, +∞) and +s �→ ˜r(s)−˜r(a) +f(s) +is continuous at a. +Moreover, since r(x) +−→ +x→+∞ 0, and f converges to a positive limit, there exist M, d > 0 such that +r(s)−˜r(a) +f(s) +< −d for all s ≥ M. Thus, for all t ≥ 0, x ∈ (a, +∞), +� x +Y (t,x) +˜r(s) − ˜r(a) +f(s) +ds ≤ +� M +a +|˜r(s) − ˜r(a)| +f(s) +ds +� +�� +� +:=C1 ++ +� x +M +˜r(s) − ˜r(a) +f(s) +ds 1(M,+∞)(x) +≤ C1 + +� x +M +r(s) − ˜r(a) +f(s) +� +�� +� +≤−d +ds 1(M,+∞)(x) + +� x +M +f ′(s) +f(s) ds 1(M,+∞)(x) +≤ C1 − d(x − M)1(M,+∞) + +� +∞ +M +|f ′(s)| +f(s) ds +� +�� +� +:=C2 +, +24 + +with C1, C2 < +∞, by the regularity of ˜r, f ∈ BV (R), and the fact that f converges to a positive +constant at infinity. +By proceeding in the same way for all x ≤ a, we show that for all x ∈ R, t ≥ 0, +n(t, x) ≤ Ce−d|x| +for some constants C, d > 0, which ensures, according to the dominated convergence theorem, +that t �→ n(t, ·) converges to x �→ n0(a)e +� x +a +tr(s)−˜r(a) +f(s) +dse +� +∞ +0 +˜r(a)−ρ(s)ds in L1(R). +4.2.3 +Two equilibria +In this section we assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0 and f ′(b) < 0 (hence +f > 0 on (a, b)). The case f ′(a) < 0, f ′(b) > 0, f < 0 on (a, b) is similar. Depending on the functions r +and n0, n will either converge to a function in L1, or converge to a Dirac mass at b. We split this result +into two propositions: the first one assumes that the support of n0 crosses a, which means that n0 > 0 in +a neighbourhood of a. The second one assumes that supp(n0) ⊂ [a, +∞), and we consider the case where +n0(a) = 0, which leads to other regular functions being reached. +Proposition 8. Let us assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0, f ′(b) < 0, and +that n0(a) > 0. Then: +• If r(b) > r(a) − f ′(a), then ρ(t) −→ +t→+∞ r(b) and n(t, ·) +⇀ +t→+∞ r(b)δb. +• If r(b) < r(a) − f ′(a), then ρ(t) −→ +t→+∞ r(a) − f ′(a), and n(t, ·) −→ +t→+∞ n in L1(R), where +n(x) := De +� x +a +˜r(s)−˜r(a) +f(s) +ds1(−∞,b), +with ˜r = r − f ′, and D > 0 is such that +� +R n(x)dx = r(a) − f ′(a). +Proof. Note that since n0 is assumed to be continuous on R, n0 > 0 on a neighbourhood of a. +• Let us assume that r(b) > r(a) − f ′(a). We again follow the three points of the method outlined in the +beginning of the subsection. +1. Let us denote O1 = (−∞, a), O2 = (a, b), O3 = (b, +∞). One easily checks that these sets satisfy +the hypotheses of Proposition 3, thanks to Lemma 3. According to Proposition 5, R1 converges +to max(0, ˜r(a)) < r(b) and R2 and R3 both converge to r(b) with an exponential speed. +2. By Proposition 4, ρ converges to r(b) with an exponential speed, and ρ1(t) = +� a +−∞ n(t, x)dx +converges to 0. +3. Let x ∈ (a, b). Using (7), we find n(t, x) = n0(Y (t, x))e +� t +0 ˜r(Y (s,x))−ρ(s))ds, for all t ≥ 0. Let +K ⊂ (a, b) be a compact set, δ ∈ +� +0, 1 +2 (r(b) − ˜r(a)) +� +, and let us denote d := r(b) − ˜r(a) − 2δ > 0. +Since ρ converges to r(b), and (Y (s, x))s≥0 converges to a uniformly on K, there exists T0 such +that for all s ≥ T0 and all x ∈ K, +ρ(s) ≥ r(b) − δ +and +˜r(Y (s, x)) ≤ ˜r(a) + δ, +Thus, +� +K +n(t, x)dx ≤ ∥n0∥∞ +� +K +e +� T0 +0 +˜r(Y (s,x))−ρ(s)dsdx e−d(t−T0) −→ +t→+∞ 0. +Let K′ be a compact subset of (b, +∞). Since n0 has compact support, there exists T0 such that +n0(Y (t, x)) = 0 for all t ≥ T0, x ∈ K′. Thus, t �→ +� +K′ n(t, x)dx converges to 0. By Proposition 1, +n(t, ·) +⇀ +t→+∞ r(b)δb. +25 + +• Let us now assume that r(b) < r(a) − f(a). +1. With the same choice for O1, O2 and O3, Proposition 5 shows that R1 and R2 converge to ˜r(a), +and that R3 converges to r(b) < ˜r(a). +2. We then apply Proposition 4 to infer that ρ converges to ˜r(a) with an exponential speed, and +ρ3(t) = +� +∞ +b +n(t, x)dx converges to 0. +3. Let x ∈ (−∞, b), t ≥ 0. By the semi-explicit expression (7), +n(t, x) = n0(Y (t, x))e +� t +0 ˜r(Y (s,x))−ρ(s))ds += n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜r(a) +f(s) +dse +� t +0 ˜r(a)−ρ(s)ds, +where we used the change of variable ‘s′ = Y (t, x)’. The latter function converges pointwise to +n0(a)e +� x +a +˜r(s)−˜r(a) +f(s) +dse +� +∞ +0 +˜r(a)−ρ(s)ds. +As for the case of a unique unstable equilibrium (proof of Proposition 7) one can find C, d ≥ 0 +such that n(t, x) ≤ Ce−d|x| for all x ≤ a. Moreover, for all x ∈ (a, b), +n(t, x) ≤ ∥n0∥∞ e +� +∞ +0 +|˜r(a)−ρ(s)|ds e +� x +a +˜r(s)−˜r(a) +f(s) +1{˜r(s)>˜r(a)}(s)ds, +which provides an L1-domination, since e +� x +a +˜r(s)−˜r(a) +f(s) += +O +x→+∞ +� +|x − b| +˜r(a)−r(b) +−f′(b) +−1 +� +, thanks to +Lemma 4. +By the dominated convergence theorem, combined with the fact that ρ converges +to ˜r(a) and ρ3 converges to 0, this ensures that n(t, ·) converges to the expected limit. +In the following proposition, we assume that n0 is C1 on (a, b) and on (b, +∞), and not necessarily on +the whole of R. +Proposition 9. Let us assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0, f ′(b) < 0, and +that supp(n0) ⊂ [a, +∞). We distinguish between several cases: +• If n0(a) > 0, then +– If r(b) > r(a) − f ′(a), then ρ(t) −→ +t→+∞ r(b), and n(t, ·) +⇀ +t→+∞ r(b)δb. +– If r(b) < r(a) − f ′(a), then ρ(t) −→ +t→+∞ r(a) − f ′(a), and n(t, ·) −→ +t→+∞ n0 in L1(R), where +n0(x) := D0e +� x +a +˜r(s)−˜r(a) +f(s) +ds1(a,b), +with ˜r = r − f ′, and D0 > 0 is such that +� +R n0(x)dx = r(a) − f ′(a). +• If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + +O +y→a+ ((y − a)α), then +– If r(b) > r(a) − (1 + α)f ′(a), then ρ(t) −→ +t→+∞ r(b), and n(t, ·) +⇀ +t→+∞ r(b)δb. +– If r(b) < r(a) − (1 + α)f ′(a), then ρ(t) −→ +t→+∞ r(a) − (1 + α)f ′(a), and n(t, ·) −→ +t→+∞ nα in L1(R), +where +nα(x) := Dα(x − a)αe +� x +a +˜r(s)−˜rα(a) +f(s) +− +α +s−a ds1(a,b), +where ˜r = r − f ′, ˜rα = r − (1 + α)f ′, and Dα > 0 is such that +� +R nα(x)dx = r(a) − (1 + α)f ′(a). +• If n0(a) = 0, and if there exists ε > 0 such that n0(y) = 0 for all y ∈ [a, a + ε], then ρ(t) −→ +t→+∞ r(b), +and n(t, ·) +⇀ +t→+∞ r(b)δb. +26 + +Proof. Since the proof of this proposition is very similar to the one of Proposition 8, we do not write it in +full detail, but we simply underline the points that must be adapted. +• In the case where n0(a) > 0, and supp(n0) ⊂ [a, +∞), the proof is the same, but by considering only +the two sets O2 = (a, b), and O3 = (b, +∞), and not O1 = (−∞, a). We easily check using Lemma 3 +that (O2, O3) satisfy the hypotheses of Lemma 3, since supp(n0) ∩ O1 = ∅ +• The case where n0(a) = 0 and the hypothesis on n0′ holds is quite similar, except that Proposition 5 +now shows that R2 converges to max (˜rα(a), r(b)), with an exponential speed (if ˜rα(a) ̸= r(b)). Thus, +we treat the case r(b) > ˜rα(a) in exactly the same way; the case r(b) < r(a) − (1 + α) is a little more +intricate: recalling that for all x ∈ (a, b), t ≥ 0, +n(t, x) = n0(Y (t, x))e +� x +Y (t,x) +˜r(s)−˜rα(a) +f(s) +dse +� t +0 ˜rα(a)−ρ(s)ds +and +(Y (t, x) − a)α = (x − a)αe− +� x +Y (t,x) +α +s−a ds, +one notes that +n(t, x) = +n0(Y (t, x)) +(Y (t, x) − a)α e +� t +0 ˜rα(a)−ρ(s)ds(x − a)αe +� x +Y (t,x) +˜r(s)−˜rα(a) +f(s) +− +α +s−a ds +converges pointwise to +Ce +� +∞ +0 +˜rα(a)−ρ(s)ds(x − a)αe +� x +a +˜r(s)−˜rα(a) +f(s) +− +α +s−a ds, +which is well-defined, since ρ converges to ˜rα(a) with an exponential speed, and s �→ ˜r(s)−˜rα(s) +f(s) +− +α +s−a +is continuous on a, as seen in the proof of Proposition 5. Moreover, +x �→ ∥ n0(·) +(· − a)α ∥∞e +� +∞ +0 +|˜rα(s)−ρ(s)|ds e +� x +a +ϕ(s) +f(s) 1{ϕ(s)≥0}ds, +with ϕ(s) = ˜r(s)−˜rα(s)−α f(s) +s−a is clearly a domination of n, and is in L1, since ϕ(b) = r(b)−˜rα(a)−f ′(b), +which implies by Lemma 4 that e +� x +a +ϕ(s) +f(s) ds = +O +x→b− +� +(b − x) +r(b)−˜rα(a) +f′(b) +−1 +� +, with r(b)−˜rα(a) +f ′(b) +> 0. +• This last point is the simplest, and is in fact analogous to the case of a single equilibrium point. +According to Proposition 5, R2 and R3 converge to r(b): we deduce the result by following the steps +of Proposition 6. +Remark. Since Proposition 9 provides a completely explicit expression for the limit functions nα, α ≥ 0, +one can easily determine their asymptotic behaviour at the boundary of the segment (a, b). Since for all +α > 0, x ∈ (a, b), +nα(x) = Dα(x − a)αe +� x +a +˜r(s)−˜rα(s) +f(s) +− +α +s−a ds, +and s �→ ˜r(s)−˜rα(a) +f(s) +− α +s is continuous on [a, b), it is clear that nα(x) = +Θ +x→a+ ((x − a)α). In particular, nα +can be extended by continuity at 0, with nα(a) = 0 if α > 0, and n0(a) ∈ (0, +∞). +Moreover, since ˜r(b)−˜rα(a)− αf(b) +b−a = r(b)−˜rα(a)−f(b), Lemma 4 ensures that nα(x) = +Θ +x→b− +� +(b − x) +r(b)−˜rα(a) +f′(b) +−1 +� +. +In particular +lim +x→b−nα(x) = +� +� +� +� +� +0 +if ˜r(b) < ˜rα(a) +l > 0 +if ˜r(b) = ˜rα(a) ++∞ +if ˜r(b) > ˜rα(a) +. +These different cases are illustrated by Figure 2. +The case where f has two roots a < b with f ′(a) < 0 and f ′(b) > 0 is symmetric to the cases here, and +thus lead to the same results, by switching a and b in the Propositions. +27 + +Figure 2: Continuous limit functions nα, for different values of α > 0, as defined in Proposition 9. In this +example, we have chosen f(x) = x(1 − x), and r(x) = b − ax (with b = 6, a = 4). With this choice, we +easily compute that, for all α ∈ [0, a − 1), and all x ∈ (0, 1) nα(x) = Dαxα(1 − x)a−α−2, for the appropriate +constant Dα. This illustrates the variety of limit functions that can be reached depending on the initial +condition, as detailed in Proposition 9. +4.3 +More than two equilibria +In this subsection, we deal with the cases where f has more than two equilibria. As evidenced by the previous +result, listing all possible scenarios when there are two roots already is cumbersome: this is why we will not +do so in a more general case, and will focus on the case where n0 is positive on the neighbourhood of the +unstable equilibrium points. The other cases can of course be treated as seen above, keeping in mind that +this changes the value of the limits reached by the R functions. +Proposition 10. Let us assume that f has a finite number of roots, which are all hyperbolic equilibrium points +for the ODE ˙u = f(u), i.e. f ′ has a sign at each root of f, and let us denote x1 +u, ..., xp +u the asymptotically +unstable equilibria, and x1 +s, ..., x1 +s the asymptotically stable one. Moreover, let us denote Mu := max{r(x1 +u) − +f ′(x1 +u), ...r(xp +u) − f ′(xp +u)}, and Ms := max{r(x1 +s), ..., r(xm +s )}, and let us assume that these two maxima are +both reached at a unique point. Lastly, let us assume that n0(xi +u) > 0 for all i ∈ {1, ..., p}. +• If Ms > Mu, then ρ(t) +−→ +t→+∞ Ms, and n(t, ·) +⇀ +t→+∞ Msδxi +s, with xi +s the unique stable equilibria such +that Ms = r(xi +s). +• If Ms < Mu, then ρ(t) −→ +t→+∞ Mu, and n(t, ·) −→ +t→+∞ ni∗ in L1, where +ni∗(x) = Ci∗e +� x +xi∗ +u +˜r(s)−˜r(xi∗ +u ) +f(s) +ds1Ii∗ (x), +with ˜r = r−f ′, i∗ the unique integer of {1, ..., p} such that ˜r(xi∗ +u ) = Mu, Ii∗ the open interval delimited +by the two stable equilibria which enclose xi∗ +u (or −∞ or +∞ if xi∗ +u is the smallest or the greatest root +of f), and Ci∗ a positive constant such that +� +Ii∗ ni∗(x)dx = Mu. +Proof. The proof of this proposition is in similar to that of Proposition 8: we denote O0, ..., Op+m, the +intervals between each roots of f, which satisfy the hypotheses of Proposition 3, according to Lemma 3, +28 + +16 +0= +14 +α=1 +α=2 +12 +α=2.5 +10 +8 +6 +4 +2 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0and, using Proposition 5, we are able to compute the limit of the function Ri, for all i ∈ {0, . . . , p + m}, +and thus determine the long-time behaviour of ρ by Proposition 4. We conclude by using the semi-explicit +expression (7) for n. +This method also allows to deal with the case where f ≡ 0 on a whole segment: we do not return to the +case f ≡ 0 on R, which has already been studied in [34] and [29], but we consider the case where f ≡ 0 on +an interval, and then becomes positive. +To make the assumption of the following proposition possible, we assume that f is C2 on (−∞, a) and +on (a, +∞), but not necessarily on the whole of R. +Proposition 11. Let us assume that there exists a ∈ R such that f ≡ 0 on (−∞, a), f > 0 on (a, +∞), +f ′(a+) > 0, and that supp(n0) = [s−, s+], with s− < a < s+. Then, +• If there exists a unique xM ∈ (s−, a) such that r(xM) = +max +x∈[s−,a] r(x), and f ′′(xM) < 0, then ρ converges +to r(xM), and n(t, ·) +⇀ +t→+∞ r(xM)δxM . +• If r|[s−,a] reaches its maximum at a (and only at a), then ρ converges to r(a), and n(t, ·) +⇀ +t→+∞ r(a)δa. +Proof. +• +1. Let us denote O1 := (s−, a), O2 := (a, +∞), which satisfy the hypothesis of Proposition 3, +according to Lemma 3. By Proposition 5, R1 converges to r(xM) and R2 converges to r(xM) − +f ′(xM). +2. From Proposition 4, ρ and ρ1 converge to r(xM), and ρ2 converges to 0. +3. Let K ⊂ [s−, a] be a compact set that does not contain xM. Thanks to the semi-explicit expression +(7), and using the fact that f ′(x) = 0 and Y (t, x) = x for all x ∈ K and all t ≥ 0, +n(t, x) = n0(x)e +� t +0 r(x)−ρ(s)ds ≤ n0(x)e +� t +0 rK−ρ(s)ds, +with rK := max +x∈K r(x) < r(xM). +Thus, +� +K +n(t, x)dx ≤ +� +K +n0(x)dxe +� t +0 rM−ρ(s)ds, +which converges to 0, since rM − ρ(s) is negative for any s large enough. This proves the result +thanks to Proposition 1. +• +1. Here we have to make a slightly more subtle choice of subsets than usual. Let ε > 0, and let us +denote Oε +1 := (s−, a − 2ε), Oε +2 := (a − 2ε, a − ε), Oε +3 := (a − ε, a), O4 := (a, +∞). We easily check +that these four sets satisfy the hypotheses of Proposition 3. Moreover, since f ≡ 0 on [s−, a] for +all i ∈ {1, 2, 3}, +Rε +i (t) = +� +Oε +i r(x)er(x)tdx +� +Oε +i er(x)tdx +. +Thus, for all t ≥ 0, i ∈ {1, 2, 3}, +min +x∈O +ε +i +r(x) ≤ Rε +i (t) ≤ max +x∈O +ε +i +r(x). +In particular, +Rε +1 ≤ +max +x∈[s−,a−2ε] r(x) +and +Rε +3 ≥ +min +x∈[a−ε,a] r(x). +Finally, Proposition 5 shows that R4 converges to r(a) − f ′(a+). +29 + +2. Since r reaches its unique maximum at a, for any ε small enough, we get +Rε +3 > Rε +1 +and +Rε +3 > +lim +t→+∞ R4(t). +Thus, according to Proposition 4, ρε +1 and ρ4 converge to 0, for all ε > 0. The choice of ε being +arbitrary, it also proves that ρε +2 converges to 0. Thus, ρ = ρε +3, and ρ = ρε +3, for all ε > 0. Since for +all t ≥ 0 +min +x∈[a−ε,a] r(x) ≤ Rε +3(t) ≤ r(a), +we prove that ρ converges to r(a) by making ε tend to 0. +3. We have proved that t �→ +� a +s− n(t, x)dx converges to r(a), that t �→ +� +∞ +a +n(t, x)dx converges to 0 +and that for all ε > 0, +� a−ε +s− +n(t, x)dx converges to 0. The hypotheses of Proposition 1 are therefore +met, which concludes the proof. +Note that the methods of Propositions 10 and 11 can be coupled to treat more complex cases, where, for +example, f ≡ 0 on several disjoint segments. +5 +Some results in higher dimensions +As seen in the previous sections, our entire method is based on the computation of the limit of the Ri +functions defined in Section 3. Unfortunately, these computations seem out of reach in the multidimensional +case Rd, d ≥ 2. +In this section, we nevertheless address the question of the possible convergence to smooth or singular +measures in higher dimensions in some specific simple cases. We first analyse how the solution support +evolves over time. This allows us to conclude that that the solution converges to a Dirac mass in the case of +a unique equilibrium which is asymptotically stable for the ODE ˙u = f(u), and provide hypotheses under +which the solution cannot converge to a smooth function. We then characterise which stationary measures +may or may not be limits for solutions of (1), before providing a criterion ensuring the existence of continuous +stationary solutions. +5.1 +Limit support +Definition 1 (Limit support). We define the limit support of n as: +σ∞ = +� +t≥0 +� +s≥t +supp (n(s, ·)). +Recalling the semi-explicit expression (7), +n(t, x) = n0(Y (t, x))e +� t +0 (r−∇·f)(Y (s,x))−ρ(s)ds, +and that for all t ≥ 0, supp +� +n0(Y (t, ·)) +� += X(t, supp(n0)), we get +σ∞ = +� +t≥0 +� +s≥t +supp (n0 (Y (s, ·))) = +� +t≥0 +� +s≥t +X (s, supp (n0)). +(28) +In the cases where we are able to determine the latter set, we gather information about possible limits for n. +Lemma 5. If the limit support of n is of measure zero, then n does not converge (weakly) to a non-zero +function in L1(Rd). +30 + +Proof. Let us argue by contradiction. By denoting ν the Lebesgue measure, let us assume that ν(σ∞) = 0, +and that n converges weakly to n ∈ L1(Rd), n ̸≡ 0. Since lim sup +t→+∞ supp (n (t, ·)) = � +t≥0 +� +s≥t +supp (n(s, ·)) ⊂ σ∞, +lim sup +t→+∞ ν (supp(n(t, ·))) ≤ ν +� +lim sup +t→+∞ supp (n(t, ·)) +� +≤ ν(σ∞) = 0, +which contradicts the initial hypothesis. +Proposition 12. Let us assume that f has a unique root, denoted x, which is globally asymptotically +stable for the ODE ˙u = f(u) over Rd, and that the set � +t≥0 +X(t, supp(n0)) is bounded. Then, n(t, ·) +⇀ +t→+∞ +r(x)δx, and ρ converges to r(x). +Proof. Since the support of n0 is compact, and x is globally asymptotically stable, we easily check, according +to (28), that σ∞ = {x}. By Lemma 1, it is hence enough to prove that ρ converges to r(x). As seen in the +proof of Lemma 3.2, ρ satisfies, for all t ≥ 0, +˙ρ(t) = +� +Rd +� +r(x) − ρ(t) +� +n(t, x)dx = +� +supp(n(t,·)) +� +r(x) − ρ(t) +� +n(t, x)dx. +Let ε > 0. Since σ∞ = {x} is the intersection of compact decreasing sets, there exists Tε > 0 such that, for +all t ≥ Tε, supp(n(t, ·)) ⊂ B(x, ε). Thus, by denoting +rε +m := +min +x∈B(x,ε) r(x) +and +rε +M := +max +x∈B(x,ε) r(x), +we get, for all t ≥ Tε, +(rε +m − ρ(t))ρ(t) ≤ ˙ρ(t) ≤ (rε +M − ρ(t))ρ(t), +which ensures that +lim inf +t→+∞ ρ(t) ≥ rε +m +and +lim sup +t→+∞ ρ(t) ≤ rε +M. +Since these inequalities hold for any ε > 0, and rε +m and rε +M both converge to r(x) when ε goes to 0, it +concludes the proof. +Because of the diversity of possible behaviours of ODE systems, it is difficult to compute the limit support +for a given f, unless very strong assumptions are made about the ODE ˙u = f(u). This is what we do in the +following proposition, motivated by a family of ODE systems commonly used in systems biology. +We say that the two-dimensional system +� +˙x1 = f1(x1, x2) +˙x2 = f2(x1, x2) +(29) +is competitive if ∂2f1 ≤ 0 and ∂1f2 ≤ 0, and cooperative if ∂2f1 ≥ 0 and ∂1f2 ≥ 0. For instance, +such systems are commonly used to model the interactions between two proteins in the context of cell +differentiation [20, 37, 18, 26], and are known to have an interesting property: trajectories either go to +∞, +or converge [24], i.e. for all x ∈ Rd, +∥Y (t, x)∥ ̸−→ +t→+∞ +∞ +⇒ +t �→ Y (t, x) converges . +(30) +Note that if the ODE (29) is competitive (or cooperative), then the reverse ODE ˙u = −f(u) is cooperative +(or competitive). This motivates the hypothesis of the following proposition. Before giving its statement, +we recall that if x is a root of f, x is called a hyperbolic equilibrium if all the eigenvalues of Jac f(x) +have a non-zero real part, and is called a repellor if all these eigenvalues have a positive real part. Lastly, +we recall that the unstable set of x is defined by {x ∈ Rd : Y (t, x) −→ +t→+∞ x}. +31 + +Proposition 13. Let us assume that f has a finite number of roots, and is such that identity (30) holds. +Then, the limit support of n is included in the closure of the union of the unstable sets of the roots of f, i.e. +by denoting x1, ....xN the roots of f, +σ∞ ⊂ +� +1≤i≤N +� +x ∈ Rd : Y (t, x) −→ +t→+∞ xi +� +. +Moreover, if all the roots of f are hyperpolic points, and if none of them is a repellor, then the limit support +of n is of measure 0. In particular, n does not converge (weakly) to a function in L1. +Proof. The inclusion is clear: by hypothesis for all x ∈ Rd such that t �→ Y (t, x) does not converge, +t �→ ∥Y (t, x)∥ goes to +∞, and since the support of n0 is bounded, the points of the limit support are +necessary in the unstable set of one of the equilibria. The second part of the proposition is a consequence +of the stable manifold theorem [33], which ensures that the unstable set of an equilibrium which is not a +repellor is a smooth manifold of dimension at most d − 1, hence a set of measure zero. We conclude with +Lemma 5. +5.2 +Stationary solutions +In this subsection, we define the stationary solution in the weak sense, which allows to include measures. +As seen in the previous section, under appropriate hypotheses on f, the presence of a repellor is necessary +to hope for solutions which converge to smooth functions. In this section, we prove that, under appropriate +hypotheses, the presence of a repellor ensures the existence of smooth stationary solutions. +Definition 2 (Weak stationary solution). Let µ be a finite positive Radon measure. We say that µ is a +weak stationary solution of equation (1) if it satisfies +∀ϕ ∈ C1 +c +� +Rd� +, +� +Rd +� +f(x).∇ϕ(x) + (r(x) − µ(Rd))ϕ(x) +� +dµ(x) = 0. +(31) +Remark. If x is a root of f, let us note that r(x)δx is a weak stationary solution of (1). +The following proposition shows, as we might expect, that convergent solutions of (1) (in the weak sense) +necessarily converge to a weak stationary solution. +Proposition 14. Let us assume that r ∈ C0(Rd), and let n(t, ·) be a solution of (1) which converges in the +weak sense in the space of Radon measure. Then its limit is a weak stationary solution of equation (1). +Proof. We let µ be the limit of n(t, ·). Let us first prove that, under these conditions, ρ(t) = +� +Rd n(t, x)dx +converges when t goes to +∞. +Let us denote ψ(t) := +� +Rd r(x)n(t, x)dx, which is non-negative, according to the non-negativity of r and +n, and converges to ψ := +� +Rd r(x)dµ(x)dx, by definition of the weak convergence, and since r ∈ C0(Rd). Let +us assume that ψ > 0. Let ε ∈ (0, ψ). Since ψ converges to ψ, and since ρ satisfies the ODE +˙ρ(t) = ψ(t) − ρ(t)2, +there exists Tε > 0 such that for all t ≥ Tε, +ψ − ε − ρ(t)2 ≤ ˙ρ(t) ≤ ψ + ε − ρ(t)2. +In other words, ρ is a super-solution of ˙u = ψ − ε − u2, and a sub-solution of ˙u = ψ + ε − u2. Since the +solutions of these equations converge to +� +ψ − ε and +� +ψ + ε respectively, +lim inf +t→+∞ ρ(t) ≥ +� +ψ − ε +and +lim sup +t→+∞ ρ(t) ≤ +� +ψ + ε. +Since these inequalities hold for any ε ∈ (0, ψ), it proves that ρ indeed converges to +� +ψ. +32 + +If ψ = 0, we prove that lim sup ρ ≤ 0 with the same method, and the non-negativity of ψ ensures that +lim inf ρ ≥ 0. +Let ϕ ∈ C1 +c +� +Rd� +, and let us denote ρ := +lim +t→+∞ρ(t). We recall that if a differentiable function converges, +then its derivative is either divergent or converges to 0. Hence, since t �→ +� +Rd ϕ(x)n(t, x)dx converges (by +hypothesis), and +d +dt +� +Rd ϕ(x)n(t, x)dx = − +� +Rd ∇ · (f(x)n(t, x)) ϕ(x)dx +� +Rd (r(x) − ρ(t))ϕ(x)n(t, x)dx += + +� +Rd f(x).∇ϕ(x)n(t, x)dx + +� +Rd (r(x) − ρ(t)) ϕ(x)n(t, x)dx +−→ +t→+∞ +� +Rd f(x).∇ϕ(x) + (r(x) − ρ)ϕ(x)dµ(x), +the equality +� +Rd f(x).∇ϕ(x) + (r(x) − ρ)ϕ(x)dµ(x) = 0 +(32) +holds for any ϕ ∈ C1 +c(Rd). +It remains to prove that µ(Rd) = ρ. If ρ = 0, then the non-negativity of n and the definition of ρ lead to +µ = 0. Let us now assume that ρ > 0, and let ε > 0. Since µ is a finite measure, r ∈ C0(Rd), and owing to +the definition of ψ and ρ, there exists K ⊂ Rd a compact set such that +• µ(K) ≥ µ(Rd) − ε +• +� +K r(x)dµ(x)dx ≥ +� +Rd r(x)dµ(x)dx − ε = ρ2 − ε. +Let ϕK ∈ C1 +c(Rd) such that ϕK ≡ 1 on K, 0 ≤ ϕ ≤ 1 on Rd. Since ∇ϕK ≡ 0 on K, +���� +� +Rd f(x).∇ϕK(x)dµ(x) +���� ≤ ∥f.∇ϕK∥∞µ(Rd\K) ≤ ε∥f.∇ϕK∥∞. +Moreover, according to the choice of ϕK +� +Rd r(x)ϕK(x)dµ(x) ∈ [ρ2 − ε, ρ2], +and +� +Rd ϕK(x)dµ(x) ∈ [µ(Rd) − ε, µ(Rd)]. +Hence, injecting these inequalities in (32), we obtain +−Cε ≤ ρ(ρ − µ(Rd)) ≤ Cε +for some C ≥ 0. Since this equality holds for any ε, and ρ is positive, it proves that µ(Rd) = ρ. +Weak stationary solutions which are smooth enough (at least in C1(Rd)) are in fact stationary solutions +in the strong sense, as defined in the following lemma, and can be further characterised. +Lemma 6. Let n ∈ C1(Rd). Then, n is a weak stationary solution of (1) if and only if for all t ≥ 0, y ∈ Rd, +� +n(X(t, y)) = e +� t +0 ˜r(X(s,y))−ρ ds n(y) +� +Rd n(x)dx = ρ +. +Proof. First, let us note that, since n ∈ C1(Rd), one can integrate by parts in the expression (31) in order to +prove that n is a weak stationary solution if and only if for any ϕ ∈ C1 +c(Rd), +� +Rd (−∇ · (f(x)n(x)) + (r(x) − ρ) n(x)) ϕ(x)dx = 0, +33 + +with ρ = +� +Rd n(x)dx, which means that n is a weak stationary solution if and only if it is a stationary solution +in the strong sense, i.e +−∇ · (f(x)n(x)) + (r(x) − ρ) n(x) = 0 +for all x ∈ Rd. +The result follows, since for any y ∈ Rd +d +dt +� +n(X(t, y))e− +� t +0 ˜r(X(s,y))−ρ ds� += +� +f(X(t, y)).∇n(X(t, y)) − (˜r(X(t, y) − ρ) n(X(t, y)) +� +e− +� t +0 ˜r(X(s,y))−ρ ds += − +� +− ∇· (f(X(t, y))n(X(t, y))) + (r(X(t, y)) − ρ)n(X(t, y)) +� +e− +� t +0 ˜r(X(s,y))−ρ ds = 0. +Lemma 6 allows us to conclude that in the case where the ODE ˙u = f(u) has a repellor with a bounded +unstable set, there exists a smooth stationary solution for (1). +Corollary 15. Let xu ∈ Rd be a repellor point for the ODE ˙x = f(x), and let us assume that +n(x) := ˜r(xu) +α +e +� +∞ +0 +˜r(Y (s,x))−˜r(xu)ds1B(x) +is well-defined, and that n ∈ C1(B)∩L1(B), where B = {x ∈ Rd : Y (t, x) −→ +t→+∞ xu} is the unstable set of xu. +Then, n is a C1 stationary solution. +Proof. For all y ∈ R, t ≥ 0, +n(X(t, y)) = ˜r(xu) +α +e +� +∞ +0 +˜r(X(t−s,x))−˜r(xu)ds = ˜r(xu) +α +e +� t +−∞ ˜r(X(s,y))−˜r(xu)ds, +with the change of variable s′ = t − s and +n(y) = ˜r(xu) +α +e +� 0 +−∞ ˜r(X(s,y))−˜r(xu)ds, +with the change of variable s′ = −s. Thus, the equality of Lemma 6 holds, which concludes the proof. +References +[1] Matthieu Alfaro, Henri Berestycki, and Ga¨el Raoul. The effect of climate shift on a species submitted +to dispersion, evolution, growth, and nonlocal competition. SIAM Journal on Mathematical Analysis, +49(1):562–596, 2017. +[2] Lu´ıs Almeida, Patrizia Bagnerini, Giulia Fabrini, Barry D Hughes, and Tommaso Lorenzi. Evolution of +cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype- +structured model. ESAIM: Mathematical Modelling and Numerical Analysis, 53(4):1157–1190, 2019. +[3] Guy Barles, Sepideh Mirrahimi, and Benoˆıt Perthame. Concentration in lotka-volterra parabolic or +integral equations: a general convergence result. Methods and Applications of Analysis, 16(3):321–340, +2009. +[4] Olivier Bonnefon, J´erˆome Coville, and Guillaume Legendre. Concentration phenomenon in some non- +local equation. arXiv preprint arXiv:1510.01971, 2015. +[5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoˆıt Perthame, Ga¨el Raoul, and +Rapha¨el Voituriez. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave +acceleration. Comptes Rendus Mathematique, 350(15-16):761–766, 2012. +34 + +[6] `Angel Calsina and S´ılvia Cuadrado. +Stationary solutions of a selection mutation model: The pure +mutation case. Mathematical Models and Methods in Applied Sciences, 15(07):1091–1117, 2005. +[7] `Angel Calsina, S´ılvia Cuadrado, Laurent Desvillettes, and Ga¨el Raoul. Asymptotics of steady states of +a selection–mutation equation for small mutation rate. Proceedings of the Royal Society of Edinburgh +Section A: Mathematics, 143(6):1123–1146, 2013. +[8] Nicolas Champagnat, R´egis Ferri`ere, and Sylvie M´el´eard. Unifying evolutionary dynamics: from indi- +vidual stochastic processes to macroscopic models. Theoretical population biology, 69(3):297–321, 2006. +[9] Nicolas Champagnat, R´egis Ferri`ere, and Sylvie M´el´eard. From individual stochastic processes to macro- +scopic models in adaptive evolution. Stochastic Models, 24(sup1):2–44, 2008. +[10] Rebecca H Chisholm, Tommaso Lorenzi, and Alexander Lorz. Effects of an advection term in nonlocal +lotka–volterra equations. Communications in mathematical sciences, 14(4):1181–1188, 2016. +[11] Rebecca H Chisholm, Tommaso Lorenzi, Alexander Lorz, Annette K Larsen, Lu´ıs Neves de Almeida, +Alexandre Escargueil, and Jean Clairambault. Emergence of drug tolerance in cancer cell populations: +an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation. Cancer +research, 75(6):930–939, 2015. +[12] Jerome Coville. Convergence to equilibrium for positive solutions of some mutation-selection model. +arXiv preprint arXiv:1308.6471, 2013. +[13] Laurent Desvillettes, Pierre Emmanuel Jabin, St´ephane Mischler, and Ga¨el Raoul. On selection dy- +namics for continuous structured populations. Communications in Mathematical Sciences, 6(3):729–747, +2008. +[14] Ulf Dieckmann and Richard Law. The dynamical theory of coevolution: a derivation from stochastic +ecological processes. Journal of mathematical biology, 34(5):579–612, 1996. +[15] Odo Diekmann, Pierre-Emanuel Jabin, St´ephane Mischler, and Benoıt Perthame. The dynamics of +adaptation: an illuminating example and a hamilton–jacobi approach. Theoretical population biology, +67(4):257–271, 2005. +[16] Ronald J DiPerna and Pierre-Louis Lions. Ordinary differential equations, transport theory and sobolev +spaces. Inventiones mathematicae, 98(3):511–547, 1989. +[17] Frank Ernesto Alvarez and Jules Guilberteau. Particle method for adaptive dynamics equations. In +preparation. +[18] Timothy S Gardner, Charles R Cantor, and James J Collins. Construction of a genetic toggle switch in +escherichia coli. Nature, 403(6767):339–342, 2000. +[19] Stefan AH Geritz, Eva Kisdi, Johan AJ Metz, et al. Evolutionarily singular strategies and the adaptive +growth and branching of the evolutionary tree. Evolutionary ecology, 12(1):35–57, 1998. +[20] Ra´ul Guantes and Juan F Poyatos. +Multistable decision switches for flexible control of epigenetic +differentiation. PLoS computational biology, 4(11):e1000235, 2008. +[21] Mats Gyllenberg and G´eza Mesz´ena. On the impossibility of coexistence of infinitely many strategies. +Journal of mathematical biology, 50(2):133–160, 2005. +[22] TG Hallam and CE Clark. Non-autonomous logistic equations as models of populations in a deteriorating +environment. Journal of Theoretical Biology, 93(2):303–311, 1981. +[23] WGS Hines. Evolutionary stable strategies: a review of basic theory. Theoretical Population Biology, +31(2):195–272, 1987. +35 + +[24] Morris W Hirsch. Systems of differential equations which are competitive or cooperative: I. limit sets. +SIAM Journal on Mathematical Analysis, 13(2):167–179, 1982. +[25] Pierre-Emmanuel Jabin and Ga¨el Raoul. On selection dynamics for competitive interactions. Journal +of mathematical biology, 63(3):493–517, 2011. +[26] Dongya Jia, Mohit Kumar Jolly, William Harrison, Marcelo Boareto, Eshel Ben-Jacob, and Herbert +Levine. Operating principles of tristable circuits regulating cellular differentiation. Physical biology, +14(3):035007, 2017. +[27] Tommaso Lorenzi, Rebecca H Chisholm, Laurent Desvillettes, and Barry D Hughes. Dissecting the dy- +namics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments. +Journal of theoretical biology, 386:166–176, 2015. +[28] Tommaso Lorenzi, Benoˆıt Perthame, and Xinran Ruan. Invasion fronts and adaptive dynamics in a +model for the growth of cell populations with heterogeneous mobility. European Journal of Applied +Mathematics, 33(4):766–783, 2022. +[29] Tommaso Lorenzi and Camille Pouchol. Asymptotic analysis of selection-mutation models in the pres- +ence of multiple fitness peaks. Nonlinearity, 33(11):5791, 2020. +[30] Alexander Lorz, Tommaso Lorenzi, Michael E Hochberg, Jean Clairambault, and Benoˆıt Perthame. Pop- +ulational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM: +Mathematical Modelling and Numerical Analysis, 47(2):377–399, 2013. +[31] Johan AJ Metz, Stefan AH Geritz, G´eza Mesz´ena, Frans JA Jacobs, and Joost S Van Heerwaarden. +Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. 1995. +[32] Philippe Michel, St´ephane Mischler, and Benoˆıt Perthame. General relative entropy inequality: an +illustration on growth models. Journal de math´ematiques pures et appliqu´ees, 84(9):1235–1260, 2005. +[33] Lawrence Perko. Differential equations and dynamical systems, volume 7. Springer Science & Business +Media, 2013. +[34] Benoˆıt Perthame. Transport equations in biology. Springer Science & Business Media, 2006. +[35] Benoˆıt Perthame and Guy Barles. +Dirac concentrations in lotka-volterra parabolic pdes. +Indiana +University Mathematics Journal, pages 3275–3301, 2008. +[36] Camille Pouchol and Emmanuel Tr´elat. +Global stability with selection in integro-differential lotka- +volterra systems modelling trait-structured populations. Journal of Biological Dynamics, 12(1):872–893, +2018. +[37] Ren´e Thomas. Laws for the dynamics of regulatory networks. International Journal of Developmental +Biology, 42(3):479–485, 2002. +[38] John J Tyson and Bela Novak. A dynamical paradigm for molecular cell biology. Trends in Cell Biology, +30(7):504–515, 2020. +[39] Roderick Wong. Asymptotic approximations of integrals. SIAM, 2001. +[40] Jingyu Zhang, Xiao-Jun Tian, Hang Zhang, Yue Teng, Ruoyan Li, Fan Bai, Subbiah Elankumaran, and +Jianhua Xing. Tgf-β–induced epithelial-to-mesenchymal transition proceeds through stepwise activation +of multiple feedback loops. Science signaling, 7(345):ra91–ra91, 2014. +36 + diff --git a/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/load_file.txt b/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a321dd4cf179e8737676c6b3eb1f7504f94eb1e0 --- /dev/null +++ b/QdE0T4oBgHgl3EQfkQFh/content/tmp_files/load_file.txt @@ -0,0 +1,1055 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf,len=1054 +page_content='Long-time behaviour of an advection-selection equation Jules Guilberteau∗, Camille Pouchol† and Nastassia Pouradier Duteil‡ Abstract We study the long-time behaviour of the advection-selection equation ∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t)) n(t, x), ρ(t) = � Rd n(t, x)dx t ≥ 0, x ∈ Rd, with an initial condition n(0, ·) = n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this case, x �→ n(t, x) represents the density of the population characterised by a phenotypic trait x, the advection term ‘∇ · (f(x)n(t, x))’ a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term ‘(r(x) − ρ(t)) n(t, x)’ the growth of the population, which is of logistic type through the total population size ρ(t) = � Rd n(t, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the one-dimensional case x ∈ R, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Depending on the parameters n0, f and r, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the L1-function which is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Advection-selection equation We consider the asymptotic behaviour of the advection-selection equation � � � � � ∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t)) n(t, x), t ≥ 0, x ∈ Rd ρ(t) = � Rd n(t, x)dx, t ≥ 0 n(0, x) = n0(x), x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (1) This type of model typically comes up in the field of adaptive dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The aim is to understand how, among heterogeneous populations of individuals structured by a so-called continuous trait or phenotype x, the distribution of the density x �→ n(t, x) evolves over time, and which phenotypes prevail in large times t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the model above (1), the partial differential equation (PDE) takes into account advection via the term ∇ · (f(x)n(t, x)), whereby individuals follow the flow associated with f, growth via the term (r(x) − ρ(t))n(t, x), which is of logistic type through the total population size ρ(t) = � Rd n(t, x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The literature concerning so-called phenotype-structured partial differential equations for adaptive dy- namics is abundant [1, 5, 3, 7, 6, 10, 12, 15, 28, 30, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' These models usually take into account selection, which favors individuals with the most adapted traits in terms of growth, and mutations, which ∗Sorbonne Universit´e, CNRS, Universit´e Paris Cit´e, Inria, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' jules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='guilberteau@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='fr †Universit´e Paris Cit´e, FP2M, CNRS FR 2036, MAP5 UMR 8145, F-75006 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' camille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='pouchol@u-paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='fr ‡Sorbonne Universit´e, Inria, Universit´e Paris Cit´e, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France nastassia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='pouradier duteil@sorbonne-universite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='fr 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='02470v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='AP] 6 Jan 2023 induce a slight phenotypic change upon reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Mutation is often assumed to be rare and small compared to selection, [14, 19, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Models with no mutation at all have also been the subject of several studies [2, 13, 21, 25, 29, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' One way to analyse how the population adapts is to study the long-time behaviour for solutions of such PDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, determining if the population becomes monomorphic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' the solution concentrates around a certain trait, called Evolutionary Stable Strategy (ESS) [23]), or if phenotypic diversity is preserved is a fundamental question when studying such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Broadly speaking, it has been shown that selection leads to concentration (around a finite number of phenotypic traits), while mutations, on the contrary, tend to regularise solutions, and, possibly, their limits [4, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' However, less emphasis has been put on studying the effect of advection, except for the recent few examples [10, 27, 11] where most results are of numerical nature, or assume a very specific form of the functions r and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Yet, considering advection is relevant in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' From the phenomenological point of view, it may represent how the environment drives the individuals towards specific regions, as opposed to more random mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' It is also the rigorous way to model phenotype changes that are intrinsic to the individual, mediated by an ordinary differential equation (ODE) of the form ˙x(t) = f(x(t)), (2) where x(t) ∈ Rd denotes the phenotypic trait of the individual at time t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As is well known, the PDE for the density of individuals corresponding to the sole model (2) is indeed the advection equation ∂tn(t, x) + ∇ · (f(x)n(t, x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Our original motivation is that of cell differentiation, for which very refined ODE models have been developed in systems biology (see for instance [20, 37, 38, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The goal of the present article is to investigate the combined effect of selection and advection, assuming that mutations are absent or sufficiently small to be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We hence study the long-time behaviour of the PDE (1), where n0 is the initial population distribution, and ρ(t) is the size of the population at time t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The equation incorporates advection with the flow f of the corresponding ODE, and selection (or growth) through the non-linear and non-local term (r(x) − ρ(t))n(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Here, r(x) − ρ(t) can be interpreted as the fitness of individuals with trait x inside the environment created by the total population, where the individuals are in a blind competition with all the other ones, regardless of their phenotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We note that such models can rigorously be derived from stochastic individual based-models, in the limit of large populations [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the absence of differentiation (f ≡ 0), the long-time behaviour of this model has been studied in detail by Benoˆıt Perthame [34], Tommaso Lorenzi and Camille Pouchol [29], and it has been proved that, in general, solutions typically concentrate onto a single trait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This result is rather intuitive, since this model does not take mutations into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Solutions of the advection equation alone are also known to converge to weighted Dirac masses located at the roots of f which are asymptotically stable for the ODE (2) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On the contrary, when considering both selection and advection as in equation (1), the long-time behaviour is not known, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Intuitively, two antagonistic effects will compete: advection will push the solution towards the asymptotically stable equilibria of ODE (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' growth will push the solution towards regions where r is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' When coupling these two phenomena, our aim is to uncover whether the solution of (1) converges to a weighted Dirac mass, or if it converges to a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We show that both phenomena can occur, depending on the parameters n0, f and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Perhaps surprisingly, the model (1) features convergence to smooth functions even in the absence of terms modelling mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Determining which parameters lead to convergence to a continuous function seems rather intricate in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, this problem cannot be addressed with traditional entropy methods as developed in [32], since in the absence of mutations, there is no decrease of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 Main results In this paper, we thus develop a different strategy allowing to reduce this problem to the study of parameter- dependent integrals, which is mainly applied to the one-dimensional case (x ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this case, we elucidate 2 the asymptotic behaviour for a large class of parameter values, and we show that there exist many different subcases depending on the number of zeros of the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' A general statement encompassing all our results is hence rather convoluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In order to illustrate our main results, we here focus on a few example cases which highlight the main two parameter regimes encountered for the asymptotic behaviour of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that the parameter functions f, n0 and r are smooth enough, that f has a unique root (that we denote xs), and that f ′(xs) < 0 (which means that xs is asymptotically stable for ODE (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, ρ converges to r(xs), and n converges to a weighted Dirac mass at xs, when t goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, in the presence of a single asymptotically stable equilibrium point for ODE (2), the solution of PDE (1) converges to a Dirac mass at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In other words, the selection term is dominated by the advection term, which determines the point in which the solution concentrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As soon as f has at least two roots, the situation is much more complex and solutions may converge to L1 functions, as illustrated in Figure 1 and exposed in the following proposition: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that the functions f, n0 and r are smooth enough, that f has exactly two roots (that we denote xu and xs, with xu < xs), such that f ′(xu) > 0 and f ′(xs) < 0, which means that the points xu and xs are respectively asymptotically unstable and asymptotically stable for the ODE (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, let us assume that n0 has its support in [xu, xs], and that n0(xu) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, the following alternative holds: If r(xs) > r(xu) − f ′(xu), n converges to a weighted Dirac mass at xs, and ρ converges to r(xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If r(xs) < r(xu) − f ′(xu), n converges to a function in L1(xu, xs), and ρ converges to r(xu) − f ′(xu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This proposition can be interpreted as follows: since f is positive on (xu, xs), the advection term drives the solution towards xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On the other hand, since xu is an equilibrium, albeit unstable, it acts as a counterweight by controlling the speed of the transition towards xs in the neighbourhood of xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, in the case where r(xu) − f ′(xu) is large enough (r(xu) − f ′(xu) > r(xs)), the growth rate around xu is large enough to compensate for the advection term, leading to the convergence of n to a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the other case, the advection term is dominant, and n converges to a weighted Dirac mass at xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(xu) = 0, the toggle value between the two regimes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' the convergence to a smooth function or to a Dirac mass) changes, depending on how n0 vanishes at xu, and other limit functions can be reached: the complete result is detailed in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The method of analysis proposed in this article allows in fact to solve this problem for any function f with a finite number of roots, as detailed in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The case where f is equal to zero on a whole interval can also be studied with our method, as highlighted by Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='3 Discussion Open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Some limit cases of the problem remain unclear: we do not deal with the case of non- hyperbolic equilibria, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x ∈ R which satisfy f(x) = f ′(x) = 0, and we are not able to determine what happens in the case where several carrying capacities, as defined in Section 3, converge to the same maximum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This last case might lead to other asymptotic behaviours, such as convergence to a sum of weighted Dirac masses, or a sum of weighted Dirac masses and L1-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lastly, we did not manage to elucidate the equality cases (of the form r(xs) = r(xu) − f ′(xu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Furthermore, even if the framework introduced in Section 3 could theoretically be applied in any dimen- sion, computing the limits of the carrying capacities seems out of reach in the multidimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As shown by the semi-explicit expression introduced in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1, the behaviour of n is closely linked to that of the solutions of ODE ˙x = f(x), which suggests that other asymptotic behaviours, such as convergence to a limit cycle, or chaotic behaviours (if the dimension is greater than or equal to 3) might occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' These behaviours may be excluded by making specific assumptions regarding the function f, for example by requiring in the 2D case that ODE ˙x = f(x) be competitive or cooperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Additionally if the roots of f are hyperbolic and none of them is a repellor, then n cannot converge to a L1-function (Proposition 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Nevertheless, the question of the asymptotic limit of n in this case remains open, and might be, in the presence of a saddle point, a singular measure which is not a sum of weighted Dirac masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This situation is commonplace for some applications, since toggle switches used to model cell differentiation phenomena are usually competitive or cooperative ODE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3 Figure 1: The two possible regimes of convergence stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In both cases, we have chosen f(x) = x(1 − x), n0 ≡ 6, and we work on the segment (0, 1) (hence xu = 0, xs = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The three figures above (in red) show the time evolution of the solution in the case where r(x) = 6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='5x (and thus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='5 = r(1) > r(0)−f ′(0) = 5), which implies, according to Proposition 2, that the solution converges to a weighted Dirac mass at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The three figures below (in blue) show the time evolution of the solution in the case where r(x) = 6 − 4x, (and thus 2 = r(1) < r(0) − f ′(0) = 5), which implies that the solution converges to a continuous function in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The black dashed curve represents this limit function, which can explicitly be computed (see Proposition 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' A natural generalisation for the model would be to model mutations, either by means of a Laplacian term or an integral term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Because of their smoothing effect, convergence to Dirac masses will typically be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The method developed in this paper does not seem to handle such cases well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' However, it is an interesting perspective to tackle the asymptotic behaviour with entropy methods when mutations are added [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' From the numerical point of view, we have proved that the solution of this equation could be approximated with a particle method, with which we obtained the plots of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The details of the scheme, and the proof of its convergence will be published in a forthcoming article [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This paper is organised as follows: Section 2 introduces the measure-theoretic framework in which convergence is considered, and includes several important reminders regarding ODE theory which will be used throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Section 3 details the method used to determine the asymptotic behaviour of (1), and Section 4 corresponds to a direct application of this method to several examples in the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lastly, section 5 presents two results in higher dimension which allow to determine, in some specific cases, if some initial solution can lead to a convergence to a smooth function or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4 T=1 16 14 12 10 - 8 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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='02 Framework and reminders We consider the asymptotic behaviour of the integro-differential PDE � � � � � ∂tn(t, x) + ∇ · (f(x)n(t, x)) = (r(x) − ρ(t))n(t, x), t ≥ 0, x ∈ Rd ρ(t) = � Rd n(t, x)dx, t ≥ 0 n(0, x) = n0(x), x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (1) All along the article, we make the following regularity hypotheses f is Lipschitz-continuous, and is in C2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' r is positive, is in L1(Rd) ∩ C1(Rd), and goes to zero when ∥x∥ goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us note that these hypotheses imply that r is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' n0 is in C1 c(Rd) (the space of C1 functions with a compact support), is non-negative and is not the zero function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Whenever possible, we will indicate whether these hypotheses can be weakened for a given specific result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If not specified, it will be assumed that these three hypotheses hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' From the modelling point of view, they can be justified as follows: n0 denoting the initial density, it is reasonable to consider that a bounded range of phenotypic traits is initially represented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' the hypothesis on r at +∞ is made in order to prevent an unlikely proliferation of individuals with more and more extreme (∥x∥ → +∞) phenotypic traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Under the above hypotheses, we can prove that there exists a unique solution n ∈ C � R+, L1(R) � for this Cauchy problem by coupling the well-known method of characteristics for the advection equation [16] with the method applied in [34] for the case f ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We do not elaborate further here on the issue of existence and uniqueness, that will be addressed in a more general framework in an upcoming article [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since we are concerned with the long-time behaviour of the PDE (1) and we expect to obtain convergence either to Dirac masses or to regular functions, the space of Radon measures is a natural setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We start with a few usual reminders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 The space of Radon measures We recall that the space of finite Radon measures can be identified with the topological dual space of Cc(Rd), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' the space of continuous functions on Rd with a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, we say that a sequence of finite Radon measures (µk)k∈N weakly converges to a finite Radon measure µ (denoted uk ⇀ µ) if ∀ϕ ∈ Cc(Rd), � Rd ϕ(x)dµk(x) −→ k→+∞ � Rd ϕ(x)dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this article, we will be confronted mainly with convergence to Dirac masses or to L1 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' It is clear that the convergence in L1 to a certain function implies the weak convergence to this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The following standard lemma provides a sufficient condition to prove the weak convergence to a single Dirac mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For completeness, we provide a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let u : R+ × Rd → R be a non-negative mapping such that u(t, ·) ∈ L1(Rd) for all t ≥ 0, and u(t, ·) is compactly supported, uniformly in t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We assume that there exists x ∈ Rd, such that for all compact set Kx which does not contain x, � Kx u(t, x)dx −→ t→+∞ 0, and that there exists Vx a compact neighbourhood of x and C ∈ R such that � Vx u(t, x)dx −→ t→+∞ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, u(t, ·) ⇀ t→+∞ Cδx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ϕ ∈ Cc(Rd), and let K be a compact set such that, for all t ≥ 0, supp(u(t, ·)) ∪ Vx ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, ���� � Rd ϕ(x)u(t, x)dx − Cϕ(x) ���� = ���� � K ϕ(x)u(t, x)dx − � K ϕ(x)u(t, x)dx + � K ϕ(x)u(t, x)dx − Cϕ(x) ���� ≤ � K |ϕ(x) − ϕ(x)|u(t, x)dx + |ϕ(x)| ���� � K u(t, x)dx − C ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 5 The second term tends to 0 since K contains Vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' It remains to prove that t �→ � Rd |ϕ(x) − ϕ(x)|u(t, x)dx converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ϕ is continuous, there exists Bx a neighbourhood of x, which can be chosen as a subset of Vx, such that |ϕ(x) − ϕ(x)| ≤ ε, for all x ∈ Bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all t ≥ 0, � K |ϕ(x) − ϕ(x)|u(t, x)dx = � K\\Bx |ϕ(x) − ϕ(x)|u(t, x)dx + � Bx |ϕ(x) − ϕ(x)|u(t, x)dx ≤ 2∥ϕ∥∞ � K\\Bx u(t, x)dx + ε � Bx u(t, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This concludes the proof, since t �→ � K\\Vx u(t, x)dx converges to zero and for any t large enough, � Bx u(t, x)dx ≤ � Vx u(t, x)dx ≤ C + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 General statement regarding the characteristics curves We are led to consider the characteristics curves associated with the advection term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this section, we introduce some notations and state some classical results from ODE theory, that will prove to be useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since f is assumed to be Lipschitz-continuous, the global Cauchy-Lipschitz theorem ensures the global existence on R+ and the uniqueness of the characteristic curves related to f defined for all y ∈ Rd as the solution to the ODE � ˙X(t, y) = f(X(t, y)) t ≥ 0 X(0, y) = y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (3) It is well-known that for all t ≥ 0, y �→ X(t, y) is a C1-diffeomorphism between Rd and itself [16], and that the inverse function of X(t, ·), that we denote x �→ Y (t, x), is the unique solution of � ˙Y (t, x) = −f(Y (t, x)) t ≥ 0 Y (0, x) = x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (4) Moreover, Liouville’s formula states that for all t ≥ 0 and y ∈ Rd, det (JacyX(t, y)) = e � t 0 ∇·f(X(s,y))ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (5) It follows from the uniqueness of solutions to (3) that for all 0 ≤ s ≤ t, X(s, Y (t, x)) = Y (t − s, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (6) Specific results in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us note that the behaviour of the characteristic curves is particularly simple in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Indeed, an elementary ODE analysis shows that for all x, y ∈ R, t �→ X(t, y) and t �→ Y (t, x) are monotonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This implies that these characteristic curves either converge to a root of f, or go to ±∞ as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' More precisely, if f has a finite number of roots, then for all y ∈ R such that f(y) > 0, t �→ X(t, y) converges to the closest root of f which is greater than y, or to +∞ if y is greater than the greatest root of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Similarly, for all y ∈ R such that f(y) < 0, t �→ X(t, y) converges to the closest root of f which is lesser y, and to −∞ if y is lesser the smallest root of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, if each of these roots are hyperbolic equilibrium points for the ODE ˙x = f(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' if f ′(x) ̸= 0 for all x root of f, then a given root of f is either asymptotically unstable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' f ′(x) > 0), which implies that its basin of attraction is limited to itself, or asymptotically stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' f ′(x) < 0), which implies that its basin of attraction in an open interval containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lastly, let us recall that under these hypotheses, the convergence to an asymptotically stable point happens with an exponential speed, which means that for all y ∈ R, x root of f, X(t, y) −→ t→+∞ x ⇒ ∃δy > 0 : X(t, y) − x = O t→+∞(e−δy t) Since the reverse characteristic curves satisfy (4), the same results hold for Y (t, x), provided that we replace f by −f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In brief, the asymptotically stable equilibria become unstable for the reverse ODE, and vice versa, and if t �→ X(t, y) is increasing (respectively decreasing), then t �→ Y (t, x) is decreasing (respectively increasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 6 3 Resolution method The method of resolution to determine the asymptotic behaviour of n that we propose here is based on the following two propositions, which are developed in the following two subsections, respectively: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all t ≥ 0, x ∈ Rd, we can express n(t, x) as a function which only depends on t, x, on the functions n0, f and r, on the inverse characteristic curves Y (t, x), and on the population size ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Therefore, knowing the limit of Y (t, x) and ρ(t) as t goes to +∞ is enough to understand the long-time behaviour of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The population size ρ is the solution of a non-autonomous ODE, and its long-time behaviour may be inferred from the limit of some parameter-dependent integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Combining these two propositions allows us to reduce the study of the asymptotic behaviour of n to that of parameter-dependent integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Semi-explicit expression of the solution According to the definition of the characteristic curves (3), for all t ≥ 0 and all y ∈ Rd, d dtn(t, X(t, y)) = � r(X(t, y)) − ∇ · f(X(t, y)) − ρ(t) � n(t, X(t, y)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' n(t, X(t, y)) = e � t 0 (r(X(s,y))−∇·f(X(s,y))−ρ(s))dsn0(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Replacing y by Y (t, x) in this last expression, we get a semi-explicit expression for n, which is expressed as a function of t, x and ρ: n(t, x) = n0(Y (t, x))e � t 0 ((r−∇·f)(X(s,Y (t,x)))−ρ(s))ds = n0(Y (t, x))e � t 0 ((r−∇·f)(Y (s,x))−ρ(s))ds, (7) The second equality holds according to equality (6) and the change of variable s′ = t − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Beyond the non-negativity of n, this semi-explicit expression shows that determining the asymptotic behaviour of ρ and Y is enough to uncover that of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the following section, we show that ρ is the solution of a non-autonomous ODE, and that its asymptotic behaviour is related to that of parameter-dependent integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This expression also provides exhaustive information about the support of of n(t, ·): indeed, it ensures that for all t ≥ 0, supp (n(t, ·)) = supp � n0 ◦ Y (t, ·) � = X � t, supp � n0�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (8) Since n0 is assumed to have a compact support, then so does n(t, ·) for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We recall that a set E ⊂ Rd is said to be positively invariant for the ODE ˙x = f(u) if for all t ≥ 0, X(t, E) ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' With this definition in mind, it becomes clear, according to (8), that if supp � n0� is positively invariant for the ODE ˙x = f(x), then supp (n(t, ·)) ⊂ supp � n0� , for all t ≥ 0, and, more generally, that if there exists E ⊂ Rd a set which is positively invariant for this ODE such that supp � n0� ⊂ E, then supp (n(t, ·)) ⊂ E, for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, even if PDE (1) is defined for all x ∈ Rd, if the support of n0 is included in a compact subset of Rd which is positively invariant, then everything happens as if we were working in this compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, the functions f and r do not need to be defined outside this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 ODE satisfied by the population size Let us start with a basic lemma which ensures that the population size ρ does not blow up as t tends to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 2 (Bounds on ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ρ be defined as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then for all t ≥ 0, ρ(t) ≤ max (∥r∥∞, ρ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (8), since, n0 is assumed to have a compact support, n(t, ·) has a compact support for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, when integrating the fist line of (1), the advection term vanishes, and we get ˙ρ(t) = � Rd � r(x) − n(t, x) � n(t, x)dx ≤ (∥r∥∞ − ρ(t)) ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In other words, ρ is a sub-solution of the logistic ODE ˙u = (∥r∥∞ − u) u, which proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the remainder of this section, we show that ρ is in fact the solution to a non-autonomous logistic equation, which can be written in different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In order to lighten the future expressions, we now denote ˜r := r − ∇ · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let E ⊂ Rd be any measurable subset of Rd, and let us denote ρE(t) := � ε n(t, x)dx, which is well-defined and bounded, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By integrating the semi-explicit expression (7) of n over E, we obtain the equality ρE(t) = SE(t)e− � t 0 ρ(s)ds, (9) where SE(t) := � E n0(Y (t, x))e � t 0 ˜r(Y (s,x))dsdx is a function which only depends on the parameters f, r and n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This function is well-defined, and differen- tiable, thanks to our regularity assumptions, and since for all t ≥ 0 n0(Y (t, ·)) has compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, under the hypothesis that for all t ≥ 0, SE(t) > 0, we obtain ln (ρE(t)) = ln (SE(t)) − � t 0 ρ(s)ds, and finally, by differentiating and multiplying by ρE on both sides, ˙ρE(t) = � ˙SE(t) SE(t) − ρ(t) � ρE(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (10) At this stage, one might be tempted to choose E = Rd to obtain, denoting S := SRd(t), ˙ρ(t) = � ˙S(t) S(t) − ρ(t) � ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (11) This proves that ρ is the solution to a non-autonomous logistic equation, and the study of such equations [22] proves that if the time-dependant carrying capacity t �→ ˙S(t) S(t) converges, then ρ converges to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Unfortunately, computing the limit of t �→ ˙S(t) S(t) is intricate (except in very specific cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This brings us to introducing a more general framework, which involves simpler functions whose limit can be computed (at least in the case x ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The idea is to partition the space Rd into several well-chosen subsets, and to consider the size of the population on each of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As seen above, to obtain equations of the type (10), we must be cautious when choosing these subsets in order for the corresponding functions SE to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' All this leads us the following proposition: 8 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let U ⊂ Rd be a set such that X(R+ × supp(n0)) ⊂ U (12) and let (Oi)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=',N} be a finite family of open subsets of U such that (i) ∀i ̸= j, Oi ∩ Oj = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ii) ν � U\\ N� i=1 Oi � = 0, where ν denotes the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iii) ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='N}, ∀t ≥ 0, X � t, supp � n0� � ∩ Oi ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, by denoting for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} ρi(t) := � Oi n(t, x)dx, (13) Si(t) := � Oi n0(Y (t, x))e � t 0 ˜r(Y (s,x))dsdx, (14) Ri(t) := ˙Si(t) Si(t), (15) the following equation holds: � � � � � � � ˙ρi(t) = (Ri(t) − ρ(t)) ρi(t) ∀t ≥ 0, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} ρ(t) = N � i=0 ρi(t) ∀t ≥ 0 ρi(0) > 0 ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (16) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Note that a sufficient condition for the third condition (iii) to hold is the following: for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}, there exists xi in the closure of Oi such that f(xi) = 0 and n0(xi) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As a consequence of the discussion at the beginning of this section, it is enough to prove that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} and all t ≥ 0, Si(t) > 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all t ≥ 0, ρ(t) = N � i=1 ρi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' First, notice that hypothesis (iii) is equivalent to supp � n0� ∩Y (t, Oi) ̸= ∅ for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} and all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, Oi is an open set, which ensures, thanks to the continuity of n0, that {x ∈ Oi : n0(Y (t, x)) > 0} has a positive measure for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This proves the first point by definition of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ρi(0) = Si(0), we also infer ρi(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The second point is due to hypothesis (12): Indeed, for any t ≥ 0, according to the semi-explicit expression of n provided by (7), n(t, x) = 0 if Y (t, x) /∈ supp(n0) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' if x /∈ X(t, supp(n0)), which ensures that ρ(t) = � Rd n(t, x)dx = � U n(t, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The first two hypotheses satisfied by the sets Oi ensure that ρ(t) = N � i=1 ρi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof of the remark: Let xi be a root of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' A classical ODE result ensures that for all t ≥ 0, x ∈ Rd, ∥Y (t, x) − xi∥ ≤ eLt∥x − xi∥, with L > 0 the Lipschitz constant of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since n0(xi) > 0 and n0 is continuous, there exists ε > 0 such that B(xi, ε) ⊂ supp(n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let t ≥ 0, x ∈ Oi ∩B(xi, εe−Lt/2) (such a point does exist, by definition of the closure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, Y (t, x) ∈ B(xi, ε) ⊂ supp(n0), which ensures that x ∈ X � t, supp � n0� � , and thus concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 9 In the one-dimensional case, assuming that f has a finite number of roots, an efficient choice for the sets Oi is to take the segments between the roots of f which interseect the support of n0, as the following result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let x ∈ R and assume that f : R → R has a finite number of roots, that we denote x1 < x2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' < xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote O0 := (−∞, x1), Oi := (xi, xi+1), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N − 1}, ON := (xN, +∞), and, among these segments, let us consider Oi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='OiM those which have an non-empty intersection with supp � n0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, the set U := � 1≤j≤M Oij and the family of sets � Oij � 1≤j≤M satisfy the hypotheses of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By applying the results stated at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2, we note that for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}, Oi is positively invariant for the ODE ˙x = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y ∈ supp(n0) ⊂ U, t ≥ 0, X(t, y) ∈ U, which ensures that X(R+ × supp(n0)) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, the same results show that for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', M}, X(t, supp(n0) ∩ Oij) ⊂ Oij, and thus that X(t, supp(n0)) ∩ Oij ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The other two points are automatically satisfied, thanks to the definition of U and the sets Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 3 shows us that ρ satisfies ODE (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Our next result shows that the long-time behaviour of this ODE depends on the long-time behaviour of the functions Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, it states that if all the functions Ri converge, then ρ converges to the maximum of their limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Before stating the result, we introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For any function g : R+ → R, we denote: g := lim inf t→+∞ g(t) and g := lim sup t→+∞ g(t), and we say that g converges to l ∈ R with an exponential speed if there exist δ > 0 such that g(t) − l = O t→+∞ � e−δt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The coupled system of ODEs (16) has the following properties: (i) For all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} and all t ≥ 0, ρi(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ii) ρ ≥ min 1≤i≤N � Ri � and ρ ≤ max 1≤i≤N � Ri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iii) Let j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If there exists i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} such that Rj < Ri, then ρj(t) −→ t→+∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iv) Let us assume that there exists l ∈ R+ ∪ {+∞}, and a non empty set I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} (where potentially I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}) such that for all i ∈ I, Ri(t) −→ t→+∞ l, and Rj < l for all j /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, ρ(t) −→ t→+∞ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (v) Under the hypotheses of (iv), if moreover 0 < l < +∞ and for all i ∈ I the function Ri converges to l with an exponential speed, then ρ converges to l with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (i) According to the first line of ODE (16), ρi(t) = e � t 0 Ri(s)−ρ(s)dsρi(0), which is positive according to the third line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ii) If min 1≤i≤N � Ri � = 0, there is nothing to prove: we assume min 1≤i≤N � Ri � > 0 and let m < min 1≤i≤N � Ri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' There exists Tm ≥ 0 such that for all t ≥ Tm, and all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}, Ri(t) ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus ˙ρ(t) = N � i=1 ˙ρi(t) = N � i=1 (Ri(t) − ρ(t)) ρi(t) ≥ (m − ρ(t)) ρ(t), which means that ρ is a super-solution of a logistic equation which converges to m, and thus that ρ ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since this inequality holds for any m < min 1≤i≤N � Ri � it proves that ρ ≥ min 1≤i≤N � Ri � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By proceeding in the same way with the limit superior, we get the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 10 (iii) Let i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} such that Rj < Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The latter inequality is written with the convention that if Ri = +∞, then Rj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Using the first point, ρj, ρi > 0 on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We can compute d dt ln � ρi(t) ρj(t) � = Ri(t) − Rj(t) > ε, for a certain ε > 0 and t large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, ρ(t) ≥ ρi(t) ≥ Ceεtρj(t), for a certain constant C > 0, which yields ˙ρj(t) ≤ � sup t>0 Rj(t) − Ceεtρj(t) � ρj(t), with sup t>0 Rj(t) < +∞ by hypothesis, and thus ρj goes to zero as t goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iv) Let us denote ρJ := � j /∈I ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (This first step is not necessary in the case I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to the previous property, ρJ converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By denoting ˜Ri := Ri − ρJ, we can thus rewrite system (16) as: � � � � � � � ˙ρi(t) = � ˜Ri(t) − ρI(t) � ρi(t) ∀t ≥ 0, ∀i ∈ I ρI(t) = � i∈I ρi(t) ∀t ≥ 0 ρi(0) > 0 ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', N} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Applying Property (ii) to this new system proves the desired result, since min i∈I � ˜Ri � = max i∈I � ˜Ri � = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (v) Let l ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to the previous point, ρ is bounded by two positive constants (and so is ρI), that we denote ρm < ρM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Using the same argument as in the proof of the third point, one proves that for all j /∈ I, there exists ε > 0 such that ρJ(t) ≤ Ce−εtρM, and thus that ρJ converges to 0 with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, it remains to prove that the convergence of ρI to l also occurs with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, there exists C, δ > 0 such that for all t ≥ 0, � i∈I | ˜Ri(t) − l| ≤ Ce−δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, by denoting C′ := C∥ρI(·) − l∥∞ ρM, we find d dt 1 2 � ρI(t) − l �2 = (ρI(t) − l) � i∈I (( ˜Ri(t) − l) − (ρI(t) − l))ρi(t) ≤ C′e−δt − ρm (ρI(t) − l)2 , which concludes the proof, according to Gr¨onwall’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4 Results in the one-dimensional case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Asymptotic behaviour of the carrying capacities As evidenced by the previous section and in particular by Proposition 4, the long-time behaviour of ρ is completely determined by that of the functions Ri, which we call carrying capacities by analogy with the logistic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As their definition suggests, computing the limit of these functions is a delicate issue: this section is dedicated to these computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The multidimensional case seems out of reach with this method, because, as we shall see, we use a change of variable that requires to be working in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 11 In order to simplify the notations, we will now denote R instead of RE or Ri, when there is no ambiguity as to which sets we are working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We are thus interested in the asymptotic behaviour of the function R(t) = ˙S(t) S(t), with S(t) = � E n0(Y (t, x))e � t 0 ˜r(Y (s,x))dsdx, (17) where E ⊂ Rd is an open set which satisfies supp(n0) ∩ Y (t, E) ̸= ∅ for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' First, let us note that for all l ∈ R, R(t) − l = d dt � S(t)e−lt� S(t)e−lt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (18) Thus, in order to prove that R converges to l ∈ R with an exponential speed, it in enough to prove that: (a) lim inf t→+∞ S(t)e−lt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) t �→ eδt d dt � S(t)e−lt� is bounded for a certain δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Indeed, we immediately deduce from (18), and the fact that S is positive, according to its definition (14), that these two hypotheses imply that for any δ′ ∈ (0, δ), R(t) − l = O t→+∞ � e−δ′t� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Integral formulae for the carrying capacities This section aims at listing several alternative formulae of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the following section, we will use one or the other, depending on the studied case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We recall that S is defined as S(t) = � E n0(Y (t, x))e � t 0 ˜r(Y (s,x))dsdx, (19) with ˜r := r − ∇ · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As seen in the first section, for any t ≥ 0, x �→ Y (t, x) is a C1-diffeomorphism from E to Y (t, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, the change of variable y = Y (t, x), and Liouville’s formula which ensures that |det(Jac(Y (t, x)))| = e � t 0 −∇· f(Y (s,x))ds provide a second expression for S, namely S(t) = � Y (t,E) n0(y)e � t 0 r(X(s,y))dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (20) Moreover, in the one-dimensional case x ∈ R, if E is an interval on which f ̸= 0, then for all y ∈ E, t �→ X(t, y) is also a C1-diffeomorphism from (0, t) to (y, X(t, y)) or (X(t, y), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This allows us to make the change of variable s′ = Y (s, x) and s′ = X(s, y) in the two expressions for S, thereby obtaining two new formulations S(t) = � E n0(Y (t, x))e � x Y (t,x) ˜r(s) f(s) dsdx = � Y (t,E) n0(y)e � X(t,y) y r(s) f(s) dsdy, (21) and, in the same way, for all l ∈ R, S(t)e−lt = � E n0(Y (t, x))e � x Y (t,x) ˜r(s)−l f(s) dsdx = � Y (t,E) n0(y)e � X(t,y) y r(s)−l f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (22) Likewise, by differentiating expressions (19) and (20), we are led to several formulae for d dt � S(t)e−lt� , namely d dt � S(t)e−lt� = � E m(Y (t, x))e � t 0 ˜r(Y (s,x))−l dsdx = � E m(y)e � t 0 r(X(s,y))−l dsdx, (23) 12 with m(y) := n0(y) (˜r(y) − l ) − f(y)n0′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (24) In the one-dimensional case x ∈ R, assuming that E is an interval in which f ̸= 0, we get the additional expressions d dt � S(t)e−lt� = � E m(Y (t, x))e � x Y (t,x) ˜r(s)−l f(s) dsdx = � E m(y)e � X(t,y) y r(s)−l f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (25) Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' in the particular one-dimensional case where E is an interval such that Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' E) = E for all t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' and f ̸= 0 on E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (which is the case if E is an interval delimited by two consecutive roots of f) one can differentiate (20) to get d dt � S(t)e−lt� = � E n0(y) (r(X(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' y)) − l) e � t 0 r(X(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='y))−l dsdy (26) and the second expression of (22) to get d dt � S(t)e−lt� = � E n0(y) (r(X(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' y)) − l) e � X(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='y) y r(s)−l f(s) dsdy = � E n0(Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x))(r(x) − l)e � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) ˜r(s)−l f(s) dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (27) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 An important estimate The lemma stated in this section will be crucial in computing limits of the relevant parameter-dependent integrals in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let x0 ∈ R ∪ {±∞}, and h and g be two functions defined in the neighbourhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If there exist C1, C2 > 0 such that C1|g(x)| ≤ |h(x)| ≤ C2|g(x)| for any x close enough to x0, we write h(x) = Θ x→x0(g(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to the definition of Θ, is is clear that for any x0 ∈ R ∪ {±∞}, g, h defined in the neighbourhood of x0, f such that h(x) = Θ x→x0(g(x)), h is integrable near x0 if and only if g is integrable near x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let x0, y ∈ R, with x0 ̸= y, and let β ∈ C2([x0, y]) such that β(y) = 0, β′(y) ̸= 0 and β ̸= 0 on [x0, y), and α ∈ C1([x0, y]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, e � x x0 α(s) β(s) ds = Θ x→y � |y − x| α(y) β′(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to the regularity of α and β, for all s ∈ (x0, y), α(s) = α(y) + O(s − y), and β(s) = (s − y)β′(y) + O((s − y)2) Thus, α(s) β(s) − α(y) (s − y)β′(y) = α(s)(s − y)β′(y) − α(y)β(s) β(s)(s − y)β′(y) = O(s − y)2 β′(y)2(s − y)2 + O((s − y)3) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, e � x x0 α(s) β(s) ds = e � x x0 α(y) β′(y) 1 s−y +O(1)ds = eO(1)|y − x| α(y) β′(y) , which proves the result of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='3 Asymptotic behaviour of the carrying capacity in one dimension We here focus on the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We recall that we assume that n0 ∈ Cc(R), f ∈ C2(R) ∩ Lip(R), r ∈ C1(R) ∩ L1(R), and that r(x) goes to 0 as x goes to ±∞ In this section, we further assume that f ∈ BV(R), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e f ′ ∈ L1(R), and that f converges to a non-zero limit at ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In order to apply Proposition 4 (as explained in Lemma 3), the most insightful division is to consider each segment between the roots of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, we must first compute the limit of the function R when the chosen set E is such a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' To be more precise, we must therefore distinguish between several cases, depending on whether the considered interval is bounded (delimited by two consecutive roots of f) or not (delimited by the smallest or the greatest root of f), and the sign of the derivative at these boundary roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In fact, when n0 vanishes at a given root a, the limit may depend on how fast n0 vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' on the value α > 0 such that n0(y) vanishes like (y − a)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For our method of proof to accommodate this case, we will need to make a slightly stronger assumption involving the derivative of n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We will see in the next section that a slight change in the limit of R may have a drastic impact on the long-time behaviour of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We also deal with cases where f does not have any root (which ensures, as one might expect, that R converges to 0), and the case where f is zero on a whole interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, this result can be seen as a generalisation of the one stated in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In each case, we assume that E ∩ supp(n0) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (i) If E = (a, +∞), f < 0 on E, f(a) = 0 and f ′(a) < 0, then R converges to r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ii) If E = (−∞, a), f > 0 on E, f(a) = 0 and f ′(a) < 0, then R converges to r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iii) If E = (a, +∞), f > 0 on E, f(a) = 0, f ′(a) > 0, then If n0(a) > 0, then – If r(a) − f ′(a) > 0, then R converges to r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) − f ′(a) < 0, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + O y→a+((y − a)α), then – If r(a) − (1 + α)f ′(a) > 0, then R converges to r(a) − (1 + α)f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) − (1 + α)f ′(a) < 0, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a, a + ε], then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iv) If E = (−∞, a), f < 0 on E, f(a) = 0, f ′(a) > 0, then If n0(a) > 0, then – If r(a) − f ′(a) > 0, then R converges to r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) − f ′(a) < 0, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = −Cα(a−y)α−1 + O y→a−((a−y)α), then – If r(a) − (1 + α)f ′(a) > 0, then R converges to r(a) − (1 + α)f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) − (1 + α)f ′(a) < 0, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a − ε, a], then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (v) If E = (a, b), f > 0 on (a, b), f(a) = f(b) = 0, f ′(a) > 0, f ′(b) < 0, then If n0(a) > 0, then – If r(b) > r(a) − f ′(a), then R converges to r(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(b) < r(a) − f ′(a), then R converges to r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + O y→a+((y − a)α), then 14 – If r(b) > r(a) − (α + 1)f ′(a), then R converges to r(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(b) < r(a) − (α + 1)f ′(a), then R converges to r(a) − (α + 1)f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exists ε > 0 such that n0(·) = 0 on [a, a + ε], then R converges to r(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (vi) If E = (a, b), f < 0 on (a, b), f(a) = f(b) = 0, f ′(a) < 0, f ′(b) > 0, then If n0(b) > 0, then – If r(a) > r(b) − f ′(b), then R converges to r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) < r(b) − f ′(b), then R converges to r(b) − f ′(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(b) = 0, and if there exist C, α > 0 such that n0′(y) = −Cα(b − y)α−1 + O y→b−((b − y)α), then – If r(a) > r(b) − (α + 1)f ′(b), then R converges to r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(a) < r(b) − (α + 1)f ′(b), then R converges to r(b) − (α + 1)f ′(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(b) = 0, and if there exists ε > 0 such that n0(·) = 0 on [b − ε, b], then R converges to r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (vii) If E = R, and f > 0 on R, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (viii) If E = R, and f < 0 on R, then R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ix) If E is a interval in which f ≡ 0, and n0 > 0, and arg max E r = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', xp} ⊂ E, with r′(xi) = 0, r′′(xi) < 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', p}, then, R converges to r := max x∈E r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, except in this last case, R converges with an exponential speed whenever it does not converge to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As explained at the beginning of this section, whenever we show that R converges with an exponential speed, we must prove successively that (a) lim inf t→+∞ S(t)e−lt > 0 (b) t �→ eδt d dt � S(t)e−lt� is bounded for a certain δ > 0, where l is the expected limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Fatou’s lemma, the point (a) can be proven by showing that the integrand involved in the expression of S (which depends on the chosen formula) converges pointwise to a non-negative function which is positive on a set of positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Depending on the case, we will use different expressions for S and S′ among those determined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In order to lighten the proof, we assume without loss of generality that a = 0 and b = 1, and we denote ˜r = r − f ′ and ˜rα := ˜r − αf ′ = r − (α + 1)f ′ for α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover since the cases (ii), (iv), (vi) and (viii) are symmetric to the cases (i), (iii), (v) and (vii) respec- tively, we omit their proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (i) Note that, according to the hypotheses satisfied by f, for all y ∈ (0, +∞), t �→ X(t, y) converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (a) According to (20), S(t)e−r(0)t = � +∞ 0 n0(y)e � t 0 r(X(s,y))−r(0)dsdy = � M 0 n0(y)e � t 0 r(X(s,y))−r(0)dsdy, for a certain M > 0, since n0 has a compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since f ′(0) < 0, there exist C, δ > 0 such that X(t, y) ≤ Ce−δt for all y ∈ [0, M], t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This proves that for all y ∈ [0, M], s �→ r(X(s, y)) − r(0) is integrable on (0, +∞), and thus that y �→ n0(y)e � +∞ 0 r(X(s,y))−r(0)ds is well-defined on [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since this function is positive on a sub-interval of [0, M], its integral on this segment is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, t �→ n0(y)e � t 0 r(X(s,y))−r(0)ds converges pointwise to this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 15 (b) As seen in the first point, there exist C, δ > 0 such that for all y ∈ [0, M] and all t ≥ 0, 0 ≤ X(t, y) ≤ Ce−δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, using expression (26), and the mean value theorem, ����eδt d dt � S(t)e−r(0)t� ���� = eδt ���� � M 0 n0(y) � r(X(t, y)) − r(0) � e � t 0 r(X(t,y))−r(0)dsdy ���� ≤ 2∥n0∥∞∥r∥L∞(0,M)C � M 0 e � t 0 |r(X(s,y))−r(0)|dsdy ≤ ∥n0∥∞∥r′∥L∞(0,M)CMe � t 0 C∥r′∥L∞(0,M)e−δsds which is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (iii) Note that, according to the hypothesis on f, for all x, y ∈ (0, +∞), t �→ X(t, y) is increasing and goes to +∞, and t �→ Y (t, x) is decreasing and converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that n0(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We distinguish two cases: – Case r(0) − f ′(0) > 0: (a) According to (22), S(t)e−˜r(0)t = � E n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(0) f(s) dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all x ∈ (0, +∞), n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(0) f(s) ds −→ t→+∞ n0(0)e � x 0 ˜r(s)−˜r(0) f(s) ds, which is well defined since s �→ ˜r(s)−˜r(0) f(s) is continuous on [0, x), thanks to the regularity of r and f, and positive, since n0(0) > 0 by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) Let δ ∈ � 0, min(˜r(0), f ′(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since δ − ˜r(0) < 0, r goes to 0 at +∞ and f is positive, we can find M ≥ 0 such that r(s)−˜r(0)+δ f(s) ≤ 0 for all s ∈ [M, +∞), and supp � n0� ∩E ⊂ [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all t ≥ 0, and all y ∈ (0, M), � X(t,y) y r(s) − ˜r(0) + δ f(s) ds ≤ � M y ���� r(s) − ˜r(0) + δ f(s) ����ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (25), eδt d dt � S(t)e−˜r(0)t� = � +∞ 0 m(y)e � X(t,y) y r(s)−˜r(0)+δ f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, since supp(m) ∩ E = supp(n0) ∩ E ⊂ [0, M], and by the previous inequality, ����eδt d dt � S(t)e−˜r(0)t� ���� ≤ � M 0 |m(y)|e � M y |r(s)−˜r(0)+δ| f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since m(y) = n0(y)(˜r(y) − ˜r(0)) − f(y)n0′(y), |m(y)| = O y→0(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since |r(0) − ˜r(0) + δ| = f ′(0) + δ, Lemme 4 yields e � M y |r(s)−˜r(0)+δ| f(s) ds = O y→0(y−1−δ/f ′(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Therefore, |m(y)|e � M y |r(s)−˜r(0)+δ| f(s) ds = O y→0(y−δ/f ′(0)), and is thus integrable since δ < f ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 16 – Case r(0) − f ′(0) < 0: in this case, we do not show that convergence occurs with an expo- nential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, we do not prove the two points as before, but simply that lim sup t→+∞ S(t) > 0 and lim t→+∞S′(t) = 0, which will imply, by definition of R (17), that R converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (22), S(t) = � +∞ 0 n0(y)e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, f converges to a positive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y > 0, there exist εy > 0 such that f(s) > εy, for all s ≥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y > 0, � +∞ y r(s) f(s)ds ≤ 1 εy ∥r∥L1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This implies that y �→ n0(y)e � +∞ y r(s) f(s) ds is well defined on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, this function is positive at any y such that n0(y) > 0, hence its integral is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Finally, t �→ n0(y)e � X(t,y) y r(s) f(s) ds converges to this function pointwise,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Owing to (26) (with l = 0), S′(t) = � supp(n0) n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, there exist ε, M > 0 such that f(s) ≥ ε for all s ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y > 0, � X(t,y) y r(s) f(s)ds ≤ � +∞ y r(s) f(s)ds ≤ � +∞ M r(s) f(s)ds � �� � ≤ ∥r∥L1 ε + � M y r(s) f(s)ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since r ∈ L1(R+), by hypothesis, this proves that there exists a constant K > 0 such that for all t ≥ 0, y > 0, ����n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) ds ���� ≤ ∥n0∥∞∥r∥∞eKe � M y r(s) f(s) ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By virtue of Lemma 4, this last quantity in integrable, since e � M y r(s) f(s) ds = O y→0 � y− r(0) f′(0) � , with r(0) < f ′(0), by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since t �→ r(X(t, y)) converges to 0 as t goes to +∞ for any y > 0, n0(y) r(X(t, y)) e � X(t,y) y r(s) f(s) dsdy converges to 0 pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to the dominated convergence theorem, S′ thus converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that n0(a) = 0, and that the hypothesis of the theorem regarding n0′ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We follow exactly the same steps and use the same formulae as in the case ‘n0(0) > 0’, by adapting the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We distinguish again two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – Case ˜rα(0) > 0: (a) According to (22), S(t)e−˜rα(0)t = � E n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(0) f(s) dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all x ∈ (0, +∞), n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜rα(0) f(s) ds = n0(Y (t, x)) Y (t, x)α e � x Y (t,x) ˜r(s)−˜r(0) f(s) ds Y (t, x)αe � x Y (t,x) αf′(0) f(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 17 Let x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On the one hand, n0(Y (t, x)) Y (t, x)α e � x Y (t,x) ˜r(s)−˜r(0) f(s) ds −→ t→+∞ Ce � x 0 ˜r(s)−˜r(0) f(s) ds which is well defined since s �→ ˜r(s)−˜r(0) f(s) is continuous on [0, x), according to the regularity assumptions on r and f, and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' by rewriting Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x)αe � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) −αf′(0) f(s) ds = eα(ln(Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x))−ln(x))xαe � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) αf′(0) f(s) ds = xαe � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) αf′(0) f(s) − α s ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' and by noting that s �→ αf ′(0) f(s) − α s is continuous at 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' since αf ′(0) f(s) − α s = αf ′(0)s − αf(s) sf(s) = αf ′(0)s − αf ′(0)s + f ′′(0)/2s2 + o(s2) f ′(0)s2 + o(s2) −→ s→0 −αf ′′(0) 2f ′(0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' we show that Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x)αe � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) −αf′(0) f(s) ds −→ t→+∞ xαe � x 0 αf′(0) f(s) − α s ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' which is also well defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) Let δ ∈ � 0, min(˜rα(0), f ′(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since δ−˜rα(0) < 0, r goes to 0 at +∞ and f is positive, we can find M ≥ 0 such that r(s)−˜rα(0)+δ f(s) ≤ 0 for all s ∈ [M, +∞), and supp � n0� ⊂ [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all t ≥ 0, and all y ∈ (0, M), � X(t,y) y r(s) − ˜rα(0) + δ f(s) ds ≤ � M y ���� r(s) − ˜rα(0) + δ f(s) ����ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (25), eδt d dt � S(t)e−˜rα(0)t� = � +∞ 0 m(y)e � X(t,y) y r(s)−˜rα(0)+δ f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, since supp(m) ∩ E = supp(n0) ∩ E ⊂ [0, M], and thanks to the previous inequality, ����eδt d dt � S(t)e−˜rα(0)t� ���� ≤ � M 0 |m(y)|e � M y |r(s)−˜rα(0)+δ| f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us prove that this integral is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' First, let us note that m(y) = n0(y)(˜r(y) − ˜rα(0)) − f(y)n0′(y) = O y→0+(yα+1) Indeed, since n0(y) = Cyα + O y→0+(yα+1) and n0′(y) = Cαyα−1 + O y→0+(yα), |m(y)| yα+1 ≤ n0(y) yα |˜r(y) − ˜r(0)| y + |αf ′(0)n0(y) − f(y)n0′(y)| yα+1 ≤ n0(y) yα ∥˜r′∥∞ + |Cαf ′(0)yα − Cαf ′(0)yα + O(yα+1)| yα+1 = O y→0+(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 18 Moreover, according to Lemma 4, since |r(0) − ˜rα(0) + δ| = (α + 1)f ′(0) + δ, e � M y |r(s)−˜rα(0)+δ| f(s) ds = O y→0+(y−α−1−δ/f ′(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Therefore, |m(y)|e � M y |r(s)−˜r(0)+δ| f(s) ds = O y→0+(y−δ/f ′(0)), and is thus integrable since δ < f ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – Case ˜rα(0) < 0: again, we just prove that lim sup t→+∞ S(t) > 0 and lim t→+∞S′(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (22), S(t) = � +∞ 0 n0(y)e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, f converges to a positive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y > 0, there exist εy > 0 such that f(s) > εy, for all s ≥ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, for all y > 0, � +∞ y r(s) f(s)ds ≤ 1 εy ∥r∥L1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This ensures that y �→ n0(y)e � +∞ y r(s) f(s) ds is well defined on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, this function is positive for every y such that n0(y) > 0, which ensures that its integral is positive, and t �→ n0(y)e � X(t,y) y r(s) f(s) ds converges to this function pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (26), (with l = 0), S′(t) = � supp(n0) n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, there exist ε, M > 0 such that f(s) ≥ ε for all s ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y > 0, � X(t,y) y r(s) f(s)ds ≤ � +∞ y r(s) f(s)ds ≤ � +∞ M r(s) f(s)ds � �� � ≤ ∥r∥L1 ε + � M y r(s) f(s)ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since r ∈ L1(R+), this proves that there exist a constant K > 0 such that for all t ≥ 0, y > 0, ����n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) ds ���� ≤ ∥r∥∞eKn0(y)e � M y r(s) f(s) ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis, and according to Lemma 4, n0(y) = O y→0+(yα) and e � M y r(s) f(s) ds = O y→0+ � y− r(0) f′(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, n0(y)e � M y r(s) f(s) ds = O y→0+ � yα− r(0) f′(0) � , with α− r(0) f ′(0) > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since t �→ r(X(t, y)) converges to 0 as t goes to +∞ for any y > 0, n0(y) r(X(t, y)) e � X(t,y) y r(s) f(s) dsdy converges to 0 pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By the dominated convergence theorem, S′ thus converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We can prove this point exactly as we treat the case f > 0 on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We therefore leave it to the reader and refer to the proof of (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 19 (v) Let us note that, for any x, y ∈ (0, 1), t �→ X(t, y) is increasing and converges to 1, and t �→ Y (t, x) is decreasing and converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that n0(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We distinguish again between two cases: – Case r(1) > ˜r(0): (a) Let us use the second expression (22) for S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' S(t)e−r(1)t = � 1 0 e � X(t,y) y r(s)−r(1) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all y ∈ (0, 1), n0(y)e � X(t,y) y r(s)−r(1) f(s) ds −→ t→+∞ n0(y)e � 1 y r(s)−r(1) f(s) ds, which is well-defined for all y ∈ (0, 1), since s �→ r(s)−r(1) f(s) is continuous on (0, 1], and positive on a set of non-zero measure, since it is positive where n0 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) Let δ ∈ � 0, min (r(1) − ˜r(0), −f ′(1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ˜r(0) − r(1) + δ < 0, there exists m ∈ (0, 1) such that ˜r(s) − r(1) + δ for all s ∈ (0, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all x ∈ (0, 1), t ≥ 0, � x Y (t,x) ˜r(s) − r(1) + δ f(s) ds ≤ � x m |˜r(s) − r(1) + δ| f(s) ds 1(m,1)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, using expression (27), eδt d dt � S(t)e−r(1)t� = � 1 0 n0 (Y (t, x)) (r(x) − r(1)) e � x Y (t,x) ˜r(s)−r(1)+δ f(s) dsdx ≤ ∥n0∥ � 1 0 |r(x) − r(1)|e � x m |˜r(s)−r(1)+δ| f(s) ds 1(m,1)(x)dx < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This last integral is finite since |˜r(1) − r(1) + δ| = −f ′(1) + δ, and thus e � x a |˜r(s)−˜r(0)| f(s) ds = O x→1 � |x − 1| δ f′(1) −1� , by Lemma 4) |r(x)−r(1)| = O x→1|x−1|, and δ f ′(1) > −1 by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – Case ˜r(0) > r(1): (a) Using (22), we find S(t)e−˜r(0)t = � 1 0 n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(0) f(s) dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all x ∈ (0, 1), n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(0) f(s) ds −→ t→+∞ n0(0)e � x 0 ˜r(s)−˜r(0) f(s) ds, which is well- defined since s �→ ˜r(s)−˜r(0) f(s) is continuous on [0, 1), and positive by hypothesis on n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) Let δ ∈ � 0, min(˜r(0) − r(1), f ′(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since r(1) − ˜r(0) + δ < 0, there exists M ∈ (0, 1) such that r(s) − ˜r(0) + δ < 0 for all s ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y ∈ (0, 1), t ≥ 0, � X(t,y) y r(s)−˜r(0)+δ f(s) ≤ � M y |r(s)−˜r(0)+δ| f(s) ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, ����eδt d dt � S(t)e−˜r(0)t� ���� = ���� � 1 0 m(y)e � X(t,y) y r(s)−˜r(0)+δ f(s) ds ���� ≤ � 1 0 |m(y)|e � M y |r(s)−˜r(0)+δ| f(s) ds 1(0,M)(y)dy, which is a finite integral, since |r(0) − ˜r(0) + δ| = f ′(0) + δ, and thus e � b y |r(s)−˜r(0)+δ| f(s) ds = O y→0 � y− δ f′(0) −1� (by Lemma 4), m(y) = n0(y) (˜r(y) − ˜r(0)) − f(y)n0′(y) = O y→0 (y), and δ f ′(0) < 1 thanks to our choice for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 20 Let us assume that n0(a) = 0, and that the hypothesis on n0′ of the theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As usual, we distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – Case r(1) > ˜rα(0): (a) This first point is exactly the same as in the case n0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us use the second expres- sion (22) for S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' S(t)e−r(1)t = � 1 0 e � X(t,y) y r(s)−r(1) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all y ∈ (0, 1), n0(y)e � X(t,y) y r(s)−r(1) f(s) ds −→ t→+∞ n0(y)e � 1 y r(s)−r(1) f(s) ds, which is well-defined for all y ∈ (0, 1), since s �→ r(s)−r(1) f(s) is continuous on (0, 1], and positive on a set of measure non-zero, since it is positive where n0 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (b) Let δ ∈ � 0, min (r(1) − ˜rα(0), −f ′(1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' First, let us note that we can rewrite Y (t, x)α = eln(Y (t,x))−ln(x)xα = xαe � x Y (t,x) − α s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, by using expression (27), we get eδt d dt � S(t)e−r(1)t� = � 1 0 n0 (Y (t, x)) (r(x) − r(1)) e � x Y (t,x) ˜r(s)−r(1)+δ f(s) dsdx = � 1 0 n0(Y (t, x)) Y (t, x)α xα(r(x) − r(1))e � x Y (t,x) ϕ(s) f(s) dsdx, with ϕ(s) := ˜r(s) − r(1) + δ − αf(s) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By hypothesis on n0, f and r, ˜n0 : y �→ n0(y) yα and ϕ are both continuous on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since ϕ(0) = ˜rα(0) − r(1) + δ < 0, there exists ε ∈ (0, 1) such that ϕ(s) < 0 for all s ∈ [0, ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, ����eδt d dt � S(t)e−r(1)t� ���� ≤ ∥ ˜n0∥∞ � 1 0 |r(x) − r(1)|xαe � x ε |ϕ(s)| f(s) ds1(ε,1)(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' since |ϕ(1)| = δ − f ′(1), Lemma 4 yields e � x ε |ϕ(s)| f(s) ds = O x→1(|x − 1| δ f′(1) −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since |r(x) − r(1)| = O x→1(x) and δ f ′(1) > −1 (by hypothesis on δ), this proves that this last integral is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – Case ˜rα(0) > r(1): (a) According to (22), S(t)e−˜rα(0)t = � 1 0 n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜rα(0) f(s) dsdx = � 1 0 n0(Y (t, x)) Y (t, x)α Y (t, x)αe � x Y (t,x) ˜r(s)−˜rα(0) f(s) dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By rewriting Y (t, x)α = xαe− � x Y (t,x) α s ds, we get S(t)e−˜rα(0)t = � 1 0 n0(Y (t, x)) Y (t, x)α xαe � x Y (t,x) ˜r(s)−˜rα(0) f(s) − α s dsdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since n0(Y (t,x)) Y (t,x)α xαe � x Y (t,x) ˜r(s)−˜rα(0) f(s) − α s ds converges pointwise to C xα e � x 0 ˜r(s)−˜rα(0) f(s) − α s ds, which is well-defined, since s �→ ˜r(s)−˜rα(0) f(s) − α s is continuous at 0 and positive, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 21 (b) Let δ ∈ � 0, min(˜rα(0) − r(1), f ′(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since r(1) − ˜rα(0) + δ < 0, there exists M ∈ (0, 1) such that r(s) − ˜rα(0) + δ < 0 for all s ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y ∈ (0, 1), t ≥ 0, � X(t,y) y r(s) − ˜rα(0) + δ f(s) ≤ � M y |r(s) − ˜rα(0) + δ| f(s) ds 1(0,M)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, using expression (25), we get ����eδt d dt � S(t)e−˜r(0)t� ���� = ���� � 1 0 m(y)e � X(t,y) y r(s)−˜r(0)+δ f(s) ds ���� ≤ � 1 0 |m(y)|e � M y |r(s)−˜r(0)+δ| f(s) ds 1(0,M)(y)dy, which is a finite integral, since |r(0) − ˜rα(0) + δ| = (1 + α)f ′(0) + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 4 leads to e � b y |r(s)−˜r(0)+δ| f(s) ds = O y→0 � y− δ f′(0) −α−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The integrability follows from m(y) = n0(y) (˜r(y) − ˜rα(0)) − f(y)n0′(y) = O y→0 � yα+1� (as seen previously), and δ f ′(0) < 1 thanks to our choice for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We prove this case with exactly the same arguments that for the case of a unique root which is asymptotically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We therefore apply the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (vii) In this case, since f > 0, X(t, y) −→ t→+∞ +∞, for all y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us prove that lim inf t→+∞ S(t) > 0 and lim t→+∞ S(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (21), S(t) = � supp(n0) n0(y)e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The integrand n0(y)e � X(t,y) y r(s) f(s) ds converges pointwise to n0(y)e � +∞ y r(s) f(s) ds, which is well defined (with values in [0, +∞]), and positive for all y ∈ supp(n0), since r f is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to (27), S′(t) = � supp(n0) n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) dsdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since f is continuous, positive, and converges to positive constants at ±∞, ε := min s∈R f(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all y ∈ R, t ≥ 0, ����n0(y)r(X(t, y))e � X(t,y) y r(s) f(s) ds ���� ≤ ∥n0∥∞∥r∥∞e ∥r∥L1 ε < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Combined with the fact that r(X(t, y)) converges to 0 as t goes to +∞ pointwise, we deduce that S′ converges to 0 by the dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (ix) Since f ≡ 0 on E, Y (t, x) = x for all (t, x) ∈ R+ × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, according to formula (20), S(t) = � E n0(x)er(x)tdx and S′(t) = � E n0(x)r(x)er(x)tdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Laplace’s formula (see [39]), S(t) ∼ t→+∞ √ 2π � p � i=1 n0(xi) � |r′′(xi)| � ert √ t 22 and S′(t) ∼ t→+∞ √ 2π � p � i=1 n0(xi)r(xi) � |r′′(xi)| � ert √ t = √ 2π r � p � i=1 n0(xi) � |r′′(xi)| � ert √ t ∼ t→+∞ r S(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, R(t) = S′(t) S(t) −→ t→+∞ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 Applications Summary of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The method that we propose in order to study the asymptotic behaviour of PDE (1) can be summarised by the following three steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Choose an appropriate family of set (Oi) which satisfies the assumptions of Proposition 12, and such that we can compute the asymptotic behaviour of the functions Ri: a good choice when f has a finite number of roots is to take the interval between the roots, as suggested in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Use Proposition 4 in order to determine the limit of ρ, and its speed of convergence when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Use the semi-explicit expression of n provided by equation (7), and eventually Proposition 1 to deduce the asymptotic behaviour of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In each of the following subsections, we apply the three points detailed in this summary to study the asymptotic behaviour of n in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Remark regarding the regularity of parameter functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='3, we make the further assumptions that f ∈ BV(R), and that f converges to a non-zero limit at ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, we easily check that all the results of this previous section remain true if we assume that n0 is C1 on each interval between the roots of f, and not necessarily on the whole of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As far as f is concerned, it is enough to assume that it is globally Lipschitz, and C2 only on a neighbourhood of its roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' It will sometimes be advisable to make these two additional assumptions: we will indicate this at the beginning of each statement whenever this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Case of a unique stable equilibrium We start by assuming that f has a unique root (denoted a), which is asymptotically stable for the ODE ˙u = f(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this case, solutions converge to a weighted Dirac mass at a, regardless of the functions r and n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The weight in front of the Dirac mass is determined by the value of r at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Note that this result can be generalised to higher dimensions, see Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has a unique root (denoted a), and that f ′(a) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, ρ converges to r(a) and n(t, ·) ⇀ t→+∞ r(a)δa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We apply the three points detailed in the summary: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote O1 := (−∞, a), O2 := (a, +∞), which satisfy the assumptions of Proposition 3, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By proposition 5, R1 and R2 both converge to r(a) (with an exponential speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 4, ρ converges to r(a) with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to to the semi-explicit expression (7), n(t, x) = n0(Y (t, x))e � t 0 ˜r(Y (s,x))−ρ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ∥Y (t, x)∥ −→ t→+∞ +∞ for all x ∈ Rd\\{a}, and n0 has a compact support, there exists T0 such that n(t, x) = 0 for all t ≥ T0, x ∈ Rd\\[a − δ, a + δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ρ(t) = � supp(n0) n(t, x)dx converges to r(a), Propositions 1 allows us to conclude that n(t, ·) ⇀ t→+∞ r(a)δa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 Case of a unique unstable equilibrium We now assume that f has a unique root (denoted a) which is asymptotically unstable for the ODE ˙u = f(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Under theses hypotheses, the growth term can counterbalance the advection term: there exist two regimes of convergence, depending on how r(a) and f ′(a) compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has a unique root (denoted a), and that f ′(a) > 0, Then: If r(a) < f ′(a), then ρ(t) −→ t→+∞ 0 and n(t, ·) −→ t→+∞ 0 in L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If r(a) > f ′(a), and n0(a) > 0, then ρ(t) −→ t→+∞ r(a) − f ′(a), and n(t, ·) −→ t→+∞ n in L1(R), where n(x) := Ce � x a ˜r(s)−˜r(a) f(s) ds, with ˜r = r − f ′ and C such that � R n(x)dx = r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We apply the three points detailed in the summary: Let us assume that r(a) < f ′(a): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote O1 := (−∞, a), O2 := (a, +∞), which satisfy the assumptions of Proposition 3, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 5 shows that R1 and R2 both converge to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 4, ρ converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We immediately deduce from the previous point that n(t, ·) −→ t→+∞ 0 in L1(R), by definition of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that r(a) > f ′(a): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' With the same choice for O1 and O2, Proposition 5 shows that R1 and R2 both converge to r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 4, ρ converges to ˜r(a) with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By the semi-explicit expression (7), n(t, x) = n0(Y (t, x))e � t 0 ˜r(Y (s,x))−ρ(s)ds = n0(Y (t, x))e � t 0 ˜r(Y (s,x))−˜r(a)dse � t 0 ˜r(a)−ρ(s)ds = n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(a) f(s) dse � t 0 ˜r(a)−ρ(s)ds (we use the change of variable s′ = Y (s, x) in the first integral to get this last expression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, n(t, ·) converges pointwise to x �→ n0(a)e � x a ˜r(s)−˜r(a) f(s) dse � +∞ 0 ˜r(a)−ρ(s)ds, which is well-defined, since ρ converges to ˜r(a) with an exponential speed, f > 0 on (a, +∞) and s �→ ˜r(s)−˜r(a) f(s) is continuous at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since r(x) −→ x→+∞ 0, and f converges to a positive limit, there exist M, d > 0 such that r(s)−˜r(a) f(s) < −d for all s ≥ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all t ≥ 0, x ∈ (a, +∞), � x Y (t,x) ˜r(s) − ˜r(a) f(s) ds ≤ � M a |˜r(s) − ˜r(a)| f(s) ds � �� � :=C1 + � x M ˜r(s) − ˜r(a) f(s) ds 1(M,+∞)(x) ≤ C1 + � x M r(s) − ˜r(a) f(s) � �� � ≤−d ds 1(M,+∞)(x) + � x M f ′(s) f(s) ds 1(M,+∞)(x) ≤ C1 − d(x − M)1(M,+∞) + � +∞ M |f ′(s)| f(s) ds � �� � :=C2 , 24 with C1, C2 < +∞, by the regularity of ˜r, f ∈ BV (R), and the fact that f converges to a positive constant at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By proceeding in the same way for all x ≤ a, we show that for all x ∈ R, t ≥ 0, n(t, x) ≤ Ce−d|x| for some constants C, d > 0, which ensures, according to the dominated convergence theorem, that t �→ n(t, ·) converges to x �→ n0(a)e � x a tr(s)−˜r(a) f(s) dse � +∞ 0 ˜r(a)−ρ(s)ds in L1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='3 Two equilibria In this section we assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0 and f ′(b) < 0 (hence f > 0 on (a, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The case f ′(a) < 0, f ′(b) > 0, f < 0 on (a, b) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Depending on the functions r and n0, n will either converge to a function in L1, or converge to a Dirac mass at b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We split this result into two propositions: the first one assumes that the support of n0 crosses a, which means that n0 > 0 in a neighbourhood of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The second one assumes that supp(n0) ⊂ [a, +∞), and we consider the case where n0(a) = 0, which leads to other regular functions being reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0, f ′(b) < 0, and that n0(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then: If r(b) > r(a) − f ′(a), then ρ(t) −→ t→+∞ r(b) and n(t, ·) ⇀ t→+∞ r(b)δb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If r(b) < r(a) − f ′(a), then ρ(t) −→ t→+∞ r(a) − f ′(a), and n(t, ·) −→ t→+∞ n in L1(R), where n(x) := De � x a ˜r(s)−˜r(a) f(s) ds1(−∞,b), with ˜r = r − f ′, and D > 0 is such that � R n(x)dx = r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Note that since n0 is assumed to be continuous on R, n0 > 0 on a neighbourhood of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that r(b) > r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We again follow the three points of the method outlined in the beginning of the subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote O1 = (−∞, a), O2 = (a, b), O3 = (b, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' One easily checks that these sets satisfy the hypotheses of Proposition 3, thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to Proposition 5, R1 converges to max(0, ˜r(a)) < r(b) and R2 and R3 both converge to r(b) with an exponential speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 4, ρ converges to r(b) with an exponential speed, and ρ1(t) = � a −∞ n(t, x)dx converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let x ∈ (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Using (7), we find n(t, x) = n0(Y (t, x))e � t 0 ˜r(Y (s,x))−ρ(s))ds, for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let K ⊂ (a, b) be a compact set, δ ∈ � 0, 1 2 (r(b) − ˜r(a)) � , and let us denote d := r(b) − ˜r(a) − 2δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ρ converges to r(b), and (Y (s, x))s≥0 converges to a uniformly on K, there exists T0 such that for all s ≥ T0 and all x ∈ K, ρ(s) ≥ r(b) − δ and ˜r(Y (s, x)) ≤ ˜r(a) + δ, Thus, � K n(t, x)dx ≤ ∥n0∥∞ � K e � T0 0 ˜r(Y (s,x))−ρ(s)dsdx e−d(t−T0) −→ t→+∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let K′ be a compact subset of (b, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since n0 has compact support, there exists T0 such that n0(Y (t, x)) = 0 for all t ≥ T0, x ∈ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, t �→ � K′ n(t, x)dx converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 1, n(t, ·) ⇀ t→+∞ r(b)δb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 25 Let us now assume that r(b) < r(a) − f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' With the same choice for O1, O2 and O3, Proposition 5 shows that R1 and R2 converge to ˜r(a), and that R3 converges to r(b) < ˜r(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We then apply Proposition 4 to infer that ρ converges to ˜r(a) with an exponential speed, and ρ3(t) = � +∞ b n(t, x)dx converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let x ∈ (−∞, b), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By the semi-explicit expression (7), n(t, x) = n0(Y (t, x))e � t 0 ˜r(Y (s,x))−ρ(s))ds = n0(Y (t, x))e � x Y (t,x) ˜r(s)−˜r(a) f(s) dse � t 0 ˜r(a)−ρ(s)ds, where we used the change of variable ‘s′ = Y (t, x)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The latter function converges pointwise to n0(a)e � x a ˜r(s)−˜r(a) f(s) dse � +∞ 0 ˜r(a)−ρ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As for the case of a unique unstable equilibrium (proof of Proposition 7) one can find C, d ≥ 0 such that n(t, x) ≤ Ce−d|x| for all x ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, for all x ∈ (a, b), n(t, x) ≤ ∥n0∥∞ e � +∞ 0 |˜r(a)−ρ(s)|ds e � x a ˜r(s)−˜r(a) f(s) 1{˜r(s)>˜r(a)}(s)ds, which provides an L1-domination, since e � x a ˜r(s)−˜r(a) f(s) = O x→+∞ � |x − b| ˜r(a)−r(b) −f′(b) −1 � , thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By the dominated convergence theorem, combined with the fact that ρ converges to ˜r(a) and ρ3 converges to 0, this ensures that n(t, ·) converges to the expected limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the following proposition, we assume that n0 is C1 on (a, b) and on (b, +∞), and not necessarily on the whole of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has exactly two roots, a < b, which satisfy f ′(a) > 0, f ′(b) < 0, and that supp(n0) ⊂ [a, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We distinguish between several cases: If n0(a) > 0, then – If r(b) > r(a) − f ′(a), then ρ(t) −→ t→+∞ r(b), and n(t, ·) ⇀ t→+∞ r(b)δb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(b) < r(a) − f ′(a), then ρ(t) −→ t→+∞ r(a) − f ′(a), and n(t, ·) −→ t→+∞ n0 in L1(R), where n0(x) := D0e � x a ˜r(s)−˜r(a) f(s) ds1(a,b), with ˜r = r − f ′, and D0 > 0 is such that � R n0(x)dx = r(a) − f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exist C, α > 0 such that n0′(y) = Cα(y − a)α−1 + O y→a+ ((y − a)α), then – If r(b) > r(a) − (1 + α)f ′(a), then ρ(t) −→ t→+∞ r(b), and n(t, ·) ⇀ t→+∞ r(b)δb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' – If r(b) < r(a) − (1 + α)f ′(a), then ρ(t) −→ t→+∞ r(a) − (1 + α)f ′(a), and n(t, ·) −→ t→+∞ nα in L1(R), where nα(x) := Dα(x − a)αe � x a ˜r(s)−˜rα(a) f(s) − α s−a ds1(a,b), where ˜r = r − f ′, ˜rα = r − (1 + α)f ′, and Dα > 0 is such that � R nα(x)dx = r(a) − (1 + α)f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If n0(a) = 0, and if there exists ε > 0 such that n0(y) = 0 for all y ∈ [a, a + ε], then ρ(t) −→ t→+∞ r(b), and n(t, ·) ⇀ t→+∞ r(b)δb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 26 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since the proof of this proposition is very similar to the one of Proposition 8, we do not write it in full detail, but we simply underline the points that must be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In the case where n0(a) > 0, and supp(n0) ⊂ [a, +∞), the proof is the same, but by considering only the two sets O2 = (a, b), and O3 = (b, +∞), and not O1 = (−∞, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We easily check using Lemma 3 that (O2, O3) satisfy the hypotheses of Lemma 3, since supp(n0) ∩ O1 = ∅ The case where n0(a) = 0 and the hypothesis on n0′ holds is quite similar, except that Proposition 5 now shows that R2 converges to max (˜rα(a), r(b)), with an exponential speed (if ˜rα(a) ̸= r(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, we treat the case r(b) > ˜rα(a) in exactly the same way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' the case r(b) < r(a) − (1 + α) is a little more intricate: recalling that for all x ∈ (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' n(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x) = n0(Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x))e � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) ˜r(s)−˜rα(a) f(s) dse � t 0 ˜rα(a)−ρ(s)ds and (Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x) − a)α = (x − a)αe− � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) α s−a ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' one notes that n(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x) = n0(Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x)) (Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' x) − a)α e � t 0 ˜rα(a)−ρ(s)ds(x − a)αe � x Y (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='x) ˜r(s)−˜rα(a) f(s) − α s−a ds converges pointwise to Ce � +∞ 0 ˜rα(a)−ρ(s)ds(x − a)αe � x a ˜r(s)−˜rα(a) f(s) − α s−a ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' which is well-defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' since ρ converges to ˜rα(a) with an exponential speed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' and s �→ ˜r(s)−˜rα(s) f(s) − α s−a is continuous on a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' as seen in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, x �→ ∥ n0(·) (· − a)α ∥∞e � +∞ 0 |˜rα(s)−ρ(s)|ds e � x a ϕ(s) f(s) 1{ϕ(s)≥0}ds, with ϕ(s) = ˜r(s)−˜rα(s)−α f(s) s−a is clearly a domination of n, and is in L1, since ϕ(b) = r(b)−˜rα(a)−f ′(b), which implies by Lemma 4 that e � x a ϕ(s) f(s) ds = O x→b− � (b − x) r(b)−˜rα(a) f′(b) −1 � , with r(b)−˜rα(a) f ′(b) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This last point is the simplest, and is in fact analogous to the case of a single equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' According to Proposition 5, R2 and R3 converge to r(b): we deduce the result by following the steps of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since Proposition 9 provides a completely explicit expression for the limit functions nα, α ≥ 0, one can easily determine their asymptotic behaviour at the boundary of the segment (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since for all α > 0, x ∈ (a, b), nα(x) = Dα(x − a)αe � x a ˜r(s)−˜rα(s) f(s) − α s−a ds, and s �→ ˜r(s)−˜rα(a) f(s) − α s is continuous on [a, b), it is clear that nα(x) = Θ x→a+ ((x − a)α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, nα can be extended by continuity at 0, with nα(a) = 0 if α > 0, and n0(a) ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since ˜r(b)−˜rα(a)− αf(b) b−a = r(b)−˜rα(a)−f(b), Lemma 4 ensures that nα(x) = Θ x→b− � (b − x) r(b)−˜rα(a) f′(b) −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular lim x→b−nα(x) = � � � � � 0 if ˜r(b) < ˜rα(a) l > 0 if ˜r(b) = ˜rα(a) +∞ if ˜r(b) > ˜rα(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' These different cases are illustrated by Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The case where f has two roots a < b with f ′(a) < 0 and f ′(b) > 0 is symmetric to the cases here, and thus lead to the same results, by switching a and b in the Propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 27 Figure 2: Continuous limit functions nα, for different values of α > 0, as defined in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this example, we have chosen f(x) = x(1 − x), and r(x) = b − ax (with b = 6, a = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' With this choice, we easily compute that, for all α ∈ [0, a − 1), and all x ∈ (0, 1) nα(x) = Dαxα(1 − x)a−α−2, for the appropriate constant Dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This illustrates the variety of limit functions that can be reached depending on the initial condition, as detailed in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='3 More than two equilibria In this subsection, we deal with the cases where f has more than two equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As evidenced by the previous result, listing all possible scenarios when there are two roots already is cumbersome: this is why we will not do so in a more general case, and will focus on the case where n0 is positive on the neighbourhood of the unstable equilibrium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The other cases can of course be treated as seen above, keeping in mind that this changes the value of the limits reached by the R functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has a finite number of roots, which are all hyperbolic equilibrium points for the ODE ˙u = f(u), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' f ′ has a sign at each root of f, and let us denote x1 u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', xp u the asymptotically unstable equilibria, and x1 s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', x1 s the asymptotically stable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, let us denote Mu := max{r(x1 u) − f ′(x1 u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='r(xp u) − f ′(xp u)}, and Ms := max{r(x1 s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', r(xm s )}, and let us assume that these two maxima are both reached at a unique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lastly, let us assume that n0(xi u) > 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If Ms > Mu, then ρ(t) −→ t→+∞ Ms, and n(t, ·) ⇀ t→+∞ Msδxi s, with xi s the unique stable equilibria such that Ms = r(xi s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If Ms < Mu, then ρ(t) −→ t→+∞ Mu, and n(t, ·) −→ t→+∞ ni∗ in L1, where ni∗(x) = Ci∗e � x xi∗ u ˜r(s)−˜r(xi∗ u ) f(s) ds1Ii∗ (x), with ˜r = r−f ′, i∗ the unique integer of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', p} such that ˜r(xi∗ u ) = Mu, Ii∗ the open interval delimited by the two stable equilibria which enclose xi∗ u (or −∞ or +∞ if xi∗ u is the smallest or the greatest root of f), and Ci∗ a positive constant such that � Ii∗ ni∗(x)dx = Mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The proof of this proposition is in similar to that of Proposition 8: we denote O0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=', Op+m, the intervals between each roots of f, which satisfy the hypotheses of Proposition 3, according to Lemma 3, 28 16 0= 14 α=1 α=2 12 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='5 10 8 6 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='0and, using Proposition 5, we are able to compute the limit of the function Ri, for all i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' , p + m}, and thus determine the long-time behaviour of ρ by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We conclude by using the semi-explicit expression (7) for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This method also allows to deal with the case where f ≡ 0 on a whole segment: we do not return to the case f ≡ 0 on R, which has already been studied in [34] and [29], but we consider the case where f ≡ 0 on an interval, and then becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' To make the assumption of the following proposition possible, we assume that f is C2 on (−∞, a) and on (a, +∞), but not necessarily on the whole of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that there exists a ∈ R such that f ≡ 0 on (−∞, a), f > 0 on (a, +∞), f ′(a+) > 0, and that supp(n0) = [s−, s+], with s− < a < s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, If there exists a unique xM ∈ (s−, a) such that r(xM) = max x∈[s−,a] r(x), and f ′′(xM) < 0, then ρ converges to r(xM), and n(t, ·) ⇀ t→+∞ r(xM)δxM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If r|[s−,a] reaches its maximum at a (and only at a), then ρ converges to r(a), and n(t, ·) ⇀ t→+∞ r(a)δa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote O1 := (s−, a), O2 := (a, +∞), which satisfy the hypothesis of Proposition 3, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Proposition 5, R1 converges to r(xM) and R2 converges to r(xM) − f ′(xM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' From Proposition 4, ρ and ρ1 converge to r(xM), and ρ2 converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let K ⊂ [s−, a] be a compact set that does not contain xM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thanks to the semi-explicit expression (7), and using the fact that f ′(x) = 0 and Y (t, x) = x for all x ∈ K and all t ≥ 0, n(t, x) = n0(x)e � t 0 r(x)−ρ(s)ds ≤ n0(x)e � t 0 rK−ρ(s)ds, with rK := max x∈K r(x) < r(xM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, � K n(t, x)dx ≤ � K n0(x)dxe � t 0 rM−ρ(s)ds, which converges to 0, since rM − ρ(s) is negative for any s large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This proves the result thanks to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Here we have to make a slightly more subtle choice of subsets than usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ε > 0, and let us denote Oε 1 := (s−, a − 2ε), Oε 2 := (a − 2ε, a − ε), Oε 3 := (a − ε, a), O4 := (a, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We easily check that these four sets satisfy the hypotheses of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, since f ≡ 0 on [s−, a] for all i ∈ {1, 2, 3}, Rε i (t) = � Oε i r(x)er(x)tdx � Oε i er(x)tdx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, for all t ≥ 0, i ∈ {1, 2, 3}, min x∈O ε i r(x) ≤ Rε i (t) ≤ max x∈O ε i r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, Rε 1 ≤ max x∈[s−,a−2ε] r(x) and Rε 3 ≥ min x∈[a−ε,a] r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Finally, Proposition 5 shows that R4 converges to r(a) − f ′(a+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since r reaches its unique maximum at a, for any ε small enough, we get Rε 3 > Rε 1 and Rε 3 > lim t→+∞ R4(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, according to Proposition 4, ρε 1 and ρ4 converge to 0, for all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The choice of ε being arbitrary, it also proves that ρε 2 converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, ρ = ρε 3, and ρ = ρε 3, for all ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since for all t ≥ 0 min x∈[a−ε,a] r(x) ≤ Rε 3(t) ≤ r(a), we prove that ρ converges to r(a) by making ε tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We have proved that t �→ � a s− n(t, x)dx converges to r(a), that t �→ � +∞ a n(t, x)dx converges to 0 and that for all ε > 0, � a−ε s− n(t, x)dx converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The hypotheses of Proposition 1 are therefore met, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Note that the methods of Propositions 10 and 11 can be coupled to treat more complex cases, where, for example, f ≡ 0 on several disjoint segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 5 Some results in higher dimensions As seen in the previous sections, our entire method is based on the computation of the limit of the Ri functions defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Unfortunately, these computations seem out of reach in the multidimensional case Rd, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this section, we nevertheless address the question of the possible convergence to smooth or singular measures in higher dimensions in some specific simple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We first analyse how the solution support evolves over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This allows us to conclude that that the solution converges to a Dirac mass in the case of a unique equilibrium which is asymptotically stable for the ODE ˙u = f(u), and provide hypotheses under which the solution cannot converge to a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We then characterise which stationary measures may or may not be limits for solutions of (1), before providing a criterion ensuring the existence of continuous stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='1 Limit support Definition 1 (Limit support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We define the limit support of n as: σ∞ = � t≥0 � s≥t supp (n(s, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Recalling the semi-explicit expression (7), n(t, x) = n0(Y (t, x))e � t 0 (r−∇·f)(Y (s,x))−ρ(s)ds, and that for all t ≥ 0, supp � n0(Y (t, ·)) � = X(t, supp(n0)), we get σ∞ = � t≥0 � s≥t supp (n0 (Y (s, ·))) = � t≥0 � s≥t X (s, supp (n0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (28) In the cases where we are able to determine the latter set, we gather information about possible limits for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If the limit support of n is of measure zero, then n does not converge (weakly) to a non-zero function in L1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By denoting ν the Lebesgue measure, let us assume that ν(σ∞) = 0, and that n converges weakly to n ∈ L1(Rd), n ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since lim sup t→+∞ supp (n (t, ·)) = � t≥0 � s≥t supp (n(s, ·)) ⊂ σ∞, lim sup t→+∞ ν (supp(n(t, ·))) ≤ ν � lim sup t→+∞ supp (n(t, ·)) � ≤ ν(σ∞) = 0, which contradicts the initial hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has a unique root, denoted x, which is globally asymptotically stable for the ODE ˙u = f(u) over Rd, and that the set � t≥0 X(t, supp(n0)) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, n(t, ·) ⇀ t→+∞ r(x)δx, and ρ converges to r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since the support of n0 is compact, and x is globally asymptotically stable, we easily check, according to (28), that σ∞ = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' By Lemma 1, it is hence enough to prove that ρ converges to r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As seen in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2, ρ satisfies, for all t ≥ 0, ˙ρ(t) = � Rd � r(x) − ρ(t) � n(t, x)dx = � supp(n(t,·)) � r(x) − ρ(t) � n(t, x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since σ∞ = {x} is the intersection of compact decreasing sets, there exists Tε > 0 such that, for all t ≥ Tε, supp(n(t, ·)) ⊂ B(x, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, by denoting rε m := min x∈B(x,ε) r(x) and rε M := max x∈B(x,ε) r(x), we get, for all t ≥ Tε, (rε m − ρ(t))ρ(t) ≤ ˙ρ(t) ≤ (rε M − ρ(t))ρ(t), which ensures that lim inf t→+∞ ρ(t) ≥ rε m and lim sup t→+∞ ρ(t) ≤ rε M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since these inequalities hold for any ε > 0, and rε m and rε M both converge to r(x) when ε goes to 0, it concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Because of the diversity of possible behaviours of ODE systems, it is difficult to compute the limit support for a given f, unless very strong assumptions are made about the ODE ˙u = f(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This is what we do in the following proposition, motivated by a family of ODE systems commonly used in systems biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We say that the two-dimensional system � ˙x1 = f1(x1, x2) ˙x2 = f2(x1, x2) (29) is competitive if ∂2f1 ≤ 0 and ∂1f2 ≤ 0, and cooperative if ∂2f1 ≥ 0 and ∂1f2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For instance, such systems are commonly used to model the interactions between two proteins in the context of cell differentiation [20, 37, 18, 26], and are known to have an interesting property: trajectories either go to +∞, or converge [24], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' for all x ∈ Rd, ∥Y (t, x)∥ ̸−→ t→+∞ +∞ ⇒ t �→ Y (t, x) converges .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (30) Note that if the ODE (29) is competitive (or cooperative), then the reverse ODE ˙u = −f(u) is cooperative (or competitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' This motivates the hypothesis of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Before giving its statement, we recall that if x is a root of f, x is called a hyperbolic equilibrium if all the eigenvalues of Jac f(x) have a non-zero real part, and is called a repellor if all these eigenvalues have a positive real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lastly, we recall that the unstable set of x is defined by {x ∈ Rd : Y (t, x) −→ t→+∞ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 31 Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that f has a finite number of roots, and is such that identity (30) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, the limit support of n is included in the closure of the union of the unstable sets of the roots of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' by denoting x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='.xN the roots of f, σ∞ ⊂ � 1≤i≤N � x ∈ Rd : Y (t, x) −→ t→+∞ xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, if all the roots of f are hyperpolic points, and if none of them is a repellor, then the limit support of n is of measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In particular, n does not converge (weakly) to a function in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The inclusion is clear: by hypothesis for all x ∈ Rd such that t �→ Y (t, x) does not converge, t �→ ∥Y (t, x)∥ goes to +∞, and since the support of n0 is bounded, the points of the limit support are necessary in the unstable set of one of the equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The second part of the proposition is a consequence of the stable manifold theorem [33], which ensures that the unstable set of an equilibrium which is not a repellor is a smooth manifold of dimension at most d − 1, hence a set of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We conclude with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='2 Stationary solutions In this subsection, we define the stationary solution in the weak sense, which allows to include measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' As seen in the previous section, under appropriate hypotheses on f, the presence of a repellor is necessary to hope for solutions which converge to smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In this section, we prove that, under appropriate hypotheses, the presence of a repellor ensures the existence of smooth stationary solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Definition 2 (Weak stationary solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let µ be a finite positive Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We say that µ is a weak stationary solution of equation (1) if it satisfies ∀ϕ ∈ C1 c � Rd� , � Rd � f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕ(x) + (r(x) − µ(Rd))ϕ(x) � dµ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' (31) Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If x is a root of f, let us note that r(x)δx is a weak stationary solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The following proposition shows, as we might expect, that convergent solutions of (1) (in the weak sense) necessarily converge to a weak stationary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that r ∈ C0(Rd), and let n(t, ·) be a solution of (1) which converges in the weak sense in the space of Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then its limit is a weak stationary solution of equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We let µ be the limit of n(t, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us first prove that, under these conditions, ρ(t) = � Rd n(t, x)dx converges when t goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us denote ψ(t) := � Rd r(x)n(t, x)dx, which is non-negative, according to the non-negativity of r and n, and converges to ψ := � Rd r(x)dµ(x)dx, by definition of the weak convergence, and since r ∈ C0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us assume that ψ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ε ∈ (0, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ψ converges to ψ, and since ρ satisfies the ODE ˙ρ(t) = ψ(t) − ρ(t)2, there exists Tε > 0 such that for all t ≥ Tε, ψ − ε − ρ(t)2 ≤ ˙ρ(t) ≤ ψ + ε − ρ(t)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In other words, ρ is a super-solution of ˙u = ψ − ε − u2, and a sub-solution of ˙u = ψ + ε − u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since the solutions of these equations converge to � ψ − ε and � ψ + ε respectively, lim inf t→+∞ ρ(t) ≥ � ψ − ε and lim sup t→+∞ ρ(t) ≤ � ψ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since these inequalities hold for any ε ∈ (0, ψ), it proves that ρ indeed converges to � ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 32 If ψ = 0, we prove that lim sup ρ ≤ 0 with the same method, and the non-negativity of ψ ensures that lim inf ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ϕ ∈ C1 c � Rd� , and let us denote ρ := lim t→+∞ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' We recall that if a differentiable function converges, then its derivative is either divergent or converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, since t �→ � Rd ϕ(x)n(t, x)dx converges (by hypothesis), and d dt � Rd ϕ(x)n(t, x)dx = − � Rd ∇ · (f(x)n(t, x)) ϕ(x)dx � Rd (r(x) − ρ(t))ϕ(x)n(t, x)dx = + � Rd f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕ(x)n(t, x)dx + � Rd (r(x) − ρ(t)) ϕ(x)n(t, x)dx −→ t→+∞ � Rd f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕ(x) + (r(x) − ρ)ϕ(x)dµ(x), the equality � Rd f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕ(x) + (r(x) − ρ)ϕ(x)dµ(x) = 0 (32) holds for any ϕ ∈ C1 c(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' It remains to prove that µ(Rd) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' If ρ = 0, then the non-negativity of n and the definition of ρ lead to µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let us now assume that ρ > 0, and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since µ is a finite measure, r ∈ C0(Rd), and owing to the definition of ψ and ρ, there exists K ⊂ Rd a compact set such that µ(K) ≥ µ(Rd) − ε � K r(x)dµ(x)dx ≥ � Rd r(x)dµ(x)dx − ε = ρ2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let ϕK ∈ C1 c(Rd) such that ϕK ≡ 1 on K, 0 ≤ ϕ ≤ 1 on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since ∇ϕK ≡ 0 on K, ���� � Rd f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕK(x)dµ(x) ���� ≤ ∥f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕK∥∞µ(Rd\\K) ≤ ε∥f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇ϕK∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Moreover, according to the choice of ϕK � Rd r(x)ϕK(x)dµ(x) ∈ [ρ2 − ε, ρ2], and � Rd ϕK(x)dµ(x) ∈ [µ(Rd) − ε, µ(Rd)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Hence, injecting these inequalities in (32), we obtain −Cε ≤ ρ(ρ − µ(Rd)) ≤ Cε for some C ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Since this equality holds for any ε, and ρ is positive, it proves that µ(Rd) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Weak stationary solutions which are smooth enough (at least in C1(Rd)) are in fact stationary solutions in the strong sense, as defined in the following lemma, and can be further characterised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let n ∈ C1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, n is a weak stationary solution of (1) if and only if for all t ≥ 0, y ∈ Rd, � n(X(t, y)) = e � t 0 ˜r(X(s,y))−ρ ds n(y) � Rd n(x)dx = ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' First, let us note that, since n ∈ C1(Rd), one can integrate by parts in the expression (31) in order to prove that n is a weak stationary solution if and only if for any ϕ ∈ C1 c(Rd), � Rd (−∇ · (f(x)n(x)) + (r(x) − ρ) n(x)) ϕ(x)dx = 0, 33 with ρ = � Rd n(x)dx, which means that n is a weak stationary solution if and only if it is a stationary solution in the strong sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='e −∇ · (f(x)n(x)) + (r(x) − ρ) n(x) = 0 for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The result follows, since for any y ∈ Rd d dt � n(X(t, y))e− � t 0 ˜r(X(s,y))−ρ ds� = � f(X(t, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='∇n(X(t, y)) − (˜r(X(t, y) − ρ) n(X(t, y)) � e− � t 0 ˜r(X(s,y))−ρ ds = − � − ∇· (f(X(t, y))n(X(t, y))) + (r(X(t, y)) − ρ)n(X(t, y)) � e− � t 0 ˜r(X(s,y))−ρ ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Lemma 6 allows us to conclude that in the case where the ODE ˙u = f(u) has a repellor with a bounded unstable set, there exists a smooth stationary solution for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Let xu ∈ Rd be a repellor point for the ODE ˙x = f(x), and let us assume that n(x) := ˜r(xu) α e � +∞ 0 ˜r(Y (s,x))−˜r(xu)ds1B(x) is well-defined, and that n ∈ C1(B)∩L1(B), where B = {x ∈ Rd : Y (t, x) −→ t→+∞ xu} is the unstable set of xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Then, n is a C1 stationary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' For all y ∈ R, t ≥ 0, n(X(t, y)) = ˜r(xu) α e � +∞ 0 ˜r(X(t−s,x))−˜r(xu)ds = ˜r(xu) α e � t −∞ ˜r(X(s,y))−˜r(xu)ds, with the change of variable s′ = t − s and n(y) = ˜r(xu) α e � 0 −∞ ˜r(X(s,y))−˜r(xu)ds, with the change of variable s′ = −s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Thus, the equality of Lemma 6 holds, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' References [1] Matthieu Alfaro, Henri Berestycki, and Ga¨el Raoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The effect of climate shift on a species submitted to dispersion, evolution, growth, and nonlocal competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' SIAM Journal on Mathematical Analysis, 49(1):562–596, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [2] Lu´ıs Almeida, Patrizia Bagnerini, Giulia Fabrini, Barry D Hughes, and Tommaso Lorenzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype- structured model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' ESAIM: Mathematical Modelling and Numerical Analysis, 53(4):1157–1190, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [3] Guy Barles, Sepideh Mirrahimi, and Benoˆıt Perthame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Concentration in lotka-volterra parabolic or integral equations: a general convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Methods and Applications of Analysis, 16(3):321–340, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [4] Olivier Bonnefon, J´erˆome Coville, and Guillaume Legendre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Concentration phenomenon in some non- local equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' arXiv preprint arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='01971, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [5] Emeric Bouin, Vincent Calvez, Nicolas Meunier, Sepideh Mirrahimi, Benoˆıt Perthame, Ga¨el Raoul, and Rapha¨el Voituriez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Comptes Rendus Mathematique, 350(15-16):761–766, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 34 [6] `Angel Calsina and S´ılvia Cuadrado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Stationary solutions of a selection mutation model: The pure mutation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Mathematical Models and Methods in Applied Sciences, 15(07):1091–1117, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [7] `Angel Calsina, S´ılvia Cuadrado, Laurent Desvillettes, and Ga¨el Raoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Asymptotics of steady states of a selection–mutation equation for small mutation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143(6):1123–1146, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [8] Nicolas Champagnat, R´egis Ferri`ere, and Sylvie M´el´eard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Unifying evolutionary dynamics: from indi- vidual stochastic processes to macroscopic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Theoretical population biology, 69(3):297–321, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [9] Nicolas Champagnat, R´egis Ferri`ere, and Sylvie M´el´eard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' From individual stochastic processes to macro- scopic models in adaptive evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Stochastic Models, 24(sup1):2–44, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [10] Rebecca H Chisholm, Tommaso Lorenzi, and Alexander Lorz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Effects of an advection term in nonlocal lotka–volterra equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Communications in mathematical sciences, 14(4):1181–1188, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [11] Rebecca H Chisholm, Tommaso Lorenzi, Alexander Lorz, Annette K Larsen, Lu´ıs Neves de Almeida, Alexandre Escargueil, and Jean Clairambault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Cancer research, 75(6):930–939, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [12] Jerome Coville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Convergence to equilibrium for positive solutions of some mutation-selection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' arXiv preprint arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content='6471, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [13] Laurent Desvillettes, Pierre Emmanuel Jabin, St´ephane Mischler, and Ga¨el Raoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On selection dy- namics for continuous structured populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Communications in Mathematical Sciences, 6(3):729–747, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [14] Ulf Dieckmann and Richard Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The dynamical theory of coevolution: a derivation from stochastic ecological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of mathematical biology, 34(5):579–612, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [15] Odo Diekmann, Pierre-Emanuel Jabin, St´ephane Mischler, and Benoıt Perthame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' The dynamics of adaptation: an illuminating example and a hamilton–jacobi approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Theoretical population biology, 67(4):257–271, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [16] Ronald J DiPerna and Pierre-Louis Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Ordinary differential equations, transport theory and sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Inventiones mathematicae, 98(3):511–547, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [17] Frank Ernesto Alvarez and Jules Guilberteau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Particle method for adaptive dynamics equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' In preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [18] Timothy S Gardner, Charles R Cantor, and James J Collins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Construction of a genetic toggle switch in escherichia coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Nature, 403(6767):339–342, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [19] Stefan AH Geritz, Eva Kisdi, Johan AJ Metz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Evolutionary ecology, 12(1):35–57, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [20] Ra´ul Guantes and Juan F Poyatos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Multistable decision switches for flexible control of epigenetic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' PLoS computational biology, 4(11):e1000235, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [21] Mats Gyllenberg and G´eza Mesz´ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On the impossibility of coexistence of infinitely many strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of mathematical biology, 50(2):133–160, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [22] TG Hallam and CE Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Non-autonomous logistic equations as models of populations in a deteriorating environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of Theoretical Biology, 93(2):303–311, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [23] WGS Hines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Evolutionary stable strategies: a review of basic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Theoretical Population Biology, 31(2):195–272, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 35 [24] Morris W Hirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Systems of differential equations which are competitive or cooperative: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' limit sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' SIAM Journal on Mathematical Analysis, 13(2):167–179, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [25] Pierre-Emmanuel Jabin and Ga¨el Raoul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' On selection dynamics for competitive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of mathematical biology, 63(3):493–517, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [26] Dongya Jia, Mohit Kumar Jolly, William Harrison, Marcelo Boareto, Eshel Ben-Jacob, and Herbert Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Operating principles of tristable circuits regulating cellular differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Physical biology, 14(3):035007, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [27] Tommaso Lorenzi, Rebecca H Chisholm, Laurent Desvillettes, and Barry D Hughes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Dissecting the dy- namics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of theoretical biology, 386:166–176, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [28] Tommaso Lorenzi, Benoˆıt Perthame, and Xinran Ruan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Invasion fronts and adaptive dynamics in a model for the growth of cell populations with heterogeneous mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' European Journal of Applied Mathematics, 33(4):766–783, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [29] Tommaso Lorenzi and Camille Pouchol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Asymptotic analysis of selection-mutation models in the pres- ence of multiple fitness peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Nonlinearity, 33(11):5791, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [30] Alexander Lorz, Tommaso Lorenzi, Michael E Hochberg, Jean Clairambault, and Benoˆıt Perthame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Pop- ulational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' ESAIM: Mathematical Modelling and Numerical Analysis, 47(2):377–399, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [31] Johan AJ Metz, Stefan AH Geritz, G´eza Mesz´ena, Frans JA Jacobs, and Joost S Van Heerwaarden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [32] Philippe Michel, St´ephane Mischler, and Benoˆıt Perthame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' General relative entropy inequality: an illustration on growth models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal de math´ematiques pures et appliqu´ees, 84(9):1235–1260, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [33] Lawrence Perko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Differential equations and dynamical systems, volume 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Springer Science & Business Media, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [34] Benoˆıt Perthame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Transport equations in biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Springer Science & Business Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [35] Benoˆıt Perthame and Guy Barles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Dirac concentrations in lotka-volterra parabolic pdes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Indiana University Mathematics Journal, pages 3275–3301, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [36] Camille Pouchol and Emmanuel Tr´elat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Global stability with selection in integro-differential lotka- volterra systems modelling trait-structured populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Journal of Biological Dynamics, 12(1):872–893, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [37] Ren´e Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Laws for the dynamics of regulatory networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' International Journal of Developmental Biology, 42(3):479–485, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [38] John J Tyson and Bela Novak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' A dynamical paradigm for molecular cell biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Trends in Cell Biology, 30(7):504–515, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [39] Roderick Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Asymptotic approximations of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' SIAM, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' [40] Jingyu Zhang, Xiao-Jun Tian, Hang Zhang, Yue Teng, Ruoyan Li, Fan Bai, Subbiah Elankumaran, and Jianhua Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Tgf-β–induced epithelial-to-mesenchymal transition proceeds through stepwise activation of multiple feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' Science signaling, 7(345):ra91–ra91, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE0T4oBgHgl3EQfkQFh/content/2301.02470v1.pdf'} diff --git a/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf 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Ashour‡ +N.S. Aybat†§ +Constantino M. Lagoa⋆ +⋆Dep. of Electrical Engineering, Pennsylvania State University, State College, PA 16801, USA. +‡Wireless R&D Department, Qualcomm Technologies, Inc, San Diego, CA 92121, USA. +†Department of Industrial and Manufacturing Engineering, Pennsylvania State University, State College, PA +16801, USA. +Email: { oms46@psu.edu, mashour@qti.qualcomm.com, nsa10@psu.edu, cml18@psu.edu } +Abstract +Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing. +Computations over sparse structures are less complex compared to their dense counterparts, and their +storage consumes less space. This paper proposes a heuristic method for retrieving sparse approximate +solutions of optimization problems via minimizing the ℓp quasi-norm, where 0 < p < 1. An iterative +two-block ADMM algorithm for minimizing the ℓp quasi-norm subject to convex constraints is proposed. +For p = s/q < 1, s, q ∈ Z+, the proposed algorithm requires solving for the roots of a scalar +degree 2q polynomial as opposed to applying a soft thresholding operator in the case of ℓ1. The +merit of that algorithm relies on its ability to solve the ℓp quasi-norm minimization subject to any +convex set of constraints. However, it suffers from low speed, due to a convex projection step in +each iteration, and the lack of mathematical convergence guarantee. We then aim to vanquish these +shortcomings by relaxing the assumption on the constraints set to be the set formed due to convex +and differentiable, with Lipschitz continuous gradient, functions, i.e. specifically, polytope sets. Using a +proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm speed +while proving its convergence. We then present various applications where the proposed algorithm excels, +namely, matrix rank minimization, sparse signal reconstruction from noisy measurements, sparse binary +classification, and system identification. The results demonstrate the significant gains obtained by the +proposed algorithm compared to those via ℓ1 minimization. +Keywords— Sparsity, compressed sensing, rank minimization, ADMM, Proximal gradient method. + +2 +I. INTRODUCTION +A. Motivation +In numerical analysis and scientific computing, a sparse matrix/array is the one with many of its elements being +zeros. The number of zeros divided by the total number of elements is called sparsity. Sparse data is often easier to +store and process. Hence, techniques for deriving sparse solutions and exploiting them have attracted the attention +of many researchers in various engineering fields like machine learning, signal processing, and control theory. +The taxonomy of sparsity can be studied through the rank minimization problem (RMP). It has been lately +considered in many engineering applications including control design and system identification. This is because the +notions of complexity and system order can be closely related to the matrix rank. The RMP can be formulated as +follows: +min +X +Rank(X), +s.t. +X ∈ M, +(1) +where X ∈ Rm×n and M ⊂ Rm×n is a convex set. The problem (1) in its generality is NP-hard [1]. Therefore, +polynomial time algorithms for solving large-scale problems of the form in (1) are not currently known. Hence, +currently adopted methods for solving such problems are approximate and structured heuristics. +A special case of RMP is the sparse vector recovery (SVR) problem involving ℓ0 pseudo-norm minimization +given by: +min +x +∥x∥0 , +s.t. +x ∈ V, +(2) +where x ∈ Rn, V ⊂ Rn is a closed convex set and ∥·∥0 counts the number of the non-zero elements of its argument. +From the definition of the rank being the number of non-zero singular values of a matrix, it can be easily realized +that (1) is a generalized form of (2). +Various works – which will be discussed in the next section in more detail – have explored efficient solution +techniques for the problems in (1) and (2) independently using Schatten-p and ℓp quasi-norm relaxations respectively. +However, all of these methods either assume a specific structure for the convex set M in (1) or only work for the +special case in (2); hence, they lack generality. Using the fact that the Schatten-p quasi-norm is the ℓp quasi-norm +of the matrix singular values, we aim to design an efficient heuristic method based on Schatten-p relaxation for +solving both problems in a unified manner. To achieve this, we first propose an algorithm for solving ℓp quasi-norm +relaxation of the SVR problem in (2), and next, exploiting the fact that (2) is a special case of (1), we then employ +the derived ℓp quasi-norm minimization algorithm as a building block for the desired generalized algorithm for the +rank minimization problem. +B. Related work +1) Sparse Vector Recovery: As discussed, a sparse solution is defined as the one having the minimum +number of non-zero components while satisfying certain constraints such as a system of linear equations +[2]. +Since many signals are either sparse or compressible, SVR problem has found applications in object recognition, +classification and compressed sensing problems, see, e.g., [3]–[5]. In [6], the authors discussed the concept of the +sparse representation of signals and systems, where they reviewed the theoretical and empirical results on sparse + +3 +optimization, and discussed sufficient conditions needed for uniqueness, stability and computational practicability. +Different applications for the SVR problem are explored in [6] and it is argued that in certain denoising and +compression tasks, the methods for sparse optimization provide state of the art solutions. +The problem of constructing a sparse solution to undetermined linear systems has received great attention. In +[7], the authors surveyed the existing algorithms for sparse approximation, namely; greedy methods [8], [9], the +methods based on convex relaxation [4], [5], [10], [11], those involving non-convex optimization [12], [13], Bayesian +framework [14], [15], and requiring brute force [16]. They also discussed the computational requirements of each +algorithm and their relation to each other. +Sparse optimization problems of the form min{f(x) + µg(x)} have been extensively studied in the literature, +where g(x) is a sparsity inducing function, e.g., ℓ1-norm, f is a loss function on measurement errors, e.g., f(x) = +∥Ax − b∥2, and µ > 0 is a trade-off parameter between data-fidelity and sparsity. In [2], the authors considered +sparse recovery problem from a set of corrupted measurements. For g(·) = ∥·∥1, they established a sufficient +condition for the exact sparse signal recovery, i.e., restricted isometry property (RIP). Motivated by the fact that +∥x∥p +p → ∥x∥0 as p → 0, it is natural to consider the above problem with g set to ℓp quasi-norm for p ∈ (0, 1). +Hence, the authors in [17] presented theoretical results demonstrating the ability of the ℓp quasi-norm to recover +sparse signals from noisy measurements. Under more relaxed RIP conditions, they showed that the ℓp quasi-norm +provides better theoretical guarantees in terms of stability and robustness than the ℓ1 minimization. In [12], the +problem of SVR via the ℓp quasi-norm minimization from small number of linear measurements of the target +signal was considered. This setting is important in applications where data acquisition is difficult or expensive. +However, the proposed approach in [12] has limited applicability due to its long reconstruction time compared +to the ℓ1 norm. In [18], the authors exploited Fourier-based algorithms for convex optimization to solve sparse +signals reconstruction problem via the ℓp quasi-norm minimization. They showed that their approach combines the +construction abilities of the non-convex methods with the speed of the convex ones. In [19] the authors proposed +an approach, for sparse reconstruction, replacing the non-convex function with a quadratic convex one. In [20], +an alternating direction method of multipliers (ADMM) based algorithm that enforces both sparsity and group +sparsity using non-convex regularization is presented. An iterative half thresholding algorithm for fast solution of +the ℓ0.5 regularization is proposed in [21]. The authors proved the existence of the resolvent of gradient of ||x||0.5 +0.5 +, calculated its analytic expression, and derived a thresholding representation of solutions for ℓ0.5 regularization. In +[22], the convergence of the iterative half thresholding algorithm is studied, where it was shown that, under certain +conditions, the half thresholding algorithm converges, to a local minimizer of the regularized problem, with a linear +convergence rate. Conditions for the convergence of an ADMM algorithm that solves the problem of minimizing +the sum of a smooth function with a bounded Hessian and a non-smooth function are derived in [23]. In [24], the +convergence of ADMM for minimizing a non-convex and possibly non-smooth objective function subject to equality +constraints is analyzed. The developed convergence guarantee covers a variety of non-convex objectives including +piece-wise linear functions, ℓp quasi-norm and Schatten-p quasi-norm (0 < p < 1), while allowing non-convex +constraints as well. + +4 +2) Rank minimization: In [25], the authors aimed at determining the least order dynamic output feedback, +using same formulation as in (1), which stabilizes a given linear time invariant system. They found that minimizing +the trace instead of the rank results in a Semi-Definite Program (SDP) that can be solved efficiently. However, their +solution was only applicable for symmetric and square matrices. In [26], a generalization of the latter approach +was introduced which is based on replacing the rank in the objective function with the summation of the singular +values of the matrix, i.e., the nuclear norm. They showed that this leads to the convex envelope of the non- +convex rank objective and boils down to the original trace heuristic when the decision matrix is a symmetric +positive semi-definite (PSD) matrix. Finally, effectiveness of the approach was shown using a frequency domain +system identification problem. In [27], another heuristic based on the logarithm of the determinant was presented +as a surrogate for the rank minimization over the subspace of PSD matrices, and the authors showed that this +formulation can be solved using a sequence of trace minimization problems. The authors also extended their +heuristic to handle matrices that are not necessarily PSD. In [28], the authors also studied the existing trace and log +determinant heuristics for approximating (1). They discussed the applications of these heuristics for computing a +low-rank approximation of 1) covariance matrices for a given dataset so that one can obtain a simple data model, +easy to interpret, which is especially important in statistics and signal processing; 2) Hankel matrices arising in +system identification of a time invariant, low-order system for given output realizations; and 3) matrices appearing +in various other problems including H∞ and reduced-order µ-synthesis with constant scaling and problems with +inertia constraints. +Although the nuclear norm is the tightest convex substitute for the non-convex rank function, one of its major +shortcomings is that it treats all the singular values equally in order to be able to preserve the convexity. Therefore, +this restricts its performance in applications where the singular values need to be treated differently, e.g., particularly +in image denoising. In [29], the authors proposed an iterative re-weighted nuclear norm heuristic to avoid this +problem and analyzed its convergence. They also proposed a gradient-based algorithm and applied it to a low-order +system identification problem. Experimental results showed that the re-weighted nuclear norm leads to a lower order +model than the nuclear norm itself. In [30], the solution of weighted nuclear norm (WNN) problem was analyzed +under different circumstances where the weights could be in a non-ascending, arbitrary or a non-descending order. +The authors applied their proposed WNN algorithm to an image denoising problem by exploiting the image non- +local self-similarity. Numerical results showed that the proposed WNN algorithm outperforms many of the state of +the art denoising methods in terms of both quantitative measures and visual quality. +Another method, inspired by the success of the ℓp quasi-norm (0 < p < 1) for sparse signal reconstruction, +is to enforce low-rank structure by using the Schatten-p quasi-norm, which is defined as the ℓp quasi-norm of +the singular values. In [31], the authors considered the matrix completion problem, which deals with constructing +a low-rank matrix, given a subset of its entries. Instead of minimizing the nuclear norm, the authors proposed +a Schatten-p quasi-norm formulation, for which they came up with an algorithm and studied its convergence +properties. In each iteration, the sub-problem that needs to be solved has a closed-form solution, which makes +it fast and suitable for large-scale problems. To improve the robustness of the solutions to matrix completion +problem, in [32] Schatten-p quasi-norm for low-rank recovery was combined with ℓp quasi-norm (0 < p ≤ 1) + +5 +of the prediction errors on the observed entries. The authors proposed an algorithm based on the ADMM, which +performed better in their numerical experiments than other completion methods like fixed-point continuation and +accelerated proximal gradient singular-value thresholding. In [33], the authors extended the theoretical recovery +results previously developped for the nuclear norm to Schatten-p quasi-norm using a weaker version of the RIP +assumption; they showed that the minimum rank solution can be recovered by solving the Schatten-p quasi-norm +minimization problem. In [34], the authors developed an iterative re-weighted least squares algorithm to solve an +unconstrained ℓp minimization problem. The algorithm, and analysis, are extended to include the low rank recovery +problem. Another non-convex approaches for matrix optimization problems involving sparsity are developed, by +means of a generalized shrinkage operation, in [35]. These approaches are applied to the decomposition of video +into low rank and sparse components, which is able to separate moving objects from the stationary background +better than in the convex case. +C. Contributions +Despite the good performance of the various algorithms discussed in I-B1 and I-B2 for solving different +relaxations of (2) and (1), they are all based on a specific structure of the considered convex constraint set; +therefore, they are problem specific and lack generality. In this work, we present a general-purpose method based +on projections onto the constraint set, for which we assume only closed convexity as the specific structure. [36] +and [37] examine the characteristics of the projection on the constraint set. In the latter, the projection technique +of every given point on ℓp balls is assumed to be known a priori, whereas in the former, the issue is solved while +omitting an important coupling condition for the polynomial equations. +First, based on our previous work in [38], we propose an ADMM algorithm (pQN-ADMM) to solve the ℓp +quasi-norm relaxation of (2). At each iteration, the bottleneck operation is to compute Euclidean projections on +to some particular convex and non-convex sets. The proposed algorithm possesses two important properties: 1) +Its computational complexity is similar to ℓ1 minimization algorithms except for the additional effort of solving +for the roots of a polynomial; 2) No specific structure for the convex set is required. Numerical results, using +a SVR and binary classification examples, are presented to show the competitive performance of our algorithm +against the ℓ1 minimization approach. We then extend the proposed algorithm to solve the relaxation of (1) based +on the Schatten-p quasi-norm – here, we exploit the fact that minimizing the ℓp-norm of the vector of singular +values is equivalent to minimizing the Schatten-p quasi-norm. We consider two different numerical examples: 1) +We formulate time domain system identification problem for minimum order system detection and solve it using the +pQN-ADMM approach and compare our recovery results against the nuclear norm minimization approach of [26]; +2) We consider a matrix completion problem, where the goal is to recover an unknown low-rank matrix based on a +small fraction of observed entries. Numerical results in both examples show that our method is competitive against +some of the state-of-the-art algorithms in terms of both the detected system order (for the system identification +problem) and the rank of the matrix recovered (for the matrix completion problem). +Finally, since the derived algorithm depends on a computationally expensive convex projection step in every +iteration, we aim to develop a faster algorithm with a mathematical convergence guarantee. Considering only a + +6 +subset of problems where the constraint set is a polytope, we utilize concepts from the proximal gradient (PG) +method to derive a fast algorithm and prove that it converges with a rate O( 1 +K ), where K is the iteration budget +given to the algorithm. +II. NOTATIONS AND BASIC DEFINITIONS +Unless otherwise specified, we denote vectors with lowercase boldface letters, i.e., x, with i-th entry as xi, while +matrices are in uppercase, i.e. X, with (i, j)-th entry as xi,j. For an integer n ∈ Z+, [n] +∆= {1, . . ., n}. 1 represents +a vector of all entries equal to 1, while +1G(.) is an indicator function to the set G, i.e., it evaluates to zero if its +argument belongs to the set G and is +∞ otherwise. +For a vector x ∈ Rn, the general ℓp norm is defined as +∥x∥p +∆= ( +� +i∈[n] +|xi|p) +1 +p . +(3) +For convenience, we let ∥x∥ be the well known euclidean norm, i.e., p = 2. When 0 < p < 1, the expression in +(3) is termed as the quasi-norm satisfying the same axioms of the norm except the triangular inequality making it +a non-convex function. +Definition 1. Let X : V −→ W be a linear operator between two normed spaces equipped with ℓp norm, +p ∈ [1, ∞). The induced p-norm is defined as, +∥X∥p +∆= sup +v̸=0 +�∥Xv∥p +∥v∥p +� +. +(4) +A special case of (4) is when p = 2, known as the spectral radius, which can be shown to be the square root of +the maximum eigen value of XHX, where XH is the complex conjugate of the transpose of X, i.e., X⊤. In the +rest of the analysis, we will drop the subscript 2 in the spectral norm notation and only refer to it with ∥.∥. +For a matrix X ∈ Rm×n, the Lp,q entry-wise norm is defined as, +∥X∥p,q +∆= ( +� +j∈[n] +( +� +i∈[m] +|xij|p) +q +p ) +1 +q . +(5) +A special case of (5) is when p = q = 2, known as the Frobenius norm, which were refer to by ∥.∥f. +Definition 2. Let H1 ⊂ Rn and H2 ⊂ Rm be two separable Hilbert spaces and X ∈ Rm×n be a linear compact +operator from H1 to H2, the Schatten-p norm of X is then defined as, +∥X∥p +∆= ( +� +i∈[min{m,n}] +σi(X)p) +1 +p , +(6) +where σi(X) is the i-th singular value of the matrix X. +When p = 1, equation (6) yields to the nuclear norm which is the convex envelope of the rank function. +Throughout the paper, we consider a non-convex relaxation for the rank function, specifically p = 1/2, and compare +its performance with the nuclear norm case in the results section. + +7 +We define ⌈·⌉ as the ceiling operator, vec(X) ∈ Rmn as a vector formed by stacking the columns of the +matrix X ∈ Rm×n and Hankel(.) as an operator that outputs a Hankel matrix constructed from the applied vector +arguments. +III. +SPARSE VECTOR RECOVERY ALGORITHM +A. Problem Formulation +This section develops a method for approximating the solution of (2) using the following relaxation, +min +x +∥x∥p +p , +s.t. +x ∈ V, +(7) +where p ∈ (0, 1] and V is a closed convex set. Problem (7) is convex for p = 1; hence, can be solved to optimality +efficiently. However, the problem becomes non-convex when p < 1. An epigraph equivalent formulation of (7) is +obtained by introducing the variable t = [ti]i∈[n]: +min +x,t +1⊤t, +s.t. +ti ≥ |xi|p, +i ∈ [n], +x ∈ V. +(8) +Let X ⊂ R2 denote the epigraph of the scalar function |x|p, i.e., X = {(x, t) ∈ R2 : t ≥ |x|p}, which is a +non-convex set for p < 1. Then, (8) can be cast as +min +x,t +� +i∈[n] +1X (xi, ti) + 1⊤t, +s.t. +x ∈ V. +(9) +ADMM exploits the structure of the problem to split the optimization over the variables via iteratively solving fairly +simple subproblems. In particular, we introduce auxiliary variables y = [yi]i∈[n] and z = [zi]i∈[n] and obtain an +ADMM equivalent formulation of (9) given by: +min +x,t,y,z +� +i∈[n] +1X (xi, ti) + +1Y(y) + 1⊤z, +s.t. +x = y : λ, +t = z : θ, +(10) +where Y is the 0-sublevel set of f, i.e., Y = {y ∈ Rn : y ∈ V}. The dual variables associated with the constraints +x = y and t = z are λ and θ, respectively. Hence, the Lagrangian function corresponding to (10) augmented with +a quadratic penalty on the violation of the equality constraints with penalty parameter ρ > 0, is given by: +Lρ(x, t, y, z, λ, θ) = +� +i∈[n] +1X (xi, ti) + +1Y(y) + 1⊤z ++ λ⊤(x − y) + θ⊤(t − z) + ρ +2 +� +∥x − y∥2 + ∥t − z∥2� +. +(11) +Considering the two block variables (x, t) and (y, z), ADMM [39] consists of the following iterations: +(x, t)k+1 += +argmin +x,t +Lρ(x, t, yk, zk, λk, θk) +(12) +(y, z)k+1 += +argmin +y,z +Lρ(xk+1, tk+1, y, z, λk, θk) +(13) +λk+1 += +λk + ρ(xk+1 − yk+1) +(14) +θk+1 += +θk + ρ(tk+1 − zk+1). +(15) + +8 +Algorithm 1 ADMM (ρ > 0) +1: Initialize: y0, z0, λ0, θ0 +2: for k ≥ 0 do +3: +(xi, ti)k+1 ← ΠX +� +yk +i − λk +i +ρ , zk +i − θk +i +ρ +� +, ∀i ∈ [n] +4: +yk+1 ← ΠY +� +xk+1 + λk +ρ +� +5: +zk+1 ← tk+1 + θk−1 +ρ +6: +λk+1 ← λk + ρ(xk+1 − yk+1) +7: +θk+1 ← θk + ρ(tk+1 − zk+1). +According to the expression of the augmented Lagrangian function in (11), it follows from (12) that the variables +x and t are updated via solving the following non-convex problem +min +x,t +∥x − yk + λk +ρ ∥2 + ∥t − zk + θk +ρ ∥2 +s.t. +(xi, ti) ∈ X, +i ∈ [n]. +(16) +Exploiting the separable structure of (16), one immediately concludes that (16) can be split into n independent +2-dimensional problems that can be solved in parallel, i.e., for each i ∈ [n], +(xi, ti)k+1 = ΠX +� +yk +i − λk +i +ρ , zk +i − θk +i +ρ +� +, +(17) +where ΠX (.) denotes the Euclidean projection operator onto the set X. Furthermore, (11) and (13) imply that y +and z are independently updated as follows: +yk+1 += +ΠY +� +xk+1 + λk +ρ +� +(18) +zk+1 += +tk+1 + θk − 1 +ρ +. +(19) +Algorithm 1 summarizes the proposed ADMM algorithm. It is clear that z, λ, and θ merit closed-form updates. +However, updating (x, t) requires solving n non-convex problems. Our strategy for dealing with this issue is +presented in the section that follows. +B. Non-convex Projection +In this part, we present the method used to tackle the non-convex projection problem required to update x and +t. +Among the advantages of the proposed algorithm is that it is amenable to decentralization. As it is clear from +(17), x and t can be updated element-wise via performing a projection operation onto the non-convex set X, one +for each i ∈ [n]. The n projection problems can be run independently in parallel. We now outline the proposed +idea for solving one such projection, i.e., we suppress the dependence on the index of the entry of x and t. For +(¯x, ¯t) ∈ R2, ΠX (¯x, ¯t) entails solving +min +x,t +g(x, t) ≜ (t − ¯t)2 + (x − ¯x)2, +s.t. +t ≥ |x|p. +(20) + +9 +If ¯t ≥ |¯x|p, then trivially ΠX (¯x, ¯t) = (¯x, ¯t). Thus, we focus on the case in which ¯t < |¯x|p. The following proposition +states the necessary optimality conditions for (20). +Proposition 1. Let ¯t < |¯x|p, and (x∗, t∗) be an optimal solution of (20). Then, the following properties are satisfied +(a) sign(x∗) = sign(¯x), +(b) t∗ ≥ ¯t, +(c) |x∗|p ≥ ¯t, +(d) t∗ = |x∗|p. +Proof. We prove the statements by contradiction as follows: +(a) Suppose that sign(x∗) ̸= sign(¯x), then +|x∗ − ¯x|=|x∗ − 0|+|¯x − 0| > |¯x − 0|, +(21) +i.e., (x∗−¯x)2 >(0−¯x)2. Hence, g(x∗, t∗)−g(0, t∗)>0. Moreover, the feasibility of (x∗, t∗) implies that t∗ > 0. +Thus, (0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts the +optimality of (x∗, t∗). +(b) Assume that t∗ < ¯t. Then, +g(x∗, t∗) − g(x∗, ¯t) = (t∗ − ¯t)2 > 0. +(22) +Furthermore, by the feasibility of (x∗, t∗), we have |x∗|p ≤ t∗ < ¯t. Thus, (x∗, ¯t) is feasible and attains a lower +objective value than that attained by (x∗, t∗). This contradicts the optimality of (x∗, t∗). +(c) Suppose that |x∗|p < ¯t, i.e., +− ¯t +1 +p < x∗ < ¯t +1 +p . +(23) +We now consider two cases, ¯x > 0 and ¯x < 0. First, let ¯x > 0. Then, we have by (a) and (23) that 0 < x∗ < ¯t +1 +p . +Since ¯t < |¯x|p, i.e., (¯x, ¯t) /∈ X, therefore ¯t +1 +p < ¯x and hence, 0 < x∗ < ¯t +1 +p < ¯x. Pick x0 > 0 such that |x0|p = ¯t, +i.e., x0 = ¯t +1 +p . Then clearly, x∗ < x0 < ¯x. Thus, we have +g(x∗, t∗) − g(x0, t∗) = (x∗ − ¯x)2 − (x0 − ¯x)2 > 0, +(24) +where the last inequality follows the just proven identity that x∗ < x0 < ¯x. Moreover, we have |x0|p = ¯t ≤ t∗ by +(b). Thus, (x0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts +the optimality of (x∗, t∗). On the other hand, let ¯x < 0. Then, we have by (a) and (23) that −¯t +1 +p < x∗ < 0. +Since ¯t < |¯x|p, i.e., (¯x, ¯t) /∈ X, then ¯t +1 +p < |¯x|, i.e., ¯x < −¯t +1 +p . Therefore, ¯x < −¯t +1 +p < x∗. Pick x0 < 0 such that +|x0|p = ¯t, i.e., x0 = −¯t +1 +p . Then, (24) also holds when ¯x < 0. Note that |x0|p = ¯t ≤ t∗ by (b). Thus, (x0, t∗) +is feasible and attains a lower objective value than that attained by (x∗, t∗). This contradicts the optimality of +(x∗, t∗). +(d) The feasibility of (x∗, t∗) eliminates the possibility that t∗ < |x∗|p. Now let t∗ > |x∗|p and pick t0 = |x∗|p. +Then, ¯t ≤ |x∗|p = t0 < t∗, where the first inequality follows from (c). Then, 0 ≤ t0 − ¯t < t∗ − ¯t. Thus, we +have +g(x∗, t∗) − g(x∗, t0) = (t∗ − ¯t)2 − (t0 − ¯t)2 > 0, +(25) + +10 +Algorithm 2 Non-convex projection (p = s +q < 1) +1: R ← roots{a2q + s +q(a2s − ¯tas) − |¯x|aq} +2: ¯R ← R \ {complex numbers and negative reals in R} +3: T ← {(rq, rs) : r ∈ ¯R} +4: (ˆx, t∗) ← argmin {g(x, t) : (x, t) ∈ T } +5: x∗ ← sign(¯x)ˆx +Furthermore, the feasibility of (x∗, t0) follows trivially from the choice of t0. Thus, (x∗, t0) is feasible and +attains a lower objective value than that attained by (x∗, t∗). This contradicts the optimality of (x∗, t∗). +This concludes the proof. +We now make use of the fact that for (20), an optimal solution (x∗, t∗) satisfies t∗ = |x∗|p and hence, (20) +reduces to solving +min +x +(|x|p − ¯t)2 + (x − ¯x)2. +(26) +The first order necessary optimality condition for (26) implies the following: +p|x∗|p−1sign(x∗)(|x∗|p − ¯t) + x∗ − ¯x = 0. +(27) +By the symmetry of the function |x|p, without loss of generality, assume that x∗ > 0 and let 0 < p = s +q < 1 for +some s, q ∈ Z+. A change of variables aq = x∗ plugged in (27) shows that finding an optimal solution for (20) +reduces to finding a root of the following scalar degree 2q polynomial: +a2q + s +q +� +a2s − ¯tas� +− ¯xaq. +(28) +Thus, to find ΠX (¯x, ¯t), solve for a root a∗ of the polynomial in (28) such that (a∗q, a∗s) minimizes g(x, t). Algorithm +2 summarizes the method we use to solve problem (20). In case ¯x = 0, we set x∗ = t∗ = 0. If the set ¯R is empty, +we set x∗ = 0 and t∗ = (¯t)+. +C. Convex Projection +The convex projection for y-update in (18) can be formulated as the following convex optimization problem +yk+1 = argmin +y +�����y − (xk+1 + λk +ρ ) +����� +2 +, +s.t. +y ∈ V, +(29) +where ∥.∥ is the euclidean norm. Convex problems can be solved by a variety of contemporary methods including +bundle methods [40], sub-gradient projection [41], interior point methods [42], and ellipsoid methods [43]. The +efficiency of optimization techniques rely mainly on exploiting the structure of the constraint set. As mentioned +in I-C, to be general, we aim to solve the problem in (7) with no assumptions on the set V, other than it being +closed and convex. That said, if possible, through exploiting the structure of V, one should be able to reduce the +computational complexity of solving (29). + +11 +Remark 1. As per our knowledge, none of the existing literature considered the convergence of an ADMM algorithm +for solving the general problem in (7). As discussed in I-B1, on one hand, the work in [23] studied the convergence +of ADMM under mild assumptions. However, assuming V has a particular form, these assumptions hold only if the +function f defining the the constraint set V = {x : f(x) ≤ 0} in (7) is Lipschitz differentiable. On the other hand, +[44] studied the convergence of a non ADMM algorithm to solve (7) while assuming that the global optimal for +each update step can be found efficiently. +IV. RANK MINIMIZATION ALGORITHM +We consider the same problem as in (1) and propose a method for approximating its solution efficiently. The +Schatten-p heuristic of (1) can be written as +min +X +∥X∥p +p +∆= +L +� +i=1 +|σi(X)|p, +s.t. +X ∈ M, +(30) +where L = min(m, n) and σi(X) is the ith singular value of X. When p = 1, problem (30) is a convex one +which is eventually the nuclear norm heuristic. We consider a non-convex case where 0 < p < 1, which has the +corresponding epi-graph form, +min +X,t +1⊤t, +s.t. +|σi(X)|p ≤ ti, +i ∈ {1, . . .L}, +X ∈ M, +(31) +such that t = [ti]i∈[L]. Defining the epi-graph set ˚ +X for the function σ(X), where ˚ +X +∆= {(σ(X), t)∈R2 :|σ(X)|p ≤ +t} ⊆ R2, the problem in (31) can be written as, +min +X,t +1⊤t + +1M(X) + +L +� +i=1 +1 ˚ +X (σi(X), ti). +(32) +In order to structure the problem in a from that ADMM can exploit, we introduce the auxiliary variables +Y ∈ Rm×n and z = [zi]i∈[L] which makes the problem in (32) be, +min +X,t,Y,z +1⊤z + +1V(Y) + +L +� +i=1 +1 ˚ +X (σi(X), ti), +s.t. +X = Y : Λ, +t = z : θ, +(33) +such that Λ, θ are the dual variables associated with X and t respectively. Similar to (11), the Lagrangian function +associated with (33) augmented with a quadratic penalty for the equality constraint violation with a parameter +ρ > 0, is +Lρ(X, Y, t, z, Λ, θ)=1⊤z+ +1M(Y)+ +L +� +i=1 +1 ˚ +X (σi(X), ti) ++T r{Λ⊤(X−Y)}+θ⊤(t−z)+ ρ +2(∥X−Y∥2 +f +∥t−z∥2), +(34) + +12 +where T r{.} is the trace operator. Considering the 2-tuples (X, t) and (Y, z), the ADMM iterations is, +(X, t)k+1 = argmin +X,t +Lρ(X, Yk, t, zk, Λk, θk), +(35) +Yk+1 = argmin +Y +Lρ(Xk+1, Y, tk+1, zk, Λk, θk), +(36) +zk+1 = argmin +z +Lρ(Xk+1, Yk+1, tk+1, z, Λk, θk), +(37) +Λk+1 = Λk + ρ(Xk+1 − Yk+1), +(38) +θk+1 = θk + ρ(tk+1 − zk+1). +(39) +A. (X, t) update +By completing the square and with some simple algebra, it can be shown that the problem in (35) is equivalent +to +min +X,t +��X − ¯Xk��2 +f + +��t − ¯tk��2 , +s.t. +|σi(X)|p ≤ ti, +i ∈ {1, . . . L}, +(40) +where ¯Xk ∆= Yk − Λk +ρ +and ¯tk ∆= zk − θk +ρ . For an ease of notations, we will drop the iteration index k. Assume +that X = PΣQ⊤ and ¯X = U∆V⊤ is the singular value decomposition (SVD) of X and ¯X respectively. Where +Σ, ∆ ∈ RL×L are diagonal matrices with the singular values associated X and ¯X while P, U ∈ Rm×L and +Q, V ∈ Rn×L are the unitary matrices. By applying the same steps as in Theorem 3 of [45], we can write the first +term of (40) after dropping k as, +��X − ¯X +��2 +f = +��PΣQ⊤ − U∆V⊤��2 +f += +��PΣQ⊤��2 +f + +��U∆V⊤��2 +f − 2T {X⊤ ¯X} +(a) += T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{QΣ⊤P⊤U∆V⊤} +(b) +≥ T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{Σ⊤∆}=∥Σ−∆∥2 +f , +(41) +where (a) is because P⊤P = Q⊤Q = U⊤U = V⊤V = IL×L with IL×L being an identity matrix of size L, +and exploiting the circular property of the trace while (b) holds is from the main result of [46]. In order to make +��X − ¯Xk��2 +f achieve its derived lower bound, we set P = U and Q = V. +Henceforth, the problem in (40) will be equivalent to, +min +X,t +∥x − ¯x∥2 + ∥t − ¯t∥2 , +s.t. +|xi|p ≤ ti, +i ∈ {1, . . . L}, +(42) +where x = [xi]i∈[L] and ¯x = [¯xi]i∈[L] are the vectors of singular values of the matrices X and ¯X respectively. +The optimal solution X∗ for (40) can be calculated by finding the optimal x∗ of (42) and then X∗ = UΣ∗VT , +where Σ∗ =diag(x∗) and diag(.) is an operator that converts a vector to its corresponding diagonal matrix. Since +the problem in (42) is separable, we drop the index i and only consider solving +min +x,t +(x − ¯x)2 + (t − ¯t)2, +s.t. +|x|p ≤ t. +(43) + +13 +It can be realized that (43) is the same as (20), hence, its optimal solution can be found by applying algorithm +2. +B. (Y, z) update +After updating (X, t) while fixing Λ and θ, the problem in (36) can be written as, +Yk+1 =argmin +Y +�����Y−(Xk+1+ Λk +ρ ) +����� +2 +f +, s.t. +Y ∈ M, +(44) +which is clearly a convex optimization problem representing the projection of the point Xk+1 + Λk +ρ on the set M +and can be solved by various known class of algorithms as discussed in section III-C. +Upon updating Y, the z update in (37) is +zk+1 = argmin +z +1⊤z + ρ +2 +�����z − (tk+1 + θk +ρ ) +����� +2 +, +(45) +which has the closed-form solution z = tk+1 + θk−1 +ρ +. +V. PROXIMAL GRADIENT ALGORITHM +The SVR algorithm deals with the ℓp relaxation of (2) without assuming any specific structure for V, other than +being closed and convex. Indeed, the algorithm only requires the Euclidean projections onto V as in (29). However, +this approach suffers from two pitfalls: 1) high computational complexity per iteration as a result of solving (29) +in every iteration, and 2) the lack of convergence guarantees. +In this section, we consider a sub-class of problems with a specific structure for the convex set of the form +V = {x : f(x) ≤ 0}, where f(x) = ∥Ax − b∥2 − ǫ for some given ǫ ≥ 0, A ∈ Rm×n and b ∈ Rm. Note that f(x) +is a convex function with Lipschitz continuous gradient. i.e., f is L-smooth: ∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥ for +all x, y ∈ Rn and L ≜ ∥A∥2. Specifically, in order to solve +min +x +∥x∥p +p , +s.t. +f(x) ≤ 0, +(46) +we aim to develop an efficient algorithm with some convergence guarantees for the following Lagrangian relaxation: +min +x +F(x) +∆= ∥x∥p +p + µ +2 f(x), +(47) +where µ ≥ 0 is the dual multiplier that captures the trade-off between solution sparsity and fidelity. +A canonical problem for the regularized risk minimization has the following form: +min +x +g(x) + h(x) +(48) +where h is an L-smooth loss function and g is a a regularizer term. When both g and h are convex, the proximal +gradient (PG) algorithm [47] can compute a solution to (48) through iteratively taking PG steps, i.e., xk+1 = +proxg/λ(xk − ∇h(xk)/L) where proxg/λ(.) +∆= argminx g(x) + λ +2 ∥x − ·∥2, for some constant λ. When g is +convex, prox operation is well-defined; thus, the PG step can be computed. + +14 +Comparing both (47) and (48), the convexity assumption of g(x) in (48) is not satisfied for ∥x∥p +p in (47). When +the regularizer is a continuous nonconvex function, the proximal map proxg/λ may not exist, let alone it can be +computed in closed form. On the other hand, for ∥x∥p +p, using similar arguments for the non-convex projection step +introduced in subsection III-B, we aim to derive an analytical solution that can be computed efficiently. Indeed, +assuming p ∈ (0, 1) is a positive rational number, the proposed method for computing the proximal map of ∥x∥p +p +involves finding the roots of a polynomial of order 2q, where q ∈ Z+ such that p = s/q for some s ∈ Z+. +Since f is L-smooth, for all x, y ∈ Rn, we have +f(x) ≤ f(y) + ∇f(y)⊤(x − y) + L +2 ∥x − y∥2 . +(49) +Given xk, replacing f(x) with the upper bound in (49) for y = xk, the prox-gradient operation naturally arises as +follows: +xk+1 = argmin +X +∥x∥p +p + µ +2 [f(xk) + ∇f(xk)⊤(x − xk) ++ L +2 +��x − xk��2]. +(50) +By completing the square, (50) yields to +xk+1 = argmin +X +∥x∥p +p + µL +4 +����x − +� +xk − 1 +L∇f(xk) +����� +2 +. +(51) +Defining ¯xk ∆= xk − 1 +L∇f(xk), (51) can be rewritten as +xk+1 = argmin +X +∥x∥p +p + µL +4 +��x − ¯xk��2 += argmin +X +n +� +i=1 +|xi|p + µL +4 (xi − ¯xk +i )2, +(52) +which is clearly a separable structure in the entries of x. Therefore, for each i ∈ [n], we have +xk+1 +i +=argmin +xi +|xi|p+ µL +4 (xi−¯xk +i )2 = prox¯g/ µL +2 (¯xk +i ), +(53) +where ¯g : R → R+ such that ¯g(t) = |t|p for some positive rational p ∈ (0, 1). +Next, we consider a generic form of (53), i.e., given some ¯t ∈ R, we would like to compute +t∗ = argmin +t +{|t|p + µL +4 (t − ¯t)2}. +(54) +The first-order optimality condition for (54) can be written as +p|t∗|p−1sign(t∗) + µL +2 (t∗ − ¯t) = 0. +(55) +Using similar arguments with those in section III-B for Proposition 1, we can conclude that the optimal solution t∗ +attains the property that sign(t∗) = sign(¯t). Without loss of generality, exploiting the symmetry of the function ¯g, +we only consider the case when ¯t > 0; hence, the optimal solution t∗ is the smallest positive root of the following +polynomial: +p|t∗|p−1 + µL +2 (t∗ − ¯t) = 0. +(56) + +15 +Algorithm 3 Accelerated PG algorithm +1: Initialize: µ, s = 1, q = 2, l, x0, x1, k = 1. +2: repeat +3: +yk = xk + k−1 +k+2(xk − xk−1) +4: +∆k = maxt=max{1,k−l},...,k F(xt) +5: +if F(yk) ≤ ∆k then: +6: +vk = yk +7: +else: +8: +vk = xk +9: +¯xk = vk − 1 +L∇f(vk) +10: +for i ∈ [n] do: +11: +solve a2q − ¯xiaq + +2s +qµLas = 0 +12: +xk+1 +i += a∗q +13: +k = k + 1 +14: until convergence +As in (28), suppose 0 < p = s +q < 1 for some s, q ∈ Z+. Using the change of variables a ≜ (t∗) +1 +q , (56) reduces to +finding the roots of a polynomial of degree 2q: +a2q − ¯taq + 2s +qµLas = 0. +(57) +To efficiently solve (46), we will use Algorithm 3, which is an implementation of nonconvex inexact accelerated +proximal gradient (APG) descent method proposed in [48, Algorithm 2]. To summarize, [48, Algorithm 2] is +designed to solve composite problems of the form in (48) assuming that h is L-smooth and g is proper lower- +semicontinuous such that F ≜ h + g is bounded from below and coercive, i.e., lim∥∥→∞ F() = +∞ – note that +there is no assumption regarding neither h nor g to be convex. The key points enhancing both practical behavior +of and theoretical guarantees for [48, Algorithm 2] can be summarized as given below: +• An extrapolation yk is generated as introduced in [49] for the APG algorithm (step 3). +• Steps 4 through 9 allow non monotone update of the objective. F(yk) is checked with respect to the maximum +of the latest l objective values. The gradient step is adjusted according to this (step 9). This permits yk to +occasionally increase the objective and makes F(yk) be less than the maximum of the objective value of the +latest l iterations. +• Steps 11 and 12 are the solution of the PG step using the non-convex projection method. +In the next part, we show that algorithm 3 converges to a critical point and it exhibits a convergence rate of O( 1 +k), +where k is the iteration budget that is given to the algorithm. + +16 +Definition 3. ( [50]) The Frechet sub-differential of F at x is +ˆ∂F(x) +∆= +� +u : lim +y̸=x lim +y→x +F(y) − F(x) − u⊤(y − x) +∥y − x∥ +≥ 0 +� +. +(58) +The sub-differential of F at x is +∂F(x) +∆= {u : ∃xk → x, F(xk) → F(x) and uk ∈ ˆ∂F(xk) +→ u as k → ∞}. +(59) +Definition 4. ( [50]) x is a critical point of F if 0 ∈ ∂g(x) + ∇h(x). +By comparing (48) and (47), it can be realized that the functions g(x) and h(x) in definition 4 are equal to +∥x∥p +p and µ +2 f(x) respectively. +Theorem 1. The sequence xk generated from algorithm 3 has at least one limit point and all the generated limit +points are critical points of (47). Moreover, the algorithm converges with rate O( 1 +K ), where K is the iteration +budget given to the algorithm. +Proof. It can easily be verified that our problem in (47) satisfies all required assumptions for Algorithm 3. Indeed, +1) The function g(x) = ∥x∥p +p is a proper and lower semi-continuous function. +2) The gradient of h(x) = µ +2 f(x) is ¯L-Lipschitz smooth, i.e., ∥∇h(x) − ∇h(y)∥ ≤ ¯L ∥x − y∥ for all x, y ∈ Rn, +with ¯L = µ +2 L. +3) F(x)=g(x)+h(x) is bounded from below, i.e., F(x)≥0. +4) lim∥x∥→∞ F(x) = ∞. +5) The introduced non-convex projection method is an exact solution for the proximal gradient step. This is +because it is based on finding the roots of a polynomial of order 2q in equation (57). +Therefore, the assumptions required for theorem 4.1 for critical point convergence and proposition 4.3 for the rate +of convergence in [48] are satisfied which then completes the proof. +Remark 2. The global convergence of several exact iterative methods that solve (48) has been explored, under the +framework of Kurdyka–Lojasiewicz (KL) theory, in various additional literature including [50]–[54]. Other work +(see [55] and references therein) considered the linear convergence of non-exact algorithms with relaxations on +the assumptions of KL theory, however, it is difficult to verify that the sequence generated by algorithm 3 satisfies +the relaxed assumptions stated in [55]. +VI. NUMERICAL RESULTS FOR SVR PROBLEM +In this section, we present two numerical examples for the p-quasi-norm ADMM (pQN-ADMM) from algorithm 1 +and the non-convex projection from algorithm 2. For both examples, the pQN-ADMM algorithm result is compared +with the ℓ1 objective function solution from MOSEK solver [56]. The two examples include; i) Sparse signal +reconstruction from noisy measurements, where the pQN-ADMM algorithm is also compared with another ℓ0.5 +quasi-norm minimization based algorithm, named ℓ0.5-FL, described in [57]. ii) Binary classification using support +vector machines (SVM). + +17 +10-4 +10-3 +10-2 +10-1 +100 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +Fig. 1: Effect of noise variance on the sparsity of solutions obtained by pQN-ADMM algorithm, +ℓ0.5-FL algorithm and ℓ1 norm minimization. +A. Sparse Signal Reconstruction +Let n = 210 and m = n/4, randomly construct the sparse binary matrix, M ∈ Rm× n +2 , with a few number of +ones in each column. The number of ones in each column of M is generated independently and randomly in the +range of integers between 10 and 20, and their locations are randomly chosen independently for each column. Let +U = [M, −M], which is the vertical concatenation of the matrix M and its negative. Following the same setup in +[58], the column orthogonality in U is not satisfied. Let xopt ∈ Rn be a reference signal with ∥xopt∥0 = ⌈0.2n⌉, +where the non-zero locations are chosen uniformly at random with the values following a zero mean, unit variance +Gaussian distribution. Let v = Uxopt+n be the allowable measurement, where n ∈ Rm is a Gaussian random vector +with zero mean and co-variance matrix σ2Im×m, where I is the identity matrix. The sparse vector is reconstructed +from v by solving (7) with V = {x : ∥Ux − v∥/∥v∥ − ǫ ≤ 0}. Figure 1 plots the relation between the sparsity +level and the noise variance for ℓ1 norm minimization, ℓ0.5-FL quasi-norm and pQN-ADMM solutions. A threshold +value of 10−6 was used where the threshold is a value below which the entry of the solution vector is considered to +be zero. Depending on the noise variance σ2, the value of ǫ was chosen to make the problem feasible. The reported +result is the average of 100 independent random runs. It can be realized that pQN-ADMM algorithm produces a +sparser solution than its counter baselines for different values of σ2. On increasing σ2, the sparsity level for all +methods decreases. This is due to the increased scarcity of information on the original signal in the realization +vector which makes the reconstruction process less accurate. + +18 +B. Binary Classification +In this part, we build an email spam classifier based on support vector machines. We use a subset of the +training set used in the SpamAssassin Public Corpus [59]. Let {(uj, vj)}j∈[m] be the training set of feature vectors +uj ∈ {0, 1}n with corresponding labels vj ∈ {−1, 1} identifying whether the email is spam or not. We highlight +the effectiveness of our method in designing an email spam detector using the least number of words. Following +[60], we maintain a dictionary of n = 1899 words. For a given email j ∈ [m], the ith entry of uj is 1 if word +wi, i ∈ [n] of the dictionary is in email j, and is 0 otherwise. We aim to build a linear classifier with the decision +rule ˆv = sign(u⊤x), where u is the feature vector of the email in question and x is a vector of the classifier +coefficients with the first entry being the bias term. The main aim is to build a classifier that detects whether an +email is a spam or not, using the least number of words from the dictionary and achieving a high training data +accuracy. To achieve this objective, we solve (7) with M = {x : 1 +m +� +j∈[m] +� +1 − vju⊤ +j x +�+ − ǫ ≤ 0}. +It can be clearly realized that the training set accuracy is controlled by ǫ. Algorithm 1 was run for p = 0.5, 2000 +training emails and various values for ǫ. For each value of ǫ, the algorithm was terminated after 100 iterations and +performance tested on 1000 emails. For comparison purpose, the problem was also solved with the ℓ1 norm convex +relaxation under the same setup. In figure 2a, we plot the number of non-zero entries in the optimal classifier from +both the pQN-ADMM and ℓ1 solutions vs different values of ǫ. +We used a threshold of 10−4, where the threshold is defined as in section VI-A. It can be realized from figure 2a +that the pQN-ADMM solution outperforms the ℓ1 in terms of the number of words used for legitimacy detection. +When the value of ǫ increases, the number of required words decreases for both ℓ0.5 and ℓ1 problems. This outlines +the trade-off between the sparsity level of the classifier and its accuracy, i.e., small values of ǫ enforces a low +classification error in expense of a less sparse solution. The corresponding training and test set accuracies for the +obtained classifiers are plotted in Fig. 2b. Both figures 2a and 2b depict the performance of the pQN-ADMM +solution from algorithm 1 in terms of the sparsity level while maintaining nearly the same level of accuracy as the +ℓ1 solution for both the training and test sets. +VII. NUMERICAL RESULTS FOR RMP PROBLEM +A. Time domain system identification +In this part, we apply the derived pQN-ADMM approach on a time domain system identification example. In +that example, input is applied to randomly generated systems with a known order. Using the outputs corresponding +to these systems, the minimum rank/order system is derived and results are compared to nuclear norm heuristic in +[26]. +We consider a discrete time stable Single Input Single Output (SISO) system with an input u ∈ RT, where T +represents the number of input samples, i.e., input time span. We assume an impulse response of a fixed number +of samples n. The corresponding system output is y ∈ Rm. However, we assume that only noisy realizations, ˆy, of +the output can be considered, such that; ˆy +∆= y + z = h ⊛ u + z , where h ∈ Rn is the system’s original impulse +response, z ∈ Rm is a random vector with entries drawn independently from samples of a uniform distribution +on the range [−0.25, 0.25], i.e., zi ∼ U[−0.25, 0.25], while ⊛ denotes the convolution operator. From the window + +19 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +50 +100 +150 +200 +250 +Number of selected words +(a) Number of words selected for classification +versus ǫ for pQN-ADMM and ℓ1 norm. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +65 +70 +75 +80 +85 +90 +95 +100 +Correct percentage +(b) Training and test set accuracies versus ǫ +for pQN-ADMM and ℓ1 norm. +Fig. 2: Binary classification numerical results. +property of the convolution, m = n + T − 1. Assume that ui, hi and yi are the ith components of the vectors u, h +and y respectively. The three components are related to each other by convolution through yi = �∞ +j=−∞ hjui−j +which is a linear relation. Hence, let T ∈ Rm×n be the Toeplitz matrix formed by the input u , it can be easily +seen that h ⊛ u = hT ⊤. Assume that x ∈ Rn is an impulse response variable and let X ∈ Rn×n be a Hankel +matrix formed by the entries of x. From [29], [61]–[63], the minimum order time domain system identification +problem can be formulated as, +min +x,X +RankX, +(60a) +s.t. +X = Hankel(x), +(60b) +��ˆy − xT ⊤��2 ≤ ǫ, +(60c) +(60b) ensures that X is a Hankel matrix and (60c) holds to make the result by applying the input, u, to the optimal +impulse response, x, fit the available noisy data, ˆy, in a non-trivial sense. Defining the convex set C +∆={X∈Rn×n : +��ˆy−hT ⊤��2−ǫ ≤ 0, X=Hankel(x)}, (60) can be cast as, +min +x,X +Rank(X), +s.t. +X ∈ C, +(61) +which is clearly identical to the problem in (1). The problem was solved using the same pQN-ADMM approach +discussed in section IV. +We let T = m = 50 and n = 40. Note that m < T + n − 1, which is a reasonable assumption as in some +practical applications, one is allowed only a specific window to realize the output. We consider the simulation for +10 different original system orders, i.e., η = 2 : 2 : 10. An input vector, u, is generated, where the elements of u +are independent and follow a uniform distribution on the interval [−5, 5]. For each η; 1) 50 random stable systems +are generated using the command ’drss’ in MATLAB. 2) The generated input is applied to each system to get the +corresponding noisy output ˆy. 3) Given the output ˆy, the problem in (60) is solved and the corresponding system’s + +20 +2 +4 +6 +8 +10 +Original system order +0 +5 +10 +15 +20 +Average rank +Threshold=10-4 +2 +4 +6 +8 +10 +Original system order +0 +5 +10 +15 +20 +Average rank +Threshold=10-5 +Fig. 3: Average rank vs original system order. Red and cyan colors are for the nuclear norm and +pQN-ADMM algorithm respectively. +η=2 +η=6 +η=10 +Nuclear norm +2.3907 +6.6668 +7.2572 +pQN-ADMM +0.5292 +0.9042 +1.0861 +TABLE I: Standard deviation for threshold=10−4 +rank is calculated using singular value decomposition. 4) The results are averaged out to get the corresponding +average rank to each original η. +Figure 3 shows the average rank for the the nuclear norm and pQN-ADMM heuristics. The results are for +two different values of thresholds, where the threshold is defined as the value below which the singular value is +considered to be zero. It can be realized that the introduced pQN-ADMM approach outperforms the nuclear norm +one for both values of thresholds. Moreover, when the threshold value decreases from 10−4 to 10−5, the behavior +of the pQN-ADMM remains the same. However, the average rank for the nuclear norm increases. This proves the +robustness of the derived pQN-ADMM in comparison to the nuclear norm one. Tables I and II show the standard +deviation of the algorithms. It can be seen that the standard deviation is the same for the pQN-ADMM when +η=2 +η=6 +η=10 +Nuclear norm +6.9877 +11.2638 +11.7854 +pQN-ADMM +0.5325 +0.9113 +1.0861 +TABLE II: Standard deviation for threshold=10−5 + +21 +changing the threshold, however, it increases for the nuclear norm as the threshold value decreases. +B. Matrix Completion Example +In this section, we apply our algorithm (pQN-ADMM) to a matrix completion example and compare the +result to the matrix iterative re-weighted least squares (MatrixIRLS) [64], [65], truncated iterative re-weighted +unconstrained Lq (tIRucLq) [34] and iterative re-weighted least squares (sIRLS-p & IRLS-p) [66] algorithms. The +matrix completion problem is a special case of the low rank minimization where a linear transform takes a few +random entries of an ambiguous matrix X ∈ Rm×n. Given only these entries, the goal is to approximate X and +find the missing ones. The matrix completion problem with low rank recovery can be approximated by, +min +X +∥X∥p +p , +s.t. +∥A(X) − b∥ ≤ ǫ, +(62) +where A : Rm×n → Rq is a linear map with q ≪ mn and b ∈ Rq. In order to apply the mentioned algorithms, the +linear transform A(X) will be rewritten as Avec(X), where A ∈ Rq×mn and vec(X) ∈ Rmn is a vector formed +by stacking the columns of the matrix X. +A random matrix M ∈ Rm×n with rank r is created using the following method: 1) M = MLM⊤ +R, where +ML ∈ Rm×r and MR ∈ Rn×r. 2) The entries of both ML and MR are i.i.d Gaussian random variables with zero +mean and unit variance. Let ˆ +M = M + Z, where Z ∈ Rm×n is a Gaussian noise with each entry being an i.i.d +Gaussian random variable with zero mean and variance σ2. The vector b is then created by selecting random q +elements from vec( ˆ +M). Since b = Avec( ˆ +M), one can easily construct the matrix A which is a sparse matrix where +each row is composed of a value 1 at the index of the corresponding selected entry in the vector b while the rest +are zeros. We set m = n = 100, r = 5 and p = 0.5. Let dr = r(m+n−r) denotes the dimension of the set of rank +r matrices and define s = +q +mn as the sampling ratio. We assume that s = 0.195 which yields to q = 1950. It can +be realized that dr +q < 1. We set σ = 0.1 and let the algorithms terminate if a budget of 1000 iterations is reached. +In order to compare the results from different algorithms, we consider the average of 50 runs for two measures: a) +the relative Frobenius distance (RFD) to the matrix M, b) the relative error to singular (REtS) values of M. +In figures 4a and 4b, we report the average RFD and REtS values for all the algorithms. Despite that all the +baselines are designed to exploit the specific structure of the matrix completion problem, described in (62), while +the proposed pQN-ADMM doesn’t, it is competitive against them all. This in turns shows the effectiveness of the +pQN-ADMM algorithm in solving the rank minimization problems without requiring any prior information about +the structure of the associated convex set. +VIII. NUMERICAL RESULTS FOR THE NONCONVEX ACCELERATED PROXIMAL +GRADIENT (APG) ALGORITHM +In this subsection, we present numerical results for the APG method, displayed in Algorithm 3. Following the +same procedure in [67], we first generate the target signal x∗ through +x∗ +i = + + + +Θ(1) +i 103Θ(2) +i , +∀ i ∈ Λ, +0, +∀ i ∈ [n] \ Λ; +(63) + +22 +0 +200 +400 +600 +800 +1000 +Number of iterations +10-2 +10-1 +100 +IRLS-p +sIRLS-p +MatrixIRLS +tIRucLq +pQN-ADMM +(a) RFD to M. +1 +2 +3 +4 +5 +Index of singular value +0 +0.02 +0.04 +0.06 +0.08 +0.1 +IRLS-p +sIRLS-p +MatrixIRLS +tIRucLq +pQN-ADMM +(b) REtS values of M. +Fig. 4: The RFD and REtS average values. +where the design parameters Λ ⊂ [n], and Θ(1) +i , Θ(2) +i +for i ∈ Λ are chosen as follows: +1) the index set Λ ⊂ [n] is constructed by selecting a subset of [n] with cardinality s uniformly at random; +2) {Θ(1) +i }i∈Λ are independent, identically distributed (IID) Bernoulli random variables taking values ±1 with +equal probability; +3) {Θ(2) +i }i∈Λ are IID uniform [0, 1] random variables. +The measurement matrix A ∈ Rm×n is a partial Discrete Cosine Transform (DCT) matrix with rows correspond- +ing to m < n frequency, where these m indices are chosen uniformly at random from [n]. The noisy measurement +vector b ∈ Rm is then set to be b = A(x∗ + ǫ1) + ǫ2, where ǫ1 ∼ N(0, σ2 +1) and ǫ2 ∼ N(0, σ2 +2) are the input and +realization noises. +In our experiments, n = 4096, s = ⌈0.5m⌉ and the PG algorithm memory to 5, i.e., l = 5. Following the +medium noise setup in [68], we set σ1 = 0.005, σ2 = 0.001. +For f(x) = ∥Ax − b∥2, we have L = 2∥A∥2. We perform our experiment for various values of m, i.e., number +of noisy measurements, and µ, i.e., trade-off parameter, see (47). For each (m, µ) selection, in order to capture +the inherent statistical variation of the problem, we generate 20 random instances of the triplet (x∗, A, b) and each +random instance is solved by Algorithm 3. We reported the average performance. We terminated Algorithm 3 when +the relative error between consecutive iterates satisfies +��xk − xk−1�� / +��xk−1�� ≤ 10−5 for the first time. +In our experiments, we compared solving (47) for p = 0.5 against p = 1, i.e., against ℓ1-optimization for sparse +recovery. On one hand, when p = 0.5, i.e., for ℓ0.5 minimization, we solve (47) using Algorithm 3, called ℓ0.5 exact, +and using the algorithm 2 of [69], which we call ℓ0.5 approx. On the other hand, when p = 1, ℓ1-minimization +problem is a convex one and we adopt the FISTA algorithm of [70]. The solution is denoted by ¯x while the target +signal, from (63), by x∗. In Algorithm 3, x0 is set to a zero vector while x1 is the ℓ1 norm solution. +Figures 5 and 6 highlight the relation between the average error and sparsity vs µ for different values of n/m. It +can be realized that the average error (sparsity) decreases (increases) on increasing µ. For small values of µ, more +weight is given to the loss function, which emphasizes the ℓ0 quasi-norm minimization, and hence the sparsity level, + +23 +Fig. 5: Average error vs µ for different values of n/m. Yellow and cyan shades are the standard +deviations for the exact and approximate ℓ0.5 quasi-norms respectively. +as in figure 6, is low. However, for high values of µ, more weight is assigned to the minimization of the regularization +term, which solves ∥Ax − b∥2, and hence the error decreases, as shown in figure 3, with a corresponding increase +in the sparsity. It can be realized that the ℓ0.5 solutions always outperforms the ℓ1 one with very slight difference +between the exact and the approximate ones. +Figure 7 highlights the statistics of the number of iterations used until convergence for both the ℓ0.5 exact and +approximate algorithms. It can be realized that with a sufficient number of available realizations, n/m = 8 and +n/m = 16, both algorithms approximately consume the same number of iterations. However, when the number +of available realizations decreases, n/m = 32 and higher, our exact proximal solution requires significantly less +number of iterations to converge. This conclusion, along with figures 5 and 6 findings, indicates that our algorithm +not only finds a similar solution to the approximate method, but also converges with a fewer number of iterations. + +n/m=8 +n/m=16 +n/m=32 +0 +0 +0 +l1 +l1 +-1 +- l0.5 exact +-1 +lo0.5 exact +-1 +- l0.5 exact +l0.5 approx +l0.5approx +l0.5 approx +2 +-2 +-2 +all +-3 +-3 +-3 +-4 +-4 +5 +-5 +-5 +-6 +-6 +-6 +-7 +-7 +-7 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +u +n/m=64 +n/m=128 +n/m=256 +0 +0 +l1 +l1 +-1 +lo.5exact +-1 +l0.5 exact +-1 +l0.5 exact +l0.5approx +0.5 approx +0.5approx +-2 +-2 +-2 +-3 +gl +-3 +3 +-4 +-4 +4 +-5 +-5 +.5 +-6 +-6 +-6 +-7 +-7 +-7 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +u24 +Fig. 6: Sparsity vs µ for different values of n/m. Yellow and cyan shades are the standard +deviations for the exact and approximate ℓ0.5 quasi-norms respectively. +IX. CONCLUSION +In this study, we presented a non-convex ADMM algorithm (pQN-ADMM) to solve the ℓp norm minimization +problem. The algorithm has a similar complexity to that of the ℓ1 minimization in addition to solving the roots of a +polynomial for the non-convex projection. Our algorithm can also be considered as a general procedure for solving +ℓp problems as no specific structure for the convex constraint set was assumed and a convex projection on that set +was done for variables update. Applying sparse signal recover and binary classification examples, our method was +found to outperform the ℓ1 minimization in terms of the sparsity of the generated solution. In addition, we studied +the problem of solving a non-convex relaxation of RMPs using Schatten-p quasi-norm. This relaxation was shown +to be the ℓp minimization of the singular values of the variable matrix and hence the primary developed algorithm +could be used. Showing the numerical results, the pQN-ADMM was found to be less sensitive to the threshold +decrease in time domain system identification problems. Additionally, the pQN-ADMM method was shown to be + +n/m=8 +n/m=16 +n/m=32 +5 +5 +5 +l1 +lo.5 exact +l0.5exact +4. +l0.5 exact +C0.5approx +0.5 approx +-0.5 approx +3 +3 +3 +x*lo +2 +区 +2 +[x* +2 +区 +1 +0 +0 +0 +-1 +-1 +-1 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +μ +n/m=64 +n/m=128 +n/m=256 +6 +6 +10 +5 +l0.5 exact +5 +lo.5exact +8 +lo0.5 exact +L0.5approx +0.5approx +0.5 approx +4 +4 +6 +[x o +3 +3 +[x*10 +重 +[x*| +4 +2 +区 +2 +2 +0 +0 +0 +-1 +-2 +7 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +u25 +Fig. 7: Iterations count vs µ for different values of n/m. +competitive against various other baselines when solving the matrix completion problem. +REFERENCES +[1] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, no. 1, pp. 49–95, 1996. +[2] E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, +pp. 4203–4215, 2005. +[3] J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE +Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210–227, 2009. +[4] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete +frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006. +[5] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. +[6] A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of +signals and images,” SIAM review, vol. 51, no. 1, pp. 34–81, 2009. + +n/m=8 +n/m=16 +n/m=32 +6000 +6000 +6000 +l0.5 exact +l0.5 exact +l0.5 exact +5000 +0.5 approx +5000 +l0.5approx +5000 +0.5approx +count +4000 +count +4000 +count +4000 +Iterations +erations +Iterations +3000 +3000 +3000 +2000 +2000 +2000 +1000 +1000 +1000 +0 +0 +0 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +u +n/m=64 +n/m=128 +n/m=256 +6000 +6000 +6000 +l0.5 exact +l0.5 exact +l0.5 exact +5000 +l0.5approx +5000 +l0.5 approx +5000 +lo.5 approx +A +Iterations count +4000 +4000 +count +4000 +rations +3000 +3000 +rations +3000 +2000 +2000 +2000 +1000 +1000 +1000 +0 +0 +0 +0 +10 +20 +30 +0 +10 +20 +30 +0 +10 +20 +30 +u +u26 +[7] J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proceedings of the +IEEE, vol. 98, no. 6, pp. 948–958, 2010. +[8] S. G. Mallat and Zhifeng Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Transactions on Signal +Processing, vol. 41, no. 12, pp. 3397–3415, 1993. +[9] J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Transactions on Information Theory, +vol. 50, no. 10, pp. 2231–2242, 2004. +[10] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM review, vol. 43, no. 1, +pp. 129–159, 2001. +[11] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society: Series B +(Methodological), vol. 58, no. 1, pp. 267–288, 1996. +[12] R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol. 14, +no. 10, pp. 707–710, 2007. +[13] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in 2008 IEEE International +Conference on Acoustics, Speech and Signal Processing, pp. 3869–3872, IEEE, 2008. +[14] D. P. Wipf and B. D. Rao, “Sparse bayesian learning for basis selection,” IEEE Transactions on Signal Processing, vol. 52, +no. 8, pp. 2153–2164, 2004. +[15] P. Schniter, L. C. Potter, and J. Ziniel, “Fast bayesian matching pursuit,” in 2008 Information Theory and Applications +Workshop, pp. 326–333, 2008. +[16] A. Miller, Subset selection in regression. CRC Press, 2002. +[17] R. Saab, R. Chartrand, and O. Yilmaz, “Stable sparse approximations via nonconvex optimization,” in 2008 IEEE +International Conference on Acoustics, Speech and Signal Processing, pp. 3885–3888, 2008. +[18] R. Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in 2009 IEEE +International Symposium on Biomedical Imaging: From Nano to Macro, pp. 262–265, 2009. +[19] N. Mourad and J. P. Reilly, “Minimizing nonconvex functions for sparse vector reconstruction,” IEEE Transactions on +Signal Processing, vol. 58, no. 7, pp. 3485–3496, 2010. +[20] R. Chartrand and B. Wohlberg, “A nonconvex ADMM algorithm for group sparsity with sparse groups,” in 2013 IEEE +International Conference on Acoustics, Speech and Signal Processing, pp. 6009–6013, 2013. +[21] Z. Xu, X. Chang, F. Xu, and H. Zhang, “l1/2 regularization: A thresholding representation theory and a fast solver,” IEEE +Transactions on Neural Networks and Learning Systems, vol. 23, no. 7, pp. 1013–1027, 2012. +[22] J. Zeng, S. Lin, Y. Wang, and Z. Xu, “l1/2 regularization: Convergence of iterative half thresholding algorithm,” IEEE +Transactions on Signal Processing, vol. 62, no. 9, pp. 2317–2329, 2014. +[23] G. Li and T. K. Pong, “Global convergence of splitting methods for nonconvex composite optimization,” SIAM Journal +on Optimization, vol. 25, no. 4, pp. 2434–2460, 2015. +[24] Y. Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” Journal of Scientific +Computing, vol. 78, no. 1, pp. 29–63, 2019. +[25] M. Mesbahi, “On the semi-definite programming solution on the least order dynamic output feedback synthesis,” 1999. +[26] M. Fazel, H. Hindi, and S. P. Boyd, “A rank minimization heuristic with application to minimum order system +approximation,” in Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148), vol. 6, pp. 4734–4739 +vol.6, 2001. +[27] M. Fazel, H. Hindi, and S. P. Boyd, “Log-det heuristic for matrix rank minimization with applications to hankel and +euclidean distance matrices,” in Proceedings of the 2003 American Control Conference, 2003., vol. 3, pp. 2156–2162 +vol.3, 2003. + +27 +[28] M. Fazel, H. Hindi, and S. Boyd, “Rank minimization and applications in system theory,” in Proceedings of the 2004 +American Control Conference, vol. 4, pp. 3273–3278 vol.4, 2004. +[29] K. Mohan and M. Fazel, “Reweighted nuclear norm minimization with application to system identification,” in Proceedings +of the 2010 American Control Conference, pp. 2953–2959, 2010. +[30] S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” in +2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2862–2869, 2014. +[31] F. Nie, H. Huang, and C. Ding, “Low-rank matrix recovery via efficient Schatten p-norm minimization,” in Twenty-sixth +AAAI conference on artificial intelligence, 2012. +[32] F. Nie, H. Wang, H. Huang, and C. Ding, “Joint schatten p-norm and lp norm robust matrix completion for missing value +recovery,” Knowledge and Information Systems, vol. 42, no. 3, pp. 525–544, 2015. +[33] L. Liu, W. Huang, and D.-R. Chen, “Exact minimum rank approximation via Schatten p-norm minimization,” Journal of +Computational and Applied Mathematics, vol. 267, pp. 218–227, 2014. +[34] M.-J. Lai, Y. Xu, and W. Yin, “Improved iteratively reweighted least squares for unconstrained smoothed ℓq minimization,” +SIAM Journal on Numerical Analysis, vol. 51, no. 2, pp. 927–957, 2013. +[35] R. Chartrand, “Nonconvex splitting for regularized low-rank + sparse decomposition,” IEEE Transactions on Signal +Processing, vol. 60, no. 11, pp. 5810–5819, 2012. +[36] M. D. Gupta and S. Kumar, “Non-convex p-norm projection for robust sparsity,” in 2013 IEEE International Conference +on Computer Vision, pp. 1593–1600, 2013. +[37] S. Bahmani and B. Raj, “A unifying analysis of projected gradient descent for ℓp-constrained least squares,” Applied and +Computational Harmonic Analysis, vol. 34, no. 3, pp. 366–378, 2013. +[38] M. E. Ashour, C. M. Lagoa, and N. S. Aybat, “Lp quasi-norm minimization,” in 2019 53rd Asilomar Conference on +Signals, Systems, and Computers, pp. 726–730, 2019. +[39] S. Boyd, N. Parikh, and E. Chu, Distributed optimization and statistical learning via the alternating direction method of +multipliers. Now Publishers Inc, 2011. +[40] C. Helmberg and F. Rendl, “A spectral bundle method for semidefinite programming,” SIAM Journal on Optimization, +vol. 10, no. 3, pp. 673–696, 2000. +[41] A. Beck and M. Teboulle, “Mirror descent and nonlinear projected subgradient methods for convex optimization,” +Operations Research Letters, vol. 31, no. 3, pp. 167–175, 2003. +[42] Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming. SIAM, 1994. +[43] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization: analysis, algorithms, and engineering applications. +SIAM, 2001. +[44] Y. Xu and W. Yin, “A globally convergent algorithm for nonconvex optimization based on block coordinate update,” +Journal of Scientific Computing, vol. 72, no. 2, pp. 700–734, 2017. +[45] Z. Zha, X. Zhang, Y. Wu, Q. Wang, X. Liu, L. Tang, and X. Yuan, “Non-convex weighted lp nuclear norm based ADMM +framework for image restoration,” Neurocomputing, vol. 311, pp. 209–224, 2018. +[46] L. Mirsky, “A trace inequality of John von Neumann,” Monatshefte f¨ur mathematik, vol. 79, no. 4, pp. 303–306, 1975. +[47] N. Parikh and S. Boyd, “Proximal algorithms,” Foundations and Trends in optimization, vol. 1, no. 3, pp. 127–239, 2014. +[48] Q. Yao, J. T. Kwok, F. Gao, W. Chen, and T.-Y. Liu, “Efficient inexact proximal gradient algorithm for nonconvex +problems,” arXiv preprint arXiv:1612.09069, 2016. +[49] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal +on imaging sciences, vol. 2, no. 1, pp. 183–202, 2009. + +28 +[50] H. Attouch, J. Bolte, and B. F. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: proximal +algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Mathematical Programming, vol. 137, +no. 1, pp. 91–129, 2013. +[51] H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, “Proximal alternating minimization and projection methods for +nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality,” Mathematics of operations research, +vol. 35, no. 2, pp. 438–457, 2010. +[52] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth +problems,” Mathematical Programming, vol. 146, no. 1, pp. 459–494, 2014. +[53] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for +nonsmooth optimization,” SIAM Journal on Optimization, vol. 23, no. 2, pp. 1126–1153, 2013. +[54] P. Tseng and S. Yun, “A coordinate gradient descent method for nonsmooth separable minimization,” Mathematical +Programming, vol. 117, no. 1, pp. 387–423, 2009. +[55] Y. Hu, C. Li, K. Meng, and X. Yang, “Linear convergence of inexact descent method and inexact proximal gradient +algorithms for lower-order regularization problems,” Journal of Global Optimization, vol. 79, no. 4, pp. 853–883, 2021. +[56] M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0., 2019. +[57] S. Foucart and M.-J. Lai, “Sparsest solutions of under-determined linear systems via ℓq-minimization for 0 < q ≤ 1,” +Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 395–407, 2009. +[58] D. Ge, X. Jiang, and Y. Ye, “A note on the complexity of ℓp minimization,” Mathematical programming, vol. 129, no. 2, +pp. 285–299, 2011. +[59] SpamAssassin Public Corpus, http://spamassassin.apache.org. +[60] Andrew Ng, Machine learning MOOC, https://www.coursera.org/learn/machine-learning. +[61] M. Sznaier, M. Ayazoglu, and T. Inanc, “Fast structured nuclear norm minimization with applications to set membership +systems identification,” IEEE Transactions on Automatic Control, vol. 59, no. 10, pp. 2837–2842, 2014. +[62] Z. Liu and L. Vandenberghe, “Semidefinite programming methods for system realization and identification,” in Proceedings +of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, +pp. 4676–4681, 2009. +[63] M. Fazel, T. K. Pong, D. Sun, and P. Tseng, “Hankel matrix rank minimization with applications to system identification +and realization,” SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 3, pp. 946–977, 2013. +[64] C. K¨ummerle and C. Mayrink Verdun, “Escaping saddle points in ill-conditioned matrix completion with a scalable +second order method,” in Workshop on Beyond First Order Methods in ML Systems at the 37th International Conference +on Machine Learning, 2020. +[65] C. K¨ummerle and C. Mayrink Verdun, “A scalable second order method for ill-conditioned matrix completion from few +samples,” in International Conference on Machine Learning (ICML), 2021. +[66] K. Mohan and M. Fazel, “Iterative reweighted algorithms for matrix rank minimization,” Journal of Machine Learning +Research, vol. 13, no. 110, pp. 3441–3473, 2012. +[67] N. S. Aybat and G. Iyengar, “A first-order augmented Lagrangian method for compressed sensing,” SIAM Journal on +Optimization, vol. 22, no. 2, pp. 429–459, 2012. +[68] E. T. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for l1-regularized minimization with applications +to compressed sensing,” CAAM TR07-07, Rice University, vol. 43, p. 44, 2007. +[69] C. O’Brien and M. D. Plumbley, “Inexact proximal operators for ℓp-Quasi-norm minimization,” in 2018 IEEE International +Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4724–4728, 2018. + +29 +[70] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal +on imaging sciences, vol. 2, no. 1, pp. 183–202, 2009. + diff --git a/SdFLT4oBgHgl3EQfPS8r/content/tmp_files/load_file.txt b/SdFLT4oBgHgl3EQfPS8r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..418169ce450db9a895cfd8d8fa5a215548126e91 --- /dev/null +++ b/SdFLT4oBgHgl3EQfPS8r/content/tmp_files/load_file.txt @@ -0,0 +1,1166 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf,len=1165 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='12027v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='SP] 27 Jan 2023 1 Lp Quasi-norm Minimization: Algorithm and Applications Omar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='Sleem⋆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ashour‡ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Aybat†§ Constantino M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Lagoa⋆ ⋆Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' of Electrical Engineering, Pennsylvania State University, State College, PA 16801, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ‡Wireless R&D Department, Qualcomm Technologies, Inc, San Diego, CA 92121, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' †Department of Industrial and Manufacturing Engineering, Pennsylvania State University, State College, PA 16801, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Email: { oms46@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='edu, mashour@qti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='qualcomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='com, nsa10@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='edu, cml18@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='edu } Abstract Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This paper proposes a heuristic method for retrieving sparse approximate solutions of optimization problems via minimizing the ℓp quasi-norm, where 0 < p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' An iterative two-block ADMM algorithm for minimizing the ℓp quasi-norm subject to convex constraints is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For p = s/q < 1, s, q ∈ Z+, the proposed algorithm requires solving for the roots of a scalar degree 2q polynomial as opposed to applying a soft thresholding operator in the case of ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The merit of that algorithm relies on its ability to solve the ℓp quasi-norm minimization subject to any convex set of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, it suffers from low speed, due to a convex projection step in each iteration, and the lack of mathematical convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We then aim to vanquish these shortcomings by relaxing the assumption on the constraints set to be the set formed due to convex and differentiable, with Lipschitz continuous gradient, functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' specifically, polytope sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Using a proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm speed while proving its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We then present various applications where the proposed algorithm excels, namely, matrix rank minimization, sparse signal reconstruction from noisy measurements, sparse binary classification, and system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The results demonstrate the significant gains obtained by the proposed algorithm compared to those via ℓ1 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Keywords— Sparsity, compressed sensing, rank minimization, ADMM, Proximal gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Motivation In numerical analysis and scientific computing, a sparse matrix/array is the one with many of its elements being zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The number of zeros divided by the total number of elements is called sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sparse data is often easier to store and process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, techniques for deriving sparse solutions and exploiting them have attracted the attention of many researchers in various engineering fields like machine learning, signal processing, and control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The taxonomy of sparsity can be studied through the rank minimization problem (RMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It has been lately considered in many engineering applications including control design and system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This is because the notions of complexity and system order can be closely related to the matrix rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The RMP can be formulated as follows: min X Rank(X), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X ∈ M, (1) where X ∈ Rm×n and M ⊂ Rm×n is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The problem (1) in its generality is NP-hard [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Therefore, polynomial time algorithms for solving large-scale problems of the form in (1) are not currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, currently adopted methods for solving such problems are approximate and structured heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A special case of RMP is the sparse vector recovery (SVR) problem involving ℓ0 pseudo-norm minimization given by: min x ∥x∥0 , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' x ∈ V, (2) where x ∈ Rn, V ⊂ Rn is a closed convex set and ∥·∥0 counts the number of the non-zero elements of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' From the definition of the rank being the number of non-zero singular values of a matrix, it can be easily realized that (1) is a generalized form of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Various works – which will be discussed in the next section in more detail – have explored efficient solution techniques for the problems in (1) and (2) independently using Schatten-p and ℓp quasi-norm relaxations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, all of these methods either assume a specific structure for the convex set M in (1) or only work for the special case in (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' hence, they lack generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Using the fact that the Schatten-p quasi-norm is the ℓp quasi-norm of the matrix singular values, we aim to design an efficient heuristic method based on Schatten-p relaxation for solving both problems in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' To achieve this, we first propose an algorithm for solving ℓp quasi-norm relaxation of the SVR problem in (2), and next, exploiting the fact that (2) is a special case of (1), we then employ the derived ℓp quasi-norm minimization algorithm as a building block for the desired generalized algorithm for the rank minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Related work 1) Sparse Vector Recovery: As discussed, a sparse solution is defined as the one having the minimum number of non-zero components while satisfying certain constraints such as a system of linear equations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since many signals are either sparse or compressible, SVR problem has found applications in object recognition, classification and compressed sensing problems, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [6], the authors discussed the concept of the sparse representation of signals and systems, where they reviewed the theoretical and empirical results on sparse 3 optimization, and discussed sufficient conditions needed for uniqueness, stability and computational practicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Different applications for the SVR problem are explored in [6] and it is argued that in certain denoising and compression tasks, the methods for sparse optimization provide state of the art solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The problem of constructing a sparse solution to undetermined linear systems has received great attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [7], the authors surveyed the existing algorithms for sparse approximation, namely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' greedy methods [8], [9], the methods based on convex relaxation [4], [5], [10], [11], those involving non-convex optimization [12], [13], Bayesian framework [14], [15], and requiring brute force [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They also discussed the computational requirements of each algorithm and their relation to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sparse optimization problems of the form min{f(x) + µg(x)} have been extensively studied in the literature, where g(x) is a sparsity inducing function, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', ℓ1-norm, f is a loss function on measurement errors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', f(x) = ∥Ax − b∥2, and µ > 0 is a trade-off parameter between data-fidelity and sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [2], the authors considered sparse recovery problem from a set of corrupted measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For g(·) = ∥·∥1, they established a sufficient condition for the exact sparse signal recovery, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', restricted isometry property (RIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Motivated by the fact that ∥x∥p p → ∥x∥0 as p → 0, it is natural to consider the above problem with g set to ℓp quasi-norm for p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, the authors in [17] presented theoretical results demonstrating the ability of the ℓp quasi-norm to recover sparse signals from noisy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Under more relaxed RIP conditions, they showed that the ℓp quasi-norm provides better theoretical guarantees in terms of stability and robustness than the ℓ1 minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [12], the problem of SVR via the ℓp quasi-norm minimization from small number of linear measurements of the target signal was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This setting is important in applications where data acquisition is difficult or expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, the proposed approach in [12] has limited applicability due to its long reconstruction time compared to the ℓ1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [18], the authors exploited Fourier-based algorithms for convex optimization to solve sparse signals reconstruction problem via the ℓp quasi-norm minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They showed that their approach combines the construction abilities of the non-convex methods with the speed of the convex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [19] the authors proposed an approach, for sparse reconstruction, replacing the non-convex function with a quadratic convex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [20], an alternating direction method of multipliers (ADMM) based algorithm that enforces both sparsity and group sparsity using non-convex regularization is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' An iterative half thresholding algorithm for fast solution of the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 regularization is proposed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The authors proved the existence of the resolvent of gradient of ||x||0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 , calculated its analytic expression, and derived a thresholding representation of solutions for ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [22], the convergence of the iterative half thresholding algorithm is studied, where it was shown that, under certain conditions, the half thresholding algorithm converges, to a local minimizer of the regularized problem, with a linear convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Conditions for the convergence of an ADMM algorithm that solves the problem of minimizing the sum of a smooth function with a bounded Hessian and a non-smooth function are derived in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [24], the convergence of ADMM for minimizing a non-convex and possibly non-smooth objective function subject to equality constraints is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The developed convergence guarantee covers a variety of non-convex objectives including piece-wise linear functions, ℓp quasi-norm and Schatten-p quasi-norm (0 < p < 1), while allowing non-convex constraints as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4 2) Rank minimization: In [25], the authors aimed at determining the least order dynamic output feedback, using same formulation as in (1), which stabilizes a given linear time invariant system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They found that minimizing the trace instead of the rank results in a Semi-Definite Program (SDP) that can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, their solution was only applicable for symmetric and square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [26], a generalization of the latter approach was introduced which is based on replacing the rank in the objective function with the summation of the singular values of the matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', the nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They showed that this leads to the convex envelope of the non- convex rank objective and boils down to the original trace heuristic when the decision matrix is a symmetric positive semi-definite (PSD) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Finally, effectiveness of the approach was shown using a frequency domain system identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [27], another heuristic based on the logarithm of the determinant was presented as a surrogate for the rank minimization over the subspace of PSD matrices, and the authors showed that this formulation can be solved using a sequence of trace minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The authors also extended their heuristic to handle matrices that are not necessarily PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [28], the authors also studied the existing trace and log determinant heuristics for approximating (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They discussed the applications of these heuristics for computing a low-rank approximation of 1) covariance matrices for a given dataset so that one can obtain a simple data model, easy to interpret, which is especially important in statistics and signal processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) Hankel matrices arising in system identification of a time invariant, low-order system for given output realizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' and 3) matrices appearing in various other problems including H∞ and reduced-order µ-synthesis with constant scaling and problems with inertia constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Although the nuclear norm is the tightest convex substitute for the non-convex rank function, one of its major shortcomings is that it treats all the singular values equally in order to be able to preserve the convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Therefore, this restricts its performance in applications where the singular values need to be treated differently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', particularly in image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [29], the authors proposed an iterative re-weighted nuclear norm heuristic to avoid this problem and analyzed its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' They also proposed a gradient-based algorithm and applied it to a low-order system identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Experimental results showed that the re-weighted nuclear norm leads to a lower order model than the nuclear norm itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [30], the solution of weighted nuclear norm (WNN) problem was analyzed under different circumstances where the weights could be in a non-ascending, arbitrary or a non-descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The authors applied their proposed WNN algorithm to an image denoising problem by exploiting the image non- local self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Numerical results showed that the proposed WNN algorithm outperforms many of the state of the art denoising methods in terms of both quantitative measures and visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Another method, inspired by the success of the ℓp quasi-norm (0 < p < 1) for sparse signal reconstruction, is to enforce low-rank structure by using the Schatten-p quasi-norm, which is defined as the ℓp quasi-norm of the singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [31], the authors considered the matrix completion problem, which deals with constructing a low-rank matrix, given a subset of its entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Instead of minimizing the nuclear norm, the authors proposed a Schatten-p quasi-norm formulation, for which they came up with an algorithm and studied its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In each iteration, the sub-problem that needs to be solved has a closed-form solution, which makes it fast and suitable for large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' To improve the robustness of the solutions to matrix completion problem, in [32] Schatten-p quasi-norm for low-rank recovery was combined with ℓp quasi-norm (0 < p ≤ 1) 5 of the prediction errors on the observed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The authors proposed an algorithm based on the ADMM, which performed better in their numerical experiments than other completion methods like fixed-point continuation and accelerated proximal gradient singular-value thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [33], the authors extended the theoretical recovery results previously developped for the nuclear norm to Schatten-p quasi-norm using a weaker version of the RIP assumption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' they showed that the minimum rank solution can be recovered by solving the Schatten-p quasi-norm minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In [34], the authors developed an iterative re-weighted least squares algorithm to solve an unconstrained ℓp minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The algorithm, and analysis, are extended to include the low rank recovery problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Another non-convex approaches for matrix optimization problems involving sparsity are developed, by means of a generalized shrinkage operation, in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' These approaches are applied to the decomposition of video into low rank and sparse components, which is able to separate moving objects from the stationary background better than in the convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Contributions Despite the good performance of the various algorithms discussed in I-B1 and I-B2 for solving different relaxations of (2) and (1), they are all based on a specific structure of the considered convex constraint set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' therefore, they are problem specific and lack generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In this work, we present a general-purpose method based on projections onto the constraint set, for which we assume only closed convexity as the specific structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [36] and [37] examine the characteristics of the projection on the constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In the latter, the projection technique of every given point on ℓp balls is assumed to be known a priori, whereas in the former, the issue is solved while omitting an important coupling condition for the polynomial equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' First, based on our previous work in [38], we propose an ADMM algorithm (pQN-ADMM) to solve the ℓp quasi-norm relaxation of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' At each iteration, the bottleneck operation is to compute Euclidean projections on to some particular convex and non-convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The proposed algorithm possesses two important properties: 1) Its computational complexity is similar to ℓ1 minimization algorithms except for the additional effort of solving for the roots of a polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) No specific structure for the convex set is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Numerical results, using a SVR and binary classification examples, are presented to show the competitive performance of our algorithm against the ℓ1 minimization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We then extend the proposed algorithm to solve the relaxation of (1) based on the Schatten-p quasi-norm – here, we exploit the fact that minimizing the ℓp-norm of the vector of singular values is equivalent to minimizing the Schatten-p quasi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We consider two different numerical examples: 1) We formulate time domain system identification problem for minimum order system detection and solve it using the pQN-ADMM approach and compare our recovery results against the nuclear norm minimization approach of [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) We consider a matrix completion problem, where the goal is to recover an unknown low-rank matrix based on a small fraction of observed entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Numerical results in both examples show that our method is competitive against some of the state-of-the-art algorithms in terms of both the detected system order (for the system identification problem) and the rank of the matrix recovered (for the matrix completion problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Finally, since the derived algorithm depends on a computationally expensive convex projection step in every iteration, we aim to develop a faster algorithm with a mathematical convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Considering only a 6 subset of problems where the constraint set is a polytope, we utilize concepts from the proximal gradient (PG) method to derive a fast algorithm and prove that it converges with a rate O( 1 K ), where K is the iteration budget given to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' NOTATIONS AND BASIC DEFINITIONS Unless otherwise specified, we denote vectors with lowercase boldface letters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', x, with i-th entry as xi, while matrices are in uppercase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X, with (i, j)-th entry as xi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For an integer n ∈ Z+, [n] ∆= {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1 represents a vector of all entries equal to 1, while 1G(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=') is an indicator function to the set G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', it evaluates to zero if its argument belongs to the set G and is +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For a vector x ∈ Rn, the general ℓp norm is defined as ∥x∥p ∆= ( � i∈[n] |xi|p) 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (3) For convenience, we let ∥x∥ be the well known euclidean norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When 0 < p < 1, the expression in (3) is termed as the quasi-norm satisfying the same axioms of the norm except the triangular inequality making it a non-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let X : V −→ W be a linear operator between two normed spaces equipped with ℓp norm, p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The induced p-norm is defined as, ∥X∥p ∆= sup v̸=0 �∥Xv∥p ∥v∥p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (4) A special case of (4) is when p = 2, known as the spectral radius, which can be shown to be the square root of the maximum eigen value of XHX, where XH is the complex conjugate of the transpose of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', X⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In the rest of the analysis, we will drop the subscript 2 in the spectral norm notation and only refer to it with ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For a matrix X ∈ Rm×n, the Lp,q entry-wise norm is defined as, ∥X∥p,q ∆= ( � j∈[n] ( � i∈[m] |xij|p) q p ) 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (5) A special case of (5) is when p = q = 2, known as the Frobenius norm, which were refer to by ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='∥f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let H1 ⊂ Rn and H2 ⊂ Rm be two separable Hilbert spaces and X ∈ Rm×n be a linear compact operator from H1 to H2, the Schatten-p norm of X is then defined as, ∥X∥p ∆= ( � i∈[min{m,n}] σi(X)p) 1 p , (6) where σi(X) is the i-th singular value of the matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When p = 1, equation (6) yields to the nuclear norm which is the convex envelope of the rank function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Throughout the paper, we consider a non-convex relaxation for the rank function, specifically p = 1/2, and compare its performance with the nuclear norm case in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 7 We define ⌈·⌉ as the ceiling operator, vec(X) ∈ Rmn as a vector formed by stacking the columns of the matrix X ∈ Rm×n and Hankel(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=') as an operator that outputs a Hankel matrix constructed from the applied vector arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' SPARSE VECTOR RECOVERY ALGORITHM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Problem Formulation This section develops a method for approximating the solution of (2) using the following relaxation, min x ∥x∥p p , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' x ∈ V, (7) where p ∈ (0, 1] and V is a closed convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Problem (7) is convex for p = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' hence, can be solved to optimality efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, the problem becomes non-convex when p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' An epigraph equivalent formulation of (7) is obtained by introducing the variable t = [ti]i∈[n]: min x,t 1⊤t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ti ≥ |xi|p, i ∈ [n], x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (8) Let X ⊂ R2 denote the epigraph of the scalar function |x|p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', X = {(x, t) ∈ R2 : t ≥ |x|p}, which is a non-convex set for p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, (8) can be cast as min x,t � i∈[n] 1X (xi, ti) + 1⊤t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (9) ADMM exploits the structure of the problem to split the optimization over the variables via iteratively solving fairly simple subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In particular, we introduce auxiliary variables y = [yi]i∈[n] and z = [zi]i∈[n] and obtain an ADMM equivalent formulation of (9) given by: min x,t,y,z � i∈[n] 1X (xi, ti) + 1Y(y) + 1⊤z, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' x = y : λ, t = z : θ, (10) where Y is the 0-sublevel set of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', Y = {y ∈ Rn : y ∈ V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The dual variables associated with the constraints x = y and t = z are λ and θ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, the Lagrangian function corresponding to (10) augmented with a quadratic penalty on the violation of the equality constraints with penalty parameter ρ > 0, is given by: Lρ(x, t, y, z, λ, θ) = � i∈[n] 1X (xi, ti) + 1Y(y) + 1⊤z + λ⊤(x − y) + θ⊤(t − z) + ρ 2 � ∥x − y∥2 + ∥t − z∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (11) Considering the two block variables (x, t) and (y, z), ADMM [39] consists of the following iterations: (x, t)k+1 = argmin x,t Lρ(x, t, yk, zk, λk, θk) (12) (y, z)k+1 = argmin y,z Lρ(xk+1, tk+1, y, z, λk, θk) (13) λk+1 = λk + ρ(xk+1 − yk+1) (14) θk+1 = θk + ρ(tk+1 − zk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (15) 8 Algorithm 1 ADMM (ρ > 0) 1: Initialize: y0, z0, λ0, θ0 2: for k ≥ 0 do 3: (xi, ti)k+1 ← ΠX � yk i − λk i ρ , zk i − θk i ρ � , ∀i ∈ [n] 4: yk+1 ← ΠY � xk+1 + λk ρ � 5: zk+1 ← tk+1 + θk−1 ρ 6: λk+1 ← λk + ρ(xk+1 − yk+1) 7: θk+1 ← θk + ρ(tk+1 − zk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' According to the expression of the augmented Lagrangian function in (11), it follows from (12) that the variables x and t are updated via solving the following non-convex problem min x,t ∥x − yk + λk ρ ∥2 + ∥t − zk + θk ρ ∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (xi, ti) ∈ X, i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (16) Exploiting the separable structure of (16), one immediately concludes that (16) can be split into n independent 2-dimensional problems that can be solved in parallel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', for each i ∈ [n], (xi, ti)k+1 = ΠX � yk i − λk i ρ , zk i − θk i ρ � , (17) where ΠX (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=') denotes the Euclidean projection operator onto the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Furthermore, (11) and (13) imply that y and z are independently updated as follows: yk+1 = ΠY � xk+1 + λk ρ � (18) zk+1 = tk+1 + θk − 1 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (19) Algorithm 1 summarizes the proposed ADMM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It is clear that z, λ, and θ merit closed-form updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, updating (x, t) requires solving n non-convex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Our strategy for dealing with this issue is presented in the section that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Non-convex Projection In this part, we present the method used to tackle the non-convex projection problem required to update x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Among the advantages of the proposed algorithm is that it is amenable to decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' As it is clear from (17), x and t can be updated element-wise via performing a projection operation onto the non-convex set X, one for each i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The n projection problems can be run independently in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We now outline the proposed idea for solving one such projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', we suppress the dependence on the index of the entry of x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For (¯x, ¯t) ∈ R2, ΠX (¯x, ¯t) entails solving min x,t g(x, t) ≜ (t − ¯t)2 + (x − ¯x)2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' t ≥ |x|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (20) 9 If ¯t ≥ |¯x|p, then trivially ΠX (¯x, ¯t) = (¯x, ¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, we focus on the case in which ¯t < |¯x|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The following proposition states the necessary optimality conditions for (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let ¯t < |¯x|p, and (x∗, t∗) be an optimal solution of (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, the following properties are satisfied (a) sign(x∗) = sign(¯x), (b) t∗ ≥ ¯t, (c) |x∗|p ≥ ¯t, (d) t∗ = |x∗|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We prove the statements by contradiction as follows: (a) Suppose that sign(x∗) ̸= sign(¯x), then |x∗ − ¯x|=|x∗ − 0|+|¯x − 0| > |¯x − 0|, (21) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', (x∗−¯x)2 >(0−¯x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, g(x∗, t∗)−g(0, t∗)>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Moreover, the feasibility of (x∗, t∗) implies that t∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, (0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This contradicts the optimality of (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (b) Assume that t∗ < ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, g(x∗, t∗) − g(x∗, ¯t) = (t∗ − ¯t)2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (22) Furthermore, by the feasibility of (x∗, t∗), we have |x∗|p ≤ t∗ < ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, (x∗, ¯t) is feasible and attains a lower objective value than that attained by (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This contradicts the optimality of (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (c) Suppose that |x∗|p < ¯t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', − ¯t 1 p < x∗ < ¯t 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (23) We now consider two cases, ¯x > 0 and ¯x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' First, let ¯x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, we have by (a) and (23) that 0 < x∗ < ¯t 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since ¯t < |¯x|p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', (¯x, ¯t) /∈ X, therefore ¯t 1 p < ¯x and hence, 0 < x∗ < ¯t 1 p < ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Pick x0 > 0 such that |x0|p = ¯t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', x0 = ¯t 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then clearly, x∗ < x0 < ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, we have g(x∗, t∗) − g(x0, t∗) = (x∗ − ¯x)2 − (x0 − ¯x)2 > 0, (24) where the last inequality follows the just proven identity that x∗ < x0 < ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Moreover, we have |x0|p = ¯t ≤ t∗ by (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, (x0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This contradicts the optimality of (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On the other hand, let ¯x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, we have by (a) and (23) that −¯t 1 p < x∗ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since ¯t < |¯x|p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', (¯x, ¯t) /∈ X, then ¯t 1 p < |¯x|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', ¯x < −¯t 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Therefore, ¯x < −¯t 1 p < x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Pick x0 < 0 such that |x0|p = ¯t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', x0 = −¯t 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, (24) also holds when ¯x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Note that |x0|p = ¯t ≤ t∗ by (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, (x0, t∗) is feasible and attains a lower objective value than that attained by (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This contradicts the optimality of (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (d) The feasibility of (x∗, t∗) eliminates the possibility that t∗ < |x∗|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Now let t∗ > |x∗|p and pick t0 = |x∗|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, ¯t ≤ |x∗|p = t0 < t∗, where the first inequality follows from (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Then, 0 ≤ t0 − ¯t < t∗ − ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, we have g(x∗, t∗) − g(x∗, t0) = (t∗ − ¯t)2 − (t0 − ¯t)2 > 0, (25) 10 Algorithm 2 Non-convex projection (p = s q < 1) 1: R ← roots{a2q + s q(a2s − ¯tas) − |¯x|aq} 2: ¯R ← R \\ {complex numbers and negative reals in R} 3: T ← {(rq, rs) : r ∈ ¯R} 4: (ˆx, t∗) ← argmin {g(x, t) : (x, t) ∈ T } 5: x∗ ← sign(¯x)ˆx Furthermore, the feasibility of (x∗, t0) follows trivially from the choice of t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Thus, (x∗, t0) is feasible and attains a lower objective value than that attained by (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This contradicts the optimality of (x∗, t∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We now make use of the fact that for (20), an optimal solution (x∗, t∗) satisfies t∗ = |x∗|p and hence, (20) reduces to solving min x (|x|p − ¯t)2 + (x − ¯x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (26) The first order necessary optimality condition for (26) implies the following: p|x∗|p−1sign(x∗)(|x∗|p − ¯t) + x∗ − ¯x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (27) By the symmetry of the function |x|p, without loss of generality, assume that x∗ > 0 and let 0 < p = s q < 1 for some s, q ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A change of variables aq = x∗ plugged in (27) shows that finding an optimal solution for (20) reduces to finding a root of the following scalar degree 2q polynomial: a2q + s q � a2s − ¯tas� − ¯xaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (28) Thus, to find ΠX (¯x, ¯t), solve for a root a∗ of the polynomial in (28) such that (a∗q, a∗s) minimizes g(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Algorithm 2 summarizes the method we use to solve problem (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In case ¯x = 0, we set x∗ = t∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' If the set ¯R is empty, we set x∗ = 0 and t∗ = (¯t)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Convex Projection The convex projection for y-update in (18) can be formulated as the following convex optimization problem yk+1 = argmin y �����y − (xk+1 + λk ρ ) ����� 2 , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' y ∈ V, (29) where ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='∥ is the euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Convex problems can be solved by a variety of contemporary methods including bundle methods [40], sub-gradient projection [41], interior point methods [42], and ellipsoid methods [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The efficiency of optimization techniques rely mainly on exploiting the structure of the constraint set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' As mentioned in I-C, to be general, we aim to solve the problem in (7) with no assumptions on the set V, other than it being closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' That said, if possible, through exploiting the structure of V, one should be able to reduce the computational complexity of solving (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 11 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' As per our knowledge, none of the existing literature considered the convergence of an ADMM algorithm for solving the general problem in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' As discussed in I-B1, on one hand, the work in [23] studied the convergence of ADMM under mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, assuming V has a particular form, these assumptions hold only if the function f defining the the constraint set V = {x : f(x) ≤ 0} in (7) is Lipschitz differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On the other hand, [44] studied the convergence of a non ADMM algorithm to solve (7) while assuming that the global optimal for each update step can be found efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' RANK MINIMIZATION ALGORITHM We consider the same problem as in (1) and propose a method for approximating its solution efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The Schatten-p heuristic of (1) can be written as min X ∥X∥p p ∆= L � i=1 |σi(X)|p, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X ∈ M, (30) where L = min(m, n) and σi(X) is the ith singular value of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When p = 1, problem (30) is a convex one which is eventually the nuclear norm heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We consider a non-convex case where 0 < p < 1, which has the corresponding epi-graph form, min X,t 1⊤t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' |σi(X)|p ≤ ti, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='L}, X ∈ M, (31) such that t = [ti]i∈[L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Defining the epi-graph set ˚ X for the function σ(X), where ˚ X ∆= {(σ(X), t)∈R2 :|σ(X)|p ≤ t} ⊆ R2, the problem in (31) can be written as, min X,t 1⊤t + 1M(X) + L � i=1 1 ˚ X (σi(X), ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (32) In order to structure the problem in a from that ADMM can exploit, we introduce the auxiliary variables Y ∈ Rm×n and z = [zi]i∈[L] which makes the problem in (32) be, min X,t,Y,z 1⊤z + 1V(Y) + L � i=1 1 ˚ X (σi(X), ti), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X = Y : Λ, t = z : θ, (33) such that Λ, θ are the dual variables associated with X and t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Similar to (11), the Lagrangian function associated with (33) augmented with a quadratic penalty for the equality constraint violation with a parameter ρ > 0, is Lρ(X, Y, t, z, Λ, θ)=1⊤z+ 1M(Y)+ L � i=1 1 ˚ X (σi(X), ti) +T r{Λ⊤(X−Y)}+θ⊤(t−z)+ ρ 2(∥X−Y∥2 f +∥t−z∥2), (34) 12 where T r{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='} is the trace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Considering the 2-tuples (X, t) and (Y, z), the ADMM iterations is, (X, t)k+1 = argmin X,t Lρ(X, Yk, t, zk, Λk, θk), (35) Yk+1 = argmin Y Lρ(Xk+1, Y, tk+1, zk, Λk, θk), (36) zk+1 = argmin z Lρ(Xk+1, Yk+1, tk+1, z, Λk, θk), (37) Λk+1 = Λk + ρ(Xk+1 − Yk+1), (38) θk+1 = θk + ρ(tk+1 − zk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (39) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (X, t) update By completing the square and with some simple algebra, it can be shown that the problem in (35) is equivalent to min X,t ��X − ¯Xk��2 f + ��t − ¯tk��2 , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' |σi(X)|p ≤ ti, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' L}, (40) where ¯Xk ∆= Yk − Λk ρ and ¯tk ∆= zk − θk ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For an ease of notations, we will drop the iteration index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Assume that X = PΣQ⊤ and ¯X = U∆V⊤ is the singular value decomposition (SVD) of X and ¯X respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Where Σ, ∆ ∈ RL×L are diagonal matrices with the singular values associated X and ¯X while P, U ∈ Rm×L and Q, V ∈ Rn×L are the unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' By applying the same steps as in Theorem 3 of [45], we can write the first term of (40) after dropping k as, ��X − ¯X ��2 f = ��PΣQ⊤ − U∆V⊤��2 f = ��PΣQ⊤��2 f + ��U∆V⊤��2 f − 2T {X⊤ ¯X} (a) = T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{QΣ⊤P⊤U∆V⊤} (b) ≥ T r{Σ⊤Σ}+T r{∆⊤∆}−2T r{Σ⊤∆}=∥Σ−∆∥2 f , (41) where (a) is because P⊤P = Q⊤Q = U⊤U = V⊤V = IL×L with IL×L being an identity matrix of size L, and exploiting the circular property of the trace while (b) holds is from the main result of [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In order to make ��X − ¯Xk��2 f achieve its derived lower bound, we set P = U and Q = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Henceforth, the problem in (40) will be equivalent to, min X,t ∥x − ¯x∥2 + ∥t − ¯t∥2 , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' |xi|p ≤ ti, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' L}, (42) where x = [xi]i∈[L] and ¯x = [¯xi]i∈[L] are the vectors of singular values of the matrices X and ¯X respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The optimal solution X∗ for (40) can be calculated by finding the optimal x∗ of (42) and then X∗ = UΣ∗VT , where Σ∗ =diag(x∗) and diag(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=') is an operator that converts a vector to its corresponding diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since the problem in (42) is separable, we drop the index i and only consider solving min x,t (x − ¯x)2 + (t − ¯t)2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' |x|p ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (43) 13 It can be realized that (43) is the same as (20), hence, its optimal solution can be found by applying algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (Y, z) update After updating (X, t) while fixing Λ and θ, the problem in (36) can be written as, Yk+1 =argmin Y �����Y−(Xk+1+ Λk ρ ) ����� 2 f , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Y ∈ M, (44) which is clearly a convex optimization problem representing the projection of the point Xk+1 + Λk ρ on the set M and can be solved by various known class of algorithms as discussed in section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Upon updating Y, the z update in (37) is zk+1 = argmin z 1⊤z + ρ 2 �����z − (tk+1 + θk ρ ) ����� 2 , (45) which has the closed-form solution z = tk+1 + θk−1 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' PROXIMAL GRADIENT ALGORITHM The SVR algorithm deals with the ℓp relaxation of (2) without assuming any specific structure for V, other than being closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Indeed, the algorithm only requires the Euclidean projections onto V as in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, this approach suffers from two pitfalls: 1) high computational complexity per iteration as a result of solving (29) in every iteration, and 2) the lack of convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In this section, we consider a sub-class of problems with a specific structure for the convex set of the form V = {x : f(x) ≤ 0}, where f(x) = ∥Ax − b∥2 − ǫ for some given ǫ ≥ 0, A ∈ Rm×n and b ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Note that f(x) is a convex function with Lipschitz continuous gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', f is L-smooth: ∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥ for all x, y ∈ Rn and L ≜ ∥A∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Specifically, in order to solve min x ∥x∥p p , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' f(x) ≤ 0, (46) we aim to develop an efficient algorithm with some convergence guarantees for the following Lagrangian relaxation: min x F(x) ∆= ∥x∥p p + µ 2 f(x), (47) where µ ≥ 0 is the dual multiplier that captures the trade-off between solution sparsity and fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A canonical problem for the regularized risk minimization has the following form: min x g(x) + h(x) (48) where h is an L-smooth loss function and g is a a regularizer term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When both g and h are convex, the proximal gradient (PG) algorithm [47] can compute a solution to (48) through iteratively taking PG steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', xk+1 = proxg/λ(xk − ∇h(xk)/L) where proxg/λ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=') ∆= argminx g(x) + λ 2 ∥x − ·∥2, for some constant λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When g is convex, prox operation is well-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' thus, the PG step can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 14 Comparing both (47) and (48), the convexity assumption of g(x) in (48) is not satisfied for ∥x∥p p in (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When the regularizer is a continuous nonconvex function, the proximal map proxg/λ may not exist, let alone it can be computed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On the other hand, for ∥x∥p p, using similar arguments for the non-convex projection step introduced in subsection III-B, we aim to derive an analytical solution that can be computed efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Indeed, assuming p ∈ (0, 1) is a positive rational number, the proposed method for computing the proximal map of ∥x∥p p involves finding the roots of a polynomial of order 2q, where q ∈ Z+ such that p = s/q for some s ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since f is L-smooth, for all x, y ∈ Rn, we have f(x) ≤ f(y) + ∇f(y)⊤(x − y) + L 2 ∥x − y∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (49) Given xk, replacing f(x) with the upper bound in (49) for y = xk, the prox-gradient operation naturally arises as follows: xk+1 = argmin X ∥x∥p p + µ 2 [f(xk) + ∇f(xk)⊤(x − xk) + L 2 ��x − xk��2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (50) By completing the square, (50) yields to xk+1 = argmin X ∥x∥p p + µL 4 ����x − � xk − 1 L∇f(xk) ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (51) Defining ¯xk ∆= xk − 1 L∇f(xk), (51) can be rewritten as xk+1 = argmin X ∥x∥p p + µL 4 ��x − ¯xk��2 = argmin X n � i=1 |xi|p + µL 4 (xi − ¯xk i )2, (52) which is clearly a separable structure in the entries of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Therefore, for each i ∈ [n], we have xk+1 i =argmin xi |xi|p+ µL 4 (xi−¯xk i )2 = prox¯g/ µL 2 (¯xk i ), (53) where ¯g : R → R+ such that ¯g(t) = |t|p for some positive rational p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Next, we consider a generic form of (53), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', given some ¯t ∈ R, we would like to compute t∗ = argmin t {|t|p + µL 4 (t − ¯t)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (54) The first-order optimality condition for (54) can be written as p|t∗|p−1sign(t∗) + µL 2 (t∗ − ¯t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (55) Using similar arguments with those in section III-B for Proposition 1, we can conclude that the optimal solution t∗ attains the property that sign(t∗) = sign(¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Without loss of generality, exploiting the symmetry of the function ¯g, we only consider the case when ¯t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' hence, the optimal solution t∗ is the smallest positive root of the following polynomial: p|t∗|p−1 + µL 2 (t∗ − ¯t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (56) 15 Algorithm 3 Accelerated PG algorithm 1: Initialize: µ, s = 1, q = 2, l, x0, x1, k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2: repeat 3: yk = xk + k−1 k+2(xk − xk−1) 4: ∆k = maxt=max{1,k−l},.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=',k F(xt) 5: if F(yk) ≤ ∆k then: 6: vk = yk 7: else: 8: vk = xk 9: ¯xk = vk − 1 L∇f(vk) 10: for i ∈ [n] do: 11: solve a2q − ¯xiaq + 2s qµLas = 0 12: xk+1 i = a∗q 13: k = k + 1 14: until convergence As in (28), suppose 0 < p = s q < 1 for some s, q ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Using the change of variables a ≜ (t∗) 1 q , (56) reduces to finding the roots of a polynomial of degree 2q: a2q − ¯taq + 2s qµLas = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (57) To efficiently solve (46), we will use Algorithm 3, which is an implementation of nonconvex inexact accelerated proximal gradient (APG) descent method proposed in [48, Algorithm 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' To summarize, [48, Algorithm 2] is designed to solve composite problems of the form in (48) assuming that h is L-smooth and g is proper lower- semicontinuous such that F ≜ h + g is bounded from below and coercive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', lim∥∥→∞ F() = +∞ – note that there is no assumption regarding neither h nor g to be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The key points enhancing both practical behavior of and theoretical guarantees for [48, Algorithm 2] can be summarized as given below: An extrapolation yk is generated as introduced in [49] for the APG algorithm (step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Steps 4 through 9 allow non monotone update of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' F(yk) is checked with respect to the maximum of the latest l objective values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The gradient step is adjusted according to this (step 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This permits yk to occasionally increase the objective and makes F(yk) be less than the maximum of the objective value of the latest l iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Steps 11 and 12 are the solution of the PG step using the non-convex projection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In the next part, we show that algorithm 3 converges to a critical point and it exhibits a convergence rate of O( 1 k), where k is the iteration budget that is given to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 16 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ( [50]) The Frechet sub-differential of F at x is ˆ∂F(x) ∆= � u : lim y̸=x lim y→x F(y) − F(x) − u⊤(y − x) ∥y − x∥ ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (58) The sub-differential of F at x is ∂F(x) ∆= {u : ∃xk → x, F(xk) → F(x) and uk ∈ ˆ∂F(xk) → u as k → ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (59) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ( [50]) x is a critical point of F if 0 ∈ ∂g(x) + ∇h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' By comparing (48) and (47), it can be realized that the functions g(x) and h(x) in definition 4 are equal to ∥x∥p p and µ 2 f(x) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The sequence xk generated from algorithm 3 has at least one limit point and all the generated limit points are critical points of (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Moreover, the algorithm converges with rate O( 1 K ), where K is the iteration budget given to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can easily be verified that our problem in (47) satisfies all required assumptions for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Indeed, 1) The function g(x) = ∥x∥p p is a proper and lower semi-continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) The gradient of h(x) = µ 2 f(x) is ¯L-Lipschitz smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', ∥∇h(x) − ∇h(y)∥ ≤ ¯L ∥x − y∥ for all x, y ∈ Rn, with ¯L = µ 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3) F(x)=g(x)+h(x) is bounded from below, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', F(x)≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4) lim∥x∥→∞ F(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 5) The introduced non-convex projection method is an exact solution for the proximal gradient step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This is because it is based on finding the roots of a polynomial of order 2q in equation (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Therefore, the assumptions required for theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='1 for critical point convergence and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='3 for the rate of convergence in [48] are satisfied which then completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The global convergence of several exact iterative methods that solve (48) has been explored, under the framework of Kurdyka–Lojasiewicz (KL) theory, in various additional literature including [50]–[54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Other work (see [55] and references therein) considered the linear convergence of non-exact algorithms with relaxations on the assumptions of KL theory, however, it is difficult to verify that the sequence generated by algorithm 3 satisfies the relaxed assumptions stated in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' NUMERICAL RESULTS FOR SVR PROBLEM In this section, we present two numerical examples for the p-quasi-norm ADMM (pQN-ADMM) from algorithm 1 and the non-convex projection from algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For both examples, the pQN-ADMM algorithm result is compared with the ℓ1 objective function solution from MOSEK solver [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The two examples include;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' i) Sparse signal reconstruction from noisy measurements, where the pQN-ADMM algorithm is also compared with another ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 quasi-norm minimization based algorithm, named ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5-FL, described in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ii) Binary classification using support vector machines (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 17 10-4 10-3 10-2 10-1 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='26 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1: Effect of noise variance on the sparsity of solutions obtained by pQN-ADMM algorithm, ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5-FL algorithm and ℓ1 norm minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sparse Signal Reconstruction Let n = 210 and m = n/4, randomly construct the sparse binary matrix, M ∈ Rm× n 2 , with a few number of ones in each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The number of ones in each column of M is generated independently and randomly in the range of integers between 10 and 20, and their locations are randomly chosen independently for each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let U = [M, −M], which is the vertical concatenation of the matrix M and its negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Following the same setup in [58], the column orthogonality in U is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let xopt ∈ Rn be a reference signal with ∥xopt∥0 = ⌈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2n⌉, where the non-zero locations are chosen uniformly at random with the values following a zero mean, unit variance Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let v = Uxopt+n be the allowable measurement, where n ∈ Rm is a Gaussian random vector with zero mean and co-variance matrix σ2Im×m, where I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The sparse vector is reconstructed from v by solving (7) with V = {x : ∥Ux − v∥/∥v∥ − ǫ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Figure 1 plots the relation between the sparsity level and the noise variance for ℓ1 norm minimization, ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5-FL quasi-norm and pQN-ADMM solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A threshold value of 10−6 was used where the threshold is a value below which the entry of the solution vector is considered to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Depending on the noise variance σ2, the value of ǫ was chosen to make the problem feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The reported result is the average of 100 independent random runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that pQN-ADMM algorithm produces a sparser solution than its counter baselines for different values of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On increasing σ2, the sparsity level for all methods decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This is due to the increased scarcity of information on the original signal in the realization vector which makes the reconstruction process less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Binary Classification In this part, we build an email spam classifier based on support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We use a subset of the training set used in the SpamAssassin Public Corpus [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let {(uj, vj)}j∈[m] be the training set of feature vectors uj ∈ {0, 1}n with corresponding labels vj ∈ {−1, 1} identifying whether the email is spam or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We highlight the effectiveness of our method in designing an email spam detector using the least number of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Following [60], we maintain a dictionary of n = 1899 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For a given email j ∈ [m], the ith entry of uj is 1 if word wi, i ∈ [n] of the dictionary is in email j, and is 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We aim to build a linear classifier with the decision rule ˆv = sign(u⊤x), where u is the feature vector of the email in question and x is a vector of the classifier coefficients with the first entry being the bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The main aim is to build a classifier that detects whether an email is a spam or not, using the least number of words from the dictionary and achieving a high training data accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' To achieve this objective, we solve (7) with M = {x : 1 m � j∈[m] � 1 − vju⊤ j x �+ − ǫ ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be clearly realized that the training set accuracy is controlled by ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Algorithm 1 was run for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5, 2000 training emails and various values for ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For each value of ǫ, the algorithm was terminated after 100 iterations and performance tested on 1000 emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For comparison purpose, the problem was also solved with the ℓ1 norm convex relaxation under the same setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In figure 2a, we plot the number of non-zero entries in the optimal classifier from both the pQN-ADMM and ℓ1 solutions vs different values of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We used a threshold of 10−4, where the threshold is defined as in section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized from figure 2a that the pQN-ADMM solution outperforms the ℓ1 in terms of the number of words used for legitimacy detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' When the value of ǫ increases, the number of required words decreases for both ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 and ℓ1 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This outlines the trade-off between the sparsity level of the classifier and its accuracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', small values of ǫ enforces a low classification error in expense of a less sparse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The corresponding training and test set accuracies for the obtained classifiers are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Both figures 2a and 2b depict the performance of the pQN-ADMM solution from algorithm 1 in terms of the sparsity level while maintaining nearly the same level of accuracy as the ℓ1 solution for both the training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' NUMERICAL RESULTS FOR RMP PROBLEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Time domain system identification In this part, we apply the derived pQN-ADMM approach on a time domain system identification example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In that example, input is applied to randomly generated systems with a known order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Using the outputs corresponding to these systems, the minimum rank/order system is derived and results are compared to nuclear norm heuristic in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We consider a discrete time stable Single Input Single Output (SISO) system with an input u ∈ RT, where T represents the number of input samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', input time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We assume an impulse response of a fixed number of samples n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The corresponding system output is y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, we assume that only noisy realizations, ˆy, of the output can be considered, such that;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ˆy ∆= y + z = h ⊛ u + z , where h ∈ Rn is the system’s original impulse response, z ∈ Rm is a random vector with entries drawn independently from samples of a uniform distribution on the range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='25], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', zi ∼ U[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='25], while ⊛ denotes the convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' From the window 19 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 0 50 100 150 200 250 Number of selected words (a) Number of words selected for classification versus ǫ for pQN-ADMM and ℓ1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 65 70 75 80 85 90 95 100 Correct percentage (b) Training and test set accuracies versus ǫ for pQN-ADMM and ℓ1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2: Binary classification numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' property of the convolution, m = n + T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Assume that ui, hi and yi are the ith components of the vectors u, h and y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The three components are related to each other by convolution through yi = �∞ j=−∞ hjui−j which is a linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hence, let T ∈ Rm×n be the Toeplitz matrix formed by the input u , it can be easily seen that h ⊛ u = hT ⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Assume that x ∈ Rn is an impulse response variable and let X ∈ Rn×n be a Hankel matrix formed by the entries of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' From [29], [61]–[63], the minimum order time domain system identification problem can be formulated as, min x,X RankX, (60a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X = Hankel(x), (60b) ��ˆy − xT ⊤��2 ≤ ǫ, (60c) (60b) ensures that X is a Hankel matrix and (60c) holds to make the result by applying the input, u, to the optimal impulse response, x, fit the available noisy data, ˆy, in a non-trivial sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Defining the convex set C ∆={X∈Rn×n : ��ˆy−hT ⊤��2−ǫ ≤ 0, X=Hankel(x)}, (60) can be cast as, min x,X Rank(X), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' X ∈ C, (61) which is clearly identical to the problem in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The problem was solved using the same pQN-ADMM approach discussed in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We let T = m = 50 and n = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Note that m < T + n − 1, which is a reasonable assumption as in some practical applications, one is allowed only a specific window to realize the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We consider the simulation for 10 different original system orders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', η = 2 : 2 : 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' An input vector, u, is generated, where the elements of u are independent and follow a uniform distribution on the interval [−5, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For each η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1) 50 random stable systems are generated using the command ’drss’ in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) The generated input is applied to each system to get the corresponding noisy output ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3) Given the output ˆy, the problem in (60) is solved and the corresponding system’s 20 2 4 6 8 10 Original system order 0 5 10 15 20 Average rank Threshold=10-4 2 4 6 8 10 Original system order 0 5 10 15 20 Average rank Threshold=10-5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3: Average rank vs original system order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Red and cyan colors are for the nuclear norm and pQN-ADMM algorithm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' η=2 η=6 η=10 Nuclear norm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='3907 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='6668 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2572 pQN-ADMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='9042 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='0861 TABLE I: Standard deviation for threshold=10−4 rank is calculated using singular value decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4) The results are averaged out to get the corresponding average rank to each original η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Figure 3 shows the average rank for the the nuclear norm and pQN-ADMM heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The results are for two different values of thresholds, where the threshold is defined as the value below which the singular value is considered to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that the introduced pQN-ADMM approach outperforms the nuclear norm one for both values of thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Moreover, when the threshold value decreases from 10−4 to 10−5, the behavior of the pQN-ADMM remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, the average rank for the nuclear norm increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This proves the robustness of the derived pQN-ADMM in comparison to the nuclear norm one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tables I and II show the standard deviation of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be seen that the standard deviation is the same for the pQN-ADMM when η=2 η=6 η=10 Nuclear norm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='9877 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='2638 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='7854 pQN-ADMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='9113 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='0861 TABLE II: Standard deviation for threshold=10−5 21 changing the threshold, however, it increases for the nuclear norm as the threshold value decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Matrix Completion Example In this section, we apply our algorithm (pQN-ADMM) to a matrix completion example and compare the result to the matrix iterative re-weighted least squares (MatrixIRLS) [64], [65], truncated iterative re-weighted unconstrained Lq (tIRucLq) [34] and iterative re-weighted least squares (sIRLS-p & IRLS-p) [66] algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The matrix completion problem is a special case of the low rank minimization where a linear transform takes a few random entries of an ambiguous matrix X ∈ Rm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Given only these entries, the goal is to approximate X and find the missing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The matrix completion problem with low rank recovery can be approximated by, min X ∥X∥p p , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ∥A(X) − b∥ ≤ ǫ, (62) where A : Rm×n → Rq is a linear map with q ≪ mn and b ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In order to apply the mentioned algorithms, the linear transform A(X) will be rewritten as Avec(X), where A ∈ Rq×mn and vec(X) ∈ Rmn is a vector formed by stacking the columns of the matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A random matrix M ∈ Rm×n with rank r is created using the following method: 1) M = MLM⊤ R, where ML ∈ Rm×r and MR ∈ Rn×r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) The entries of both ML and MR are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='d Gaussian random variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let ˆ M = M + Z, where Z ∈ Rm×n is a Gaussian noise with each entry being an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='d Gaussian random variable with zero mean and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The vector b is then created by selecting random q elements from vec( ˆ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Since b = Avec( ˆ M), one can easily construct the matrix A which is a sparse matrix where each row is composed of a value 1 at the index of the corresponding selected entry in the vector b while the rest are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We set m = n = 100, r = 5 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Let dr = r(m+n−r) denotes the dimension of the set of rank r matrices and define s = q mn as the sampling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We assume that s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='195 which yields to q = 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that dr q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We set σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='1 and let the algorithms terminate if a budget of 1000 iterations is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In order to compare the results from different algorithms, we consider the average of 50 runs for two measures: a) the relative Frobenius distance (RFD) to the matrix M, b) the relative error to singular (REtS) values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In figures 4a and 4b, we report the average RFD and REtS values for all the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Despite that all the baselines are designed to exploit the specific structure of the matrix completion problem, described in (62), while the proposed pQN-ADMM doesn’t, it is competitive against them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This in turns shows the effectiveness of the pQN-ADMM algorithm in solving the rank minimization problems without requiring any prior information about the structure of the associated convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' NUMERICAL RESULTS FOR THE NONCONVEX ACCELERATED PROXIMAL GRADIENT (APG) ALGORITHM In this subsection, we present numerical results for the APG method, displayed in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Following the same procedure in [67], we first generate the target signal x∗ through x∗ i = \uf8f1 \uf8f2 \uf8f3 Θ(1) i 103Θ(2) i , ∀ i ∈ Λ, 0, ∀ i ∈ [n] \\ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (63) 22 0 200 400 600 800 1000 Number of iterations 10-2 10-1 100 IRLS-p sIRLS-p MatrixIRLS tIRucLq pQN-ADMM (a) RFD to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1 2 3 4 5 Index of singular value 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='1 IRLS-p sIRLS-p MatrixIRLS tIRucLq pQN-ADMM (b) REtS values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4: The RFD and REtS average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' where the design parameters Λ ⊂ [n], and Θ(1) i , Θ(2) i for i ∈ Λ are chosen as follows: 1) the index set Λ ⊂ [n] is constructed by selecting a subset of [n] with cardinality s uniformly at random;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2) {Θ(1) i }i∈Λ are independent, identically distributed (IID) Bernoulli random variables taking values ±1 with equal probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3) {Θ(2) i }i∈Λ are IID uniform [0, 1] random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The measurement matrix A ∈ Rm×n is a partial Discrete Cosine Transform (DCT) matrix with rows correspond- ing to m < n frequency, where these m indices are chosen uniformly at random from [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The noisy measurement vector b ∈ Rm is then set to be b = A(x∗ + ǫ1) + ǫ2, where ǫ1 ∼ N(0, σ2 1) and ǫ2 ∼ N(0, σ2 2) are the input and realization noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In our experiments, n = 4096, s = ⌈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5m⌉ and the PG algorithm memory to 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Following the medium noise setup in [68], we set σ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='005, σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For f(x) = ∥Ax − b∥2, we have L = 2∥A∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We perform our experiment for various values of m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', number of noisy measurements, and µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', trade-off parameter, see (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For each (m, µ) selection, in order to capture the inherent statistical variation of the problem, we generate 20 random instances of the triplet (x∗, A, b) and each random instance is solved by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We reported the average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' We terminated Algorithm 3 when the relative error between consecutive iterates satisfies ��xk − xk−1�� / ��xk−1�� ≤ 10−5 for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In our experiments, we compared solving (47) for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 against p = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', against ℓ1-optimization for sparse recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On one hand, when p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', for ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 minimization, we solve (47) using Algorithm 3, called ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact, and using the algorithm 2 of [69], which we call ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' On the other hand, when p = 1, ℓ1-minimization problem is a convex one and we adopt the FISTA algorithm of [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The solution is denoted by ¯x while the target signal, from (63), by x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In Algorithm 3, x0 is set to a zero vector while x1 is the ℓ1 norm solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Figures 5 and 6 highlight the relation between the average error and sparsity vs µ for different values of n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that the average error (sparsity) decreases (increases) on increasing µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' For small values of µ, more weight is given to the loss function, which emphasizes the ℓ0 quasi-norm minimization, and hence the sparsity level, 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 5: Average error vs µ for different values of n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yellow and cyan shades are the standard deviations for the exact and approximate ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 quasi-norms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' as in figure 6, is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, for high values of µ, more weight is assigned to the minimization of the regularization term, which solves ∥Ax − b∥2, and hence the error decreases, as shown in figure 3, with a corresponding increase in the sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 solutions always outperforms the ℓ1 one with very slight difference between the exact and the approximate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Figure 7 highlights the statistics of the number of iterations used until convergence for both the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact and approximate algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' It can be realized that with a sufficient number of available realizations, n/m = 8 and n/m = 16, both algorithms approximately consume the same number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' However, when the number of available realizations decreases, n/m = 32 and higher, our exact proximal solution requires significantly less number of iterations to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This conclusion, along with figures 5 and 6 findings, indicates that our algorithm not only finds a similar solution to the approximate method, but also converges with a fewer number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' n/m=8 n/m=16 n/m=32 0 0 0 l1 l1 1 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 1 lo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 1 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 2 2 2 all 3 3 3 4 4 5 5 5 6 6 6 7 7 7 0 10 20 30 0 10 20 30 0 10 20 30 u n/m=64 n/m=128 n/m=256 0 0 l1 l1 1 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5exact 1 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 1 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 2 2 2 3 gl 3 3 4 4 4 5 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 6 6 6 7 7 7 0 10 20 30 0 10 20 30 0 10 20 30 u24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 6: Sparsity vs µ for different values of n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yellow and cyan shades are the standard deviations for the exact and approximate ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 quasi-norms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' CONCLUSION In this study, we presented a non-convex ADMM algorithm (pQN-ADMM) to solve the ℓp norm minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' The algorithm has a similar complexity to that of the ℓ1 minimization in addition to solving the roots of a polynomial for the non-convex projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Our algorithm can also be considered as a general procedure for solving ℓp problems as no specific structure for the convex constraint set was assumed and a convex projection on that set was done for variables update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Applying sparse signal recover and binary classification examples, our method was found to outperform the ℓ1 minimization in terms of the sparsity of the generated solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' In addition, we studied the problem of solving a non-convex relaxation of RMPs using Schatten-p quasi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' This relaxation was shown to be the ℓp minimization of the singular values of the variable matrix and hence the primary developed algorithm could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Showing the numerical results, the pQN-ADMM was found to be less sensitive to the threshold decrease in time domain system identification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Additionally, the pQN-ADMM method was shown to be n/m=8 n/m=16 n/m=32 5 5 5 l1 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5exact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 3 3 3 x*lo 2 区 2 [x* 2 区 1 0 0 0 1 1 1 0 10 20 30 0 10 20 30 0 10 20 30 μ n/m=64 n/m=128 n/m=256 6 6 10 5 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 5 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5exact 8 lo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 4 4 6 [x o 3 3 [x*10 重 [x*| 4 2 区 2 2 0 0 0 1 2 7 0 10 20 30 0 10 20 30 0 10 20 30 u25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 7: Iterations count vs µ for different values of n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' competitive against various other baselines when solving the matrix completion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' REFERENCES [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Vandenberghe and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, “Semidefinite programming,” SIAM Review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 49–95, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Candes and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4203–4215, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wright, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ganesh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sastry, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ma, “Robust face recognition via sparse representation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 210–227, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Candes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Romberg, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 489–509, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1289–1306, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Bruckstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Donoho, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 34–81, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' n/m=8 n/m=16 n/m=32 6000 6000 6000 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 5000 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx count 4000 count 4000 count 4000 Iterations erations Iterations 3000 3000 3000 2000 2000 2000 1000 1000 1000 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 u n/m=64 n/m=128 n/m=256 6000 6000 6000 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 exact 5000 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5approx 5000 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx 5000 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='5 approx A Iterations count 4000 4000 count 4000 rations 3000 3000 rations 3000 2000 2000 2000 1000 1000 1000 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 u u26 [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tropp and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wright, “Computational methods for sparse solution of linear inverse problems,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 98, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 948–958, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mallat and Zhifeng Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3397–3415, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2231–2242, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Donoho, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Saunders, “Atomic decomposition by basis pursuit,” SIAM review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 129–159, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society: Series B (Methodological), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 267–288, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 707–710, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yin, “Iteratively reweighted algorithms for compressive sensing,” in 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3869–3872, IEEE, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wipf and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Rao, “Sparse bayesian learning for basis selection,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2153–2164, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Schniter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Potter, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ziniel, “Fast bayesian matching pursuit,” in 2008 Information Theory and Applications Workshop, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 326–333, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Miller, Subset selection in regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' CRC Press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Saab, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yilmaz, “Stable sparse approximations via nonconvex optimization,” in 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3885–3888, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand, “Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data,” in 2009 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 262–265, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [19] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mourad and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Reilly, “Minimizing nonconvex functions for sparse vector reconstruction,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3485–3496, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wohlberg, “A nonconvex ADMM algorithm for group sparsity with sparse groups,” in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 6009–6013, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [21] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Xu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zhang, “l1/2 regularization: A thresholding representation theory and a fast solver,” IEEE Transactions on Neural Networks and Learning Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1013–1027, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zeng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wang, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Xu, “l1/2 regularization: Convergence of iterative half thresholding algorithm,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2317–2329, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Li and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Pong, “Global convergence of splitting methods for nonconvex composite optimization,” SIAM Journal on Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2434–2460, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” Journal of Scientific Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 78, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 29–63, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mesbahi, “On the semi-definite programming solution on the least order dynamic output feedback synthesis,” 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hindi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, “A rank minimization heuristic with application to minimum order system approximation,” in Proceedings of the 2001 American Control Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='01CH37148), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4734–4739 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='6, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hindi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, “Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices,” in Proceedings of the 2003 American Control Conference, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2156–2162 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='3, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 27 [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hindi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, “Rank minimization and applications in system theory,” in Proceedings of the 2004 American Control Conference, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3273–3278 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='4, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mohan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, “Reweighted nuclear norm minimization with application to system identification,” in Proceedings of the 2010 American Control Conference, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2953–2959, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Gu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zuo, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Feng, “Weighted nuclear norm minimization with application to image denoising,” in 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2862–2869, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Nie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Huang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ding, “Low-rank matrix recovery via efficient Schatten p-norm minimization,” in Twenty-sixth AAAI conference on artificial intelligence, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Nie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Huang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ding, “Joint schatten p-norm and lp norm robust matrix completion for missing value recovery,” Knowledge and Information Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 525–544, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Huang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chen, “Exact minimum rank approximation via Schatten p-norm minimization,” Journal of Computational and Applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 267, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 218–227, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Lai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Xu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yin, “Improved iteratively reweighted least squares for unconstrained smoothed ℓq minimization,” SIAM Journal on Numerical Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 927–957, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chartrand, “Nonconvex splitting for regularized low-rank + sparse decomposition,” IEEE Transactions on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 5810–5819, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Gupta and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Kumar, “Non-convex p-norm projection for robust sparsity,” in 2013 IEEE International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1593–1600, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Bahmani and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Raj, “A unifying analysis of projected gradient descent for ℓp-constrained least squares,” Applied and Computational Harmonic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 366–378, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ashour, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Lagoa, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Aybat, “Lp quasi-norm minimization,” in 2019 53rd Asilomar Conference on Signals, Systems, and Computers, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 726–730, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Parikh, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chu, Distributed optimization and statistical learning via the alternating direction method of multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Now Publishers Inc, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Helmberg and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Rendl, “A spectral bundle method for semidefinite programming,” SIAM Journal on Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 673–696, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Beck and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Teboulle, “Mirror descent and nonlinear projected subgradient methods for convex optimization,” Operations Research Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 167–175, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [42] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Nesterov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Nemirovskii, Interior-point polynomial algorithms in convex programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' SIAM, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ben-Tal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Nemirovski, Lectures on modern convex optimization: analysis, algorithms, and engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' SIAM, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [44] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Xu and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yin, “A globally convergent algorithm for nonconvex optimization based on block coordinate update,” Journal of Scientific Computing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 72, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 700–734, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [45] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zha, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yuan, “Non-convex weighted lp nuclear norm based ADMM framework for image restoration,” Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 311, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 209–224, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [46] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mirsky, “A trace inequality of John von Neumann,” Monatshefte f¨ur mathematik, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 79, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 303–306, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [47] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Parikh and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Boyd, “Proximal algorithms,” Foundations and Trends in optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 127–239, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [48] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Kwok, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Gao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Chen, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Liu, “Efficient inexact proximal gradient algorithm for nonconvex problems,” arXiv preprint arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='09069, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Beck and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal on imaging sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 183–202, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 28 [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Attouch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Bolte, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Mathematical Programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 137, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 91–129, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Attouch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Bolte, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Redont, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Soubeyran, “Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality,” Mathematics of operations research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 438–457, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Bolte, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sabach, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Mathematical Programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 146, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 459–494, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Razaviyayn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hong, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM Journal on Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1126–1153, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [54] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tseng and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yun, “A coordinate gradient descent method for nonsmooth separable minimization,” Mathematical Programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 117, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 387–423, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Meng, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yang, “Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems,” Journal of Global Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 79, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 853–883, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' ApS, The MOSEK optimization toolbox for MATLAB manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [57] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Foucart and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Lai, “Sparsest solutions of under-determined linear systems via ℓq-minimization for 0 < q ≤ 1,” Applied and Computational Harmonic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 395–407, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [58] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ge, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Jiang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ye, “A note on the complexity of ℓp minimization,” Mathematical programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 129, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 285–299, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [59] SpamAssassin Public Corpus, http://spamassassin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [60] Andrew Ng, Machine learning MOOC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='coursera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content='org/learn/machine-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [61] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sznaier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Ayazoglu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Inanc, “Fast structured nuclear norm minimization with applications to set membership systems identification,” IEEE Transactions on Automatic Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2837–2842, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [62] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Liu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Vandenberghe, “Semidefinite programming methods for system realization and identification,” in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4676–4681, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [63] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Pong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Sun, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Tseng, “Hankel matrix rank minimization with applications to system identification and realization,” SIAM Journal on Matrix Analysis and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 946–977, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [64] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' K¨ummerle and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mayrink Verdun, “Escaping saddle points in ill-conditioned matrix completion with a scalable second order method,” in Workshop on Beyond First Order Methods in ML Systems at the 37th International Conference on Machine Learning, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [65] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' K¨ummerle and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mayrink Verdun, “A scalable second order method for ill-conditioned matrix completion from few samples,” in International Conference on Machine Learning (ICML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [66] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Mohan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Fazel, “Iterative reweighted algorithms for matrix rank minimization,” Journal of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 110, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 3441–3473, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [67] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Aybat and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Iyengar, “A first-order augmented Lagrangian method for compressed sensing,” SIAM Journal on Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 429–459, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [68] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Hale, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Yin, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Zhang, “A fixed-point continuation method for l1-regularized minimization with applications to compressed sensing,” CAAM TR07-07, Rice University, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 43, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 44, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' [69] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' O’Brien and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Plumbley, “Inexact proximal operators for ℓp-Quasi-norm minimization,” in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 4724–4728, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 29 [70] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Beck and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM journal on imaging sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} +page_content=' 183–202, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFLT4oBgHgl3EQfPS8r/content/2301.12027v1.pdf'} diff --git a/StE3T4oBgHgl3EQfzQsi/content/tmp_files/2301.04726v1.pdf.txt b/StE3T4oBgHgl3EQfzQsi/content/tmp_files/2301.04726v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a843a7b711c758da7cb9f9fbd546138ebd4b8127 --- /dev/null +++ b/StE3T4oBgHgl3EQfzQsi/content/tmp_files/2301.04726v1.pdf.txt @@ -0,0 +1,2041 @@ +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR +COGNATE SEQUENCES +G.-S. CHEON1, T. FORGÁCS2, AND K. TRAN2 +Abstract. Given a Sheffer sequence of polynomials, we introduce the notion of an associated +sequence called the cognate sequence. We study the relationship between the zeros of this pair +of associated sequences and show that in case of an Appell sequence, as well as a more general +family of Sheffer sequences, the zeros of the members of each sequence (for large n) are either +real, or lie on a line Re z = c. In addition to finding the zero locus, we also find the limiting +probability distribution function of such sequences. +MSC: 05A15, 05A40, 30C15, 30E15 +Key words: Sheffer sequence, cognate sequence, zero locus, limiting distribution +1. Introduction +Sequences of polynomials play a fundamental role in several fields of mathematics, including +enumerative combinatorics, functional analysis, applied mathematics, and differential equations. +Polynomial sequences have been studied extensively from many different points of view [3, 8]. Some +of the aspects recent research has focused on include their explicit formulas, generating functions, +recurrence relations, and zero distributions. +By a Sheffer sequence [12, 13] we shall mean a polynomial sequence indexed by the nonnegative +integers 0, 1, 2, . . ., in which the index of each polynomial equals its degree, satisfying conditions +related to the umbral calculus [11] in combinatorics. In this paper, given a Sheffer sequence we +introduce the notion of its cognate sequence, and study zeros of the cognate sequence of certain +Sheffer sequences. This direction of research is largely motivated by polynomial pairs defined using +recurrence relations, such as the Lucas polynomial sequences [4, 7] for example. +A Sheffer sequence {Gn(s)}∞ +n=0 is characterized by its exponential generating function +∞ +� +n=0 +Gn(s)zn +n! = g(z)esf(z) +for some (formal) power series g and f in the variable z satisfying the conditions g(0) ̸= 0, f(0) = +0 and f ′(0) ̸= 0. +By convention, we call {Gn(s)}∞ +n=0 the Sheffer sequence for the pair (g, f). +In particular, the Sheffer sequence for a pair (g, az) with a constant a ̸= 0 is called an Appell +sequence. There are a number of classical polynomial sequences that are Appell sequences, including +the Bernoulli polynomials Bn(s) for the pair ( +z +ez−1, z), the Euler polynomials En(s) for the pair +G.-S. Cheon was partially supported by the National Research Foundation of Korea (NRF) grant funded by the +Korean government (MSIP) (2016R1A5A1008055 and 2019R1A2C1007518). +The third author thanks the organizers and participants of the workshop on Optimal Point Configurations on +Manifolds hosted by the Erwin Schrödinger International Institute for Mathematics and Physics. +1 +arXiv:2301.04726v1 [math.CV] 11 Jan 2023 + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +2 +( +2 +ez+1, z), and the Hermite polynomials Hn(s) for the pair (e−z2, 2z). +Sheffer sequences form a group called the Sheffer group with the operation of umbral composition, +defined as follows (see [6]). Suppose {Gn(s)}∞ +n=0 and {Hn(s)}∞ +n=0 are Sheffer sequences for the pairs +(g, f) and (h, ℓ) respectively, given by +Gn(s) = +n +� +k=0 +an,ksk +and +Hn(s) = +n +� +k=0 +bn,ksk. +(1.1) +Then the umbral composition of Gn(s) with Hn(s), denoted by Gn ◦ Hn(s), is the sequence of +polynomials defined by +Gn ◦ Hn(s) = +n +� +k=0 +an,kHk(s) = +� +0≤ℓ≤k≤n +an,kbk,ℓsℓ. +It is shown in[6] that the Sheffer group is isomorphic to the Riordan group of exponential Riordan +matrices defined in terms of exponential generating functions as follows. Let D = [di,j]i,j≥0 be an +infinite lower triangular matrix with complex entries. If there exists a pair of exponential generating +functions +g = +� +k≥0 +gk +zk +k! and f = +� +k≥1 +fk +zk +k! +with g0 ̸= 0 and f1 ̸= 0, such that the k-th column of D has exponential generating function gf k/k! +for k = 0, 1, 2, . . ., then D is called an exponential Riordan matrix, and is denoted by [g, f]. Let R +be the set of all exponential Riordan matrices. R is a group called the (exponential) Riordan group +under usual matrix multiplication. In terms of generating functions the product is expressed by +[g, f][h, ℓ] = [gh(f), ℓ(f)] +(1.2) +where h(f) denotes composition of power series with f(0) = 0. +By definition, we see that the coefficient matrices [an,k] of {Gn(s)}∞ +n=0 and [bn,k] of {Hn(s)}∞ +n=0 +in (1.1) are exponential Riordan matrices [g, f] and [h, ℓ] respectively. Moreover, {Gn ◦ Hn(s)}∞ +n=0 +is the Sheffer sequence for the pair (gh(f), ℓ(f)). +We claim that the Riordan group R is isomorphic to the group R′ of exponential Riordan matrices +of the form [f ′/g, f]. To see this, consider the map φ : R → R′ given by φ([g, f]) = [f ′/g, f]. Then +for any A = [g, f] and B = [h, ℓ] in R, we have +φ(AB) = φ([gh(f), ℓ(f)]) = +�f ′ℓ′(f) +gh(f) , ℓ(f) +� += +�f ′ +g , f +� �ℓ′ +h , ℓ +� += φ(A)φ(B). +Hence φ is a group homomorphism. In addition, ker(φ) = {(1, z)} and clearly, φ is onto. Thus, φ +is a group isomorphism. We may thus associate to the Sheffer sequence {Gn(s)}∞ +n=0 for the pair +(g, f), its cognate sequence {Gc +n(s)}∞ +n=0 generated by the relation +f ′(z) +g(z) esf(z) = +� +n≥0 +Gc +n(s)zn +n! . +For each n, we call Gc +n(s) the cognate polynomial of Gn(s). It is natural to ask how the zeros of +Gn(s) relate to the zeros of the cognate polynomial Gc +n(s). After all, the map +{Gn(s)}∞ +n=0 +Φ +−→ {Gc +n(s)}∞ +n=0 +is a transformation on R[x], and the properties of such transformations, as they relate to the +preservation of zero locus, have been a central problem of study in the context of the Pólya-Schur + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +3 +program (see [1]) and beyond. In this paper we study a subset of such maps – or equivalently pairs +of Sheffer sequences and their cognate sequence – which preserve the symmetry type of the zero +locus of a Sheffer sequence. +The paper is organized as follows. In Section 2 we discuss Appell sequences and their cognate +sequences, building on the example of the Bernoulli polynomials. We also provide a characterization +of all Appell sequneces whose zeros exhibit the same type of symmetry as those of the Bernoulli +polynomials. In Section 3 we show that the Sheffer sequences considered in [3] along with their +cognate sequence are generated by a pair of functions of the same general form. We show that +any Sheffer sequence generated by functions of this kind consist of polynomials Hn whose zeros are +either real or lie on a line of the form Re z = c for n ≫ 1. We accomplish this in two subsections: +the first (subsection 3.1) develops the necessary asymptotic formulas for the integral representation +of the polynomials under investigation, while the second (subsection 3.2) finds the precise location +of the zeros of this sequence. The paper concludes with Section 4, in which we discuss the limiting +distribution of the zeros of the family of sequences defined in Theorem 8. +2. The zeros of Appell sequences and their cognate sequences +We begin our investigations with the zeros of the cognate sequence of the Bernoulli polynomials. +By definition, the cognate sequence {Bc +n(s)}∞ +n=0 of the Bernoulli polynomials {Bn(s)}∞ +n=0 is the +Appell sequence for the pair ( ez−1 +z +, z). It is known that all the zeros of Bernoulli polynomials Bn(s) +(n ≥ 1) are symmetrical with respect to the line Re s = 1 +2. +Our first theorem (c.f. Theorem 2) demonstrates that the location of the zeros of Bc +n(s) is closely +related to that of the zeros of Bn(s). In the proof of this result we need to make use of the following +lemma. +Lemma 1. [2, 14] Let G(s) ∈ C[s] be a polynomial all of whose zeros have positive imaginary part, +and let ¯G(s) be the polynomial whose coefficients are the complex conjugates of those of G(s). Then +all zeros of G(s) + ¯G(s) ∈ R[s] are real. +Theorem 2. For n ≥ 1 let Bc +n(s) be the cognate polynomial of the Bernoulli polynomial Bn(s). +Then all zeros of Bc +n(s) lie on the line Re s = − 1 +2. +Proof. Define Gn(s) = Bc +n +� +− 1 +2 + is +� +. Then all zeros of Gn(s) are real if and only if the real part +of every zero of Bc +n(s) is − 1 +2, or equivalently, all zeros of Bc +n(s) lie on the line Re s = − 1 +2. Thus it +suffices to show that all zeros of Gn(s) are real. By the definition of Gn(s) we have +� +n≥0 +Gn(s)zn +n! = ez − 1 +z +e(− 1 +2 +is)z = e +1 +2 z − e− 1 +2 z +z +eisz = 1 +z +� +e( 1 +2 +is)z + +� +−e(− 1 +2 +is)z�� +. +Let +e( 1 +2 +is)z = +� +n≥0 +fn(s)zn +n! +and +− e(− 1 +2 +is)z = +� +n≥0 +gn(s)zn +n! . +Since fn(s) = +� 1 +2 + is +�n, all zeros of fn(s) (and also −fn(s)) have positive imaginary part, namely +1 +2. It is easy to see that gn(s) = (−1)n+1 ¯fn(s). Hence by Lemma 1, we obtain that all zeros of +Gn(s) = fn(s) + (−1)n+1 ¯fn(s) are real, which completes the proof. +□ +The above connection between the zeros of Bernoulli polynomials and their cognate sequence +does not extended to arbitrary Appell sequences and their cognate sequences. Thus, one is naturally + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +4 +led to the problem of finding and characterizing all Sheffer sequences and their cognate sequence +whose zeros exhibit symmetries akin to that displayed by the zeros of {Bn(s)}∞ +n=0 and {Bc +n(s)}∞ +n=0. +Generally, it would be of interest to understand the relationship between the zeros of a Sheffer +sequence and its cognate sequence. +To obtain some information about the zeros of the cognate sequences of Appell sequences, we +begin with the following lemma. +Lemma 3. Let G(s) ∈ R[s] with degree n ≥ 1. Then all zeros of G(s) are symmetrical with respect +to the line Re s = − m +2 (m ∈ R) if and only if G(−s) = (−1)nG(s − m). +Proof. Suppose that all zeros of G(s) are symmetrical with respect to the line Re s = − m +2 for some +m ∈ R. First let n be even. Then we may assume that n zeros of G(s) are of the form − m +2 ± qk +(qk ∈ C, k = 1, 2, . . . , n +2 ) so that G(s) can be written as +G(s) = +n/2 +� +k=1 +� +s + m +2 + qk +� � +s + m +2 − qk +� +. +Hence +G(−s) += +n/2 +� +j=1 +� +− +� +s − m +2 − qk +�� � +− +� +s − m +2 + qk +�� += +(−1)n +n/2 +� +k=1 +� +(s − m) + m +2 − qk +� � +(s − m) + m +2 + qk +� += (−1)nG(s − m). +Now let n be odd. Then G(s) is of the form +G(s) = +� +s + m +2 +�j (n−j)/2 +� +k=1 +� +s + m +2 + qk +� � +s + m +2 − qk +� +where j ≥ 1 and j is odd. A simple computation shows that G(−s) = (−1)nG(s − m) also holds +for this case. +Conversely, suppose that G(−s) = (−1)nG(s − m) holds for some m ∈ R. Let a + bi (a, b ∈ R) +be a zero of G(s). Then a − bi = − m +2 + ((a + m +2 ) − bi) is also a zero of G(s). It follows from +0 = G(a − bi) = G +� +−m +2 + +�� +a + m +2 +� +− bi +�� += (−1)nG +� +−m +2 − +�� +a + m +2 +� +− bi +�� +that −(a + m) + bi = − m +2 − +�� +a + m +2 +� +− bi +� +is a zero of G(s), which implies that all zeros of G(s) +are symmetrical with respect to the line Re s = − m +2 . This completes the proof. +□ +Theorem 4. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). Then the following are +equivalent: +(i) For n ≥ 1, the zeros of Gn(s) are symmetric with respect to the line Re s = − g′(0) +2a +; +(ii) For n ≥ 1, Gn(−s) = (−1)nGn(s − g′(0) +a ) ; +(iii) g(z) = g(−z)eg′(0)z. + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +5 +Proof. It follows from Lemma 3 that (i) and (ii) are equivalent. Moreover, (ii) holds if and only if +g(z)e−asz += +� +n≥0 +Gn(−s)zn +n! = +� +n≥0 +(−1)nGn +� +s − g′(0) +a +� zn +n! = g(−z)e−a(s− g′(0) +a +)z += +g(−z)eg′(0)ze−asz, +which is equivalent to (iii). +□ +Theorem 5. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). If all zeros of Gn(s) for +n ≥ 1 are symmetrical with respect to the line Re s = − g′(0) +2a , then all zeros of Gc +n(s) are symmetrical +with respect to the line Re s = g′(0) +2a . +Proof. It suffices to show that Gc +n(−s) = (−1)nGc +n(s + g′(0) +a ) holds for all n ≥ 1. By Theorem 4 we +have +� +n≥0 +Gc +n(−s)zn +n! += +a +g(z)e−asz = +a +g(−z)e(−as−g′(0))z = +a +g(−z)e−a(s+ g′(0) +a +)z += +� +n≥0 +(−1)nGc +n +� +s + g′(0) +a +� zn +n! , +which implies that for all n ≥ 1, Gc +n(−s) = (−1)nGc +n(s + g′(0) +a ). The proof is complete. +□ +The following theorem shows that there are infinitely many Appell sequences satisfying the +assumptions of Theorem 5. +Theorem 6. Let {Gn(s)}n≥0 be the Appell sequence for the pair (g, az). If +g(z) = 2ρ(z)(1 + tanh(kz)), +(2.1) +where ρ(z) is any even function with ρ(0) = 1 and k ∈ R, then all the zeros of Gn(s) are symmetrical +with respect to the line Re s = − k +a for n ≥ 1. Conversely, if all the zeros of Gn(s) (n ≥ 1) are +symmetrical with respect to a vertical line in C then there exist an even function ρ(z) with ρ(0) = 1 +and k ∈ R satisfying (2.1). +Proof. Suppose g(z) = 2ρ(z)(1 + tanh(kz)) for some even function ρ(z) such that ρ(0) = 1 and +k ∈ R. Then +g(−z)e2kz += +2ρ(z)(1 + tanh(−kz))e2kz = 2ρ(z) +� +1 + e−kz − ekz +e−kz + ekz +� +e2kz += +2ρ(z) +� +1 + ekz − e−kz +ekz + e−kz +� += 2ρ(z)(1 + tanh(kz)) = g(z), +and g′(0) = 2k. +Thus, Theorem 4 (iii) holds, and hence so does (i), i.e. +the zeros of Gn(s) +(n ≥ 1) are symmetric with respect to the line Re s = − k +a. Conversely, suppose that all the zeros +of Gn(s) are symmetric with respect to a vertical line in C. Let g(z) = 2 +� +1 + � +n≥1 gnzn� +. Since +G1(s) = 2(g1 +as), this vertical line is Re s = − g1 +a . By Theorem 4, g(z) satisfies g(z) = g(−z)e2g1z. +A simple computation shows that +g(z) = g(z) + g(−z) +2 +(1 + tanh(g1z)). +Setting ρ(z) = g(z)+g(−z) +2 +yields an even function, and k = g1 ∈ R, as desired. +□ + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +6 +Remark 7. We note that if {Gn(s)}n≥0 is the Appell sequence for the pair (g(z), z) then the Appell +sequence for the pair (g(z), az) is {Gn(as)}n≥0. Thus it suffices to explore the Appell sequence for +the pair (g(z), z) when studying the zeros of the Appell sequence for the pair (g(z), az). +3. The zeros of a certain family of Sheffer sequences and their cognate sequences +We now turn our attention to the cognate sequences of Sheffer sequences previously treated in [3]. +Note – as a preview – that the symmetry of the zeros of the cognate sequence about a line remains, +and that our main result (c.f. Theorem 8) also exploits the fact that (a suitable modification of) +the non-exponential factor of the generating function is even. +In order to set up the statement of the main result, suppose z2 > z1 > 0, and let Log(·) denote the +principle logarithm. Set +f(z) = Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z) +g(z) = (z1 + z)(z2 + z). +The sequence of Sheffer polynomials {Gn(s)}∞ +n=0 for the pair (g(z), f(z)) is generated by +∞ +� +n=0 +Gn(s)zn +n! = g(z)esf(z) = (z1 − z)s(z2 − z)s(z1 + z)1−s(z2 + z)1−s. +The corresponding cognate sequence {Gc +n(s)}∞ +n=0 is the Sheffer sequence for the pair (f ′(z)/g(z), f(z)), +generated by +∞ +� +n=0 +Gc +n(s)zn +n! = f ′(z) +g(z) esf(z), +where +f ′(z) +g(z) = − +1 +(z1 + z)(z2 + z) +� +1 +z1 − z + +1 +z2 − z + +1 +z1 + z + +1 +z2 + z +� += − +2(z1 + z2) +(z1 + z)2(z2 + z)2(z1 − z)(z2 − z)(z1z2 − z2). +The reader will note that both g and f ′ +g are of the form +(z1 − z)p(z1 + z)p∗(z2 − z)q(z2 + z)q∗ m +� +i=1 +(αi − z2)pi +for appropriate values of the constants. As Theorem 8 shows, both the Sheffer sequence for the +pair (g, f), and its cognate sequence have zeros that are symmetric about a line Re z = k. This fact +about the Sheffer sequence was already addressed in [3, Theorem 5]. Theorem 8 is a generalization +of that result. +Theorem 8. Suppose m ∈ N and p, p∗, q, q∗,αi, pi, 1 ≤ i ≤ m, are real numbers such that αi > z2 +1. +Let +(3.1) +h(z) = (z1 − z)p(z1 + z)p∗(z2 − z)q(z2 + z)q∗ m +� +i=1 +(αi − z2)pi, +let +f(z) = Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z), + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +7 +and {Hn(s)}∞ +n=0 be the Sheffer sequence for the pair (h(z), f(z)). Assume that p∗−p = q∗−q := 2c. +If c + p < 0, then all the zeros of Hn(s), n ≫ 1, lie on the line s = c + it. If c + p ≥ 0, then the +same conclusion holds except for 2 ⌈c + p⌉ real zeros, each of which approaches c ± (c + p + 1 − k), +0 < k ≤ ⌈c + p⌉as n → ∞. +-4 +-3 +-2 +-1 +1 +-0.4 +-0.2 +0.2 +0.4 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-0.5 +0.5 +Figure 3.1. Zeros of Hn(s) +Example 1. Let z1 = 1, z2 = 7. The left side graph in Figure 3.1 shows the zeros of H20(s) when +p = 4, p∗ = 1, q = 2, q∗ = −1, p − p∗ = 3 = q − q∗, c + p = 11 +2 and +h(z) = (1 − z)4(1 + z)(7 − z)2(7 + z)−1(2 − z2)−1. +The right side graph shows the zeros of H20(s) when p = −4, p∗ = −1, q = −2, q∗ = 1, p − p∗ = +−3 = q − q∗, c + p = − 11 +2 and +h(z) = (1 − z)−4(1 + z)−1(7 − z)−2(7 + z)1(2 − z2)−1. +The proof of Theorem 8 is presented in the next two sections, the first of which develops the +asymptotics for an integral representation of the Hn(s)s, followed by a section on the counting of +the zeros of these polynomials on the designated locus. +3.1. The asymptotic formulas. In this section we find an integral representation for the Sheffer +polynomials Hn(c + int) described in Theorem 8, and we develop of an asymptotic formula for said +integral representation, which is uniform on the parameter range e− ln4 n/n ≪ t ≪ ln4 n/n. To +begin, let {Hn}∞ +n=0 be the Sheffer sequence for the pair (h, f) as in the statement of Theorem 8. +The substitution s = c + int and the Cauchy integral formula gives +Hn (c + int) = n! +2πi +‰ +|z|=ϵ +h(z)e(c+int)f(z) +zn+1 +dz += n! +2πi +‰ +|z|=ϵ +ψ(z)e−nφ(z,t)dz, +where +φ(z, t) = Log z − it (Log(z1 − z) + Log(z2 − z) − Log(z1 + z) − Log(z2 + z)) , +and +ψ(z) = h(z) +z +(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c . + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +8 +It follows from the definition of h(z) that, as a function of z, ψ(z)e−nφ(z,t) is analytic on the +complement of +{0} ∪ [z1, ∞) ∪ (−∞, −z1]. +The defintions of h(z) and f(z) also imply that on any circle arc CR with large radius R, +lim +R→∞ +ˆ +CR +h(z)e(c+int)f(z) +zn+1 +dz = 0, +for +n ≫ 1. +Thus, +(3.2) +Hn (c + int) = n! +2πi +ˆ +Γ1∪Γ2 +ψ(z)e−nφ(z,t)dz, +where Γ1 and Γ2 are two halves of a loop around infinity and the cuts (−∞, −z1] and [z1, ∞) with +the counter clockwise orientation. +The assumption p∗ − p = q∗ − q implies that +Figure 3.2. The loop around the cuts and infinity +h(z)(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +is even. Using the substitution z �→ −z we see that the part of the integral in (3.2) over Γ1 is equal +to +(−1)n +ˆ +Γ2 +ψ(z)e−nφ(z,−t)dz. +Making the subsequent substitution z �→ z, this integral becomes the conjugate of +(−1)n+1 +ˆ +Γ2 +ψ(z)e−nφ(z,t)dz. +We deduce that πHn(c + int) is imaginary part, or −i times the real part of the integral +(3.3) +ˆ +Γ2 +ψ(z)e−nφ(z,t)dz, +depending on whether n is even or odd. +For the sake of completeness we now present the setup and definitions developed in [3] in order to +help establish the asymptotic formula we see. To this end, let +T1 := z2 − z1 +z1 + z2 +, +T2 := z1 + z2 +4√z1z2 +, + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +9 +and +T = +� +T1 +if z2 +1 − 6z1z2 + z2 +2 ≥ 0 +T2 +if z2 +1 − 6z1z2 + z2 +2 < 0. +(3.4) +For t ∈ (0, T), the function +ζ = ζ(t) : = z1 + z2 +2 +� +it − +� +T 2 +1 − t2 + +� +1 − 2t2 − 2it +� +T 2 +1 − t2 +� +for +0 ≤ t < T1, +(3.5) +ζ = ζ(t) : = z1 + z2 +2 +� +it + i +� +t2 − T 2 +1 + +� +1 − 2t2 − 2t +� +t2 − T 2 +1 +� +for +T1 ≤ t ≤ T2, +(3.6) +is a solution of φz(z, t) = 0. +Set +g(ζ) +:= +2πψ2(ζ)e−2nφ(ζ,t) +nφz2(ζ, t) +, +(3.7) +p(ζ) +:= +ˆ +Γ2 +ψ(z)e−nφ(z,t)dz, +and consider the following intervals +I1 = +� +t| ln4 n/n ≪ t < T − ln2 n/n2/3� +, +I2 = +� +t| ln4 n/n ≪ t < T1 − ln2 n/n2/3� +, +I3 = [T1 + ln2 n/n2/3, T2 − ln2 n/n2/3], +I = +� +t|1/n2/3 ≪ T − t < ln2 n/n2/3� +. +With these definitions, the results in [3] show that as n → ∞, +p2(ζ) ∼ g(ζ) +uniformly on t ∈ I2 ∪ I3 if T = T2, and on t ∈ I1 if T = T1. Furthermore, if t ∈ I, then +p(ζ) ∼ +4cψ(ζ(T))e−nφ(ζ,t) +√ +6 +� +C3φz3(ζ(T), T) +αn(t) +uniformly in t, where +C = +� +� +� +� +� +z1+z2 +2 +� +−√2T1 + +T1 +√2T1 +√ +2T 2 +1 −1 +� +if T = T1 +√ +32(z1z2)3/4 +√ +(z1+z2)(−z2 +1+6z1z2−z2 +2) +if T = T2 +and Re αn(t) ≥ 0. If T = T2, then +p(ζ) ∼ +4dψ(ζ(T1))e−nφ(ζ,t) +√ +6 +� +D3φz3(ζ(T1), T1) +βn(t) +uniformly on 1 +n ≪ |T1 − t| ≤ ln2 n/n2/3, where +D = −(z1 + z2)√T1 +√ +2 +� +1 + +iT1 +� +1 − 2T 2 +1 +� + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +10 +and Re βn(t) ≥ 0. Before we state and prove our main estimate, we note that (3.1) implies that as +z → z1, +h(z) = (z1 − z)p (hp + O (z1 − z)) +for some hp ∈ R. In the remainder of this section, we prove the following asymptotic equivalence. +Proposition 9. Let hp and p(ζ) be as defined in the beginning of the section, and let Γ denote the +Gamma function. As n → ∞ the following asymptotic formula holds uniformly on e− ln4 n/n ≪ +t ≪ ln4 n/n: +p(ζ) ∼ +2iπhp(z2 − z1)c+int +zn−p +1 +2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int) +. +Proof. We rewrite p(ζ) as +p(ζ) = +ˆ (z− +1 ) ++∞ +ψ(z)e−nφ(z,t)dz, +where path of integration is the Hankel contour which loops around the ray [z1, ∞). The notation +z− +1 means the path goes around z1 in the negative direction. We make the substitution w = z/z1 +followed by the subsitution ez = w to arrive at the expression +p(ζ(t)) = z1 +ˆ (1−) ++∞ +ψ(wz1)e−nφ(wz1,t)dw = z1 +ˆ (0−) ++∞ +ψ(z1ez)e−nφ(z1ez,t)ezdz. +Suppose ϵ > 0 is small such that nϵ = o(1). We break the last integral into three pieces: +z1 +ˆ (ln5n/n)±iϵ +(0−) +ψ(z1ez)e−nφ(z1ez,t)ezdz ++z1 +ˆ +∞+iϵ +(ln5 n/n)+iϵ +ψ(z1ez)e−nφ(z1ez,t)ezdz ++z1 +ˆ (ln5 n/n)−iϵ ++∞−iϵ +ψ(z1ez) +2iπhp(z2 − z1)c+int +zn−p +1 +2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int) +e−nφ(z1ez,t)ezdz. +(3.8) +Recall that +(3.9) +ψ(z)e−nφ(z,t) = h(z) +zn+1 +(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +�(z1 − z)(z2 − z) +(z1 + z)(z2 + z) +�nit +, +and +h(z) = (z1 − z)p (hp + O (z1 − z)) +for z1 − z = o(1). Thus, the first integral in (3.8) is asymptotic to +zc+p+int +1 +hp(z2 − z1)c+int +zn +1 2c+intzc+int +1 +(z2 + z1)c+int +ˆ (ln5 n/n)±iϵ +(0−) +(1 − ez)c+p+int +enz +dz +∼ +zc+p+int +1 +hp(z2 − z1)c+int +zn +1 2c+intzc+int +1 +(z2 + z1)c+int +ˆ (ln5 n/n)±iϵ +(0−) +(−z)c+p+inte−nzdz += +zc+p+int +1 +hp(z2 − z1)c+int +zn +1 2c+intzc+int +1 +(z2 + z1)c+intnc+p+1+int +ˆ (ln5 n)±inϵ +(0−) +(−z)c+p+inte−zdz. +(3.10) +The following auxiliary lemma helps us further refine this estimate. + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +11 +Lemma 10. Suppose, as in the statement of Proposition 9, that e− ln4 n/n ≪ t ≪ ln4 n/n. If +nϵ = o(1), then the following asymptotic equivalence holds: +ˆ ln5 n±inϵ +(0−) +(−z)c+p+inte−zdz ∼ 2i sin π(c + p + 1 + int)Γ(c + p + 1 + int). +Proof. We apply the Hankel contour representation (see for example [9]) for the Gamma function +Γ(s) = +1 +2i sin πs +ˆ (0−) ++∞ +e−z(−z)s−1dz +(s ̸= 0, −1, −2, . . .) +to rewrite +ˆ ln5 n±inϵ +(0−) +(−z)c+p+inte−zdz +as +(3.11) +2i sin π(c+p+1+int)Γ(c+p+1+int)− +ˆ +∞+inϵ +ln5 n+inϵ +(−z)c+p+inte−zdz + +ˆ +∞−inϵ +ln5 n−inϵ +(−z)c+p+inte−zdz. +In the case e− ln4 n ≪ nt = O(1), we have +(3.12) +2i sin π(c + p + 1 + int)Γ(c + p + 1 + int) ≫ e− ln4 n. +Moreover, in this case, the second and the third terms in (3.11) are bounded by +(3.13) +ˆ +∞±inϵ +ln5 n±inϵ +|z|c+pe−nt Arg(−z)e− Re zd|z|. +Since nϵ = o(1), we have |z| = Re z + o(1), whence by the asymptotic behavior of the upper +incomplete Gamma function (see [5, 10]), the integral in (3.13) is O(e− ln5 n ln5(c+p) n). The result +in this case now follows. +If, on the other hand, nt ≫ 1, then the Stirling formula +Γ(s) = exp +� +(s − 1/2) Log s − s + 1 +2 ln 2π + O(s−1) +� +(|s| ≫ 1) +and the fact nt ≪ ln4 n imply that +|Γ(c + p + 1 + int)| += exp +�� +c + p + 1 +2 +� +ln |c + p + 1 + int| − nt Arg(c + p + 1 + int) − (c + p + 1) + 1 +2 ln 2π + O +� 1 +nt +�� +≫ exp(−π ln4 n), +from which we deduce +|2i sin π(c + p + 1 + int)Γ(c + p + 1 + int)| ≫ exp +� +nπt − π ln4 n +� +. +The claim follows from the fact that +ˆ +∞±inϵ +ln5 n±inϵ +|z|c+pe−nt Arg(−z)e− Re zd|z| = O +� +entπ−ln5 n ln5(c+p) n +� +. +The proof of Lemma 10 is complete. +□ + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +12 +We continue with the proof of Proposition 9 by finding a bound for the second integral in (3.8): +z1 +ˆ ∞+iϵ +ln5n/n+iϵ +ψ(z1ez)e−nφ(z1ez,t)ezdz. +Recall from equation (3.9) that +ψ(z1ez)e−nφ(z1ez,t) = +h(z1ez) +zn+1 +1 +e(n+1)z +(z1 − z1ez)c(z2 − z1ez)c +(z1 + z1ez)c(z2 + z1ez)c +�(z1 − z1ez)(z2 − z1ez) +(z1 + z1ez)(z2 + z1ez) +�nit +. +With z = u + iϵ, ln5 n/n ≤ u < ∞, we have +|z2 − z1ez| ≥ | Im(z2 − z1ez)| += z1eu sin ϵ +≫ ϵ, +and consequently, +(z2 − z1ez)c = O +� 1 +ϵ|c| + eu|c| +� +. +We conclude that +h(z1ez)(z2 − z1ez)c(z1 − z1ez)c +(z1 + z1ez)c(z2 + z1ez)c += +� +O +� +1 +ϵB + +n|c+p| +ln5(c+p) n +� +if z = O(1) +O(eAu) +if z ≫ 1 +for some constants A(depending on the degree of h) and B(depending on c and the number of poles +of h on [z1, ∞)). +We also note that +����� +�(z1 − z1ez)(z2 − z1ez) +(z1 + z1ez)(z2 + z1ez) +�nit����� = expnt (Arg(z1 − z1ez) + Arg(z2 − z1ez) − Arg(z1 + z1ez) − Arg(z2 + z1ez)) , +and that for z = u + iϵ and |z| ≪ 1, +Arg(z1 − z1ez) + Arg(z2 − z1ez) − Arg(z1 + z1ez) − Arg(z2 + z1ez) = Arg(−z) + O(z). +Thus, there exists a small δ (independent of n) such that if u < δ then +����� +�(z1 − z1ez)(z2 − z1ez) +(z1 + z1ez)(z2 + z1ez) +�nit����� < enπt. +For other values of u, we note that geometrically +|Arg(z1 − z1ez) − Arg(z1 + z1ez)| +is the sum of two angles of the triangle with vertices 0, z1, and (z1 + z1ez)/2, which is less than π. +The same inequality holds for +|Arg(z2 − z1ez) − Arg(z2 + z1ez)| . +We conclude that for u ≥ δ, +����� +�(z1 − z1ez)(z2 − z1ez) +(z1 + z1ez)(z2 + z1ez) +�nit����� < e2nπt. + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +13 +In order to utilize these estimates, we break the range of integration of +ˆ +∞+iϵ +ln5 n/n+iϵ +z1ψ(z1ez)e−nφ(z1ez,t)ezdz +into three pieces: (i) ln5 n/n < Re z < δ, (ii) δ ≤ Re z and Re z = O(1) and (iii) Re z ≫ 1. The +integral over the first range is +enπt +zn +1 +O +�� 1 +ϵB + +n|c| +ln5c n +� ˆ δ +ln5 n/n +e−nudu +� += enπt +nzn +1 +O +� +e− ln5 n +ϵB ++ +n|c| +ln5c ne− ln5 n +� +, +the integral over the second range is +e2πnt +zn +1 +O +�� 1 +ϵB + +n|c| +ln5c n +� e−nδ +n +� +, +while the integral over the third range is +e2πnt +zn +1 +O +�ˆ ∞ +C +e−(n+1)u+Audu +� += e2πnt +zn +1 +O +� 1 +ne−nC +� +for some large constant C. We recall that nt ≪ ln4 n. If we choose ϵ so that in addition to satisfying +the condition nϵ = o(1) we also have +1 +ϵB = O +� +exp +�ln5 n +2 +�� +, +then +z1 +ˆ +∞+iϵ +ln5 n/n+iϵ +ψ(z1ez)e−nφ(z1ez,t)ezdz = eπnt +zn +1 +O +� +exp +� +−ln5 n +2 +�� +. +With a similar argument we obtain +z1 +ˆ ln5 n/n−iϵ ++∞−iϵ +ψ(z1ez)e−nφ(z1ez,t)ezdz = eπnt +zn +1 +O +� +exp +� +−ln5 n +2 +�� +. +From equations (3.8), (3.10), Lemma 10, and the fact that +2i sin π(c + p + 1 + int)Γ(c + p + 1 + int) +� +≫ e− ln4 n +if e− ln4 n ≪ nt = O(1) +≫ exp +� +nπt − π ln4 n +4 +� +if 1 ≪ nt ≪ ln4 n +, +we conclude that +ˆ (z− +1 ) ++∞ +ψ(z)e−nφ(z,t)dz ∼ 2i sin π(c + p + 1 + int)Γ(c + p + 1 + int)zc+p+int +1 +hp(z2 − z1)c+int +zn +1 2c+intzc+int +1 +(z2 + z1)c+intnc+p+1+int +uniformly on e− ln4 n ≪ nt ≪ ln4 n. The proof of Proposition 9 is complete. +□ +Remark 11. If c+p /∈ Z+, we do not require the condition e− ln4 n ≪ nt for the estimate in equation +(3.12). Thus the asymptotics in Proposition 9 hold for nt ≪ ln4 n if c + p /∈ Z+. + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +14 +3.2. The location of the zeros of {Hn}∞ +n=0. With the asymptotic analysis complete, in this +section we establish that for all n ≫ 1, the zeros fo Hn lie on the line Re s = c (c.f. Theorem 8) +except perhaps a finite number of real zeros, whose asymptotic locations we also identify. We begin +with a technical, but crucial result. +Lemma 12. Suppose τ1 and τ2 are constant multiples of e− ln4 n/n and ln4 n/n and α is the unique +angle such that −π < α ≤ π and (c + p + 1/2)π = 2kπ + α for k ∈ Z. Let g and p be defined as in +equation (3.7). Then +(i) p(ζ(t)) ̸= 0 for τ1 ≤ t ≤ τ2, and +(ii) ∆ argτ1≤t≤τ2 p(ζ(t)) = 1 +2 lim +ξ→0 ∆ argξ≤t≤τ2 g(ζ(t)) + |c + p|π +2 ++ η, +where +(3.14) +η = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +−π/2 +if c + p < 0 +0 +if c + p ≥ 0 and α = ± π +2 +−α +if c + p ≥ 0 and − π/2 < α < π/2 +−α − π +if c + p ≥ 0 and − π < α < −π/2 +−α + π +if c + p ≥ 0 and π/2 < α ≤ π. +. +Proof. The fact that p(ζ(t)) ̸= 0 for τ1 ≤ t ≤ τ2 follows immediately from Proposition 10. To +establish the claim regarding the change of arguments, we start by recalling that +g(ζ) = 2πψ2(ζ)e−2nφ(ζ) +nφz2(ζ, t) += +2π +nφz2(ζ, t) · h2(ζ) +z2n+2 +(z1 − z)2c(z2 − z)2c +(z1 + z)2c(z2 + z)2c +�(z1 − z)(z2 − z) +(z1 + z)(z2 + z) +�2nit +. +Since z1 − z = −iz1t + O(t2), on ξ ≤ t ≤ τ2 the change in the argument of +h2(ζ)(z1 − z)2c(z2 − z)2c +(z1 + z)2c(z2 + z)2c = (z1 − z)2p∗H(t) ∼ (−z2 +1t2)p∗ + O(t3) +is o(1). Thus, using the same computations in the proof of Lemma 39 in [3], we conclude that +(3.15) +lim +ξ→0 ∆ argξ≤t≤τ2 g(ζ(t)) = 2nτ2 ln τ2(z2 − z1) +2(z1 + z2) − 2nτ2 + π +2 + o(1). +For τ1 ≤ t ≤ τ2, the change of the arguments of the factors 2c+int, (z2 − z1)c+int, (z2 + z1)c+int, +and e(c+p+1+int) ln n in the expression +2iπhp(z2 − z1)c+int +zn−p +1 +2c+int(z2 + z1)c+intnc+p+1+intΓ(−c − p − int) +are n(τ2 − τ1) ln 2, n(τ2 − τ1) ln(z2 − z1), n(τ2 − τ1) ln(z2 + z1), and n(τ2 − τ1) ln n respectively. +We next compute the change in argument of the factor Γ(−c − p − int) , τ1 ≤ t ≤ τ2, which is +given by the expression +Im Log Γ(−c − p − int)|τ2 +τ1 , +where the function Log Γ(s) is defined as +Log Γ(s) = −γs − Log s + +∞ +� +k=1 +� s +k − Log(1 + s/k) +� +. +Using the Stirling formula, +Log Γ(s) ∼ (s − 1/2) Log s − s + 1/2 Log(2π) + O(1/s), +for |s| → ∞ and | Arg s| ≤ π − δ, + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +15 +we conclude that +Im Log Γ(−c − p − inτ2) += Im ((−c − p − 1/2 − inτ2) Log(−c − p − inτ2) + nτ2 + O(1/ ln4 n) += − nτ2 ln |c + p + inτ2| + (c + p + 1/2)π +2 + nτ2 + O(1/ ln4 n). +Employing the estimate +ln |c + p + inτ2| = ln +����inτ2 +� +1 + c + p +inτ2 +����� = ln(nτ2) + O +� +1 +n2τ 2 +2 +� +, +the last expression becomes +−nτ2 ln(nτ2) + (c + p + 1/2)π +2 + nτ2 + O +� +1 +ln4 n +� +. +If −c − p > 0, then the fact that nτ1 ≍ e− ln4 n implies that +Im Log Γ(−c − p − inτ1) = O +� +e− ln4 n� +, +and consequently +∆ argτ1≤t≤τ2 Γ(−c − p − int) = −nτ2 ln(nτ2) + (c + p + 1/2)π +2 + nτ2 + O +� +1 +ln4 n +� +. +If, on the other hand, if c + p ≥ 0, then the identity +Γ(−c − p − int) = +π +sin π(c + p + 1 + int)Γ(c + p + 1 + int) +implies that +∆ argτ1≤t≤τ2 Γ(−c−p−int) = −∆ argτ1≤t≤τ2 sin π(c+p+1+int)−∆ argτ1≤t≤t2 Γ(c+p+1+int). +Using the conjugate of the gamma function, we write +−∆ argτ1≤t≤t2 Γ(c+p+1+int) = ∆ argτ1≤t≤τ2 Γ(c+p+1−int) = −nτ2 ln(nτ2)−(c+p+1/2)π +2 +nτ2+O +� +1 +ln4 n +� +. +Analyzing the change in the argument of sin π(c + p + int) requires further considerations. To this +end, recall that +sin π(c + p + 1 + int) = e−πnt+iπ(c+p+1) − eπnt−iπ(c+p+1) +2i += −eπnt−iπ(c+p+1) +2i +� +e−2πnt + 1 +� +, +and whence +Im sin π(c + p + 1 + int) = 1 +2 +� +e−πnt − eπnt� +cos π(c + p + 1). +It is immediate that if c + p + 1/2 ∈ Z, then +∆ argτ1≤t≤τ2 sin π(c + p + 1 + int) = 0. +On the other hand, if c + p + 1/2 /∈ Z, then sin π(c + p + 1 + int) /∈ R, and +∆ argτ1≤t≤τ2 sin π(c + p + 1 + int) = Arg sin π(c + p + 1 + inτ2) − Arg sin π(c + p + 1 + inτ1). + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +16 +We write +Arg sin π(c + p + 1 + inτ2) = Arg +�−eπnτ2−iπ(c+p+1) +2i +� ++ O +� +e−2πnτ2� += −α + O(e−2π ln4 n), +where α is the unique angle such that −π < α ≤ π and (c + p + 1/2)π = 2kπ + α for k ∈ Z. Note +that α is given explicitly by the formula +α = ((c + p + 3/2)π +mod 2π) − π. +If c + p ∈ Z, then the Taylor expansion of the sine function yields +Arg sin π(c + p + 1 + inτ1) = (−1)c+p+1 π +2 + O +� +e− ln4 n� +. +If c + p /∈ Z, then +Arg sin π(c + p + 1 + inτ1) = +� +� +� +� +� +� +� +O +� +e− ln4 n� +if sin π(c + p + 1) > 0 +π +if sin π(c + p + 1) < 0 and cos π(c + p + 1) > 0 +−π +if sin π(c + p + 1) < 0 and cos π(c + p + 1) < 0. +Combining these cases we conclude for any c, p ∈ R, +∆ argτ1≤t≤τ2 Γ(−c − p − int) = −nτ2 ln(nτ2) + nτ2 − |c + p|π +2 +− π +4 − η + O +� +1 +ln4 n +� +, +and finally, +∆ argτ1 0, and +also describes the asymptotic behavior of these zeros as n → ∞. +Lemma 14. Let {Hn}∞ +n=0 be as in the statement of Theorem 8. If c + p > 0, then Hn(s) has at +least 2 ⌈c + p⌉ many real zeros which approach c ± (c + p + 1 − k), 0 < k ≤ ⌈c + p⌉ as n → ∞. +Proof. For x ∈ R, the Cauchy integral formula yields +Hn(c + x) = n! +2πi +‰ +|z|=ϵ +h(z) +zn+1 +(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +�(z1 − z)(z2 − z) +(z1 + z)(z2 + z) +�x +dz +(3.17) += n! +2πi +ˆ +Γ1∪Γ2 +h(z) +zn+1 +(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +�(z1 − z)(z2 − z) +(z1 + z)(z2 + z) +�x +dz, +where Γ1 and Γ2 are two loops around two cuts (−∞, −z1] and [z1, ∞) oriented counter clockwise. +Using the substitution z �→ −z and the fact that +h(z)(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +is an even function, we see that the integral over Γ1 is equal to +(−1)n +ˆ +Γ2 +h(z) +zn+1 +(z1 − z)c−x(z2 − z)c−x +(z1 + z)c−x(z2 + z)c−x dz. +We apply Remark 11 to t = 0 and c replaced by c + x to conclude that if c + p + x /∈ Z+, then the +integral over Γ2 is asymptotic to +2ihp(z2 − z1)c+x sin π(c + p + x + 1)Γ(c + p + x + 1) +zn−p +1 +2c+x(z2 + z1)c+xnc+p+x+1 +. +With the same application to the case when c replaced by c − x, we conclude that +Hn(c + x) ∼ n!hp(z2 − z1)c+x sin π(c + p + x + 1)Γ(c + p + x + 1) +πzn−p +1 +2c+x(z2 + z1)c+xnc+p+x+1 ++ (−1)n n!hp(z2 − z1)c−x sin π(c + p − x + 1)Γ(c + p − x + 1) +πzn−p +1 +2c−x(z2 + z1)c−xnc+p−x+1 +(3.18) +if c + p ± x /∈ Z+ and x ̸= 0. For any small fixed δ > 0 (independent of n), we consider the intervals +(3.19) +Jk = [c + p + 1 − k − δ, c + p + 1 − k + δ], +0 < k < c + p + 1 − δ. +For each k, the values of sin(c + p − x + 1) when x is at the endpoints of Jk are (−1)k−1 sin δ and +(−1)k sin δ. Also at these endpoints c + p ± x /∈ Z+ (for small δ), Γ(c + p − x + 1) > 0, and the +second term of (3.18) dominates the first term when n is large. Thus, by the Intermediate Value +Theorem, each interval Jk contain at least one zero of Hn(c + x). We deduce that Hn(c + x) has at +least ⌈c + p⌉positive real zeros. The substitutions z by −z and x by −x in equation (3.17) yield +Hn(c − x) = (−1)nHn(c + x). +The result now follows from the fact that if x is a real zero of Hn(c + x), then so is −x. +□ + +ON THE ZEROS OF CERTAIN SHEFFER SEQUENCES AND THEIR COGNATE SEQUENCES +19 +We now turn our attention to the proof of Theorem 8. In addition to the number of real zeros +of Hn(s) given in Lemma 14, we will count the number of zeros of Hn(c + nit) on t ∈ (0, T) and +compare this number with the degree of Hn(s). We start with a lemma concerning the degree of +Hn(s). +Lemma 15. Let {Hn}∞ +n=0 be defined as in Theorem 8. Then for each n ≥ 0, polynomial Hn(c + x) +has degree n and the sign of its leading coefficient is (−1)n. +Proof. Since +Hn(c − x) = (−1)nHn(c + x), +it suffices to prove that Hn(c − x) has degree n, and that its leading coefficient is positive. The +generating function for Hn(c − x) is given by is +h(z)(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c (1 − z/z1)−x(1 − z/z2)−x(1 + z/z1)x(1 + z/z2)x. +For each k ∈ N, the coefficient of zk in the binomial expansion of each factor (1 − z/z1)−x, (1 − +z/z2)−x, (1 + z/z1)x, and (1 + z/z2)x is a polynomial of degree k in x with a positive leading +coefficient. Thus, given an n ∈ N, the coefficient of zn in the product +(1 − z/z1)−x(1 − z/z2)−x(1 + z/z1)x(1 + z/z2)x +is of the form +� +i+j+k+ℓ=n +pi(x)pj(x)pk(x)pℓ(x), +where each factor of each summand – and hence the entire expression – has a positive leading +coefficient, and degree equal to its index. We expand +h(z)(z1 − z)c(z2 − z)c +(z1 + z)c(z2 + z)c +as a power series in z (with constant coefficients) and deduce that Hn(c − x) has degree n, and the +sign of its leading coefficient is the same as the sign of the constant coefficient of this series which +is h(0) > 0. +□ +The final piece in accounting for all of the zeros of Hn(s) is provided by the fact (to be proven +in short order) that the total number of real zeros of Hn(s) and those on c + it (except the possible +zero at c) is at least +(3.20) +� +n − 2 +if 2 | n +n − 3 +if 2 ∤ n . +Assuming this fact, we now provide an argument to complete the proof of Theorem 8. If n is odd, +then we let x = 0 in Hn(c − x) = (−1)nHn(c + x) to conclude that c is a zero of Hn(s). It thus +remains to account for the two possible missing zeros of Hn(s) regardless of the parity of n. Since +the degree of Hn(s) is n, and the zeros of Hn(s) are symmetric about the real line and the line +c + it, it suffices to show that the possible two remaining zeros of Hn(s) are not real. Note that +Hn(c + x) has opposite signs at the endpoints of each Jk (as defined in (3.19)). Hence, Hn(c + x) +must have exactly one zero on each Jk and consequently the two remaining zeros cannot lie on any +Jk. Since on the set +(0, c + p + δ)\ +� +0 0 and the asymptotic expression in (3.18) that the sign Hn(c + x) +at x = c + p + δ is (−1)n. By Lemma 15, this is the same as the sign of limx→∞ Hn(c + x), and we +conclude that Hn(c + x) has no zero on [c + p + δ, ∞). It follows that the remaining two possible +zeros must lie on the line Re z = c, completing the proof of Theorem 8. +Remark 16. In the case c + p ∈ Z+, (3.18) implies that Hn(c + x) is nonzero on (0, 1 − δ) and its +sign is (−1)n+c+p there. +We now present the proof of the zero count of Hn(s) claimed in expression (3.20) above. Since +a lower bound for the number of real zeros of Hn(s) is provided by Lemma 14, it remains to count +the number of zeros of Hn(c + int) on |t| ∈ (0, T). We recall that for t ∈ (0, T), πHn(c + int) is +the imaginary part of −i times the real part of p(ζ(t)). It therefore suffices to compute the change +in the argument of p(ζ(t)) in order to get a lower bound on the zero count of Hn(s) on the line +Re z = c. We proceed by case analysis, depending on whether T = T2 or T = T1 (c.f. equation +(3.4)). +Case T = T2. If T = T2 and p(ζ(T1)) ̸= 0, then for some |C| < 3π/2 + o(1) and c2 ∈ R+ +∆e−ln4n/n≪t 0, the number of real zeros of +Hn(c + int) on (0, T)\{T1} is at least +� +∆e−ln4n/n≪t 0, we let Nn,ϵ(x) denote the number of zeros of Hn(c + int) +on the interval t ∈ (x, x + ϵ). It follows that the limiting probability density function at x ∈ (0, T) +is given by +lim +ϵ→0 +1 +ϵ lim +n→∞ +Nn,ϵ(x) +n +. +We note that for any x ∈ (0, T) and x ̸= T1 (if T = T2), +p2(ζ(t)) ∼ g(ζ(t)) = 2πψ2(ζ)e−2nφ(ζ,t) +nφz2(ζ, t) +uniformly on t ∈ (x, x + ϵ), and consequently +∆ argx 0, is O(1). Since +∆ argT1+ξ/n 1, all extracted tokens contain spatio-temporal +information. +For the special case of t = 1 each token +contains spatial-only information for each acquisition time +and temporal information is accounted for only through the +encoder layers. Since the computation cost of global self- +attention layers is quadratic w.r.t. the length of the token se- +quence O(N 2), choosing larger values for t, h, w can lead +to significantly reduced number of FLOPS. In our experi- +ments, however, we have found small values for t, h, w to +work much better in practice. For all presented experiments +we use a value of t = 1 motivated in part because this choice +simplifies the implementation of acquisition-time-specific +temporal position encodings, described in section 3.6. With +regards to the spatial dimensions of extracted patches we + +H/h +Conv2d(in channels=C, out channels=d +kernel size=(h, w), stride=(h, w)) +H +h +-M→ +W +t1 +tT +timehave found small values to work best for semantic segmen- +tation, which is reasonable given that small patches retain +additional spatial granularity. In the end, our tokenization +scheme is similar to ViT’s applied in parallel for each acqui- +sition as shown in Fig.3, however, at this stage, instead of +unrolling feature dimensions, we retain the spatial structure +of the original input as reshape operations will be handled +by the TSViT encoder submodules. +3.4. Encoder architecture +In the previous section we presented a motivation for us- +ing small values t, h, w for the extracted patches. Unless +other measures are taken to reduce the model’s computa- +tional cost this choice would be prohibitive for process- +ing SITS with multiple acquisition times. To avoid such +problems, we choose to factorize our inputs across their +temporal and spatial dimensions, a practice commonly em- +ployed for video processing [17,27,37,56,66,72]. We note +that all these works use a spatial-temporal factorization or- +der, which is reasonable when dealing with natural images, +given that it allows the extraction of higher level, semanti- +cally aware spatial features, whose relationship in time is +useful for scene understanding. However, we argue that in +the context of SITS, reversing the order of factorization is +a meaningful design choice for the following reasons: 1) in +contrast to natural images in which context can be useful for +recognising an object, in crop type recognition context can +provide little information, or can be misleading. This arises +from the fact that the shape of agricultural parcels, does not +need to follow its intended use, i.e. most crops can gener- +ally be cultivated independent of a field’s size or shape. Of +course there exist variations in the shapes and sizes of agri- +cultural fields [34], but these depend mostly on local agri- +cultural practices and are not expected to generalize over +unseen regions. Furthermore, agricultural parcels do not in- +herently contain sub-components or structure. Thus, know- +ing what is cultivated in a piece of land is not expected to +provide information about what grows nearby. This is in +contrast to other objects which clearly contain structure, e.g. +in human face parsing there are clear expectations about the +relative positions of various face parts. To test this hypoth- +esis we enumerate over all agricultural parcels belonging to +the most popular crop types in the T31TFM S2 tile in France +for year 2018 and take crop-type-conditional pixel counts +over a 1km square region from their centers. Then, we cal- +culate the cosine similarity of these values with uncondi- +tional pixel counts over the extent of the T31TFM tile and +find a high degree of similarity, suggesting that there are no +significant variations between these distributions; 2) a small +region in SITS is far more informative than its equivalent in +natural images, as it contains more channels than regular +RGB images (S2 imagery contains 13 bands in total) whose +intensities are averaged over a relatively large area (high- +est resolution of S2 images is 10 × 10 m2); 3) SITS for +land cover recognition do not typically contain moving ob- +jects. As a result, a timeseries of single pixel values can be +used for extracting features that are informative of a spe- +cific object part found at that particular location. Therefore, +several objects can be recognised using only information +found in a single location; plants, for example, can be recog- +nised by variations of their spectral signatures during their +growth cycle. Many works performing crop classification +do so using only temporal information in the form of time- +series of small patches [47], pixel statistics over the extent +of parcels [46] or even values from single pixels [40, 48]. +On the other hand, the spatial patterns in a single image +are uninformative of the crop type, as evidenced by the low +performance of systems relying on single images [14]. Our +encoder architecture can be seen in Fig.4(a,b). We now de- +scribe the temporal and spatial encoder submodules. +Temporal encoder Thus, we tokenize a SITS record +X ∈ RT ×H×W ×C into a set of tokens of size (NT × NH × +NW × d), as described in section 3.3 and subsequently re- +shape to ZT ∈ RNHNW ×NT ×d, to get a list of token time- +series for all patch locations. The input to the temporal en- +coder is: +Z0 +T = concat(ZTcls, ZT + PT[t, :]) ∈ RNHNW ×K+NT ×d +(5) +where PT[t, :] ∈ RNT ×d and ZTcls ∈ RK×d are re- +spectively added and prepended to all NHNW timeseries +and t ∈ RT is a vector containing all T acquisition times. +All samples are then processed in parallel by a Transformer +module. Consequently, the final feature map of the tempo- +ral encoder becomes ZL +T ∈ RNHNW ×K+NT ×d in which the +first K tokens in the temporal dimension correspond to the +prepended cls tokens. We only keep these tokens, discard- +ing the remaining NT vectors. +Spatial encoder We now transpose the first and second +dimensions in the temporal encoder output, to obtain a list +of patch features ZS ∈ RK×NHNW ×d for all output classes. +In a similar spirit, the input to the spatial encoder becomes: +Z0 +S = concat(ZScls, ZS + PS) ∈ RK×1+NHNW ×d +(6) +Where PS ∈ RNHNW ×d are respectively added to all +K spatial representations and each element of ZScls ∈ +RK×1×d is prepended to each class-specific feature map. +We note, that while in the temporal encoder cls tokens were +prepended to all patch locations, now there is a single cls +token per spatial feature map such that ZScls are used to +gather global SITS-level information. Processing with the +spatial encoder leads to a similar size output feature map +ZL +S ∈ RK×1+NHNW ×d. +3.5. Decoder architecture +The TSViT encoder architecture described in the pre- +vious section is designed as a general backbone for SITS + +Figure 4. TSViT submodules. (a) Temporal encoder. We reshape tokenized inputs, retaining the spatio-temporal structure of SITS, into +a set of timeseries for each spatial location, add temporal position encodings PT[t, :] for acquisition times t, concatenate local cls tokens +ZTcls (eq.5) and process in parallel with a Transformer. Only the first K output tokens are retained. (b) Spatial encoder. We reshape +the outputs of the temporal encoder into a set of spatial feature maps for each cls token, add spatial position encodings PS, concatenate +global cls tokens ZScls (eq.6) and process in parallel with a Transformer. (c) Segmentation head. Each local cls token is projected into hw +values denoting class-specific evidence for every pixel in a patch. All patches are then reassembled into the original image dimensions. (d) +Classification head. Global cls tokens are projected into scalar values, each denoting evidence for the presence of a specific class. +processing. To accommodate both global and dense pre- +diction tasks we design two decoder heads which feed on +different components of the encoder output. We view the +output of the encoder as ZL +S = [ZL +Sglobal|ZL +Slocal] respec- +tively corresponding to the states of the global and local +cls tokens. For image classification, we only make use of +ZL +Sglobal ∈ RK×d. We proceed, as described in sec.3.2, by +projecting each feature into a scalar value and concatenate +these values to obtain global unormalised class probabilities +as shown in Fig.4(d). Complementarily, for semantic seg- +mentation we only use ZL +Slocal ∈ RK×NHNW ×d. These +features encode information for the presence of each class +over the spatial extent of each image patch. By project- +ing each feature into hw dimensions and further reshaping +the feature dimension to (h × w) we obtain a set of class- +specific probabilities for each pixel in a patch. It is pos- +sible now to merge these patches together into an output +map (H × W × K) which represents class probabilities for +each pixel in the original image. This process is presented +schematically in Fig.4(c). +3.6. Position encodings +As described in section 3.4, positional encodings are +injected in two different locations in our proposed net- +work. First, temporal position encodings are added to all +patch tokens before processing by the temporal encoder +as shown in eq.(5). This operation aims at breaking the +permutation invariance property of MSA by introducing +time-specific position biases to all extracted patch tokens. +For crop recognition encoding the absolute temporal po- +sition of features is important as it helps identifying a +plant’s growth stage within the crop cycle. Furthermore, +the time interval between successive images in SITS varies +depending on acquisition times and other factors, such as +the degree of cloudiness or corrupted data. To introduce +acquisition-time-specific biases into the model, our tempo- +ral position encodings PT[t, :] depend directly on acquisi- +tion times t. More specifically, we make note of all the +dates t′ = [t1, t2, ..., tT ′] corresponding to the acquisition +times found in the training data and construct a lookup ta- +ble PT ∈ RT ′×d containing all learnt temporal position +encodings indexed by date. Finding the date-specific en- +codings that need to be added to patch tokens (eq.5) re- +duces to looking up appropriate indices from PT. In this +way temporal position encodings introduce a dynamic prior +of where to look at in the models’ global temporal receptive +field, rather than simply encoding the order of SITS acquisi- +tions which would discard valuable information. Following +token processing by the temporal encoder, spatial position +embeddings PS are added to the extracted cls tokens. These +are not dynamic in nature and are similar to the position en- +codings used in the original ViT architecture, with the dif- +ference that these biases are now added to K feature maps +instead of a single one. + +00·0000... +00·0 +location in time +Transformer +Transformer +location in space +patch (local) cls tokens +0-000-00 +000-0 +000-[ +0.000 +global cls tokens +Reassemble +concat(Zcls, Z + PT(t) +concat(Zcls, Z + Ps) +Zrels +Zscls +Z + PT(t) +Z+ Ps +eq.(5) +Pr[t,: +.00 +00:0 +Ps +Reshape: (K x NHNw x hw) +→(NHNw x h x w x K) +concat(*) +Reshape: (T × NH × Nw × d) -→ (NHNw × T × d) +Reshape: (NHNw × K × d) → (K × NHNw × d) +MLP: d → hw +MLP: d→1 +... +0:00 +0·0 +00-0 +(a) Temporal Encoder +(b) Spatial Encoder +(c) Segmentation Head +(d) Classification Head4. Experiments +We apply TSViT to two tasks using SITS records X ∈ +RT ×H×W ×C as inputs: classification and semantic seg- +mentation. At the object level, classification models learn +a mapping f(X) ∈ RK for the object occupying the center +of the H × W region. Semantic segmentation models learn +a mapping f(X) ∈ RH×W ×K, predicting class probabili- +ties for each pixel over the spatial extent of the SITS record. +We use an ablation study on semantic segmentation to guide +model design and hyperparameter tuning and proceed with +presenting our main results on three publicly available SITS +semantic segmentation and classification datasets. +4.1. Training and evaluation +Datasets To evaluate the performance of our proposed +semantic segmentation model we are using three publicly +available S2 land cover recognition datasets. The dataset +presented in [47] covers a densely cultivated area of interest +of 102×42 km2 north of Munich, Germany and contains 17 +distinct classes. Individual image samples cover a 240×240 +m2 area (24×24 pixels) and contain 13 bands. The PASTIS +dataset [14] contains images from four different regions in +France with diverse climate and crop distributions, span- +ning over 4000 km2 and including 18 crop types. In total, +it includes 2.4k SITS samples of size 128 × 128, each con- +taining 33-61 acquisitions and 10 image bands. Because +the PASTIS sample size is too large for efficiently train- +ing TSViT with available hardware, we split each sample +into 24 × 24 patches and retain all acquisition times for a +total of 57k samples. To accommodate a large set of ex- +periments we only use fold 1 among the five folds provided +in PASTIS. Finally, we use the T31TFM-1618 dataset [60] +which covers a densely cultivated S2 tile in France for years +2016-18 and includes 20 distinct classes. In total, it includes +140k samples of size 48 × 48, each containing 14-33 ac- +quisitions and 13 image bands. For the SITS classification +experiments, we construct the datasets from the respective +segmentation datasets. More specifically, for PASTIS we +use the provided object instance ids to extract 24 × 24 pixel +regions whose center pixel falls inside each object and use +the class of this object as the sample class. The remain- +ing two datasets contain samples of smaller spatial extent, +making the above strategy not feasible in practice. Here, we +choose to retain the samples as they are and assign the class +of the center pixel as the global class. We note that this +strategy forces us to discard samples in which the center +pixels belongs to the background class. Additional details +are provided in the supplementary material. +Implementation details For all experiments presented +we train for the same number of epochs using the provided +data splits from the respective publications for a fair com- +parison. More specifically, we train on all datasets using +the provided training sets and report results on the valida- +Ablation +Settings +mIoU +Factorization order +Spatial & Temporal +48.8 +Temporal & Spatial +78.5 +#cls tokens +1 +78.5 +K +83.6 +Position encodings +Static +80.8 +Date lookup +83.6 +Interactions between +cls tokens +Temporal +Spatial +✓ +✓ +✓ +XXX +81.5 +✓ +✓ +83.6 +Patch size +2 × 2 +84.8 +3 × 3 +83.6 +4 × 4 +81.5 +6 × 6 +79.6 +Table 1. Ablation on design choices for TSViT. All proposed +design choices are found to have a positive effect on performance. +tion sets for Germany and T31TFM-1618, and on the test +set for PASTIS. For training TSViT we use the AdamW op- +timizer [23] with a learning rate schedule which includes a +warmup period starting from zero to a maximum value 10−3 +at epoch 10, followed by cosine learning rate decay [33] +down to 5 ∗ 10−6 at the end of training. For Germany and +T31TFM-1618 we train with the above settings and report +the best performances between what we achieve and the +original studies. Since we split PASTIS, we are training +with both settings and report the best results. Overall, we +find that our settings improve model performance. We train +with a batch size of 16 or 32 and no regularization on ×2 +Nvidia Titan Xp gpus in a data parallel fashion. All mod- +els are trained with a Masked Cross-Entropy loss, masking +the effect of the background class in both training loss and +evaluation metrics. We report overall accuracy (OA), aver- +aged over pixels, and mean intersection over union (mIoU) +averaged over classes. For SITS classification, in addition +to the 1D models presented in section 2 we modify the best +performing semantic segmentation models by aggregating +extracted features across space prior to the application of a +classifier, thus, outputing a single prediction. Classification +models are trained with Focal loss [30]. We report OA and +mean accuracy (mAcc) averaged over classes. +4.2. Ablation studies +We perform an ablation study on design parameters of +our framework using the Germany dataset [47]. Starting +with a baseline TSViT with L = 4 for both encoder net- +works, a single cls token, h = w = 3, t = 1, d = +128 we successively update our design after each ablation. +Here, we present the effect of the most important design +choices; additional ablations are presented in the supple- +mentary material. Overall, we find that the order of factor- + +Dataset +Germany [47] +PASTIS [14] +T31TFM-1618 [60] +Model +Semantic segmentation (OA / mIoU) +BiCGRU [47] +91.3 / 72.3 +80.5 / 56.2 +88.6 / 57.7 +FPN-CLSTM [7] +91.8 / 73.7 +81.9 / 59.5 +88.4 / 57.8 +UNET3D [45] +92.4 / 75.2 +82.3 / 60.4 +88.4 / 57.6 +UNET3Df [60] +92.4 / 75.4 +82.1 / 60.2 +88.6 / 57.7 +UNET2D-CLSTM [45] +92.9/ 76.2 +82.7 / 60.7 +89.0 / 58.8 +U-TAE [14] +93.1 / 77.1 +82.9 / 62.4 +88.9 / 58.5 +TSViT (ours) +95.0 / 84.8 +83.4 / 65.1 (83.4/ 65.4) +90.3 / 63.1 +Model +Object classification (OA / mAcc) +TempCNN∗ [40] +89.8 / 78.4 +84.8 / 69.1 +84.7 / 62.6 +DuPLo∗ [24] +93.1 / 82.2 +84.8 / 69.4 +83.9 / 69.5 +Transformer∗ [48] +92.4/ 84.3 +84.4 / 68.1 +84.3 / 71.4 +UNET3D [45] +92.7 / 83.9 +84.8 / 70.2 +84.8 / 71.4 +UNET2D-CLSTM [45] +93.0 / 84.0 +84.7 / 70.3 +84.7 / 71.6 +U-TAE [14] +92.6 / 83.7 +84.9 / 71.8 +84.8 / 71.7 +TSViT (ours) +94.7 / 88.1 +87.1 / 75.5 +87.8 / 74.2 +Table 2. Comparison with state-of-the-art models from literature. (top) Semantic segmentation. (bottom) Object classification. ∗1D +temporal only models. We report overall accuracy (OA), mean intersection over union (mIoU) and mean accuracy (mAcc). For PASTIS +we report results for fold-1 only; average test set performance across all five folds is shown in parenthesis for direct comparison with [14]. +ization is the most important design choice in our proposed +framework. +Using a spatio-temporal factorization from +the video recognition literature performs poorly at 48.8% +mIoU. Changing the factorization order to temporo-spatial +raises performance by an absolute +29.7% to 78.5% mIoU. +Including additional cls tokens increases performance to +83.6%mIoU (+5.1%), so we proceed with using K cls to- +kens in our design. We test the effect of our date-specific +position encodings compared to a fixed set of values and +find a significant −2.8% performance drop from using fixed +size PT compared to our proposed lookup encodings. As +discussed in section 3.4 our spatial encoder blocks cross cls- +token interactions. Allowing interactions among all tokens +comes at a significant increase in compute cost, O(K2) to +O(K), and is found to decrease performance by −2.1% +mIoU. Finally, we find that smaller patch sizes generally +work better, which is reasonable given that tokens retain a +higher degree of spatial granularity and are used to predict +smaller regions. Using 2 × 2 patches raises performance by ++1.2%mIoU to 84.8% compared to 3 × 3 patches. Our fi- +nal design which is used in the main experiments presented +in Table 2 employs a temporo-spatial design with K cls to- +kens, acquisition-time-specific position encodings, 2×2 in- +put patches and four layers for both encoders. +4.3. Comparison with SOTA +In Table 2 and Fig.1, we present performance results of +our final TSViT design compared to state-of-the-art mod- +els presented in section 2. For semantic segmentation, we +find that all models from literature perform similarly, with +the BiCGRU being overall the worst performer, match- +ing CNN-based architectures only in T31TFM-1618. For +Figure 5. Visualization of predictions in Germany. The back- +ground class is shown in white, ”x” indicates a false prediction. +all datasets, TSViT outperforms previously suggested ap- +proaches by a very large margin. A visualization of predic- +tions in Germany for the top-3 performers is shown in Fig.5. +In object classification, we observe that 1D temporal mod- +els are generally outperformed by spatio-temporal models, +with the exception of the Transformer [48]. All 1D models +perform poorly in PASTIS. Again, TSViT trained for clas- +sification consistently outperforms all other approaches by +a large margin across all datasets. In both tasks, we find +smaller improvements for the pixel-averaged compared to +class-averaged metrics, which is reasonable given the large +class imbalance that characterizes the datasets. + +Ground truth +UNET3Df +UNET2D-CLSTM +TSViT +×x +×x +X +XXXXXXXX +XXX +X×> +XXXXX +×××× +××5. Conclusion +In this paper we proposed TSViT, the first fully- +attentional architecture for general SITS processing. +By +taking advantage of the Transformer’s global receptive field, +capacity to learn a rich feature space and by incorporating +inductive biases that suit SITS data, we surpass the state-of- +the-art performance by a large margin in object classifica- +tion and semantic segmentation using three publicly avail- +able land cover recognition datasets. +References +[1] European Space Agency. The sentinel missions. https:// +www.esa.int/Applications/Observing_the_ +Earth/Copernicus/The_Sentinel_missions. +Accessed: 2022-11-11. 2 +[2] European Space Agency. Sentinels for common agriculture +policy. http://esa-sen4cap.org/. Accessed: 2022- +11-11. 1 +[3] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen +Sun, Mario Luˇci´c, and Cordelia Schmid. Vivit: A video vi- +sion transformer. In 2021 IEEE/CVF International Confer- +ence on Computer Vision (ICCV), pages 6816–6826, 2021. +3 +[4] Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer +normalization. ArXiv, abs/1607.06450, 2016. 4 +[5] I. Bello, B. Zoph, Q. Le, A. Vaswani, and J. Shlens. Atten- +tion augmented convolutional networks. In 2019 IEEE/CVF +International Conference on Computer Vision (ICCV), pages +3285–3294, 2019. 3 +[6] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub- +biah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan- +tan, Pranav Shyam, Girish Sastry, Amanda Askell, Sand- +hini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom +Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, +Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric +Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack +Clark, Christopher Berner, Sam McCandlish, Alec Radford, +Ilya Sutskever, and Dario Amodei. +Language models are +few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, +M.F. Balcan, and H. Lin, editors, Advances in Neural Infor- +mation Processing Systems, volume 33, pages 1877–1901. +Curran Associates, Inc., 2020. 3 +[7] Jorge Andres Chamorro Martinez, Laura Elena Cu´e La +Rosa, Raul Queiroz Feitosa, Ieda Del’Arco Sanches, and +Patrick Nigri Happ. Fully convolutional recurrent networks +for multidate crop recognition from multitemporal image se- +quences. +ISPRS Journal of Photogrammetry and Remote +Sensing, 171:188–201, 2021. 2, 8 +[8] Christopher Conrad, Stefan Dech, Olena Dubovyk, Sebas- +tian Fritsch, Doris Klein, Fabian L¨ow, Gunther Schorcht, and +Julian Zeidler. Derivation of temporal windows for accurate +crop discrimination in heterogeneous croplands of Uzbek- +istan using multitemporal rapideye images. Computers and +Electronics in Agriculture, 103:63–74, 2014. 2 +[9] Christopher Conrad, Sebastian Fritsch, Julian Zeidler, Gerd +R¨ucker, and Stefan Dech. Per-field irrigated crop classifica- +tion in arid central asia using spot and aster data. Remote +Sensing, 2(4):1035–1056, 2010. 2 +[10] Gordon Conway. +One Billion Hungry: Can we Feed the +World? Cornell University Press, 2012. 1 +[11] Xiyang Dai, Yinpeng Chen, Jianwei Yang, Pengchuan +Zhang, Lu Yuan, and Lei Zhang. Dynamic detr: End-to-end +object detection with dynamic attention. In 2021 IEEE/CVF +International Conference on Computer Vision (ICCV), pages +2968–2977, 2021. 3 +[12] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina +Toutanova. BERT: Pre-training of deep bidirectional trans- +formers for language understanding. In Proceedings of the +2019 Conference of the North American Chapter of the As- +sociation for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), pages +4171–4186, Minneapolis, Minnesota, June 2019. Associa- +tion for Computational Linguistics. 3 +[13] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, +Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, +Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- +vain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is +worth 16x16 words: Transformers for image recognition at +scale. In International Conference on Learning Representa- +tions, 2021. 2, 3, 4 +[14] Vivien Sainte Fare Garnot and Loic Landrieu. +Panoptic +segmentation of satellite image time series with convolu- +tional temporal attention networks. In Proceedings of the +IEEE/CVF International Conference on Computer Vision +(ICCV), pages 4872–4881, October 2021. 2, 5, 7, 8 +[15] V. S. F. Garnot, L. Landrieu, S. Giordano, and N. Chehata. +Time-space tradeoff in deep learning models for crop clas- +sification on satellite multi-spectral image time series. +In +IGARSS 2019 - 2019 IEEE International Geoscience and Re- +mote Sensing Symposium, pages 6247–6250, 2019. 2 +[16] Vivien Sainte Fare Garnot, Loic Landrieu, Sebastien Gior- +dano, and Nesrine Chehata. Satellite image time series clas- +sification with pixel-set encoders and temporal self-attention. +In Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition (CVPR), June 2020. 2 +[17] Rohit Girdhar and Deva Ramanan. Attentional pooling for +action recognition. In I. Guyon, U. Von Luxburg, S. Bengio, +H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, ed- +itors, Advances in Neural Information Processing Systems, +volume 30. Curran Associates, Inc., 2017. 5 +[18] Noel +Gorelick, +Matt +Hancher, +Mike +Dixon, +Simon +Ilyushchenko, David Thau, and Rebecca Moore. +Google +earth engine: Planetary-scale geospatial analysis for every- +one. Remote Sensing of Environment, 202:18–27, 2017. Big +Remotely Sensed Data: tools, applications and experiences. +2 +[19] Pengyu Hao, Yulin Zhan, Li Wang, Zheng Niu, and Muham- +mad Shakir. Feature selection of time series modis data for +early crop classification using random forest: A case study +in Kansas, USA. Remote Sensing, 7(5):5347–5369, 2015. 2 +[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. +Deep residual learning for image recognition. In 2016 IEEE +Conference on Computer Vision and Pattern Recognition +(CVPR), pages 770–778, 2016. 2 + +[21] Dan Hendrycks and Kevin Gimpel. +Gaussian error linear +units (gelus). arXiv: Learning, 2016. 4 +[22] Dino Ienco, Raffaele Gaetano, Claire Dupaquier, and Pierre +Maurel. Land cover classification via multitemporal spatial +data by deep recurrent neural networks. IEEE Geoscience +and Remote Sensing Letters, 14(10):1685–1689, 2017. 2 +[23] Loshchilov Ilya, Hutter Frank, et al. Decoupled weight decay +regularization. Proceedings of ICLR, 2019. 7 +[24] Roberto Interdonato, Dino Ienco, Raffaele Gaetano, and +Kenji Ose. +DuPLO: A dual view point deep learning ar- +chitecture for time series classification. ISPRS Journal of +Photogrammetry and Remote Sensing, 149:91 – 104, 2019. +2, 8 +[25] Xue Jinru and Baofeng Su. Significant remote sensing veg- +etation indices: A review of developments and applications. +Journal of Sensors, 2017:1–17, 01 2017. 2 +[26] Andreas Kamilaris and Francesc X. Prenafeta-Bold´u. Deep +learning in agriculture: A survey. Computers and Electronics +in Agriculture, 147:70–90, 2018. 2 +[27] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas +Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video +classification with convolutional neural networks. In 2014 +IEEE Conference on Computer Vision and Pattern Recogni- +tion, pages 1725–1732, 2014. 5 +[28] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. +Imagenet classification with deep convolutional neural net- +works. In F. Pereira, C.J. Burges, L. Bottou, and K.Q. Wein- +berger, editors, Advances in Neural Information Processing +Systems, volume 25. Curran Associates, Inc., 2012. 2 +[29] Nataliia Kussul, Mykola Lavreniuk, Sergii Skakun, and An- +drii Shelestov. Deep learning classification of land cover and +crop types using remote sensing data. IEEE Geoscience and +Remote Sensing Letters, 14(5):778–782, 2017. 2 +[30] Tsung-Yi Lin, Priya Goyal, Ross B. Girshick, Kaiming He, +and Piotr Doll´ar. +Focal loss for dense object detection. +CoRR, abs/1708.02002, 2017. 7 +[31] Tsung-Yi Lin, Piotr Dollar, Ross Girshick, Kaiming He, +Bharath Hariharan, and Serge Belongie. +Feature pyramid +networks for object detection. In Proceedings of the IEEE +Conference on Computer Vision and Pattern Recognition +(CVPR), July 2017. 2 +[32] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, +Zheng Zhang, Stephen Lin, and Baining Guo. Swin trans- +former: Hierarchical vision transformer using shifted win- +dows. In Proceedings of the IEEE/CVF International Con- +ference on Computer Vision (ICCV), pages 10012–10022, +October 2021. 3 +[33] Ilya Loshchilov and Frank Hutter. SGDR: Stochastic gradi- +ent descent with warm restarts. In International Conference +on Learning Representations, 2017. 7 +[34] NASA. +Agricultural +patterns. +https : / / +earthobservatory.nasa.gov/images/6605/ +agricultural-patterns. last accessed on 2022-9-22. +5 +[35] United Nations. Goal 2: Zero hunger. https://www.un. +org/sustainabledevelopment/hunger/. +Ac- +cessed: 2022-11-11. 1 +[36] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan Assel- +mann. Video transformer network. In 2021 IEEE/CVF In- +ternational Conference on Computer Vision Workshops (IC- +CVW), pages 3156–3165, 2021. 3 +[37] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan As- +selmann. +Video transformer network. +In Proceedings of +the IEEE/CVF International Conference on Computer Vision +(ICCV) Workshops, pages 3163–3172, October 2021. 5 +[38] Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz +Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Im- +age transformer. In Jennifer Dy and Andreas Krause, ed- +itors, Proceedings of the 35th International Conference on +Machine Learning, volume 80 of Proceedings of Machine +Learning Research, pages 4055–4064. PMLR, 10–15 Jul +2018. 3 +[39] Charlotte Pelletier, Silvia Valero, Jordi Inglada, Nicolas +Champion, and G´erard Dedieu. Assessing the robustness of +random forests to map land cover with high resolution satel- +lite image time series over large areas. Remote Sensing of +Environment, 187:156–168, 2016. 2 +[40] Charlotte Pelletier, Geoffrey I. Webb, and Franc¸ois Petitjean. +Temporal convolutional neural network for the classification +of satellite image time series. Remote Sensing, 11(5), 2019. +2, 5, 8 +[41] Jos´e M. Pe˜na-Barrag´an, Moffatt K. Ngugi, Richard E. Plant, +and Johan Six. Object-based crop identification using multi- +ple vegetation indices, textural features and crop phenology. +Remote Sensing of Environment, 115(6):1301–1316, 2011. 2 +[42] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan +Bello, Anselm Levskaya, and Jon Shlens. Stand-alone self- +attention in vision models. In H. Wallach, H. Larochelle, +A. Beygelzimer, F. d'Alch´e-Buc, E. Fox, and R. Garnett, ed- +itors, Advances in Neural Information Processing Systems, +volume 32, pages 68–80. Curran Associates, Inc., 2019. 3 +[43] Ren´e Ranftl, Alexey Bochkovskiy, and Vladlen Koltun. Vi- +sion transformers for dense prediction. In 2021 IEEE/CVF +International Conference on Computer Vision (ICCV), pages +12159–12168, 2021. 3 +[44] Bradley C. Reed, Jesslyn F. Brown, Darrel VanderZee, +Thomas R. Loveland, James W. Merchant, and Donald O. +Ohlen. Measuring phenological variability from satellite im- +agery. Journal of Vegetation Science, 5(5):703–714, 1994. +2 +[45] Rose Rustowicz, Robin Cheong, Lijing Wang, Stefano Er- +mon, Marshall Burke, and David B. Lobell. Semantic seg- +mentation of crop type in Africa: A novel dataset and analy- +sis of deep learning methods. In CVPR Workshops, 2019. 2, +8 +[46] M. Rußwurm and M. K¨orner. +Temporal vegetation mod- +elling using long short-term memory networks for crop iden- +tification from medium-resolution multi-spectral satellite im- +ages. In 2017 IEEE Conference on Computer Vision and Pat- +tern Recognition Workshops (CVPRW), pages 1496–1504, +2017. 2, 5 +[47] Marc Rußwurm and Marco K¨orner. +Multi-temporal land +cover classification with sequential recurrent encoders. IS- +PRS International Journal of Geo-Information, 7(4):129, +Mar 2018. 2, 5, 7, 8 + +[48] Marc Rußwurm and Marco K¨orner. Self-attention for raw +optical satellite time series classification. ISPRS Journal of +Photogrammetry and Remote Sensing, 169:421 – 435, 2020. +2, 5, 8 +[49] Marc Rußwurm, Charlotte Pelletier, M. Zollner, S´ebastien +Lef`evre, and Marco K¨orner. +Breizhcrops: +A time se- +ries dataset for crop type mapping. ISPRS - International +Archives of the Photogrammetry, Remote Sensing and Spa- +tial Information Sciences, XLIII-B2-2020:1545–1551, 08 +2020. 2 +[50] Sen4CAP. +Agricultural practices. +http : / / esa - +sen4cap . org / content / agricultural - +practices. Accessed: 2022-11-11. 1 +[51] Sen4CAP. Crop diversification. http://esa-sen4cap. +org/content/crop-diversification. Accessed: +2022-11-11. 1 +[52] Pierre Sermanet, David Eigen, Xiang Zhang, Michael Math- +ieu, Rob Fergus, and Yann LeCun. +Overfeat: Integrated +recognition, localization and detection using convolutional +networks. +2014. +Publisher Copyright: © 2014 Interna- +tional Conference on Learning Representations, ICLR. All +rights reserved.; 2nd International Conference on Learning +Representations, ICLR 2014 ; Conference date: 14-04-2014 +Through 16-04-2014. 2 +[53] Evan Shelhamer, Jonathan Long, and Trevor Darrell. Fully +convolutional networks for semantic segmentation. +IEEE +Transactions on Pattern Analysis and Machine Intelligence, +39(4):640–651, 2017. 2 +[54] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Ye- +ung, Wai-kin Wong, and Wang-chun WOO. Convolutional +LSTM network: A machine learning approach for precipi- +tation nowcasting. In C. Cortes, N. Lawrence, D. Lee, M. +Sugiyama, and R. Garnett, editors, Advances in Neural In- +formation Processing Systems, volume 28, pages 802–810. +Curran Associates, Inc., 2015. 2 +[55] Sofia Siachalou, Giorgos Mallinis, and Maria Tsakiri-Strati. +A hidden Markov models approach for crop classification: +Linking crop phenology to time series of multi-sensor re- +mote sensing data. Remote Sensing, 7(4):3633–3650, 2015. +2 +[56] Karen Simonyan and Andrew Zisserman. Two-stream con- +volutional networks for action recognition in videos. +In +NIPS, 2014. 5 +[57] Karen Simonyan and Andrew Zisserman. Very deep convo- +lutional networks for large-scale image recognition. In In- +ternational Conference on Learning Representations, 2015. +2 +[58] Hwanjun Song, Deqing Sun, Sanghyuk Chun, Varun Jam- +pani, Dongyoon Han, Byeongho Heo, Wonjae Kim, and +Ming-Hsuan Yang. ViDT: An efficient and effective fully +transformer-based object detector. In International Confer- +ence on Learning Representations, 2022. 3 +[59] Robin Strudel, Ricardo Garcia, Ivan Laptev, and Cordelia +Schmid. Segmenter: Transformer for semantic segmenta- +tion. In 2021 IEEE/CVF International Conference on Com- +puter Vision (ICCV), pages 7242–7252, 2021. 3 +[60] Michail Tarasiou, Riza Alp G¨uler, and Stefanos Zafeiriou. +Context-self contrastive pre-training for crop type semantic +segmentation. +IEEE Transactions on Geoscience and Re- +mote Sensing, pages 1–1, 2022. 2, 7, 8 +[61] Michail Tarasiou and Stefanos Zafeiriou. +Deepsatdata: +Building large scale datasets of satellite images for training +machine learning models. +In IGARSS 2022 - 2022 IEEE +International Geoscience and Remote Sensing Symposium, +pages 4070–4073, 2022. 2 +[62] Michail Tarasiou and Stefanos Zafeiriou. Embedding earth: +Self-supervised contrastive pre-training for dense land cover +classification. 2022. 2 +[63] Andrew Tatem, Scott Goetz, and Simon Hay. Fifty years of +earth observation satellites. American Scientist, 96:390–398, +09 2008. 2 +[64] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- +reit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Il- +lia Polosukhin. +Attention is all you need. +In I. Guyon, +U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vish- +wanathan, and R. Garnett, editors, Advances in Neural In- +formation Processing Systems 30, pages 5998–6008. Curran +Associates, Inc., 2017. 2, 3, 4 +[65] Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, +Alan Yuille, and Liang-Chieh Chen. Axial-DeepLab: Stand- +alone axial-attention for panoptic segmentation. In ECCV, +2020. 3 +[66] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua +Lin, Xiaoou Tang, and Luc Van Gool. Temporal segment net- +works: Towards good practices for deep action recognition. +In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, +editors, Computer Vision – ECCV 2016, pages 20–36, Cham, +2016. Springer International Publishing. 5 +[67] X. Wang, R. Girshick, A. Gupta, and K. He. Non-local neu- +ral networks. In 2018 IEEE/CVF Conference on Computer +Vision and Pattern Recognition, pages 7794–7803, 2018. 3 +[68] Yuqing Wang, Zhaoliang Xu, Xinlong Wang, Chunhua Shen, +Baoshan Cheng, Hao Shen, and Huaxia Xia. +End-to-end +video instance segmentation with transformers. +In 2021 +IEEE/CVF Conference on Computer Vision and Pattern +Recognition (CVPR), pages 8737–8746, 2021. 3 +[69] Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, +Zi-Hang Jiang, Francis E.H. Tay, Jiashi Feng, and Shuicheng +Yan. Tokens-to-token vit: Training vision transformers from +scratch on imagenet. In Proceedings of the IEEE/CVF In- +ternational Conference on Computer Vision (ICCV), pages +558–567, October 2021. 3 +[70] Peng Yue, Boyi Shangguan, Lei Hu, Liangcun Jiang, Chenx- +iao Zhang, Zhipeng Cao, and Yinyin Pan. Towards a train- +ing data model for artificial intelligence in earth observation. +International Journal of Geographical Information Science, +36(11):2113–2137, 2022. 2 +[71] Zixiao Zhang, Xiaoqiang Lu, Guojin Cao, Yuting Yang, +Licheng Jiao, and Fang Liu. +Vit-yolo:transformer-based +yolo for object detection. In 2021 IEEE/CVF International +Conference on Computer Vision Workshops (ICCVW), pages +2799–2808, 2021. 3 +[72] Bolei Zhou, Alex Andonian, Aude Oliva, and Antonio Tor- +ralba. +Temporal relational reasoning in videos. +In Pro- +ceedings of the European Conference on Computer Vision +(ECCV), September 2018. 5 + diff --git a/UdE4T4oBgHgl3EQfMAzo/content/tmp_files/load_file.txt b/UdE4T4oBgHgl3EQfMAzo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9260f7216a6902ef2097d4781ca0247fe31be81 --- /dev/null +++ b/UdE4T4oBgHgl3EQfMAzo/content/tmp_files/load_file.txt @@ -0,0 +1,777 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf,len=776 +page_content='ViTs for SITS: Vision Transformers for Satellite Image Time Series Michail Tarasiou Imperial College London michail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='tarasiou10@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='uk Erik Chavez Imperial College London erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='chavez@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='uk Stefanos Zafeiriou Imperial College London s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='zafeiriou@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='uk Abstract In this paper we introduce the Temporo-Spatial Vision Transformer (TSViT), a fully-attentional model for general Satellite Image Time Series (SITS) processing based on the Vision Transformer (ViT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' TSViT splits a SITS record into non-overlapping patches in space and time which are tok- enized and subsequently processed by a factorized temporo- spatial encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We argue, that in contrast to natural im- ages, a temporal-then-spatial factorization is more intu- itive for SITS processing and present experimental evidence for this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Additionally, we enhance the model’s dis- criminative power by introducing two novel mechanisms for acquisition-time-specific temporal positional encodings and multiple learnable class tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The effect of all novel design choices is evaluated through an extensive ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our proposed architecture achieves state-of-the-art performance, surpassing previous approaches by a signifi- cant margin in three publicly available SITS semantic seg- mentation and classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All model, training and evaluation codes are made publicly available to facili- tate further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Introduction The monitoring of the Earth surface man-made impacts or activities is essential to enable the design of effective in- terventions to increase welfare and resilience of societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' One example is the sector of agriculture in which monitor- ing of crop development can help design optimum strategies aimed at improving the welfare of farmers and resilience of the food production system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The second of United Nations Sustainable Development Goals (SDG) of Ending Hunger relies on increasing the crop productivity and revenues of farmers in poor and developing countries [35] - approxi- mately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 billion people’s livelihoods depend mainly on producing crops [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Achieving SDG 2 goals requires to be Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Model and performance overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (top) TSViT archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' A more detailed schematic is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (bottom) TSViT performance compared with previous arts (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' able to accurately monitor yields and the evolution of culti- vated areas in order to measure the progress towards achiev- ing several goals, as well as to evaluate the effectiveness of different policies or interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In the European Union (EU) the Sentinel for Common Agricultural Policy program (Sen4CAP) [2] focuses on developing tools and analytics to support the verification of direct payments to farmers with underlying environmental conditionalities such as the adop- tion of environmentally-friendly [50] and crop diversifica- tion [51] practices based on real-time monitoring by the European Space Agency’s (ESA) Sentinel high-resolution arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='04944v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='CV] 12 Jan 2023 ClassiticationHead Segmentation Head Spatial Encoder Temporal Encoder Token embeddingGermany PASTIS T31TFM1618 86 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7% 66 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7% 64 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3% segmentation 428864 63 64 (%)nojw 62 62 61 60 60 58 59 58 72 56 90 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8% +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7% 76 88 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5% 74 classification mAcc(%) 86 72 84 70 82 68 66 80 69 64 78 68 62 sota architectures TSViT (ours)satellite constellation [1] to complement on site verifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Recently, the volume and diversity of space-borne Earth Observation (EO) data [63] and post-processing tools [18, 61, 70] has increased exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This wealth of resources, in combination with important developments in machine learning for computer vision [20,28,53], provides an important opportunity for the development of tools for the automated monitoring of crop development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Towards more accurate automatic crop type recognition, we introduce TSViT, the first fully-attentional1 architecture for general SITS processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' An overview of the proposed architecture can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our novel design introduces some inductive biases that make TSViT particu- larly suitable for the target domain: Satellite imagery for monitoring land surface variabil- ity boast a high revisit time leading to long temporal sequences, for example Sentinel-2 (S2) satellites have an average revisit time of 5 days resulting in 60-70 acquisitions per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To reduce the amount of com- putation we factorize input dimensions into their tem- poral and spatial components, providing intuition (sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4) and experimental evidence (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2) about why the order of factorization matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' TSViT uses a Transformer backbone [64] following the recently proposed ViT framework [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' As a result, every TSViT layer has a global receptive field in time or space, in contrast to previously proposed convolu- tional and recurrent architectures [14,24,40,45,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To make our approach more suitable for SITS mod- elling we propose a tokenization scheme for the in- put image timeseries and propose acquisition-time- specific temporal position encodings in order to extract date-aware features and to account for irregularities in SITS acquisition times (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We make modifications to the ViT framework (sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2) to enhance its capacity to gather class-specific evidence which we argue suits the problem at hand and design two custom decoder heads to accommodate both global and dense predictions (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our provided intuitions are tested through extensive abla- tion studies on design parameters presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Overall, our architecture achieves state-of-the-art perfor- mance in three publicly available datasets for classification and semantic segmentation presented in Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Crop type recognition Crop type recognition is a subcategory of land use recog- nition which involves assigning one of K crop categories 1without any convolution operations (classes) at a set of desired locations on a geospatial grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For successfully doing so modelling the temporal patterns of growth during a time period of interest has been shown to be critical [15, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' As a result, model inputs are time- series of T satellite images of spatial dimensions H × W with C channels, X ∈ RT ×H×W ×C rather than single ac- quisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' There has been a significant body of work on crop type identification found in the remote sensing liter- ature [8, 9, 19, 39, 41, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' These works typically involve multiple processing steps and domain expertise to guide the extraction of features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' NDVI [25], that can be separated into crop types by learnt classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' More re- cently, Deep Neural Networks (DNN) trained on raw op- tical data [22,26,29,46,47,62] have been shown to outper- form these approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' At the object level, (SITS classi- fication) [24, 40, 48] use 1D data of single-pixel or parcel- level aggregated feature timeseries, rather than the full SITS record, learning a mapping f : RT ×C → RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Among these works, TempCNN [40] employs a simple 1D convo- lutional architecture, while [48] use the Transformer archi- tecture [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' DuPLo [24] consists of an ensemble of CNN and RNN streams in an effort to exploit the complementar- ity of extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Finally, [16] view satellite images as un-ordered sets of pixels and calculate feature statistics at the parcel level, but, in contrast to previously mentioned approaches, their implementation requires knowledge of the object geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' At the pixel level (SITS semantic seg- mentation), models learn a mapping f(X) ∈ RH×W ×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For this task, [47] show that convolutional RNN variants (CLSTM, CGRU) [54] can automatically extract useful fea- tures from raw optical data, including cloudy images, that can be linearly separated into classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The use of CNN architectures is explored in [45] who employ two models: a UNET2D feature extractor, followed by a CLSTM tem- poral model (UNET2D-CLSTM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' and a UNET3D fully- convolutional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Both are found to achieve equivalent performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In a similar spirit, [7] use a FPN [31] feature extractor, coupled with a CLSTM temporal model (FPN- CLSTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The UNET3Df architecture [60] follows from UNET3D but uses a different decoder head more suited to contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The U-TAE architecture [14] follows a different approach, in that it employs the encoder part of a UNET2D, applied on parallel on all images, and a sub- sequent temporal attention mechanism which collapses the temporal feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' These spatial-only features are further processed by the decoder part of a UNET2D to ob- tain dense predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Self-attention in vision Convolutional [20, 28, 57] and fully-convolutional net- works (FCN) [52, 53] have been the de-facto model of choice for vision tasks over the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The convo- lution operation extracts translation-equivariant features via application of a small square kernel over the spatial extent of the learnt representation and grows the feature recep- tive field linearly over the depth of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In con- trast, the self-attention operation, introduced as the main building block of the Transformer architecture [64], uses self-similarity as a means for feature aggregation and can have a global receptive field at every layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Following the adoption of Transformers as the dominant architecture in natural language processing tasks [6,12,64], several works have attempted to exploit self-attention in vision architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Because the time complexity of self-attention scales quadratically with the size of the input, its naive implemen- tation on image data, which typically contain more pixels than text segments contain words, would be prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To bypass this issue, early works focused on improving effi- ciency by injecting self-attention layers only at specific lo- cations within a CNN [5, 67] or by constraining their re- ceptive field to a local region [38,42,65], however, in prac- tice, these designs do not scale well with available hard- ware leading to slow throughput rates, large memory re- quirements and long training times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Following a different approach, the Vision Transformer (ViT) [13], presented in further detail in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1, constitutes an effort to apply a pure Transformer architecture on image data, by propos- ing a simple, yet efficient image tokenization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Sev- eral works have drawn inspiration from ViT to develop novel attention-based architectures for vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For image recognition, [32, 69] re-introduce some of the inductive bi- ases that made CNNs successful in vision, leading to im- proved performances without the need for long pre-training schedules, [43, 59] employ Transformers for dense predic- tion, [11,58,71] for object detection and [3,36,68] for video processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Among these works, our framework is more closely related to [3] who also use a spatio-temporal fac- torization of input dimensions, and [59] who use multiple learnable tokens for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' However, we deviate significantly from [3] by introducing acquisition- time-specific temporal encodings to accommodate an un- even distribution of images in time, reverse the order of fac- torization and are interested in both global and dense pre- dictions (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Additionally, we differ from [59] in that we introduce the cls tokens as an input to the encoder in order to collapse the time dimension and obtain class- specific features, while they use them as class queries inputs to the decoder similar to their use in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We also differ sig- nificantly from [59] in terms of the decoder design as they resize the output of the penultimate layer to match the in- put size and further process that to obtain pixel-level logits, while we decode each token directly into a region matching input patch dimensions and reassemble these into a dense probability map (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Backbone architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (a) Transformer backbone, (b) ViT architecture, (c) TSViT backbone employs additional cls tokens (red), each responsible for predicting a single class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Method In this section we present the TSViT architecture in de- tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' First, we give a brief overview of the ViT (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1) which provided inspiration for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 we present our modified TSViT backbone, followed by our to- kenization scheme (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3), encoder (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4) and decoder (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Finally, in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6, we discuss several considerations behind the design of our po- sition encoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Primer on ViT Inspired by the success of Transformers in natural lan- guage processing tasks [64] the ViT [13] is an application of the Transformer architecture to images with the fewest possible modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Their framework involves the to- kenization of a 2D image X ∈ RH×W ×C to a set of patch tokens Z ∈ RN×d by splitting it into a sequence of N = ⌊ H h ⌋⌊ W w ⌋ same-size and non-overlapping patches of spatial extent (h × w) which are flattened into 1D tokens xi ∈ RhwC and linearly projected into d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Over- all, the process of token extraction is equivalent to the appli- cation of 2D convolution with kernel size (h × w) at strides (h, w) across respective dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The extracted patches are used to construct model inputs as follows: Z0 = concat(zcls, Z + P) ∈ RN+1×d (1) A set of learned positional encoding vectors P ∈ RN×d, added to Z, are employed to encode the absolute posi- tion information of each token and break the permutation invariance property of the subsequent Transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' A separate learned class (cls) token zcls ∈ Rd [12] is prepended to the linearly transformed and positionally aug- Layer Layer MSA MLP: Norm Norm pp eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (2) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (3) xL (a) Transformer backbone MLP: = d-→K concat(zcls, Z+P) Transformer eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (1) : (b) ViT MLP: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='.+ d-→1 concat(Zcls,Z+P) Transformer eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (4) (c) TsViT backboneFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' SITS Tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We embed each satellite image independently following ViT [13] mented patch tokens leading to a length N + 1 sequence of tokens Z0 which are used as model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The Trans- former backbone consists of L blocks of alternating lay- ers of Multiheaded Self-Attention (MSA) [64] and residual Multi-Layer Perceptron (MLP) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Yl = MSA(LN(Zl)) + Zl (2) Zl+1 = MLP(LN(Yl)) + Yl (3) Prior to each layer, inputs are normalized following Lay- ernorm (LN) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' MLP blocks consist of two layers of linear projection followed by GELU non-linear activations [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In contrast to CNN architectures, in which spatial dimen- sions are reduced while feature dimensions increase with layer depth, Transformers are isotropic in that all feature maps Zl ∈ R1+N×d have the same shape throughout the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' After processing by the final layer L, all tokens apart from the first one (the state of the cls token) are dis- carded and unormalized class probabilities are calculated by processing this token via a MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' A schematic representation of the ViT architecture can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Backbone architecture In the ViT architecture, the cls token progressively re- fines information gathered from all patch tokens to reach a final global representation used to derive class probabil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our TSViT backbone, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2(c), essentially follows from ViT, with few modifications in the tokeniza- tion and decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' More specifically, we introduce K (equal to the number of object classes) additional learnable cls tokens Zcls ∈ RK×d, compared to ViT which uses a single token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Z0 = concat(Zcls, Z + P) ∈ RN+K×d (4) Without deviating from ViT, all cls and positionally aug- mented patch tokens are concatenated and processed by the L layers of a Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' After the final layer, we discard all patch tokens and project each cls token into a scalar value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' By concatenating these values we obtain a length K vector of unormalised class probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This de- sign choice brings the following two benefits: 1) it increases the capacity of the cls token relative to the patch tokens, al- lowing them to store more patterns to be used by the MSA operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2) it allows for more controlled handling of the interactions between evidence gathered for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Re- garding the first point, introducing multiple cls tokens can be seen as equivalent to increasing the dimension of a sin- gle cls token to an integer multiple of the patch token di- mension dcls = kdpatch and split the cls token into k sepa- rate subspaces prior to the MSA operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In this way we can increase the capacity of the cls tokens while avoiding issues such as the need for asymmetric MSA weight matri- ces for cls and patch tokens, which would effectively more than double our model’s parameter count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Furthermore, by choosing k = K and enforcing a bijective mapping from cls tokens to class predictions, the state of each cls token be- comes more focused to a specific class with network depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In TSViT we go a step further and explicitly separate cls tokens by class after processing with the temporal encoder to allow only same-cls-token interactions in the spatial en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 we argue why this is a very useful in- ductive bias for modelling spatial relationships in crop type recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Tokenization of SITS inputs A SITS record X ∈ RT ×H×W ×C consists of a series of T satellite images of spatial dimensions H × W with C channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For the tokenization of our 3D inputs, we can extend the tokenization-as-convolution approach to 3D data and apply a 3D kernel with size (t×h×w) at stride (t, h, w) across temporal and spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In this manner N = ⌊ T t ⌋⌊ H h ⌋⌊ W w ⌋ non-overlapping tokens xi ∈ RthwC are extracted, and subsequently projected to d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Using t > 1, all extracted tokens contain spatio-temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For the special case of t = 1 each token contains spatial-only information for each acquisition time and temporal information is accounted for only through the encoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Since the computation cost of global self- attention layers is quadratic w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' the length of the token se- quence O(N 2), choosing larger values for t, h, w can lead to significantly reduced number of FLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In our experi- ments, however, we have found small values for t, h, w to work much better in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For all presented experiments we use a value of t = 1 motivated in part because this choice simplifies the implementation of acquisition-time-specific temporal position encodings, described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' With regards to the spatial dimensions of extracted patches we H/h Conv2d(in channels=C, out channels=d kernel size=(h, w), stride=(h, w)) H h M→ W t1 tT timehave found small values to work best for semantic segmen- tation, which is reasonable given that small patches retain additional spatial granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In the end, our tokenization scheme is similar to ViT’s applied in parallel for each acqui- sition as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3, however, at this stage, instead of unrolling feature dimensions, we retain the spatial structure of the original input as reshape operations will be handled by the TSViT encoder submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Encoder architecture In the previous section we presented a motivation for us- ing small values t, h, w for the extracted patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Unless other measures are taken to reduce the model’s computa- tional cost this choice would be prohibitive for process- ing SITS with multiple acquisition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To avoid such problems, we choose to factorize our inputs across their temporal and spatial dimensions, a practice commonly em- ployed for video processing [17,27,37,56,66,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We note that all these works use a spatial-temporal factorization or- der, which is reasonable when dealing with natural images, given that it allows the extraction of higher level, semanti- cally aware spatial features, whose relationship in time is useful for scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' However, we argue that in the context of SITS, reversing the order of factorization is a meaningful design choice for the following reasons: 1) in contrast to natural images in which context can be useful for recognising an object, in crop type recognition context can provide little information, or can be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This arises from the fact that the shape of agricultural parcels, does not need to follow its intended use, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' most crops can gener- ally be cultivated independent of a field’s size or shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Of course there exist variations in the shapes and sizes of agri- cultural fields [34], but these depend mostly on local agri- cultural practices and are not expected to generalize over unseen regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Furthermore, agricultural parcels do not in- herently contain sub-components or structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Thus, know- ing what is cultivated in a piece of land is not expected to provide information about what grows nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This is in contrast to other objects which clearly contain structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' in human face parsing there are clear expectations about the relative positions of various face parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To test this hypoth- esis we enumerate over all agricultural parcels belonging to the most popular crop types in the T31TFM S2 tile in France for year 2018 and take crop-type-conditional pixel counts over a 1km square region from their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Then, we cal- culate the cosine similarity of these values with uncondi- tional pixel counts over the extent of the T31TFM tile and find a high degree of similarity, suggesting that there are no significant variations between these distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2) a small region in SITS is far more informative than its equivalent in natural images, as it contains more channels than regular RGB images (S2 imagery contains 13 bands in total) whose intensities are averaged over a relatively large area (high- est resolution of S2 images is 10 × 10 m2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3) SITS for land cover recognition do not typically contain moving ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' As a result, a timeseries of single pixel values can be used for extracting features that are informative of a spe- cific object part found at that particular location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Therefore, several objects can be recognised using only information found in a single location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' plants, for example, can be recog- nised by variations of their spectral signatures during their growth cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Many works performing crop classification do so using only temporal information in the form of time- series of small patches [47], pixel statistics over the extent of parcels [46] or even values from single pixels [40, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' On the other hand, the spatial patterns in a single image are uninformative of the crop type, as evidenced by the low performance of systems relying on single images [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our encoder architecture can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We now de- scribe the temporal and spatial encoder submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Temporal encoder Thus, we tokenize a SITS record X ∈ RT ×H×W ×C into a set of tokens of size (NT × NH × NW × d), as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 and subsequently re- shape to ZT ∈ RNHNW ×NT ×d, to get a list of token time- series for all patch locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The input to the temporal en- coder is: Z0 T = concat(ZTcls, ZT + PT[t, :]) ∈ RNHNW ×K+NT ×d (5) where PT[t, :] ∈ RNT ×d and ZTcls ∈ RK×d are re- spectively added and prepended to all NHNW timeseries and t ∈ RT is a vector containing all T acquisition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All samples are then processed in parallel by a Transformer module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Consequently, the final feature map of the tempo- ral encoder becomes ZL T ∈ RNHNW ×K+NT ×d in which the first K tokens in the temporal dimension correspond to the prepended cls tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We only keep these tokens, discard- ing the remaining NT vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Spatial encoder We now transpose the first and second dimensions in the temporal encoder output, to obtain a list of patch features ZS ∈ RK×NHNW ×d for all output classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In a similar spirit, the input to the spatial encoder becomes: Z0 S = concat(ZScls, ZS + PS) ∈ RK×1+NHNW ×d (6) Where PS ∈ RNHNW ×d are respectively added to all K spatial representations and each element of ZScls ∈ RK×1×d is prepended to each class-specific feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We note, that while in the temporal encoder cls tokens were prepended to all patch locations, now there is a single cls token per spatial feature map such that ZScls are used to gather global SITS-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Processing with the spatial encoder leads to a similar size output feature map ZL S ∈ RK×1+NHNW ×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Decoder architecture The TSViT encoder architecture described in the pre- vious section is designed as a general backbone for SITS Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' TSViT submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (a) Temporal encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We reshape tokenized inputs, retaining the spatio-temporal structure of SITS, into a set of timeseries for each spatial location, add temporal position encodings PT[t, :] for acquisition times t, concatenate local cls tokens ZTcls (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5) and process in parallel with a Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Only the first K output tokens are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (b) Spatial encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We reshape the outputs of the temporal encoder into a set of spatial feature maps for each cls token, add spatial position encodings PS, concatenate global cls tokens ZScls (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6) and process in parallel with a Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (c) Segmentation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Each local cls token is projected into hw values denoting class-specific evidence for every pixel in a patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All patches are then reassembled into the original image dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (d) Classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Global cls tokens are projected into scalar values, each denoting evidence for the presence of a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To accommodate both global and dense pre- diction tasks we design two decoder heads which feed on different components of the encoder output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We view the output of the encoder as ZL S = [ZL Sglobal|ZL Slocal] respec- tively corresponding to the states of the global and local cls tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For image classification, we only make use of ZL Sglobal ∈ RK×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We proceed, as described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2, by projecting each feature into a scalar value and concatenate these values to obtain global unormalised class probabilities as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Complementarily, for semantic seg- mentation we only use ZL Slocal ∈ RK×NHNW ×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' These features encode information for the presence of each class over the spatial extent of each image patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' By project- ing each feature into hw dimensions and further reshaping the feature dimension to (h × w) we obtain a set of class- specific probabilities for each pixel in a patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' It is pos- sible now to merge these patches together into an output map (H × W × K) which represents class probabilities for each pixel in the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This process is presented schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Position encodings As described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4, positional encodings are injected in two different locations in our proposed net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' First, temporal position encodings are added to all patch tokens before processing by the temporal encoder as shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' This operation aims at breaking the permutation invariance property of MSA by introducing time-specific position biases to all extracted patch tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For crop recognition encoding the absolute temporal po- sition of features is important as it helps identifying a plant’s growth stage within the crop cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Furthermore, the time interval between successive images in SITS varies depending on acquisition times and other factors, such as the degree of cloudiness or corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To introduce acquisition-time-specific biases into the model, our tempo- ral position encodings PT[t, :] depend directly on acquisi- tion times t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' More specifically, we make note of all the dates t′ = [t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', tT ′] corresponding to the acquisition times found in the training data and construct a lookup ta- ble PT ∈ RT ′×d containing all learnt temporal position encodings indexed by date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Finding the date-specific en- codings that need to be added to patch tokens (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5) re- duces to looking up appropriate indices from PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In this way temporal position encodings introduce a dynamic prior of where to look at in the models’ global temporal receptive field, rather than simply encoding the order of SITS acquisi- tions which would discard valuable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Following token processing by the temporal encoder, spatial position embeddings PS are added to the extracted cls tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' These are not dynamic in nature and are similar to the position en- codings used in the original ViT architecture, with the dif- ference that these biases are now added to K feature maps instead of a single one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 00·0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 00·0 location in time Transformer Transformer location in space patch (local) cls tokens 0-000-00 000-0 000-[ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='000 global cls tokens Reassemble concat(Zcls, Z + PT(t) concat(Zcls, Z + Ps) Zrels Zscls Z + PT(t) Z+ Ps eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (5) Pr[t,: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='00 00:0 Ps Reshape: (K x NHNw x hw) →(NHNw x h x w x K) concat(*) Reshape: (T × NH × Nw × d) -→ (NHNw × T × d) Reshape: (NHNw × K × d) → (K × NHNw × d) MLP: d → hw MLP: d→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 0:00 0·0 00-0 (a) Temporal Encoder (b) Spatial Encoder (c) Segmentation Head (d) Classification Head4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Experiments We apply TSViT to two tasks using SITS records X ∈ RT ×H×W ×C as inputs: classification and semantic seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' At the object level, classification models learn a mapping f(X) ∈ RK for the object occupying the center of the H × W region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Semantic segmentation models learn a mapping f(X) ∈ RH×W ×K, predicting class probabili- ties for each pixel over the spatial extent of the SITS record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We use an ablation study on semantic segmentation to guide model design and hyperparameter tuning and proceed with presenting our main results on three publicly available SITS semantic segmentation and classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Training and evaluation Datasets To evaluate the performance of our proposed semantic segmentation model we are using three publicly available S2 land cover recognition datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The dataset presented in [47] covers a densely cultivated area of interest of 102×42 km2 north of Munich, Germany and contains 17 distinct classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Individual image samples cover a 240×240 m2 area (24×24 pixels) and contain 13 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The PASTIS dataset [14] contains images from four different regions in France with diverse climate and crop distributions, span- ning over 4000 km2 and including 18 crop types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In total, it includes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4k SITS samples of size 128 × 128, each con- taining 33-61 acquisitions and 10 image bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Because the PASTIS sample size is too large for efficiently train- ing TSViT with available hardware, we split each sample into 24 × 24 patches and retain all acquisition times for a total of 57k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' To accommodate a large set of ex- periments we only use fold 1 among the five folds provided in PASTIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Finally, we use the T31TFM-1618 dataset [60] which covers a densely cultivated S2 tile in France for years 2016-18 and includes 20 distinct classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In total, it includes 140k samples of size 48 × 48, each containing 14-33 ac- quisitions and 13 image bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For the SITS classification experiments, we construct the datasets from the respective segmentation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' More specifically, for PASTIS we use the provided object instance ids to extract 24 × 24 pixel regions whose center pixel falls inside each object and use the class of this object as the sample class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The remain- ing two datasets contain samples of smaller spatial extent, making the above strategy not feasible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Here, we choose to retain the samples as they are and assign the class of the center pixel as the global class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We note that this strategy forces us to discard samples in which the center pixels belongs to the background class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Additional details are provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Implementation details For all experiments presented we train for the same number of epochs using the provided data splits from the respective publications for a fair com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' More specifically, we train on all datasets using the provided training sets and report results on the valida- Ablation Settings mIoU Factorization order Spatial & Temporal 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 Temporal & Spatial 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 #cls tokens 1 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 K 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 Position encodings Static 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 Date lookup 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 Interactions between cls tokens Temporal Spatial ✓ ✓ ✓ XXX 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 ✓ ✓ 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 Patch size 2 × 2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 3 × 3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 4 × 4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 6 × 6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ablation on design choices for TSViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All proposed design choices are found to have a positive effect on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' tion sets for Germany and T31TFM-1618, and on the test set for PASTIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For training TSViT we use the AdamW op- timizer [23] with a learning rate schedule which includes a warmup period starting from zero to a maximum value 10−3 at epoch 10, followed by cosine learning rate decay [33] down to 5 ∗ 10−6 at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For Germany and T31TFM-1618 we train with the above settings and report the best performances between what we achieve and the original studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Since we split PASTIS, we are training with both settings and report the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Overall, we find that our settings improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We train with a batch size of 16 or 32 and no regularization on ×2 Nvidia Titan Xp gpus in a data parallel fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All mod- els are trained with a Masked Cross-Entropy loss, masking the effect of the background class in both training loss and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We report overall accuracy (OA), aver- aged over pixels, and mean intersection over union (mIoU) averaged over classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For SITS classification, in addition to the 1D models presented in section 2 we modify the best performing semantic segmentation models by aggregating extracted features across space prior to the application of a classifier, thus, outputing a single prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Classification models are trained with Focal loss [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We report OA and mean accuracy (mAcc) averaged over classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ablation studies We perform an ablation study on design parameters of our framework using the Germany dataset [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Starting with a baseline TSViT with L = 4 for both encoder net- works, a single cls token, h = w = 3, t = 1, d = 128 we successively update our design after each ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Here, we present the effect of the most important design choices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' additional ablations are presented in the supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Overall, we find that the order of factor- Dataset Germany [47] PASTIS [14] T31TFM-1618 [60] Model Semantic segmentation (OA / mIoU) BiCGRU [47] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 / 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 / 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 FPN-CLSTM [7] 91.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 / 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 / 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 UNET2D-CLSTM [45] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='9/ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 60.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='9 / 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 TSViT (ours) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='0 / 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4/ 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 Model Object classification (OA / mAcc) TempCNN∗ [40] 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 DuPLo∗ [24] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 / 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='9 / 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 Transformer∗ [48] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4/ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 UNET3D [45] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 UNET2D-CLSTM [45] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='0 / 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 U-TAE [14] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='9 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 TSViT (ours) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7 / 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1 / 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8 / 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Comparison with state-of-the-art models from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (top) Semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' (bottom) Object classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ∗1D temporal only models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We report overall accuracy (OA), mean intersection over union (mIoU) and mean accuracy (mAcc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For PASTIS we report results for fold-1 only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' average test set performance across all five folds is shown in parenthesis for direct comparison with [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ization is the most important design choice in our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Using a spatio-temporal factorization from the video recognition literature performs poorly at 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8% mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Changing the factorization order to temporo-spatial raises performance by an absolute +29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='7% to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5% mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Including additional cls tokens increases performance to 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='6%mIoU (+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1%), so we proceed with using K cls to- kens in our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' We test the effect of our date-specific position encodings compared to a fixed set of values and find a significant −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8% performance drop from using fixed size PT compared to our proposed lookup encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' As discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='4 our spatial encoder blocks cross cls- token interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Allowing interactions among all tokens comes at a significant increase in compute cost, O(K2) to O(K), and is found to decrease performance by −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1% mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Finally, we find that smaller patch sizes generally work better, which is reasonable given that tokens retain a higher degree of spatial granularity and are used to predict smaller regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Using 2 × 2 patches raises performance by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='2%mIoU to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='8% compared to 3 × 3 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Our fi- nal design which is used in the main experiments presented in Table 2 employs a temporo-spatial design with K cls to- kens, acquisition-time-specific position encodings, 2×2 in- put patches and four layers for both encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Comparison with SOTA In Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='1, we present performance results of our final TSViT design compared to state-of-the-art mod- els presented in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For semantic segmentation, we find that all models from literature perform similarly, with the BiCGRU being overall the worst performer, match- ing CNN-based architectures only in T31TFM-1618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' For Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Visualization of predictions in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The back- ground class is shown in white, ”x” indicates a false prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' all datasets, TSViT outperforms previously suggested ap- proaches by a very large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' A visualization of predic- tions in Germany for the top-3 performers is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In object classification, we observe that 1D temporal mod- els are generally outperformed by spatio-temporal models, with the exception of the Transformer [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All 1D models perform poorly in PASTIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Again, TSViT trained for clas- sification consistently outperforms all other approaches by a large margin across all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In both tasks, we find smaller improvements for the pixel-averaged compared to class-averaged metrics, which is reasonable given the large class imbalance that characterizes the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ground truth UNET3Df UNET2D-CLSTM TSViT ×x ×x X XXXXXXXX XXX X×> XXXXX ×××× ××5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Conclusion In this paper we proposed TSViT, the first fully- attentional architecture for general SITS processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' By taking advantage of the Transformer’s global receptive field, capacity to learn a rich feature space and by incorporating inductive biases that suit SITS data, we surpass the state-of- the-art performance by a large margin in object classifica- tion and semantic segmentation using three publicly avail- able land cover recognition datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' References [1] European Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' The sentinel missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='int/Applications/Observing_the_ Earth/Copernicus/The_Sentinel_missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Accessed: 2022-11-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [2] European Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Sentinels for common agriculture policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' http://esa-sen4cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Accessed: 2022- 11-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1 [3] Anurag Arnab, Mostafa Dehghani, Georg Heigold, Chen Sun, Mario Luˇci´c, and Cordelia Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vivit: A video vi- sion transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF International Confer- ence on Computer Vision (ICCV), pages 6816–6826, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [4] Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ArXiv, abs/1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='06450, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 4 [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Bello, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Zoph, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Le, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vaswani, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Shlens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Atten- tion augmented convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2019 IEEE/CVF International Conference on Computer Vision (ICCV), pages 3285–3294, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [6] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Sub- biah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakan- tan, Pranav Shyam, Girish Sastry, Amanda Askell, Sand- hini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Larochelle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ranzato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Hadsell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Balcan, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Lin, editors, Advances in Neural Infor- mation Processing Systems, volume 33, pages 1877–1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [7] Jorge Andres Chamorro Martinez, Laura Elena Cu´e La Rosa, Raul Queiroz Feitosa, Ieda Del’Arco Sanches, and Patrick Nigri Happ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fully convolutional recurrent networks for multidate crop recognition from multitemporal image se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 171:188–201, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 8 [8] Christopher Conrad, Stefan Dech, Olena Dubovyk, Sebas- tian Fritsch, Doris Klein, Fabian L¨ow, Gunther Schorcht, and Julian Zeidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Derivation of temporal windows for accurate crop discrimination in heterogeneous croplands of Uzbek- istan using multitemporal rapideye images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Computers and Electronics in Agriculture, 103:63–74, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [9] Christopher Conrad, Sebastian Fritsch, Julian Zeidler, Gerd R¨ucker, and Stefan Dech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Per-field irrigated crop classifica- tion in arid central asia using spot and aster data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing, 2(4):1035–1056, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [10] Gordon Conway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' One Billion Hungry: Can we Feed the World?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Cornell University Press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1 [11] Xiyang Dai, Yinpeng Chen, Jianwei Yang, Pengchuan Zhang, Lu Yuan, and Lei Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Dynamic detr: End-to-end object detection with dynamic attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 2968–2977, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [12] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional trans- formers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the As- sociation for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171–4186, Minneapolis, Minnesota, June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [13] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Syl- vain Gelly, Jakob Uszkoreit, and Neil Houlsby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' An image is worth 16x16 words: Transformers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In International Conference on Learning Representa- tions, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 3, 4 [14] Vivien Sainte Fare Garnot and Loic Landrieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Panoptic segmentation of satellite image time series with convolu- tional temporal attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 4872–4881, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 5, 7, 8 [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Garnot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Landrieu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Giordano, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Chehata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Time-space tradeoff in deep learning models for crop clas- sification on satellite multi-spectral image time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In IGARSS 2019 - 2019 IEEE International Geoscience and Re- mote Sensing Symposium, pages 6247–6250, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [16] Vivien Sainte Fare Garnot, Loic Landrieu, Sebastien Gior- dano, and Nesrine Chehata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Satellite image time series clas- sification with pixel-set encoders and temporal self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [17] Rohit Girdhar and Deva Ramanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Attentional pooling for action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Guyon, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Von Luxburg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Wallach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fergus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vishwanathan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Garnett, ed- itors, Advances in Neural Information Processing Systems, volume 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [18] Noel Gorelick, Matt Hancher, Mike Dixon, Simon Ilyushchenko, David Thau, and Rebecca Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Google earth engine: Planetary-scale geospatial analysis for every- one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing of Environment, 202:18–27, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Big Remotely Sensed Data: tools, applications and experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [19] Pengyu Hao, Yulin Zhan, Li Wang, Zheng Niu, and Muham- mad Shakir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Feature selection of time series modis data for early crop classification using random forest: A case study in Kansas, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing, 7(5):5347–5369, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [21] Dan Hendrycks and Kevin Gimpel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Gaussian error linear units (gelus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' arXiv: Learning, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 4 [22] Dino Ienco, Raffaele Gaetano, Claire Dupaquier, and Pierre Maurel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Land cover classification via multitemporal spatial data by deep recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 14(10):1685–1689, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [23] Loshchilov Ilya, Hutter Frank, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Decoupled weight decay regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Proceedings of ICLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 7 [24] Roberto Interdonato, Dino Ienco, Raffaele Gaetano, and Kenji Ose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' DuPLO: A dual view point deep learning ar- chitecture for time series classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 149:91 – 104, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 8 [25] Xue Jinru and Baofeng Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Significant remote sensing veg- etation indices: A review of developments and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Journal of Sensors, 2017:1–17, 01 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [26] Andreas Kamilaris and Francesc X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Prenafeta-Bold´u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Deep learning in agriculture: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Computers and Electronics in Agriculture, 147:70–90, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [27] Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Large-scale video classification with convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2014 IEEE Conference on Computer Vision and Pattern Recogni- tion, pages 1725–1732, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [28] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Imagenet classification with deep convolutional neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Pereira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Burges, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Bottou, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Wein- berger, editors, Advances in Neural Information Processing Systems, volume 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [29] Nataliia Kussul, Mykola Lavreniuk, Sergii Skakun, and An- drii Shelestov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Deep learning classification of land cover and crop types using remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 14(5):778–782, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [30] Tsung-Yi Lin, Priya Goyal, Ross B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Girshick, Kaiming He, and Piotr Doll´ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Focal loss for dense object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' CoRR, abs/1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='02002, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 7 [31] Tsung-Yi Lin, Piotr Dollar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Feature pyramid networks for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [32] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Swin trans- former: Hierarchical vision transformer using shifted win- dows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Con- ference on Computer Vision (ICCV), pages 10012–10022, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [33] Ilya Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' SGDR: Stochastic gradi- ent descent with warm restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In International Conference on Learning Representations, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 7 [34] NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Agricultural patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' https : / / earthobservatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='gov/images/6605/ agricultural-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' last accessed on 2022-9-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [35] United Nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Goal 2: Zero hunger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' org/sustainabledevelopment/hunger/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ac- cessed: 2022-11-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1 [36] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan Assel- mann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Video transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF In- ternational Conference on Computer Vision Workshops (IC- CVW), pages 3156–3165, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [37] Daniel Neimark, Omri Bar, Maya Zohar, and Dotan As- selmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Video transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) Workshops, pages 3163–3172, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [38] Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Im- age transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Jennifer Dy and Andreas Krause, ed- itors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 4055–4064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' PMLR, 10–15 Jul 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [39] Charlotte Pelletier, Silvia Valero, Jordi Inglada, Nicolas Champion, and G´erard Dedieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Assessing the robustness of random forests to map land cover with high resolution satel- lite image time series over large areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing of Environment, 187:156–168, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [40] Charlotte Pelletier, Geoffrey I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Webb, and Franc¸ois Petitjean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Temporal convolutional neural network for the classification of satellite image time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing, 11(5), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 5, 8 [41] Jos´e M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Pe˜na-Barrag´an, Moffatt K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ngugi, Richard E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Plant, and Johan Six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Object-based crop identification using multi- ple vegetation indices, textural features and crop phenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing of Environment, 115(6):1301–1316, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [42] Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan Bello, Anselm Levskaya, and Jon Shlens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Stand-alone self- attention in vision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Larochelle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Beygelzimer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=" d'Alch´e-Buc, E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fox, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Garnett, ed- itors, Advances in Neural Information Processing Systems, volume 32, pages 68–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [43] Ren´e Ranftl, Alexey Bochkovskiy, and Vladlen Koltun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vi- sion transformers for dense prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 12159–12168, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [44] Bradley C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Reed, Jesslyn F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Brown, Darrel VanderZee, Thomas R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Loveland, James W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Merchant, and Donald O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Ohlen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Measuring phenological variability from satellite im- agery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Journal of Vegetation Science, 5(5):703–714, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [45] Rose Rustowicz, Robin Cheong, Lijing Wang, Stefano Er- mon, Marshall Burke, and David B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Lobell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Semantic seg- mentation of crop type in Africa: A novel dataset and analy- sis of deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In CVPR Workshops, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 8 [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Rußwurm and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' K¨orner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Temporal vegetation mod- elling using long short-term memory networks for crop iden- tification from medium-resolution multi-spectral satellite im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2017 IEEE Conference on Computer Vision and Pat- tern Recognition Workshops (CVPRW), pages 1496–1504, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 5 [47] Marc Rußwurm and Marco K¨orner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Multi-temporal land cover classification with sequential recurrent encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' IS- PRS International Journal of Geo-Information, 7(4):129, Mar 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 5, 7, 8 [48] Marc Rußwurm and Marco K¨orner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Self-attention for raw optical satellite time series classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 169:421 – 435, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 5, 8 [49] Marc Rußwurm, Charlotte Pelletier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Zollner, S´ebastien Lef`evre, and Marco K¨orner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Breizhcrops: A time se- ries dataset for crop type mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spa- tial Information Sciences, XLIII-B2-2020:1545–1551, 08 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [50] Sen4CAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Agricultural practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' http : / / esa - sen4cap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' org / content / agricultural - practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Accessed: 2022-11-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1 [51] Sen4CAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Crop diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' http://esa-sen4cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' org/content/crop-diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Accessed: 2022-11-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 1 [52] Pierre Sermanet, David Eigen, Xiang Zhang, Michael Math- ieu, Rob Fergus, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Overfeat: Integrated recognition, localization and detection using convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Publisher Copyright: © 2014 Interna- tional Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2nd International Conference on Learning Representations, ICLR 2014 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Conference date: 14-04-2014 Through 16-04-2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [53] Evan Shelhamer, Jonathan Long, and Trevor Darrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fully convolutional networks for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(4):640–651, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [54] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Ye- ung, Wai-kin Wong, and Wang-chun WOO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Convolutional LSTM network: A machine learning approach for precipi- tation nowcasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Cortes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Lawrence, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Sugiyama, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Garnett, editors, Advances in Neural In- formation Processing Systems, volume 28, pages 802–810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [55] Sofia Siachalou, Giorgos Mallinis, and Maria Tsakiri-Strati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' A hidden Markov models approach for crop classification: Linking crop phenology to time series of multi-sensor re- mote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Remote Sensing, 7(4):3633–3650, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [56] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Two-stream con- volutional networks for action recognition in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In NIPS, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [57] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Very deep convo- lutional networks for large-scale image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In In- ternational Conference on Learning Representations, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [58] Hwanjun Song, Deqing Sun, Sanghyuk Chun, Varun Jam- pani, Dongyoon Han, Byeongho Heo, Wonjae Kim, and Ming-Hsuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' ViDT: An efficient and effective fully transformer-based object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In International Confer- ence on Learning Representations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [59] Robin Strudel, Ricardo Garcia, Ivan Laptev, and Cordelia Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Segmenter: Transformer for semantic segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Com- puter Vision (ICCV), pages 7242–7252, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [60] Michail Tarasiou, Riza Alp G¨uler, and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Context-self contrastive pre-training for crop type semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Re- mote Sensing, pages 1–1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 7, 8 [61] Michail Tarasiou and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Deepsatdata: Building large scale datasets of satellite images for training machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In IGARSS 2022 - 2022 IEEE International Geoscience and Remote Sensing Symposium, pages 4070–4073, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [62] Michail Tarasiou and Stefanos Zafeiriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Embedding earth: Self-supervised contrastive pre-training for dense land cover classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [63] Andrew Tatem, Scott Goetz, and Simon Hay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fifty years of earth observation satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' American Scientist, 96:390–398, 09 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [64] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Il- lia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Guyon, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Luxburg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Wallach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Fergus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vish- wanathan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Garnett, editors, Advances in Neural In- formation Processing Systems 30, pages 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2, 3, 4 [65] Huiyu Wang, Yukun Zhu, Bradley Green, Hartwig Adam, Alan Yuille, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Axial-DeepLab: Stand- alone axial-attention for panoptic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [66] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua Lin, Xiaoou Tang, and Luc Van Gool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Temporal segment net- works: Towards good practices for deep action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, Computer Vision – ECCV 2016, pages 20–36, Cham, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5 [67] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Girshick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Gupta, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Non-local neu- ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7794–7803, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [68] Yuqing Wang, Zhaoliang Xu, Xinlong Wang, Chunhua Shen, Baoshan Cheng, Hao Shen, and Huaxia Xia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' End-to-end video instance segmentation with transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 8737–8746, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [69] Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Tay, Jiashi Feng, and Shuicheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Tokens-to-token vit: Training vision transformers from scratch on imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF In- ternational Conference on Computer Vision (ICCV), pages 558–567, October 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [70] Peng Yue, Boyi Shangguan, Lei Hu, Liangcun Jiang, Chenx- iao Zhang, Zhipeng Cao, and Yinyin Pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Towards a train- ing data model for artificial intelligence in earth observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' International Journal of Geographical Information Science, 36(11):2113–2137, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 2 [71] Zixiao Zhang, Xiaoqiang Lu, Guojin Cao, Yuting Yang, Licheng Jiao, and Fang Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Vit-yolo:transformer-based yolo for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision Workshops (ICCVW), pages 2799–2808, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 3 [72] Bolei Zhou, Alex Andonian, Aude Oliva, and Antonio Tor- ralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' Temporal relational reasoning in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' In Pro- ceedings of the European Conference on Computer Vision (ECCV), September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE4T4oBgHgl3EQfMAzo/content/2301.04944v1.pdf'} diff --git a/UdFKT4oBgHgl3EQfli6N/content/tmp_files/2301.11854v1.pdf.txt b/UdFKT4oBgHgl3EQfli6N/content/tmp_files/2301.11854v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7499b25c28774252d35afc497192bbced8322da1 --- /dev/null +++ b/UdFKT4oBgHgl3EQfli6N/content/tmp_files/2301.11854v1.pdf.txt @@ -0,0 +1,1840 @@ +MNRAS 000, 1–14 (2022) +Preprint 30 January 2023 +Compiled using MNRAS LATEX style file v3.0 +GrGadget: an N-body TreePM relativistic code for cosmological +simulations +Eduardo Quintana-Miranda,1,2,3★ Pierluigi Monaco1,2,3,4 and Luca Tornatore1,2,3 +1 Dipartimento di Fisica, Sezione di Astronomia, via G.B. Tiepolo 11, I-34143 Trieste, Italy +2 INAF – Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy +3 IFPU – Institute for the Fundamental Physics of the Universe, via Beirut 2, I-34100 Trieste, Italy +4 INFN – Istituto Nazionale di Fisica Nucleare, Via Valerio 2, I-34127 Trieste, Italy +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present the merging of the Particle-Mesh (PM) relativistic Gevolution code with the TreePM Gadget-4 code, with the +aim of studying general relativity effects in cosmology. Our code, called GrGadget, is able to track the evolution of metric +perturbations in the weak field limit by using Gevolution’s implementation of a relativistic PM in the Poisson gauge. To +achieve this, starting from Gevolution we have written a C++ library called Libgevolution, that allows a code to access and +use the same abstractions and resources that Gevolution uses for its PM-only N-body simulations. The code works under the +assumption that particle interactions at short distances can be approximated as Newtonian, so that we can combine the forces +computed with a Newtonian Tree with those computed with a relativistic PM. The result is a TreePM simulation code that +represents metric perturbations at the scales where they are relevant, while resolving non-linear structures. We validate our code +by closely matching Gadget-4 forces, computed with the Tree switched off, with those computed with Libgevolution in the +Newtonian limit. With GrGadget we obtain a matter power spectrum that is compatible with Newtonian Gadget-4 at small +scales and contains GR features at large scales that are consistent with results obtained with Gevolution. We demonstrate that, +due to the better resolution of the highly non-linear regime, the representation of the relativistic fields sampled on the mesh +improves with respect to the PM-only simulations. +Key words: cosmology: theory – large-scale structure of the Universe +1 INTRODUCTION +The state of the art of precision cosmology provides a standard cos- +mological model, ΛCDM, that is consistent with most observational +evidence on large scales, but relies on the existence of a dark sector +populated by Dark Matter (DM) and Dark Energy (DE). The first +is responsible for the formation of cosmological structures such as +galaxies and their large-scale density field, while the second causes +the observed accelerated expansion of the universe in the present +epoch. Their physical nature is an open problem, since the only ev- +idence of their existence comes from their gravitational interaction +with visible matter. A possible explanation is that the dark sector is +due to a misrepresentation of gravity, that on large scales does not +follow Einstein’s General Relativity (GR), at the basis of the ΛCDM +model. +This fact has triggered a wave of interest in modifications of GR, +that can lead to extra terms that explain dark energy or dark matter +(see, e.g., Silvestri & Trodden 2009; Capozziello & De Laurentis +2012, and references therein). Such modifications must be significant +only on large scales or low density, because GR is very accurate in +predicting planetary orbits, light deflection and Doppler effects in +solar system tests and has more recently been successfully tested +★ E-mail: eduardo.quintanamiranda@phd.units.it +with the detection of gravitational waves (Abbott et al. 2016) and the +direct imaging of black hole event horizons (Event Horizon Telescope +Collaboration et al. 2019). +In order to characterize dark energy in the age of its dominance, +many projects have been planned to survey large parts of the sky and +probe the large-scale distribution of matter using galaxy clustering +and galaxy lensing, both from the ground (DES1, Krause et al. 2017; +DESI2, DESI Collaboration et al. 2016; Rubin’s LSST3, Ivezić et al. +2019; SKAO4 surveys) and from space (Euclid5, Laureijs et al. 2011; +Roman6, Spergel et al. 2015; SphereX7, Doré et al. 2014). Some of +these surveys have already started to produce a flood of data that will +soon lead to a precise characterization of the galaxy and matter den- +sity fields. A comparison of these observations to model predictions, +either using summary statistics or field-level inference, will lead to +unprecedented tests not only of the cosmological model but also +of the gravity theory behind it. With precision being guaranteed by +1 www.darkenergysurvey.org +2 www.desi.lbl.gov +3 www.lsst.org +4 www.skao.int +5 sci.esa.int/web/euclid +6 roman.gsfc.nasa.gov +7 spherex.caltech.edu +© 2022 The Authors +arXiv:2301.11854v1 [astro-ph.CO] 27 Jan 2023 + +2 +E. Quintana-Miranda et al. +the amount of available high-quality data, accuracy will be achieved +only by rigorous control of systematics, both in the data and in theory +predictions. +The highly non-linear nature of the observed density field and the +non-locality of gravity make cosmological simulations necessary to +compare the predictions of current theories with the observations +at an increasing level of accuracy. Yet, most of the widely adopted +simulation codes, like e.g. Gadget-4 (Springel et al. 2021), use New- +tonian dynamics for the evolution of matter perturbations. This is not +the ideal configuration to pass from the unobservable distribution +of matter in a periodic comoving box to the observable distribution +of light in the past light cone. Relativistic corrections can be added +a posteriori by post-processing Newtonian simulation outputs; one +specific example of this approach is the modeling of lensing due +to the distortion of null geodesics (Bartelmann & Schneider 2001), +while a more comprehensive approach to adding relativistic effects is +presented by Borzyszkowski et al. (2017). However, even though the +biases introduced by this approach are expected to be small, a fully +self-consistent approach is necessary to convincingly demonstrate +our ability of controlling theory systematics. For instance, galaxy +clustering is affected by magnification bias due to lensing, and ne- +glecting this effect induces a non-negligible bias in parameter esti- +mation (Lepori et al. 2020; Alam et al. 2021). This is even more true +when modified gravity theories are used: extensions of gravity are +typically derived in a full relativistic context, and while they influence +the Newtonian limit of gravity, the small but measurable relativistic +effects may provide smoking-gun signals of a specific class of gravity +theories. In this sense, restricting to the treatment of the Newtonian +limit of modified gravity theories (as, e.g., in Puchwein et al. 2013) +may leave out crucial observable signatures. +Two examples of fully relativistic N-body codes for the evolution of +cosmic perturbations, that integrate Einstein’s equations to follow the +motion of massive particles along their geodesics, are the Adaptive +Mesh Refinement (AMR) code Gramses (Barrera-Hinojosa & Li +2020) and the Particle-Mesh (PM) code Gevolution (Adamek et al. +2016). These have proven to be precious tools to produce accurate +cosmological predictions, like a self-consistent treatment of massive +neutrinos (Adamek et al. 2022), and to explore phenomena that were +previously overlooked, like the strength of the frame dragging field +acting on dark matter haloes (Barrera-Hinojosa et al. 2020). These +codes sample the fields in a mesh that fills the simulated volume, +but while Gramses uses an AMR scheme to increase resolution only +where it is needed, PM schemes working on a single non-adaptive +mesh are well known to be limited by memory, so they are unable to +achieve the large dynamic range required, e.g., to resolve DM halos in +large cosmological volumes. The integration of Newtonian particle +trajectories has historically been addressed with the introduction of an +oct-tree data structure (Barnes & Hut 1986), that provides a 𝑁 log 𝑁 +scaling for the computation of gravity without compromising its +accuracy. Because the integration of large-scale perturbations is very +slow in this scheme, such an oct-tree is used to compute short-range +forces, and is complemented by a Particle-Mesh (PM) code on large +scales. The resulting algorithm is commonly called TreePM, and it +is the standard gravity solver for Gadget-4. +As we will show in next Section, deviations from a pure Newtonian +approach become significant on scales that are comparable with the +Hubble horizon, so a Newtonian treatment of small-scale clustering, +performed by the Tree algorithm, would introduce a negligible error +if large scales are treated by a fully relativistic gravity solver. This +can be achieved, in a TreePM scheme, by using a relativistic PM code +for large-scale gravity, where relativistic potentials are sampled on a +small enough mesh so as to be effectively Newtonian on the scales +where the Tree code gets in. +In this paper we present an implementation of Gadget-4 that +uses a PM library, based on Gevolution relativistic code, as the +PM part of the TreePM solver. This is a step toward the construc- +tion of an ecosystem of codes and post-processing tools to perform +end-to-end simulations of future surveys, with the aim of achieving +optimal control of all systematics, including theoretical ones. The +paper is organized as follows: Section 2 gives an overview of the the- +ory of relativistic perturbations, with a focus on the approach used +in Gevolution. Section 3 gives a description of the Gadget-4 and +Gevolution codes, and describes the implementation of Libgevo- +lution and GrGadget. Section 4 presents the tests performed to +validate GrGadget, while Section 5 gives our conclusions. +2 THEORY OF RELATIVISTIC PERTURBATIONS +The success of Newtonian simulations in describing the large-scale +structure of the universe follows from the fact that, for an observer at +rest with respect to the CMB, the metric of spacetime is very close to +Friedmann-Lemaitre-Robertson-Walker’s (FLRW). Deviations from +the Newtonian approach are expected to be significant, albeit small, +on scales near the Hubble horizon, or when the energy-momentum +tensor has relativistic components like radiation or fast massive neu- +trinos. Deviations from FLRW metric are expected to be strong in +the proximity of compact objects, but this happens on scales that are +far smaller than the resolution that can be afforded in simulations of +large comoving volumes. It is thus fair to assume that the perturba- +tions to the metric are small and can be described in a weak-field +regime. This does not imply that deviations of the components of +the energy-momentum tensor from homogeneity are assumed to be +small, density perturbations can be highly non-linear: what we re- +quire is that the size of self-gravitating objects is much larger than +their gravitational radius. +The Gevolution code (see Adamek et al. 2016) models the space- +time metric with a perturbed FLRW metric in the weak field regime. +In the Poisson gauge the metric can be written as: +𝑑𝑠2 =𝑎2� +− 𝑐2 𝑑𝜏2(1 + 2Ψ) − 2𝑐 𝑑𝜏𝑑𝑥𝑖𝐵𝑖+ ++ 𝑑𝑥𝑖𝑑𝑥 𝑗 �𝛾𝑖 𝑗 (1 − 2Φ) + ℎ𝑖 𝑗 +�� +, +(1) +where 𝑎(𝜏) is the scale factor of the FLRW background, 𝜏 is the +conformal time and 𝑥𝑖 are the space coordinates. It is possible to +exploit the residual degrees of freedom of the metric to impose +the conditions 𝐵𝑖|𝑖 = 0, ℎ𝑖𝑖 = 0 and ℎ𝑖 𝑗 | 𝑗 = 0. In our notation, +repeated latin indexes denote Einstein’s summation over the spatial +coordinates 1, 2, 3 and the vertical bar subscript, e.g. 𝐵𝑖| 𝑗, denotes a +covariant derivative with respect to the affine connection that emerges +from the background spatial metric 𝛾𝑖 𝑗. +The choice of the Poisson gauge is convenient because the two +potentials Ψ and Φ are the gauge-invariant Bardeen potentials, and +in the Newtonian limit the the field Ψ can be interpreted as the +gravitational potential. In other words, this is the gauge in which the +standard N-body solver is integrating the right equations of motion +in the Newtonian limit (Chisari & Zaldarriaga 2011). +2.1 Field equations +The background, characterized by 𝑎(𝜏), is by construction a solution +of the Einstein’s equations in the presence of a homogeneous and +MNRAS 000, 1–14 (2022) + +GrGadget +3 +isotropic energy-momentum tensor ¯𝑇 𝜇𝜈: +¯𝐺𝜇𝜈 = −8𝜋𝐺 +𝑐4 +¯𝑇 𝜇𝜈 , +(2) +where ¯𝐺𝜇𝜈 is Einstein’s tensor constructed from the metric (1) with +the perturbations Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 set to zero. Applying equation (2) to +the FLRW metric one obtains Friedmann’s equations. +To solve for the perturbations of the metric, the usual procedure +consist in subtracting (2) from the full Einstein’s equations: +𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 = −8𝜋𝐺 +𝑐4 (𝑇 𝜇𝜈 − ¯𝑇 𝜇𝜈) . +(3) +The right hand side now contains the perturbation of the energy- +momentum tensor due to inhomogeneities in mass and energy dis- +tributions, while the left hand side is a very complicated non-linear +expression containing the potentials Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 and their space- +time derivatives up to second order. +To reach a tractable set of equations that we can interpret and solve +numerically, we apply the weak field assumption. The perturbations +Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 are assumed to be of order 𝜖 ≪ 1. Spatial derivatives +are known to increase their amplitude by a factor of 𝜖−1/2, accounting +for the presence of shortwave fluctuations induced by the non-linear +structure in the energy-momentum tensor, while time derivatives are +assumed to preserve the perturbation order. Then one can expand +𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 in terms of the metric perturbations, neglecting contri- +butions with order higher than 𝜖. For example: Φ is a term of order +O(𝜖), Φ,𝑖 has order O(𝜖1/2), Φ|𝑛𝑛 is a leading term (order 1, because +of the second derivative), quadratic terms like Φ,𝑛Φ,𝑛 are O(𝜖), and +a term like Φ,00 is considered as O(𝜖). This type of expansion is +known as the shortwave correction (Adamek et al. 2014). +Furthermore, experience has shown that the scalar perturbations Φ +and Ψ are generally larger than the vector and tensor perturbations 𝐵𝑖 +and ℎ𝑖 𝑗. Indeed, the scalar potentials, that are sourced by the density +perturbation Δ𝑇00, become the Newtonian potential in the Newtonian +limit, while the vector perturbation 𝐵𝑖 is sourced by Δ𝑇0𝑖, that is +small by a factor of 𝑣/𝑐 for non-relativistic matter perturbations, +and ℎ𝑖 𝑗 by Δ𝑇𝑖 𝑗, that is suppressed by a (𝑣/𝑐)2 factor. Hence, it is +fair to drop quadratic terms of 𝐵𝑖 and ℎ𝑖 𝑗 in this weak field limit +approximation. +In this approximation, from Eq. (3) it descends that its time-time +component yields a Poisson-like equation for the scalar Φ: +Φ|𝑛𝑛(1 + 4Φ) − 3H +𝑐2 Φ,0 + 3H2 +𝑐2 (𝜒 − Φ) + 3 +2Φ|𝑛Φ|𝑛 += 4𝜋𝐺𝑎2 +𝑐4 +Δ𝑇00 , +(4) +where H = 𝑎−1 𝑑𝑎 +𝑑𝜏 and 𝜒 = Φ − Ψ. From the time-space section of +eq. (3) we obtain: +− +𝐵𝑖|𝑛𝑛 +4𝑐 +− Φ,𝑖0 +𝑐2 +− H +𝑐2 (Φ,𝑖 − 𝜒,𝑖) = −4𝜋𝐺𝑎2 +𝑐4 +Δ𝑇0𝑖 , +(5) +that, taking advantage of the condition 𝐵𝑛|𝑛 = 0, can be reduced to: +− +𝐵𝑖|𝑛𝑛 +4𝑐 += −4𝜋𝐺𝑎2 +𝑐4 +𝑃⊥Δ𝑇0𝑖 , +(6) +where 𝑃⊥ is a linear operator that selects from a vector field its +divergenceless component. +The traceless part of the spatial section of eq. 3 leads to: +� +𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1 +3𝛿𝑎𝑏𝛿 𝑗𝑖 +� � +𝜒| 𝑗𝑖 − 2Φ| 𝑗𝑖 𝜒 + 4ΦΦ| 𝑗𝑖 + 2Φ| 𝑗Φ|𝑖 ++ +1 +2𝑐2 ℎ𝑖 𝑗,00 + H +𝑐2 ℎ𝑖 𝑗,0 − 1 +2 ℎ𝑖 𝑗 |𝑛𝑛 ++ 1 +2𝑐 +� 𝜕 +𝜕𝜏 + 2H +� � +𝐵𝑖 | 𝑗 + 𝐵 𝑗 |𝑖� � += +� +𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1 +3𝛿𝑎𝑏𝛿 𝑗𝑖 +� � +−8𝜋𝐺 +𝑐4 Δ𝑇𝑖 𝑗 +� +, +(7) +from which we can determine the rest of the metric degrees of free- +dom 𝜒 and ℎ𝑖 𝑗. Since the source of 𝜒 and ℎ𝑖 𝑗 are the perturbation +of the of the energy-momentum tensor Δ𝑇𝑖 𝑗, their amplitude in a +matter dominated universe is suppressed by a factor (𝑣/𝑐)2. That +is equivalent to say: since dark matter is non-relativistic, 𝜒 and ℎ𝑖 𝑗 +must be very small with respect to Φ or even 𝐵𝑖. +As a matter of fact, V1.2 of Gevolution implements an improved +expansion of the metric perturbations, that has been presented in +Adamek et al. (2017). For our tests we used the implementation +of the original expansion, the one presented above. However, the +improved expansion has been ported to Libgevolution and will be +used when analysing result on the past light cone. We do not expect +the results presented in this paper to depend on the specific expansion +used. +2.2 Motion of particles along geodesics +Massive particles move along geodesics, whose equation can be +expressed as: +𝑑𝑥𝑖 +𝑑𝜏 = +𝑐𝑝𝑖 +√︁ +(𝑚𝑐𝑎)2 + 𝑝2 + 𝑐𝐵𝑖 ++ +𝑐𝑝𝑖 +√︁ +(𝑚𝑐𝑎)2 + 𝑝2 +� +Ψ + Φ2(𝑚𝑎𝑐)2 + 𝑝2 +(𝑚𝑎𝑐)2 + 𝑝2 +� +, +(8) +𝑑𝑝𝑖 +𝑑𝜏 = − 𝑐 +� +𝑝𝑛𝐵𝑛|𝑖 + Ψ,𝑖 +√︃ +(𝑚𝑐𝑎)2 + 𝑝2 + +𝑝2Φ,𝑖 +√︁ +(𝑚𝑐𝑎)2 + 𝑝2 +� +, +(9) +where 𝑝𝑖 is the space part of the particle momentum and 𝑝 its +norm. The right hand side in the last equation is the generalized +force acting on the particles. The term proportional to Ψ,𝑖 becomes +the Newtonian force in the limit of small velocities, while 𝑝𝑛𝐵𝑛|𝑖 +represent the corrections due to frame dragging and the third term in +parenthesis is a further relativistic correction. +The energy-momentum tensor is constructed from the knowledge +of particle positions and momenta, but its computation depends on the +perturbed metric. This means that, in Eqs. (4), (6), and (7), the source +terms on the right hand sides depend on the potentials themselves. +These implicit equations may be solved starting from the potentials +at the previous time step and solving the equations iteratively until +convergence. The integration scheme that Gevolution implements +is simpler: at each time step the energy momentum tensor is computed +using the potentials from the previous step, then the Poisson equations +are solved to find the updated potentials, that will be used in the next +time step to compute the energy-momentum tensor. +The Newtonian limit is recovered when we consider Fourier modes +larger than H/𝑐 and we further neglect 𝐵𝑖 and consider Φ ≪ 1; then +equation (4) becomes: +Φ|𝑛𝑛 = 4𝜋𝐺𝑎2 +𝑐4 +Δ𝑇00 +(10) +MNRAS 000, 1–14 (2022) + +4 +E. Quintana-Miranda et al. +while (8) and (9) become: +𝑑𝑥𝑖 +𝑑𝜏 = 𝑝𝑖 +𝑚𝑎 , +(11) +𝑑𝑝𝑖 +𝑑𝜏 = −Φ,𝑖𝑚𝑐2𝑎 . +(12) +3 ALGORITHMS AND CODE INFRASTRUCTURE +3.1 Gevolution +Gevolution8 (Adamek et al. 2016) is an N-body relativistic cosmo- +logical code, written in C++ and parallelized with the MPI paradigm. +The physical theory behind this code has been described at length in +Section 2. Numerically, this code implements a PM scheme to follow +the evolution of energy-momentum tensor perturbations. As in PM +codes, the advantage of working with a single grid and using Fast +Fourier Transforms (FFTs) to solve the Poisson-like equations for the +fields is paid with a high cost in memory, of O(𝑁3) where 𝑁 is the +number of grid points per dimension. +Gevolution, can run in either Newton or General Relativity +modes. The Newtonian gravity solver inverts the Laplace operator +in the Poisson equation for the Newtonian potential, Eq. 10. When +running the General Relativity mode, the code solves Eqs. 4, 6 and +7, that require the computation of the perturbed energy-momentum +tensor. This is performed using a Cloud-In-Cell (CIC) scheme both +for the density and for particle velocities; details are given in the +presentation paper. Then the Hamiltonian forces to which particles +are subjected are computed from Eqs. 8 and 9. +Gevolution solves the field equations in Fourier space, using a +C++ library called LATfield2 to operate FFTs on classical fields in +massively parallel applications with distributed memory. LATfield2 +provides a programming interface to perform operations on the fields, +either in their real or Fourier space representations. This library im- +plements FFTs of 3-dimensional fields whose memory is distributed +among parallel processes following a 2-dimensional uniform decom- +position of space, in which each process owns in memory a portion +of the grid with a rod shape (Daverio et al. 2015). In this way LAT- +field2 overcomes the scaling limitations of a simpler 1-dimensional +domain (slab) decomposition provided by the mainstream FFTW3 +library9. FFTW3 is used, however, to compute 1D FFTs. +3.2 Gadget-4 +Gadget-410 is a state-of-the-art TreePM N-body hydrodynamical +cosmological code written in C++ (see Springel et al. 2021); it is +massively parallelized in a distributed-memory paradigm using MPI. +As in most N-body codes, gravity in Gadget-4 is represented +in the Newtonian limit, but the equations of motion are modified to +take into account the Universe expansion, obtained by integrating the +Friedmann equations separately. As mentioned above, this approach +is consistent with General Relativity in the Poisson gauge, and gives +the leading-order term of weak field expansion. This amounts to +neglecting the metric degrees of freedom 𝐵𝑖, 𝜒 and ℎ𝑖 𝑗, and is +valid on scales much smaller than the Hubble horizon. In a typical +configuration that is convenient for large cosmological volumes, the +8 https://github.com/gevolution-code +9 http://fftw.org/ +10 https://wwwmpa.mpa-garching.mpg.de/gadget4 +code solves for the forces acting on each particle, representing them +as the sum of two contributions, one due to the interactions with +nearby particles, computed with a Tree algorithm, and one due to +long-range interactions, computed with a PM algorithm.11 +The Tree algorithm works by partitioning the space into cubic +cells, called nodes; in turn, each node is recursively partitioned into +8 children nodes down to a pre-determined maximum refinement +level. A tree structure tracks the list of particles that are located +within each node. This structure is used to speed up the computation +of gravitational force on a particle: in a particle-particle integration +scheme, this force is computed by adding up a series of �𝑟 𝑚/𝑟3 terms, +one for each particle pair, but we know that the accuracy of force +evaluation does not depend strongly on the small-scale distribution +of distant particles, so in the Tree scheme the evaluation of gravity +is performed by grouping particles that belong to the same node, +under the condition that the node subtends a given aperture angle +𝜃. Particle-particle computation is then used only for the nearest +neighbours. This is equivalent to considering the leading order in a +multipole expansion of the gravity force from particles belonging to a +distant cell. While the construction of the Tree is expensive in terms +of computing time, it allows to achieve O(𝑁𝑝 log 𝑁𝑝) scaling for +the force computation, where 𝑁𝑝 is the total number of particles in +the simulation. Thus the Tree is able to compute with high accuracy +the short wavelength modes of the gravitational interaction, while +keeping the computational time low for large simulations. However, +the Tree code is slow in integrating particle motions near the initial +conditions, when the departures from homogeneity are small. This +is why it is often coupled with a PM code to speed up the first time +steps of a cosmological box. +The PM algorithm represents gravity through the gravitational po- +tential field Φ, evaluated on a Cartesian cubic mesh of fixed size. The +potential is found from the density field by solving the Poisson equa- +tion in Fourier space, while the force is computed from the gradient +of the potential, obtained with a finite differences scheme. Accord- +ing to the Nyquist-Shannon theory, this implies that the information +handled by the PM is limited to the long modes, up to the Nyquist +frequency. +To combine the forces provided by the PM and Tree codes, the +gravitational potential is split into the sum of two fields: +Φ = Φ(𝐿) + Φ(𝑆) , +(13) +where Φ(𝐿) represents long-range modes from the PM, and Φ(𝑆) +represents short-range modes from the Tree. Written in Fourier space +(tilde on top of symbols denotes a Fourier transform), the Poisson +equation reads: +˜Φ𝑘 = −4𝜋 +𝑘2 ˜𝜌𝑘 , +(14) +where 𝜌 denotes the mass density. We can split the density as a sum +of short-range and long-range terms, using Gaussian filters: +˜Φ𝑘 = −4𝜋 +𝑘2 ˜𝜌𝑘 +� +1 − exp(−𝑘2𝑟2 +𝑎) +� +− 4𝜋 +𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2 +𝑎) . +(15) +The scale 𝑟𝑎 is the one at which we split long- and short-range modes. +We can obtain Φ(𝑆) by solving the modified Poisson equation for +short modes: +˜Φ(𝑆) +𝑘 += −4𝜋 +𝑘2 ˜𝜌𝑘 +� +1 − exp(−𝑘2𝑟2 +𝑎) +� +, +(16) +11 The code can work in other configurations (a non-cosmological volume, +switching off the PM, enhancing the Tree part using multipole expansion) +that are however not relevant for this paper. +MNRAS 000, 1–14 (2022) + +GrGadget +5 +and Φ(𝐿) by solving the modified Poisson equation for long modes +˜Φ(𝐿) +𝑘 += −4𝜋 +𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2 +𝑎) . +(17) +The long-mode Poisson equation (17) is solved by the PM in +Fourier space, so the convolution with the kernel is a simple mul- +tiplication. The Tree on the other hand works in real space, hence +equation (16) has to be transformed; this can be done analytically, +yielding: +Φ(𝑆) (�𝑥) = −𝐺 +∑︁ +𝑖 +𝑚𝑖 +|�𝑥 − �𝑟𝑖| erfc +� |�𝑥 − �𝑟𝑖| +2𝑟2𝑎 +� +. +(18) +3.3 GrGadget +3.3.1 Libgevolution library +In order to have a relativistic PM code working in Gadget-4, we +developed a library that implements both the Newtonian and the rel- +ativistic PM algorithms of the monolithic Gevolution code. This +was done by forking the Gevolution github repository into Libgevo- +lution, a library that is publicly available on github12 under MIT +license. +The rationale behind the development of Libgevolution is to +encapsulate Gevolution’s resources and methods into abstract ob- +jects. This yields several benefits. Firstly, Gevolution maintenance +is eased by the logical modularization of the code, i.e. instead of a +monolitic code with a unique workflow we can divide Gevolution +into components (C++ classes and/or namespaces) with well defined +purposes. Secondly, we are allowed to re-use Gevolution compo- +nents within other applications, such as we do within Gadget-4 in +the present paper. +We give here an overview of the library; the precise signature of +all the defined functions, methods and data structures is described +in the technical documentation of the code. Libgevolution is based +on three cornerstones: (i) a particle container implemented through +the class Particles_gevolution; (ii) a PM data structure named +particle_mesh, templated on the particle container type, that can +be used either as a relativistic_pm or a newtonian_pm; (iii) an +executable application that uses the previous components to produce +N-body simulations as the original code does. particle_mesh has +to be understood as a container that is aware of the parallelization of +the tasks and distribution of memory; it holds the gravitational fields +and it allows the user to compute the forces acting on the simulation +particles. The user interface declared in particle_mesh consists of +the following functions: +• sample(...), that builds the sources (density field or energy- +momentum tensor) by sampling particle properties in the mesh; +• compute_potential(...), that solves Poisson equations to +compute the potential fields; +• compute_forces(...), that computes the forces acting on +particles. +particle_mesh is specialized to solve the Newtonian problem +or the General Relativistic problem using class inheritance; Figure 1 +illustrates the class hierarchy of Libgevolution’s particle_mesh. +The expert user will be able to specialize particle_mesh to his/her +own needs, for example by deriving a PM that solves a modified +gravity problem. +newtonian_pm is the specialization of particle_mesh that +12 https://github.com/GrGadget/gevolution-1.2 +particle_mesh +relativistic_pm +newtonian_pm +Figure 1. PM class hierarchy in Libgevolution. +contains a real LATfield2::Field scalar field ΦNewton and its +complex LATField2::Field Fourier transform +˜ΦNewton, plus +a LATField2::PlanFFT that connects ΦNewton with +˜ΦNewton +through discrete Fourier transform. relativistic_pm is the spe- +cialization of particle_mesh that contains the above quoted de- +grees of freedom of the perturbed FLRW metric, Φ, 𝐵𝑖 and 𝜒. +These are represented as real LATfield2::Field, with complex +LATField2::Field counterparts to represent their Fourier trans- +forms and a LATField2::PlanFFT for each field. +As a first testing phase, we run Libgevolution, called with a sim- +ple wrapper, and the native Gevolution code, applying them to the +same set of initial conditions, checking that the results were identical +both in the Newtonian and relativistic cases. Then we stripped down +Gadget-4 by switching off the Tree code, and compared its results +to the Newtonian results of Libgevolution. It is necessary that this +comparison gives nearly identical results if we want Libgevolution +to substitute the native PM code of Gadget-4 without loss of accu- +racy. To achieve a satisfactory match of the two PM codes we had to +change the Gevolution scheme in a few points. +We started from V1.2 of Gevolution, that implemented a first- +order version of finite differences instead of the fourth-order scheme +of Gadget-4. This resulted in a difference with Gadget-4 run on +the same initial conditions, and in a percent-level offset of the matter +power spectrum on large scales at low redshift. We upgraded the +computation of spatial derivatives to fourth order, in parallel with +the Gevolution developers that had noticed the same problem; our +implementation is equivalent the most recent issue of Gevolution +(used, e.g., in Adamek et al. 2022). The upgrade is the following: let’s +consider the gravitational potential along one direction of the mesh, +and let’s call its values Φ𝑖, where the index𝑖 denotes its position along +that direction. Its first derivative is computed with finite differences +at the first order as: +𝜕Φ𝑖 +𝜕𝑥 = Φ𝑖+1 − Φ𝑖 +ℎ ++ O(ℎ), +(19) +where ℎ is the size of the mesh cell. Fourth-order Taylor expansion +gives: +𝜕Φ𝑖 +𝜕𝑥 = 8Φ𝑖+1 − Φ𝑖−1 +12ℎ +− Φ𝑖+2 − Φ𝑖−2 +12ℎ ++ O(ℎ4) . +(20) +This has a smaller error of order O(ℎ4), so it achieves higher pre- +cision than (19) with the little cost of knowing the potential value +at the second-nearest cell, that implies a negligible communication +overhead. +Another improvement with respect to V1.2 of Gevolution, that +follows an implementation of Gadget-4, was the application of cor- +recting filters to the density in Fourier space to compensate for cloud- +in-cell (CIC) interpolation. Indeed, as discussed e.g. in Springel +(2005) or Sefusatti et al. (2016), CIC interpolation at some finite or- +der leads to some loss of power that can be compensated for in Fourier +space using suitable kernels. This was applied both to the compu- +tation of the density and to the computation of energy-momentum +tensor components in the relativistic case. +Lastly, to make the Newtonian PM scheme equivalent to that of +MNRAS 000, 1–14 (2022) + +6 +E. Quintana-Miranda et al. +Gadget-4 we changed the form of the discrete Laplacian operator in +the Poisson equation solver from its original form +∇2 → −4𝑁2 +𝐿2 +� +sin2 𝜋𝑘𝑥 +𝑁 ++ sin2 𝜋𝑘𝑦 +𝑁 ++ sin2 𝜋𝑘𝑧 +𝑁 +� +, +(21) +described in Adamek et al. (2016), equation (C.5), to the form used +in Gadget-4: +∇2 → −4𝜋2 +𝐿2 +� +𝑘2 +𝑥 + 𝑘2 +𝑦 + 𝑘2 +𝑧 +� +. +(22) +3.3.2 Calling Libgevolution from Gadget-4 +The implementation of Libgevolution in Gadget-4 was performed +as follows. We created a new PM class with a similar interface as +the original one in Gadget-4, so that it is initialized and executed +with the same functions as Gadget-4, i.e. init_periodic() and +pmforce_periodic(). A new class relativistic_pm was imple- +mented within an gadget::gevolution_api namespace, avoiding +to use the wider gadget namespace to make a clear distinction of +purpose between the original Gadget-4 code and our additional fea- +tures. This relativistic_pm class acts much like a mediator taking +information in and out of gadget simulation particles, processing +the correct units conversion and calling the methods on gevolution +namespace. Figure 2 shows a diagram that summarizes the contents +of this PM class, its relation with Gadget-4’s resources and the entry +points for gevolution’s api. +relativistic_pm consists of: +• A variable of type simparticle_handler that acts as +a wrapper for providing particle information from Gadget-4’s +simparticles global variable and writing back the data produced +by gevolution’s PM. +• A variable of type latfield_handler that takes care of cor- +rectly initializing LATfield global state. Indeed, while Gadget-4 can +run with any number of MPI processes, LATfield has limitations that +depend on the number of grid points in the PM. latfield_handler +also takes care of creating a sub-communicator from Gadget-4’s MPI +global communicator that satisfies the constraints set by LATfield. +• A variable of type gevolution::cosmology that contains the +parameters for the background evolution. +• A container of type gevolution::Particles_gevolution +that holds particle information, stored according to their location on +the PM grid. +• Variables of type gevolution::relativistic_pm and +gevolution::newtonian_pm that perform the actual PM compu- +tations, i.e. construct the sources, either density or the components +of the energy-momentum tensor, compute the gravitational potential +or the metric perturbation fields and the forces that act upon the +particles. +• The methods pm_init_periodic and pmforce_periodic, +for initialization and execution of the PM, respectively. +3.3.3 Kick and drift operators +In order to keep the Hamiltonian character of the equations of motion +in Gadget-4, we have to describe the state of each particle through its +position and momentum, not velocity. Following a leap-frog scheme, +the momentum should be updated with a kick operation using the full +relativistic Eqs. (8) and (9). However, velocities in Gadget-4 are to be +interpreted as momenta (per unit mass) of non-relativistic particles in +the Newtonian limit. Then we redefine the Gadget-4 kick and drift +operators assuming non-relativistic matter, 𝑝 ≪ 𝑚𝑐𝑎, and further +neglecting the very small contribution coming from 𝜒: +𝑑𝑥𝑖 +𝑑𝜏 = 𝑝𝑖 +𝑚𝑎 (1 + 3Φ) + 𝑐𝐵𝑖 , +(23) +𝑑𝑝𝑖 +𝑑𝜏 = − 𝑐𝑝𝑛𝐵𝑛|𝑖 − Φ,𝑖𝑚𝑐2𝑎 . +(24) +The right hand side of (24) is what we call force. +3.3.4 Adding long-range and short-range forces +To combine the forces computed with the relativistic PM and +Gadget-4’s Newtonian Tree we have extended the idea of the TreePM +coupling. From equation (13) one obtains that the force acting on a +particle in a TreePM scheme consists of two terms: +�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree +Newton] + 𝐿𝑟𝑎 [ �𝐹PM +Newton]. +(25) +The first term is the force computed using the Tree on which an +exponential high-pass filter 𝑆𝑟𝑎 is applied, leaving short-wavelength +modes. The second term corresponds to the PM force on which +the complementary low-pass filter 𝐿𝑟𝑎 is applied to leave long- +wavelength modes. The symbols 𝑆𝑎 and 𝐿𝑎 formally denote these +linear operators: +𝑆𝑟𝑎 [ 𝑓 ](�𝑟) = 1 +𝑁 +∑︁ +�𝑘 +˜𝑓�𝑘 (1 − exp(−𝑘2𝑟𝑎2)) exp(−𝑖�𝑘 · �𝑟) , +(26) +and +𝐿𝑟𝑎 [ 𝑓 ](�𝑟) = 1 +𝑁 +∑︁ +�𝑘 +˜𝑓�𝑘 exp(−𝑘2𝑟𝑎2) exp(−𝑖�𝑘 · �𝑟) . +(27) +The grid smoothing scale 𝑟𝑎 scales with the PM mesh size, and +its value is optimized in Gadget-4, in a way that will be tested +below, to minimize the impact of the two different treatments of the +gravitational force. +In order to account for the relativistic dynamics while preserving +the match between tree and PM contributions that is valid in the New- +tonian case, we choose the following strategy: Gadget-4 calls both +newtonian_pm and relativistic_pm, the Newtonian value of the +force is added to the Tree force as in a standard Newtonian simula- +tion, while the difference between the Newtonian and the relativistic +forces is added on top as a correction, but filtered on a different scale +𝑟𝑏, that we call gr-smoothing scale. Eq. (25) then becomes: +�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree +Newton] + 𝐿𝑟𝑎 [ �𝐹PM +Newton] + 𝐿𝑟𝑏 [ �𝐹PM +GR − �𝐹PM +Newton] . +(28) +The case 𝑟𝑎 = 𝑟𝑏 would correspond to simply adding the relativistic +force to the Tree: +�𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree +Newton] + 𝐿𝑟𝑏 [ �𝐹PM +GR ] . +(29) +However, while the size of 𝑟𝑎, that regulates the match between +Newtonian Tree and PM forces, is very well tested within Gadget-4, +the optimal value of 𝑟𝑏 is to be found; we will show in the next +Section that using 𝑟𝑏 larger than 𝑟𝑎 allows us to achieve percent +accuracy at small scales. +4 VALIDATION +The GrGadget code has been validated by running it on a few real- +izations of initial conditions, listed in table 1. These were generated +MNRAS 000, 1–14 (2022) + +GrGadget +7 +gadget:: +simparticles +LATfield2:: +parallel +particle_handler +read/write +latfield_handler +read/write +sim::begrun1() +sim::gravity_long_range_force() +init_periodic() +pmforce_periodic() +execute +execute +gevolution::cosmology +gevolution::Particles_gevolution +gevolution::relativistic_pm +gevolution::newtonian_pm +gadget::gevolution_api::relativistic_pm:: +Figure 2. Diagram of resource ownership and relations for Libgevolution integrated into Gadget-4’s workflow. Each solid box represent a memory resource +(an instantiation of a variable type) while the dashed boxes indicate ownership. The newly developed code, represented in the right part of the diagram denoted +with the namespace gadget::gevolution_api, consists in a class named relativistic_pm that owns a particle_handler object that reads and writes +directly into gadget::simparticles, a latfield_handler that takes care of setting up and inspect the state of LATfield2::parallel, and some types +defined in Libgevolution, that are defined in gevolution namespace, like cosmology, Particles_gevolution and relativistic_pm. The methods +sim::begrun1() and sim::gravity_long_range_force() in gadget:: interact with the relativistic_pm through their interface init_periodic() +and pmforce_periodic(). +name +𝑁𝑝 (particles) +𝑁 (PM grid points) +𝐿 (box size) +N64 +643 +64 +1 Gpc/ℎ +N256 +2563 +256 +1 Gpc/ℎ +high_res +5123 +512 +500 Mpc/ℎ +Table 1. Cosmological simulation configurations used to validate GrGadget. +with Gadget-4’s ngenic code at 𝑧 = 19, starting from a linear power +spectrum generated with CAMB13 and with cosmological parame- +ters consistent with Planck 2018 result (Planck Collaboration et al. +2020): Ω𝑏ℎ2 = 0.0223, Ω𝑐ℎ2 = 0.120, 𝐻0 = 67.3 km s−1 Mpc−1, +𝐴𝑠 = 2.097 × 10−9 and 𝑛𝑠 = 0.965. +4.1 Gevolution and Gadget-4 original codes +As already discussed in Section 3.3.1, the newtonian_pm imple- +mentation in V1.2 of Gevolution computes the Newtonian forces +differently from those obtained with Gadget-4’s PM. Before imple- +menting Libgevolution as the PM engine of Gadget-4, we need to +make the two algorithms work in the same way. +To this aim, we have run a set of simulations with the configuration +N64 (described in table 1) with a small number of particles 𝑁𝑝 = 643 +to be able to compute forces using a straightforward particle-particle +(PP) scheme, that can be taken as the true force that we are trying to +approximate. The same initial conditions at 𝑧 = 19 have been fed to +both Gadget-4 (with Tree either on or switched off to have a pure PM +run) and Gevolution (in Newtonian mode) codes. At later times, +𝑧 = 8 and 𝑧 = 0, we have written snapshots of the forces that the +simulation particles experience, separating the PM and the TreePM +components; we have then compared those to the true Newtonian +force computed with the PP scheme. The data we have obtained +are summarized in the plots shown in figure 3. We have binned +particles according to the value of the true force, then for each bin +we have computed the mean (colored lines) and standard deviation +13 https://camb.info/ +(shaded regions) of the difference between the force computed with +approximate methods (PM or TreePM) and the true value. Forces +are given in Gadget-4’s default units, which is actually acceleration, +measured in units of 10𝐻0 km/s = ℎ km2 s−2 kpc−1. The green +line shows the PM result using the original Gevolution code (the +true force is anyway computed with Gadget-4 and matched particle +by particle) while the red line is obtained from a pure PM using +Gadget-4’s original code. The black line gives the TreePM method +precision, obtained using Gadget-4. +Looking at the red and green lines (and their shaded areas) we find +two known results. Firstly, the TreePM method produces far less bias +and dispersion when estimating forces; for instance, in the left panel +of Fig. 3 the error is of the order14 of 0.1 ℎ km2 s−2 kpc−1, while in +the right panel it is larger but barely visible when compared with the +other curves. Secondly, while the PM force has low bias but a much +larger variance than the TreePM one at high redshift, at low redshift, +i.e. at higher level of non-linearity, it underestimates the value of the +Newtonian force as its magnitude increases. This underestimation is +due to the failure of PM in resolving interaction at scales smaller +than the grid resolution. +When comparing Gevolution PM and true forces, we notice an +S-shaped feature in the plot, much more visible at high redshift. +As anticipated in Section 3.3.1, this is mostly due to the first-order +interpolation used to find the gradient of the potential in the code +version that we tested. +In Fig. 4 we show the matter power spectra15 obtained at 𝑧 = 0 +from a set of larger simulations with the configuration N256 (see +table 1). The red solid line shows the result obtained with the orig- +inal Gadget-4 code with its TreePM method, while the red dotted +line shows the results obtained by switching off the Tree so that the +14 This quantification is in code units, we can take this value as a reference +for a high accuracy gravity solver. +15 In this paper all particle power spectra were computed using PowerI4 +code presented in Sefusatti et al. (2016). Unless otherwise stated, all power +spectra are computed up to the the Nyquist frequency of the PM mesh. +MNRAS 000, 1–14 (2022) + +8 +E. Quintana-Miranda et al. +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +Force (direct summation) +2 +1 +0 +1 +2 +Force (estimated) - Force (direct summation) +Force Test (z=8) +TreePM (Gadget) +Gevolution PM +Gadget PM +100 +75 +50 +25 +0 +25 +50 +75 +100 +Force (direct summation) +100 +50 +0 +50 +100 +Force (estimated) - Force (direct summation) +Force Test (z=0) +TreePM (Gadget) +Gevolution PM +Gadget PM +Figure 3. Difference of gravity force with respect to the true PP value, binned according to the true force, for N64 initial conditions, at 𝑧 = 8 (left panel) and +𝑧 = 0 (right panel). Lines represent the mean value of force difference in the bins, with colours explained in the legend; the shaded regions give the standard +deviation of the corresponding force difference. +forces are computed using the PM alone. The green lines show re- +sults obtained with the latest develop version of Gevolution that +implements higher order schemes for finite differences; the dotted +line gives results obtained with GRADIENT_ORDER=1 and is identi- +cal to the result obtained with V1.2 of Gevolution, the green solid +line uses GRADIENT_ORDER=2, that corresponds to a second-order +scheme. These power spectra show that the matter distribution in +Gevolution using first-order gradients loses power in what seems +to be a uniform trend for large-scale modes. This is a behaviour +which is not inherent to the PM nature of the code, since that type of +numerical approximation should predict very well the linear evolu- +tion at large scales; indeed, the higher-order scheme recovers power +on large scales to sub-percent accuracy. Conversely, Gadget-4’s PM +and TreePM agree very well at wavenumbers below 𝑘 ∼ 0.1ℎ/Mpc +scale, +The higher-order differentiation worsens the loss of power of +Gevolution for high values of 𝑘, that is not present in Gadget-4. +This can be explained as a consequence of the particle-to-mesh sam- +pling and mesh-to-particle interpolation described in section 3.3.1. +As discussed there, Gadget-4’s PM corrects for these effects, result- +ing in a power spectrum that degrades only at very high values of 𝑘 as +we approach the Nyquist frequency, while producing a ∼ 2 percent +overcorrection at 𝑘 ∼ 0.4 ℎ/Mpc. +After implementing the higher-order differentiation scheme, the +correction for the loss of power discussed above and the change in +the discrete Laplacian operator (Section 3.3.1), the results of native +Gadget-4 and Libgevolution PMs become indistinguishable. +4.2 Newtonian forces +We have tested our implementation of the GrGadget code by run- +ning a standard test in Gadget-4: we create an N-body configuration +in which there is a single massive particle in the entire simulation +box, while other massless test particles are placed at different dis- +tances from the first. In this setting the exact value of the force on +each particle is known, hence one can compare the numerical results +coming from the TreePM algorithm to the analytical solution. +The results are shown in figure 5, where each dot represents a +test particle. The x-axis gives the distance to the massive particle +that sources the gravitational field, in units of the PM resolution +(𝐿/𝑁), while the y-axis gives the corresponding absolute value of +the relative difference of the true and estimated forces acting on the +test particle. The red and blue lines correspond to the mean value of +force residuals, for particles binned into distance bins; the red line +denotes the statistics obtained from a simulation using Gadget-4’s +original TreePM implementation and the blue line was produced +using GrGadget, in this case with the Newtonian gravity engine. +This figure shows that the accuracy with which the TreePM code +reproduces the gravitational force is at worst at percent level on scales +of a few mesh cells, corresponding to the scale where the PM and Tree +contributions are matched, and gets very accurate in the limits where +either the Tree (small scales) or the PM (large scales) dominates. +Gadget-4’s and GrGadget’s Newtonian PMs show basically the +same accuracy, even though their PM implementations are very dif- +ferent. +In Fig. 6 we show the matter power spectra of a set of N256 +simulations (see table 1). In this case we are comparing the matter +clustering of GrGadget, in blue (with Newtonian forces for testing +purposes), against Gadget-4, in red. In agreement with the previous +test of force differences, we find that both codes produce the same +matter power spectrum up to floating point errors. This is verified +both in the case of simulations computing forces using a pure PM +and in the case of TreePM. +4.3 Relativistic simulations with GrGadget. +We present here results obtained by running GrGadget with +relativistic_pm, comparing them with the corresponding rela- +MNRAS 000, 1–14 (2022) + +GrGadget +9 +102 +103 +104 +P(k)/(Mpc3h−3) +camb +Gevolution Newton PM +Gevolution Newton PM (2nd order) +Gadget4 PM +Gadget4 TreePM +10−2 +10−1 +100 +k/(hMpc−1) +−0.10 +−0.05 +0.00 +0.05 +0.10 +(P(k) − P(k)Gadget4)/P(k)Gadget4 +Matter power spectrum z = 0.00 +Figure 4. Matter power spectrum of N256 cosmological simulations. The +lower panel shows residuals with respect to Gadget-4’s original code (in red), +used as baseline. The black line shows the linear power spectrum obtained with +CAMB. Red lines show results obtained with Gadget-4, with the Tree part +on (solid line) or switched off (dotted line). Green lines show results obtained +with Gevolution in Newtonian configuration, with finite differences at first +order (dotted line) or second order (solid line). +Figure 5. Forces due to a point source: the points are test particles located at +different distances (in units of the mesh resolution 𝐿/𝑁 ) from the source and +the lines represent the RMS of the difference between real and TreePM forces +in different distance bins. The red line corresponds to Gadget-4 original +TreePM while the blue line was obtained with GrGadget in Newtonian +mode. As for the the grid smoothing scale, the default value was used: 𝑟𝑎 = +1.25𝐿/𝑁 . For this test we have used 𝑁 = 256 and 𝐿 = 1 Gpc/ℎ. +102 +103 +104 +P(k)/(Mpc3h−3) +camb +GrGadget (Newton) PM +GrGadget (Newton) TreePM +Gadget4 PM +Gadget4 TreePM +10−2 +10−1 +100 +k/(hMpc−1) +−0.10 +−0.05 +0.00 +0.05 +0.10 +(P(k) − P(k)Gadget4)/P(k)Gadget4 +Matter power spectrum z = 0.00 +Figure 6. Matter power spectrum of four simulations starting from the same +initial conditions high_res: blue lines give results for Gadget-4 original +code, red lines give results for GrGadget. In both cases dotted lines refer to +runs with PM-only, solid lines refer to runs with full TreePM. +tivistic version of Gevolution. We expect that the power spectrum +of the matter density displays some relativistic features at large scales +due to terms preceded by H in the field equation (4), while at small +scales results should be compatible with Gadget-4’s Newtonian sim- +ulations. However, the matter power spectrum shown here is not an +observable quantity, so this comparison is just meant to give a first +validation of the results. A more thorough comparison of observ- +ables reconstructed on the past light cone will be presented in a +future paper. +Figure 7 shows the matter power spectra for a series of N256 sim- +ulations (see table 1). In this case Gevolution and GrGadget are +run in GR mode. The parameter that regulates the scale of the rela- +tivistic correction (Eqs. 28 and 29) is set to 𝑟𝑏 = 6 𝐿/𝑁 ≈ 23Mpc/ℎ, +i.e. the relativistic corrections of the PM method are smoothed at a +distances below 6 grid cells. The plot shows that relativistic PM-only +simulations, GrGadget (blue dotted line) and Gevolution (green +lines) are compatible on large scales (𝑘 < 0.03ℎ/Mpc) up to a small +percent-level difference that it is likely caused by the use of differ- +ent orders for finite difference gradient; indeed, going from first- to +second-order differences (from dotted to solid green line) the power +spectrum gets nearer to GrGadget’s fourth-order one. The plot also +confirms that our combination of Tree and PM forces in the relativis- +tic weak field limit with GrGadget (blue solid line) reproduces the +Newtonian non-linear features to sub-percent level at small scales, +that is for 𝑘 > 0.1ℎ/Mpc; here Gadget-4 (red solid line) is again our +reference for the non-linear clustering. +Being designed for the use of Fourier methods from the beginning, +Libgevolution offers an interface for the computation of the power +spectrum of the fields defined through the library’s interface. Thus +we can also extract and analyse the power spectra of the individual +components of the metric perturbations from the relativistic sim- +ulations. Figures 8 and 9 show the power spectra of the relativistic +potentials, Φ, 𝐵𝑖 and 𝜒, for a high resolution configuration high_res +(see table 1). These plots show a comparison of PM (blue lines) and +TreePM (red lines) simulations. The power spectrum of the gravita- +MNRAS 000, 1–14 (2022) + +Force Test +0.0200 +Gadget TreePM +GrGadget (Newton) TreePM +0.0175 +0.0150 +0.0125 +0.0100 +0.0075 +0.0050 +0.0025 +0.0000 +100 +101 +distance*N/L10 +E. Quintana-Miranda et al. +102 +103 +104 +P(k)/(Mpc3h−3) +camb +GrGadget TreePM +GrGadget PM +Gevolution Gr PM +Gevolution Gr PM (2nd order) +Gadget4 PM +Gadget4 TreePM +10−2 +10−1 +100 +k/(hMpc−1) +−0.10 +−0.05 +0.00 +0.05 +0.10 +(P(k) − P(k)Gadget4)/P(k)Gadget4 +Matter power spectrum z = 0.00 +Figure 7. Matter power spectrum of Gadget-4, Gevolution and GrGadget +runs, the last code being run in relativistic mode. The upper panel shows +the absolute value and the lower panel the relative difference with respect to +Gadget-4’s TreePM. The black line gives the linear matter power spectrum; +red and blue lines give Gadget-4 and GrGadget results, with full TreePM +forces (solid lines) or with the Tree switched off (dotted lines). Green lines +give Gevolution results, dotted line referring to first-order finite differences +(GRADIENT_ORDER=1) and solid line referring to second-order calculation +(GRADIENT_ORDER=2). +tional potentials converge for both methods on large scales. However, +below 1 Mpc/ℎ the PM-only simulation loses power with respect to +the TreePM one; the differences can reach up to 40% as we approach +the Nyquist frequency. This pattern is equally found for the scalar +fields Φ and 𝜒, as well as for the individual components of 𝐵𝑖. +The right plot in Fig. 8 helps to understand the reason behind +this result. Generally speaking, energy density, momentum density +and their respective density current (the components of the Energy- +Momentum tensor) are sources of the metric perturbations. Even +though those quantities, as fields, are found at discrete positions of +space defined by the mesh, their values are computed by sampling +the energy and momentum carried by the particle distribution, which +contain information on the clustering due to the short range inter- +actions (through the Tree) that goes well below the mesh resolution +𝐿/𝑁. Therefore, TreePM simulations, having power on scales well +smaller than the PM mesh, give a better representation of the source +of metric perturbation, and thus allow to recover power at frequency +modes right below Nyquist. Fig. 8 highlights the particular case of +𝑇00 (the matter density) as a source for Φ; by comparing 𝑇0 +0 with +𝑘2Φ, we are verifying the Poisson equation 𝑘2 ˜Φ ≈ ˜𝑇00 that is valid +for wavelengths below the Hubble horizon. This confirms that the +presence of small-scale clustering in the particle distribution prop- +agates to the gravitational fields up to the maximum resolution that +the PM allows. The same thing is visible in the vector modes 𝐵𝑖 and +in 𝜒 (Figure 9), where we also notice a small, few-percent mismatch +on large scales. These fields are known to give sub-percent effects on +observables, so this difference, that is likely due to some degree of +numerical mode coupling, is non considered as a problem. +In figure 10 we show how the matter power spectrum obtained +using GrGadget is affected by the choice of the gr-smoothing scale +parameter 𝑟𝑏. We have used an N256 box configuration to perform +this test, and tested values of 𝑟𝑏 = 1.5, 3, 6 in units of 𝐿/𝑁 ≈ +4 Mpc/ℎ. We find that large-scales power is independent of the value +of 𝑟𝑏 parameter; structures one scales below the PM resolution are +resolved by the Tree algorithm, hence for 𝑘 > 𝑘Nyquist there is +a convergence of all simulations to a common non-linear power +spectrum tail. It is in the medium to small scales 𝑘Nyquist > 𝑘 > +0.2 Mpc−1ℎ that we notice differences in the power spectrum above +the ∼ 1% (dashed grey line). For small values of 𝑟𝑏 (∼ 1.5 𝐿/𝑁), we +obtain discrepancies in the power spectrum at 𝑘 ∼ 0.5 Mpc−1ℎ that +can be as large as 5 percent and indicate the limitations of our force +summation scheme, Eq. (28). A value of 𝑟𝑏 = 3 𝐿/𝑁 or possibly +higher is needed to obtain a good compatibility of GrGadget and +Gadget-4 for all modes greater than 0.1 Mpc−1ℎ, where relativistic +features in the matter clustering is negligible. +The last test we present here regards the convergence of the nu- +merical results for increasing resolution. Figure 11 shows the matter +power spectrum obtained from running Gadget-4’s TreePM (red +lines), GrGadget with PM-only (blue dotted lines) and GrGadget +with TreePM (blue continuous line). These various code configu- +rations were run with different combinations of the number of grid +points per dimension 𝑁 = 256, 𝑁 = 512 and box length 𝐿 = 250, 500, +1000, 2000 Mpc/ℎ; the number of particles was fixed as 𝑁𝑝 = 𝑁3. In +all cases we have set the PM smoothing scale to 𝑟𝑎 = 1.5 𝐿/𝑁 and the +gr-smoothing scale to 𝑟𝑏 = 3 𝐿/𝑁. It can be observed with the finest +resolution, in the top plots, that there is a matching between General +Relativity and Newtonian dynamics in the small scales. Then as the +mesh size becomes coarser, in the middle plots, some discrepancies +in the power spectrum start to appear which become more evident +for even coarser meshes, in the bottom plots. This mismatch may +be caused by 𝑟𝑏 = 3 𝐿/𝑁 moving towards larger scales, so that the +assumption that PM forces are Newtonian on the small scales breaks. +Indeed, while with 𝐿/𝑁 = 1 ℎ−1 Mpc (𝑟𝑏 = 3 ℎ−1 Mpc) the scales +where relativistic effects become evident in the matter power spec- +trum and the scales where the pure PM prediction starts to deviate +from TreePM are well separated, for larger 𝐿/𝑁 values the two scales +get nearer, indicating that the assumption of pure Newtonian forces +on the mesh scale may not be very good. This conclusion is appar- +ently at variance with the discussion of Figure 10, where a larger +value of 𝑟𝑏 was preferred; however, that figure refers to 𝐿/𝑁 = 1 +and is shown at 𝑧 = 0.5, where clustering is a bit weaker. swe thus +recommend to work with mesh sizes of 𝐿/𝑁 ∼ 1 Mpc/ℎ. +5 CONCLUSIONS +We have constructed a relativistic TreePM code, that we call Gr- +Gadget, where the large-scale contribution to the gravitational force +is computed using the relativistic C++ PM library Libgevolution, +based on Gevolution code, while gravity coming from small scales +is computed by the Tree code of Gadget-4. The code works under +the assumption that, in the context of cosmological simulations, dark +matter can be treated non-relativistically and then the equations of +motion of tracer particles tend to the Newtonian limit at scales well +below the Hubble horizon. Following the Gevolution approach, we +use a weak field approximation of GR, where the perturbations of the +space-time metric with respect to FLRW background are encoded as +fields and simulated by the PM. Comparing the matter power spec- +trum from GrGadget simulations with that of original Gadget-4 +and Gevolution codes, we conclude that the code produces consis- +tent results as long as the PM cell size 𝐿/𝑁 is smaller than 2 Mpc/ℎ +and the gr-smoothing parameter is 𝑟𝑏 ≈ 3 𝐿/𝑁. +MNRAS 000, 1–14 (2022) + +GrGadget +11 +10 +30 +10 +28 +10 +26 +10 +24 +10 +22 +10 +20 +10 +18 +Pk + TreePM + PM +10 +2 +10 +1 +100 +k/(h Mpc +1) +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +(Pk +Pref)/Pref +101 +102 +103 +104 +Pk +k2 + TreePM +k2 + PM +T00 TreePM +10 +2 +10 +1 +100 +k/(h Mpc +1) +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +(Pk +Pref)/Pref +Figure 8. In the left plot: power spectrum of the metric perturbation Φ in a high_res simulation obtained with GrGadget. In the right plot: power spectrum +of 𝑘2Φ and 𝑇 00. For modes well below the Hubble horizon and small perturbations it should be verified that 𝑘2 ˜Φ ≈ ˜𝑇 00. +10 +2 +10 +1 +100 +10 +29 +10 +27 +10 +25 +10 +23 +10 +21 +10 +19 +Pk +B0 TreePM +B0 PM +10 +2 +10 +1 +100 +k/(h Mpc +1) +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +(Pk +Pref)/Pref +10 +39 +10 +37 +10 +35 +10 +33 +10 +31 +10 +29 +10 +27 +Pk + TreePM + PM +10 +2 +10 +1 +100 +k/(h Mpc +1) +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +(Pk +Pref)/Pref +Figure 9. In the left plot: power spectrum of the metric perturbation 𝐵𝑖 (the 𝑥 component) in a high_res simulation obtained with GrGadget. In the right +plot: power spectrum of 𝜒. +With respect to the pure PM implementation of Gevolution, +the predictive power of GrGadget gives an improvement even on +the scales sampled by the mesh. This is due to the fact that the +energy-momentum tensor, that sources the equations of the fields +that represent the perturbations of the metric, is computed from a +fully non-linear distribution of particles, with gravity being resolved +down to a much smaller softening length and not down to the mesh +size. This may be very useful, e.g., when assessing the possibility of +detecting the frame-dragging effect of a rotating dark-matter halo, if +not of a spiral galaxy (Bruni et al. 2014). Furthermore, this code is a +development of the widely used Gadget-4 code, and because the PM +sector of the code is called only by the computation of the gravity +force, our code can be easily extended to simulations of galaxies or +galaxy clusters by switching on the hydrodynamics, star formation +and feedback sectors. All the physics described by these sectors can +safely be treated in the Newtownian limit; one should in principle +MNRAS 000, 1–14 (2022) + +12 +E. Quintana-Miranda et al. +102 +103 +104 +P(k)/(Mpc3h−3) +camb +GrGadget TreePM (rb=6.0) +GrGadget TreePM (rb=3.0) +GrGadget TreePM (rb=1.5) +Gadget4 TreePM +10−2 +10−1 +100 +k/(hMpc−1) +−0.10 +−0.05 +0.00 +0.05 +0.10 +(P(k) − P(k)Gadget4)/P(k)Gadget4 +Nyquist freq. +Matter power spectrum z = 0.50 +Figure 10. Power spectrum of matter density for Gadget-4 and GrGadget, +on a N256 simulation configuration. The upper panel shows the absolute value +and the lower panel the relative difference with respect to Gadget-4’s TreePM. +Different shades of blue indicate different values of the gr-smoothing scale +parameter 𝑟𝑏 = 1.5, 3, 6 in units of 𝐿/𝑁 . The PM smoothing scale is 𝑟𝑎 = +1.5 𝐿/𝑁 . The power spectra in this plot are computed beyond the Nyquist +frequency to show the convergence of the matter distribution correlations for +distances below the grid resolution, the Tree regime. +add thermal energy of gas particles to the energy-momentum tensor, +but while this extension is straightforward, it is likely to provide a +negligible contribution. +This is, for our group, a further step in the construction of an +ecosystem of simulation codes and post-processing tools for model- +ing the evolution of structure in the Universe, with the aim of making +predictions for precision cosmology. Sub-percent accuracy in cos- +mological predictions, that matches the smallness of the statistical +error that will be obtained with forthcoming galaxy surveys men- +tioned in the Introduction, can only be obtained taking into account +relativistic effect (e.g. Lepori et al. 2020), and we can foresee that +a self-consistent treatment of these effects (to within the required +accuracy) will soon become the standard in cosmological simula- +tions. These effects can also be added by post-processing Newtonian +simulations, but a validation of these procedures requires validation +against a more self-consistent approach. Conversely, a large com- +munity is developing Gevolution in the direction of adding modi- +fications of gravity, whose formulation is typically worked out in a +general relativistic context. This line of development, coupled with +a Newtonian treatment of modified gravity in the Tree code, would +be precious in the formulation of tests of gravity, because relativistic +effects may hide smoking-gun features of specific classes of modified +gravity theories. +APPENDIX A: CODE SCALING +The code we presented in this work is the merging of two codes +whose behaviour in terms of run-time scaling is well-known and +characterized; since we did not modify the underlying algorithms, it +is expected that the run-time scaling of our code follows that of the +parent codes. +However, the Libgevolution’s PM is obviously different from +Gadget-4’s, and we added the translation of particles data from the +host code to the target relativistic PM. Both this facts require that +we establish the overall scaling of GrGadget in its fully-relativistic +configuration and the overhead associated to both the relativistic PM +and the interface between the two codes. +In figure A1 we show the fraction of time spent in the PM in +both the original and relativistic configurations as a function of the +grid cell size (see the caption for details). The relativistic PM is an +order of magnitude more expensive than the original Gadget-4’s +Newtonian PM, although in absolute sense it is still either negligible +or secondary in the simulation sets that have been tested (it reaches a +maximum value of 16% at highest resolution, i.e. in the 𝑁 = 512,𝐿 = +250 Mpc/ℎ). However, it scales with both the resolution and the grid +number as the original Newtonian PM does. +Figures A2 and A3 report the scaling of run time in strong and +weak scaling tests respectively for the total run time, the tree time +and the PM time (left. middle and right panels in both figures; see the +captions for details). As inferred from A1, the run-time and hence its +scaling, are dominated by the Gadget-4’s Tree section. +ACKNOWLEDGEMENTS +We thank Julian Adamek for many fruitful discussions on gevolu- +tion, Volker Springel for his comments on an early draft, Francesca +Lepori, Marco Bruni, Marco Baldi and Emilio Bellini for discus- +sions. Simulations were performed with the HOTCAT system of +INAF (Taffoni et al. 2020; Bertocco et al. 2020). PM acknowledges +partial support by a Fondo di Ricerca di Ateneo grant of University +of Trieste. +DATA AVAILABILITY +The simulation codes presented in this paper are publicly available +on github in the following path: https://github.com/GrGadget. +REFERENCES +Abbott B. P., et al., 2016, Phys. Rev. Lett., 116, 061102 +Adamek J., Durrer R., Kunz M., 2014, Classical and Quantum Gravity, 31, +234006 +Adamek J., Daverio D., Durrer R., Kunz M., 2016, Journal of Cosmology +and Astroparticle Physics, 2016, 053 +Adamek J., Durrer R., Kunz M., 2017, J. Cosmology Astropart. Phys., 2017, +004 +Adamek J., et al., 2022, arXiv e-prints, p. arXiv:2211.12457 +Alam S., et al., 2021, J. Cosmology Astropart. Phys., 2021, 050 +Barnes J., Hut P., 1986, Nature, 324, 446 +Barrera-Hinojosa C., Li B., 2020, Journal of Cosmology and Astroparticle +Physics, 2020, 007 +Barrera-Hinojosa C., Li B., Bruni M., hua He J., 2020, Vector modes in +ΛCDM: the gravitomagnetic potential in dark matter haloes from rela- +tivistic 𝑁 -body simulations (arXiv:2010.08257) +Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291 +Bertocco S., et al., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J., +Verkouter H., eds, Astronomical Society of the Pacific Conference Series +Vol. 527, Astronomical Data Analysis Software and Systems XXIX. +p. 303 (arXiv:1912.05340) +Borzyszkowski M., Bertacca D., Porciani C., 2017, MNRAS, 471, 3899 +Bruni M., Thomas D. B., Wands D., 2014, Phys. Rev. D, 89, 044010 +Capozziello S., De Laurentis M., 2012, Annalen der Physik, 524, 545 +Chisari N. E., Zaldarriaga M., 2011, Physical Review D, 83 +MNRAS 000, 1–14 (2022) + +GrGadget +13 +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +(P(k) +P(k)ref)/P(k)ref +N=256 +L/N = 1 Mpc/h +L/N = 1 Mpc/h +L/N = 1 Mpc/h +L/N = 1 Mpc/h +L/N = 1 Mpc/h +L/N = 1 Mpc/h +N=512 +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +(P(k) +P(k)ref)/P(k)ref +L/N = 2 Mpc/h +L/N = 2 Mpc/h +L/N = 2 Mpc/h +L/N = 2 Mpc/h +L/N = 2 Mpc/h +L/N = 2 Mpc/h +Gadget4 TreePM +GrGadget PM +GrGadget TreePM +10 +2 +10 +1 +100 +k/(Mpc +1h) +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 +0.075 +0.100 +(P(k) +P(k)ref)/P(k)ref +10 +2 +10 +1 +100 +k/(Mpc +1h) +L/N = 4 Mpc/h +L/N = 4 Mpc/h +L/N = 4 Mpc/h +L/N = 4 Mpc/h +L/N = 4 Mpc/h +L/N = 4 Mpc/h +Figure 11. Matter power spectrum from cosmological simulations at 𝑧 = 0 using GrGadget (the blue lines) and compared to Gadget-4 (the red line) at 𝑧 = 0. +The dotted line is obtained with a simulation in which only the PM is used to compute forces. The plots show the relative difference with respect to the power +spectrum obtained with Gadget-4. The left column corresponds to simulations with 𝑁 = 256 grid points per dimension while for the right column 𝑁 = 512. +The boxsize changes along the ranks so that for the top plots the resolution is the highest 𝐿/𝑁 ≈ 1 Mpc/ℎ, in the middle 𝐿/𝑁 ≈ 2 Mpc/ℎ and the bottom plots +correspond to 𝐿/𝑁 ≈ 4 Mpc/ℎ. In all cases 𝑟𝑎 = 1.5 𝐿/𝑁 and 𝑟𝑏 = 3 𝐿/𝑁 . The grey dashed line indicate a 1% error. +DESI Collaboration et al., 2016, arXiv e-prints, p. arXiv:1611.00036 +Daverio D., Hindmarsh M., Bevis N., 2015, Latfield2: A c++ library for +classical lattice field theory (arXiv:1508.05610) +Doré O., et al., 2014, arXiv e-prints, p. arXiv:1412.4872 +Event Horizon Telescope Collaboration et al., 2019, ApJ, 875, L1 +Ivezić Ž., et al., 2019, ApJ, 873, 111 +Krause E., et al., 2017, arXiv e-prints, p. arXiv:1706.09359 +Laureijs R., et al., 2011, arXiv e-prints, p. arXiv:1110.3193 +Lepori F., Adamek J., Durrer R., Clarkson C., Coates L., 2020, Monthly +Notices of the Royal Astronomical Society, 497, 2078–2095 +MNRAS 000, 1–14 (2022) + +14 +E. Quintana-Miranda et al. +1 +2 +3 +4 +5 +6 +7 +8 +L/N (Mpc/h) +10 +2 +10 +1 +time_pm / time_total +GR (N=256) +Newton (N=256) +GR (N=512) +Newton (N=512) +Figure A1. The fraction of PM time to the total running time. Relativistic runs +are shown in blue while Newtonian runs are shown in red, whereas symbols +distinguish the value of grid points per dimension 𝑁 (squares and circles for +𝑁 = 256 and 512 respectively). We plot the time fraction on the 𝑦–axis (log +scale) vs the mesh resolution 𝐿/𝑁 on the 𝑥–axis. +Planck Collaboration et al., 2020, A&A, 641, A6 +Puchwein E., Baldi M., Springel V., 2013, MNRAS, 436, 348 +Sefusatti E., Crocce M., Scoccimarro R., Couchman H. M. P., 2016, mnras, +460, 3624 +Silvestri A., Trodden M., 2009, Reports on Progress in Physics, 72, 096901 +Spergel D., et al., 2015, arXiv e-prints, p. arXiv:1503.03757 +Springel V., 2005, Monthly Notices of the Royal Astronomical Society, 364, +1105 +Springel V., Pakmor R., Zier O., Reinecke M., 2021, MNRAS, 506, 2871 +Taffoni G., Becciani U., Garilli B., Maggio G., Pasian F., Umana G., Smareglia +R., Vitello F., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J., +Verkouter H., eds, Astronomical Society of the Pacific Conference Series +Vol. 527, Astronomical Data Analysis Software and Systems XXIX. +p. 307 (arXiv:2002.01283) +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–14 (2022) + +GrGadget +15 +25 +50 +75 +100 +125 +150 +175 +200 +# process +25 +50 +75 +100 +125 +150 +175 +200 +speedup x processes_pivot +Strong Scalability (Total time) +N = 128, L=250 +N = 128, L=500 +N = 128, L=1000 +N = 128, L=2000 +N = 256, L=250 +N = 256, L=500 +N = 256, L=1000 +N = 256, L=2000 +N = 512, L=250 +N = 512, L=500 +N = 512, L=1000 +N = 512, L=2000 +ideal +25 +50 +75 +100 +125 +150 +175 +200 +# process +25 +50 +75 +100 +125 +150 +175 +200 +speedup x processes_pivot +Strong Scalability (PM time) +N = 128, L=250 +N = 128, L=500 +N = 128, L=1000 +N = 128, L=2000 +N = 256, L=250 +N = 256, L=500 +N = 256, L=1000 +N = 256, L=2000 +N = 512, L=250 +N = 512, L=500 +N = 512, L=1000 +N = 512, L=2000 +ideal +25 +50 +75 +100 +125 +150 +175 +200 +# process +25 +50 +75 +100 +125 +150 +175 +200 +speedup x processes_pivot +Strong Scalability (Tree time) +N = 128, L=250 +N = 128, L=500 +N = 128, L=1000 +N = 128, L=2000 +N = 256, L=250 +N = 256, L=500 +N = 256, L=1000 +N = 256, L=2000 +N = 512, L=250 +N = 512, L=500 +N = 512, L=1000 +N = 512, L=2000 +ideal +Figure A2. Strong-scaling test. We present the code scaling as the number 𝑃 of MPI tasks is increased while running the same simulation set-up. All the results +refer to GrGadget, i.e. to the configuration with fully-relativistic PM. On the 𝑥–axis 𝑃 increases from 24 to 192, by ×2 steps. On the 𝑦–axis we report the +speed-up (normalized so that the ideal speed-up for 𝑃 = 1 is 1) for the total running time, the time spent in the PM and the time spent in the Tree on the Left, +Middle and Right panels respectively. Note that the ideal behaviour (black dotted line) would result in a linear speed-up. The PM data includes the translation of +particles data from Gadget-4 to Libgevolution. We show the results for 𝑁 = 128, 256 and 512 (solid, dashed and dot–dashed lines respectively) for 4 different +box sizes (i.e. mass resolutions), 𝐿 = 250, 500, 1000 and 2000 Mpc/ℎ (circles, squares and stars respectively). See the discussion in Appendix A for details. +25 +50 +75 +100 +125 +150 +175 +200 +# process +1.0 +1.1 +1.2 +1.3 +1.4 +time / time_pivot +Weak scalability (Total time) +N = 128->256, L = 250->500 +N = 128->256, L = 500->1000 +N = 128->256, L = 1000->2000 +N = 256->512, L = 250->500 +N = 256->512, L = 500->1000 +N = 256->512, L = 1000->2000 +ideal +25 +50 +75 +100 +125 +150 +175 +200 +# process +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +time / time_pivot +Weak scalability (PM time) +N = 128->256, L = 250->500 +N = 128->256, L = 500->1000 +N = 128->256, L = 1000->2000 +N = 256->512, L = 250->500 +N = 256->512, L = 500->1000 +N = 256->512, L = 1000->2000 +ideal +25 +50 +75 +100 +125 +150 +175 +200 +# process +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +1.30 +1.35 +time / time_pivot +Weak scalability (Tree time) +N = 128->256, L = 250->500 +N = 128->256, L = 500->1000 +N = 128->256, L = 1000->2000 +N = 256->512, L = 250->500 +N = 256->512, L = 500->1000 +N = 256->512, L = 1000->2000 +ideal +Figure A3. Weak-scaling test. We present the code scaling as the number 𝑃 of MPI tasks is increased for a proportionally increasing problem, then keeping +constant the particles–per–task occupancy. All the results refer to GrGadget, i.e. to the configuration with fully-relativistic PM. On the 𝑥–axis 𝑃 increases from +24 to 192 with only 2 test cases. On the 𝑦–axis we report the speed-up for the total running time, the time spent in the PM and the time spent in the Tree on the +Left, Middle and Right panels respectively. Note that the ideal behaviour would result in a constant running time (horizontal dotted black line). The PM data +includes the translation of particles data from Gadget-4 to Libgevolution. We show the results for two cases: from 𝑁 = 128, to 𝑁 = 256 (solid lines with +circles), and from 𝑁 = 256, to 𝑁 = 512 (dashed lines with squares). Each of the two cases has been run for three different box sizes (i.e. mass resolutions): +𝐿 = 250 → 𝐿 = 500 Mpc/ℎ, 𝐿 = 500 → 𝐿 = 1000 Mpc/ℎ and 𝐿 = 1000 → 𝐿 = 2000 Mpc/ℎ (red, green and blue colors respectively). See the discussion in +Appendix A for details. +MNRAS 000, 1–14 (2022) + diff --git a/UdFKT4oBgHgl3EQfli6N/content/tmp_files/load_file.txt b/UdFKT4oBgHgl3EQfli6N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..614efadeaea83e414c0b1303a08fb21e3b2d6ea6 --- /dev/null +++ b/UdFKT4oBgHgl3EQfli6N/content/tmp_files/load_file.txt @@ -0,0 +1,909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf,len=908 +page_content='MNRAS 000, 1–14 (2022) Preprint 30 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 GrGadget: an N-body TreePM relativistic code for cosmological simulations Eduardo Quintana-Miranda,1,2,3★ Pierluigi Monaco1,2,3,4 and Luca Tornatore1,2,3 1 Dipartimento di Fisica, Sezione di Astronomia, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Tiepolo 11, I-34143 Trieste, Italy 2 INAF – Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Trieste, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Tiepolo 11, I-34143 Trieste, Italy 3 IFPU – Institute for the Fundamental Physics of the Universe, via Beirut 2, I-34100 Trieste, Italy 4 INFN – Istituto Nazionale di Fisica Nucleare, Via Valerio 2, I-34127 Trieste, Italy Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present the merging of the Particle-Mesh (PM) relativistic Gevolution code with the TreePM Gadget-4 code, with the aim of studying general relativity effects in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Our code, called GrGadget, is able to track the evolution of metric perturbations in the weak field limit by using Gevolution’s implementation of a relativistic PM in the Poisson gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To achieve this, starting from Gevolution we have written a C++ library called Libgevolution, that allows a code to access and use the same abstractions and resources that Gevolution uses for its PM-only N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The code works under the assumption that particle interactions at short distances can be approximated as Newtonian, so that we can combine the forces computed with a Newtonian Tree with those computed with a relativistic PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The result is a TreePM simulation code that represents metric perturbations at the scales where they are relevant, while resolving non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We validate our code by closely matching Gadget-4 forces, computed with the Tree switched off, with those computed with Libgevolution in the Newtonian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' With GrGadget we obtain a matter power spectrum that is compatible with Newtonian Gadget-4 at small scales and contains GR features at large scales that are consistent with results obtained with Gevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We demonstrate that, due to the better resolution of the highly non-linear regime, the representation of the relativistic fields sampled on the mesh improves with respect to the PM-only simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Key words: cosmology: theory – large-scale structure of the Universe 1 INTRODUCTION The state of the art of precision cosmology provides a standard cos- mological model, ΛCDM, that is consistent with most observational evidence on large scales, but relies on the existence of a dark sector populated by Dark Matter (DM) and Dark Energy (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The first is responsible for the formation of cosmological structures such as galaxies and their large-scale density field, while the second causes the observed accelerated expansion of the universe in the present epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Their physical nature is an open problem, since the only ev- idence of their existence comes from their gravitational interaction with visible matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A possible explanation is that the dark sector is due to a misrepresentation of gravity, that on large scales does not follow Einstein’s General Relativity (GR), at the basis of the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This fact has triggered a wave of interest in modifications of GR, that can lead to extra terms that explain dark energy or dark matter (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Silvestri & Trodden 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Capozziello & De Laurentis 2012, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Such modifications must be significant only on large scales or low density, because GR is very accurate in predicting planetary orbits, light deflection and Doppler effects in solar system tests and has more recently been successfully tested ★ E-mail: eduardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='quintanamiranda@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='it with the detection of gravitational waves (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2016) and the direct imaging of black hole event horizons (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In order to characterize dark energy in the age of its dominance, many projects have been planned to survey large parts of the sky and probe the large-scale distribution of matter using galaxy clustering and galaxy lensing, both from the ground (DES1, Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' DESI2, DESI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Rubin’s LSST3, Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' SKAO4 surveys) and from space (Euclid5, Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Roman6, Spergel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' SphereX7, Doré et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Some of these surveys have already started to produce a flood of data that will soon lead to a precise characterization of the galaxy and matter den- sity fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A comparison of these observations to model predictions, either using summary statistics or field-level inference, will lead to unprecedented tests not only of the cosmological model but also of the gravity theory behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' With precision being guaranteed by 1 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='darkenergysurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='org 2 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='desi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='gov 3 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='lsst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='org 4 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='skao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='int 5 sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='int/web/euclid 6 roman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='gov 7 spherex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='edu © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='11854v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='CO] 27 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the amount of available high-quality data, accuracy will be achieved only by rigorous control of systematics, both in the data and in theory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The highly non-linear nature of the observed density field and the non-locality of gravity make cosmological simulations necessary to compare the predictions of current theories with the observations at an increasing level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Yet, most of the widely adopted simulation codes, like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Gadget-4 (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2021), use New- tonian dynamics for the evolution of matter perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is not the ideal configuration to pass from the unobservable distribution of matter in a periodic comoving box to the observable distribution of light in the past light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Relativistic corrections can be added a posteriori by post-processing Newtonian simulation outputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' one specific example of this approach is the modeling of lensing due to the distortion of null geodesics (Bartelmann & Schneider 2001), while a more comprehensive approach to adding relativistic effects is presented by Borzyszkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, even though the biases introduced by this approach are expected to be small, a fully self-consistent approach is necessary to convincingly demonstrate our ability of controlling theory systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For instance, galaxy clustering is affected by magnification bias due to lensing, and ne- glecting this effect induces a non-negligible bias in parameter esti- mation (Lepori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is even more true when modified gravity theories are used: extensions of gravity are typically derived in a full relativistic context, and while they influence the Newtonian limit of gravity, the small but measurable relativistic effects may provide smoking-gun signals of a specific class of gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this sense, restricting to the treatment of the Newtonian limit of modified gravity theories (as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', in Puchwein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2013) may leave out crucial observable signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Two examples of fully relativistic N-body codes for the evolution of cosmic perturbations, that integrate Einstein’s equations to follow the motion of massive particles along their geodesics, are the Adaptive Mesh Refinement (AMR) code Gramses (Barrera-Hinojosa & Li 2020) and the Particle-Mesh (PM) code Gevolution (Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These have proven to be precious tools to produce accurate cosmological predictions, like a self-consistent treatment of massive neutrinos (Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2022), and to explore phenomena that were previously overlooked, like the strength of the frame dragging field acting on dark matter haloes (Barrera-Hinojosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These codes sample the fields in a mesh that fills the simulated volume, but while Gramses uses an AMR scheme to increase resolution only where it is needed, PM schemes working on a single non-adaptive mesh are well known to be limited by memory, so they are unable to achieve the large dynamic range required, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', to resolve DM halos in large cosmological volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The integration of Newtonian particle trajectories has historically been addressed with the introduction of an oct-tree data structure (Barnes & Hut 1986), that provides a 𝑁 log 𝑁 scaling for the computation of gravity without compromising its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Because the integration of large-scale perturbations is very slow in this scheme, such an oct-tree is used to compute short-range forces, and is complemented by a Particle-Mesh (PM) code on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The resulting algorithm is commonly called TreePM, and it is the standard gravity solver for Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As we will show in next Section, deviations from a pure Newtonian approach become significant on scales that are comparable with the Hubble horizon, so a Newtonian treatment of small-scale clustering, performed by the Tree algorithm, would introduce a negligible error if large scales are treated by a fully relativistic gravity solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This can be achieved, in a TreePM scheme, by using a relativistic PM code for large-scale gravity, where relativistic potentials are sampled on a small enough mesh so as to be effectively Newtonian on the scales where the Tree code gets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this paper we present an implementation of Gadget-4 that uses a PM library, based on Gevolution relativistic code, as the PM part of the TreePM solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is a step toward the construc- tion of an ecosystem of codes and post-processing tools to perform end-to-end simulations of future surveys, with the aim of achieving optimal control of all systematics, including theoretical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The paper is organized as follows: Section 2 gives an overview of the the- ory of relativistic perturbations, with a focus on the approach used in Gevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Section 3 gives a description of the Gadget-4 and Gevolution codes, and describes the implementation of Libgevo- lution and GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Section 4 presents the tests performed to validate GrGadget, while Section 5 gives our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2 THEORY OF RELATIVISTIC PERTURBATIONS The success of Newtonian simulations in describing the large-scale structure of the universe follows from the fact that, for an observer at rest with respect to the CMB, the metric of spacetime is very close to Friedmann-Lemaitre-Robertson-Walker’s (FLRW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Deviations from the Newtonian approach are expected to be significant, albeit small, on scales near the Hubble horizon, or when the energy-momentum tensor has relativistic components like radiation or fast massive neu- trinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Deviations from FLRW metric are expected to be strong in the proximity of compact objects, but this happens on scales that are far smaller than the resolution that can be afforded in simulations of large comoving volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' It is thus fair to assume that the perturba- tions to the metric are small and can be described in a weak-field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This does not imply that deviations of the components of the energy-momentum tensor from homogeneity are assumed to be small, density perturbations can be highly non-linear: what we re- quire is that the size of self-gravitating objects is much larger than their gravitational radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The Gevolution code (see Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2016) models the space- time metric with a perturbed FLRW metric in the weak field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In the Poisson gauge the metric can be written as: 𝑑𝑠2 =𝑎2� − 𝑐2 𝑑𝜏2(1 + 2Ψ) − 2𝑐 𝑑𝜏𝑑𝑥𝑖𝐵𝑖+ + 𝑑𝑥𝑖𝑑𝑥 𝑗 �𝛾𝑖 𝑗 (1 − 2Φ) + ℎ𝑖 𝑗 �� , (1) where 𝑎(𝜏) is the scale factor of the FLRW background, 𝜏 is the conformal time and 𝑥𝑖 are the space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' It is possible to exploit the residual degrees of freedom of the metric to impose the conditions 𝐵𝑖|𝑖 = 0, ℎ𝑖𝑖 = 0 and ℎ𝑖 𝑗 | 𝑗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In our notation, repeated latin indexes denote Einstein’s summation over the spatial coordinates 1, 2, 3 and the vertical bar subscript, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 𝐵𝑖| 𝑗, denotes a covariant derivative with respect to the affine connection that emerges from the background spatial metric 𝛾𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The choice of the Poisson gauge is convenient because the two potentials Ψ and Φ are the gauge-invariant Bardeen potentials, and in the Newtonian limit the the field Ψ can be interpreted as the gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In other words, this is the gauge in which the standard N-body solver is integrating the right equations of motion in the Newtonian limit (Chisari & Zaldarriaga 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 Field equations The background, characterized by 𝑎(𝜏), is by construction a solution of the Einstein’s equations in the presence of a homogeneous and MNRAS 000, 1–14 (2022) GrGadget 3 isotropic energy-momentum tensor ¯𝑇 𝜇𝜈: ¯𝐺𝜇𝜈 = −8𝜋𝐺 𝑐4 ¯𝑇 𝜇𝜈 , (2) where ¯𝐺𝜇𝜈 is Einstein’s tensor constructed from the metric (1) with the perturbations Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Applying equation (2) to the FLRW metric one obtains Friedmann’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To solve for the perturbations of the metric, the usual procedure consist in subtracting (2) from the full Einstein’s equations: 𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 = −8𝜋𝐺 𝑐4 (𝑇 𝜇𝜈 − ¯𝑇 𝜇𝜈) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (3) The right hand side now contains the perturbation of the energy- momentum tensor due to inhomogeneities in mass and energy dis- tributions, while the left hand side is a very complicated non-linear expression containing the potentials Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 and their space- time derivatives up to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To reach a tractable set of equations that we can interpret and solve numerically, we apply the weak field assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The perturbations Ψ, Φ, 𝐵𝑖, ℎ𝑖 𝑗 are assumed to be of order 𝜖 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Spatial derivatives are known to increase their amplitude by a factor of 𝜖−1/2, accounting for the presence of shortwave fluctuations induced by the non-linear structure in the energy-momentum tensor, while time derivatives are assumed to preserve the perturbation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Then one can expand 𝐺𝜇𝜈 − ¯𝐺𝜇𝜈 in terms of the metric perturbations, neglecting contri- butions with order higher than 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For example: Φ is a term of order O(𝜖), Φ,𝑖 has order O(𝜖1/2), Φ|𝑛𝑛 is a leading term (order 1, because of the second derivative), quadratic terms like Φ,𝑛Φ,𝑛 are O(𝜖), and a term like Φ,00 is considered as O(𝜖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This type of expansion is known as the shortwave correction (Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Furthermore, experience has shown that the scalar perturbations Φ and Ψ are generally larger than the vector and tensor perturbations 𝐵𝑖 and ℎ𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Indeed, the scalar potentials, that are sourced by the density perturbation Δ𝑇00, become the Newtonian potential in the Newtonian limit, while the vector perturbation 𝐵𝑖 is sourced by Δ𝑇0𝑖, that is small by a factor of 𝑣/𝑐 for non-relativistic matter perturbations, and ℎ𝑖 𝑗 by Δ𝑇𝑖 𝑗, that is suppressed by a (𝑣/𝑐)2 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Hence, it is fair to drop quadratic terms of 𝐵𝑖 and ℎ𝑖 𝑗 in this weak field limit approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this approximation, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (3) it descends that its time-time component yields a Poisson-like equation for the scalar Φ: Φ|𝑛𝑛(1 + 4Φ) − 3H 𝑐2 Φ,0 + 3H2 𝑐2 (𝜒 − Φ) + 3 2Φ|𝑛Φ|𝑛 = 4𝜋𝐺𝑎2 𝑐4 Δ𝑇00 , (4) where H = 𝑎−1 𝑑𝑎 𝑑𝜏 and 𝜒 = Φ − Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' From the time-space section of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (3) we obtain: − 𝐵𝑖|𝑛𝑛 4𝑐 − Φ,𝑖0 𝑐2 − H 𝑐2 (Φ,𝑖 − 𝜒,𝑖) = −4𝜋𝐺𝑎2 𝑐4 Δ𝑇0𝑖 , (5) that, taking advantage of the condition 𝐵𝑛|𝑛 = 0, can be reduced to: − 𝐵𝑖|𝑛𝑛 4𝑐 = −4𝜋𝐺𝑎2 𝑐4 𝑃⊥Δ𝑇0𝑖 , (6) where 𝑃⊥ is a linear operator that selects from a vector field its divergenceless component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The traceless part of the spatial section of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 3 leads to: � 𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1 3𝛿𝑎𝑏𝛿 𝑗𝑖 � � 𝜒| 𝑗𝑖 − 2Φ| 𝑗𝑖 𝜒 + 4ΦΦ| 𝑗𝑖 + 2Φ| 𝑗Φ|𝑖 + 1 2𝑐2 ℎ𝑖 𝑗,00 + H 𝑐2 ℎ𝑖 𝑗,0 − 1 2 ℎ𝑖 𝑗 |𝑛𝑛 + 1 2𝑐 � 𝜕 𝜕𝜏 + 2H � � 𝐵𝑖 | 𝑗 + 𝐵 𝑗 |𝑖� � = � 𝛿 𝑗 𝑏𝛿𝑎𝑖 − 1 3𝛿𝑎𝑏𝛿 𝑗𝑖 � � −8𝜋𝐺 𝑐4 Δ𝑇𝑖 𝑗 � , (7) from which we can determine the rest of the metric degrees of free- dom 𝜒 and ℎ𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Since the source of 𝜒 and ℎ𝑖 𝑗 are the perturbation of the of the energy-momentum tensor Δ𝑇𝑖 𝑗, their amplitude in a matter dominated universe is suppressed by a factor (𝑣/𝑐)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' That is equivalent to say: since dark matter is non-relativistic, 𝜒 and ℎ𝑖 𝑗 must be very small with respect to Φ or even 𝐵𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As a matter of fact, V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 of Gevolution implements an improved expansion of the metric perturbations, that has been presented in Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For our tests we used the implementation of the original expansion, the one presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, the improved expansion has been ported to Libgevolution and will be used when analysing result on the past light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We do not expect the results presented in this paper to depend on the specific expansion used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 Motion of particles along geodesics Massive particles move along geodesics, whose equation can be expressed as: 𝑑𝑥𝑖 𝑑𝜏 = 𝑐𝑝𝑖 √︁ (𝑚𝑐𝑎)2 + 𝑝2 + 𝑐𝐵𝑖 + 𝑐𝑝𝑖 √︁ (𝑚𝑐𝑎)2 + 𝑝2 � Ψ + Φ2(𝑚𝑎𝑐)2 + 𝑝2 (𝑚𝑎𝑐)2 + 𝑝2 � , (8) 𝑑𝑝𝑖 𝑑𝜏 = − 𝑐 � 𝑝𝑛𝐵𝑛|𝑖 + Ψ,𝑖 √︃ (𝑚𝑐𝑎)2 + 𝑝2 + 𝑝2Φ,𝑖 √︁ (𝑚𝑐𝑎)2 + 𝑝2 � , (9) where 𝑝𝑖 is the space part of the particle momentum and 𝑝 its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The right hand side in the last equation is the generalized force acting on the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The term proportional to Ψ,𝑖 becomes the Newtonian force in the limit of small velocities, while 𝑝𝑛𝐵𝑛|𝑖 represent the corrections due to frame dragging and the third term in parenthesis is a further relativistic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The energy-momentum tensor is constructed from the knowledge of particle positions and momenta, but its computation depends on the perturbed metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This means that, in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (4), (6), and (7), the source terms on the right hand sides depend on the potentials themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These implicit equations may be solved starting from the potentials at the previous time step and solving the equations iteratively until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The integration scheme that Gevolution implements is simpler: at each time step the energy momentum tensor is computed using the potentials from the previous step, then the Poisson equations are solved to find the updated potentials, that will be used in the next time step to compute the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The Newtonian limit is recovered when we consider Fourier modes larger than H/𝑐 and we further neglect 𝐵𝑖 and consider Φ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' then equation (4) becomes: Φ|𝑛𝑛 = 4𝜋𝐺𝑎2 𝑐4 Δ𝑇00 (10) MNRAS 000, 1–14 (2022) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' while (8) and (9) become: 𝑑𝑥𝑖 𝑑𝜏 = 𝑝𝑖 𝑚𝑎 , (11) 𝑑𝑝𝑖 𝑑𝜏 = −Φ,𝑖𝑚𝑐2𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (12) 3 ALGORITHMS AND CODE INFRASTRUCTURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 Gevolution Gevolution8 (Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2016) is an N-body relativistic cosmo- logical code, written in C++ and parallelized with the MPI paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The physical theory behind this code has been described at length in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Numerically, this code implements a PM scheme to follow the evolution of energy-momentum tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As in PM codes, the advantage of working with a single grid and using Fast Fourier Transforms (FFTs) to solve the Poisson-like equations for the fields is paid with a high cost in memory, of O(𝑁3) where 𝑁 is the number of grid points per dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Gevolution, can run in either Newton or General Relativity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The Newtonian gravity solver inverts the Laplace operator in the Poisson equation for the Newtonian potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' When running the General Relativity mode, the code solves Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4, 6 and 7, that require the computation of the perturbed energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is performed using a Cloud-In-Cell (CIC) scheme both for the density and for particle velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' details are given in the presentation paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Then the Hamiltonian forces to which particles are subjected are computed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Gevolution solves the field equations in Fourier space, using a C++ library called LATfield2 to operate FFTs on classical fields in massively parallel applications with distributed memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' LATfield2 provides a programming interface to perform operations on the fields, either in their real or Fourier space representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This library im- plements FFTs of 3-dimensional fields whose memory is distributed among parallel processes following a 2-dimensional uniform decom- position of space, in which each process owns in memory a portion of the grid with a rod shape (Daverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this way LAT- field2 overcomes the scaling limitations of a simpler 1-dimensional domain (slab) decomposition provided by the mainstream FFTW3 library9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' FFTW3 is used, however, to compute 1D FFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 Gadget-4 Gadget-410 is a state-of-the-art TreePM N-body hydrodynamical cosmological code written in C++ (see Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' it is massively parallelized in a distributed-memory paradigm using MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As in most N-body codes, gravity in Gadget-4 is represented in the Newtonian limit, but the equations of motion are modified to take into account the Universe expansion, obtained by integrating the Friedmann equations separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As mentioned above, this approach is consistent with General Relativity in the Poisson gauge, and gives the leading-order term of weak field expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This amounts to neglecting the metric degrees of freedom 𝐵𝑖, 𝜒 and ℎ𝑖 𝑗, and is valid on scales much smaller than the Hubble horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In a typical configuration that is convenient for large cosmological volumes, the 8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='com/gevolution-code 9 http://fftw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='org/ 10 https://wwwmpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='mpa-garching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='de/gadget4 code solves for the forces acting on each particle, representing them as the sum of two contributions, one due to the interactions with nearby particles, computed with a Tree algorithm, and one due to long-range interactions, computed with a PM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='11 The Tree algorithm works by partitioning the space into cubic cells, called nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' in turn, each node is recursively partitioned into 8 children nodes down to a pre-determined maximum refinement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A tree structure tracks the list of particles that are located within each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This structure is used to speed up the computation of gravitational force on a particle: in a particle-particle integration scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' this force is computed by adding up a series of �𝑟 𝑚/𝑟3 terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' one for each particle pair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' but we know that the accuracy of force evaluation does not depend strongly on the small-scale distribution of distant particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' so in the Tree scheme the evaluation of gravity is performed by grouping particles that belong to the same node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' under the condition that the node subtends a given aperture angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Particle-particle computation is then used only for the nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is equivalent to considering the leading order in a multipole expansion of the gravity force from particles belonging to a distant cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' While the construction of the Tree is expensive in terms of computing time, it allows to achieve O(𝑁𝑝 log 𝑁𝑝) scaling for the force computation, where 𝑁𝑝 is the total number of particles in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Thus the Tree is able to compute with high accuracy the short wavelength modes of the gravitational interaction, while keeping the computational time low for large simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, the Tree code is slow in integrating particle motions near the initial conditions, when the departures from homogeneity are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is why it is often coupled with a PM code to speed up the first time steps of a cosmological box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The PM algorithm represents gravity through the gravitational po- tential field Φ, evaluated on a Cartesian cubic mesh of fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The potential is found from the density field by solving the Poisson equa- tion in Fourier space, while the force is computed from the gradient of the potential, obtained with a finite differences scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Accord- ing to the Nyquist-Shannon theory, this implies that the information handled by the PM is limited to the long modes, up to the Nyquist frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To combine the forces provided by the PM and Tree codes, the gravitational potential is split into the sum of two fields: Φ = Φ(𝐿) + Φ(𝑆) , (13) where Φ(𝐿) represents long-range modes from the PM, and Φ(𝑆) represents short-range modes from the Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Written in Fourier space (tilde on top of symbols denotes a Fourier transform), the Poisson equation reads: ˜Φ𝑘 = −4𝜋 𝑘2 ˜𝜌𝑘 , (14) where 𝜌 denotes the mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We can split the density as a sum of short-range and long-range terms, using Gaussian filters: ˜Φ𝑘 = −4𝜋 𝑘2 ˜𝜌𝑘 � 1 − exp(−𝑘2𝑟2 𝑎) � − 4𝜋 𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2 𝑎) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (15) The scale 𝑟𝑎 is the one at which we split long- and short-range modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We can obtain Φ(𝑆) by solving the modified Poisson equation for short modes: ˜Φ(𝑆) 𝑘 = −4𝜋 𝑘2 ˜𝜌𝑘 � 1 − exp(−𝑘2𝑟2 𝑎) � , (16) 11 The code can work in other configurations (a non-cosmological volume, switching off the PM, enhancing the Tree part using multipole expansion) that are however not relevant for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) GrGadget 5 and Φ(𝐿) by solving the modified Poisson equation for long modes ˜Φ(𝐿) 𝑘 = −4𝜋 𝑘2 ˜𝜌𝑘 exp(−𝑘2𝑟2 𝑎) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (17) The long-mode Poisson equation (17) is solved by the PM in Fourier space, so the convolution with the kernel is a simple mul- tiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The Tree on the other hand works in real space, hence equation (16) has to be transformed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' this can be done analytically, yielding: Φ(𝑆) (�𝑥) = −𝐺 ∑︁ 𝑖 𝑚𝑖 |�𝑥 − �𝑟𝑖| erfc � |�𝑥 − �𝑟𝑖| 2𝑟2𝑎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (18) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 GrGadget 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 Libgevolution library In order to have a relativistic PM code working in Gadget-4, we developed a library that implements both the Newtonian and the rel- ativistic PM algorithms of the monolithic Gevolution code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This was done by forking the Gevolution github repository into Libgevo- lution, a library that is publicly available on github12 under MIT license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The rationale behind the development of Libgevolution is to encapsulate Gevolution’s resources and methods into abstract ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This yields several benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Firstly, Gevolution maintenance is eased by the logical modularization of the code, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' instead of a monolitic code with a unique workflow we can divide Gevolution into components (C++ classes and/or namespaces) with well defined purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Secondly, we are allowed to re-use Gevolution compo- nents within other applications, such as we do within Gadget-4 in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We give here an overview of the library;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the precise signature of all the defined functions, methods and data structures is described in the technical documentation of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Libgevolution is based on three cornerstones: (i) a particle container implemented through the class Particles_gevolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (ii) a PM data structure named particle_mesh, templated on the particle container type, that can be used either as a relativistic_pm or a newtonian_pm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (iii) an executable application that uses the previous components to produce N-body simulations as the original code does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' particle_mesh has to be understood as a container that is aware of the parallelization of the tasks and distribution of memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' it holds the gravitational fields and it allows the user to compute the forces acting on the simulation particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The user interface declared in particle_mesh consists of the following functions: sample(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='), that builds the sources (density field or energy- momentum tensor) by sampling particle properties in the mesh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' compute_potential(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='), that solves Poisson equations to compute the potential fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' compute_forces(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='), that computes the forces acting on particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' particle_mesh is specialized to solve the Newtonian problem or the General Relativistic problem using class inheritance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figure 1 illustrates the class hierarchy of Libgevolution’s particle_mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The expert user will be able to specialize particle_mesh to his/her own needs, for example by deriving a PM that solves a modified gravity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' newtonian_pm is the specialization of particle_mesh that 12 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='com/GrGadget/gevolution-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 particle_mesh relativistic_pm newtonian_pm Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' PM class hierarchy in Libgevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' contains a real LATfield2::Field scalar field ΦNewton and its complex LATField2::Field Fourier transform ˜ΦNewton, plus a LATField2::PlanFFT that connects ΦNewton with ˜ΦNewton through discrete Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' relativistic_pm is the spe- cialization of particle_mesh that contains the above quoted de- grees of freedom of the perturbed FLRW metric, Φ, 𝐵𝑖 and 𝜒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These are represented as real LATfield2::Field, with complex LATField2::Field counterparts to represent their Fourier trans- forms and a LATField2::PlanFFT for each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As a first testing phase, we run Libgevolution, called with a sim- ple wrapper, and the native Gevolution code, applying them to the same set of initial conditions, checking that the results were identical both in the Newtonian and relativistic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Then we stripped down Gadget-4 by switching off the Tree code, and compared its results to the Newtonian results of Libgevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' It is necessary that this comparison gives nearly identical results if we want Libgevolution to substitute the native PM code of Gadget-4 without loss of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To achieve a satisfactory match of the two PM codes we had to change the Gevolution scheme in a few points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We started from V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 of Gevolution, that implemented a first- order version of finite differences instead of the fourth-order scheme of Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This resulted in a difference with Gadget-4 run on the same initial conditions, and in a percent-level offset of the matter power spectrum on large scales at low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We upgraded the computation of spatial derivatives to fourth order, in parallel with the Gevolution developers that had noticed the same problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' our implementation is equivalent the most recent issue of Gevolution (used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', in Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The upgrade is the following: let’s consider the gravitational potential along one direction of the mesh, and let’s call its values Φ𝑖, where the index𝑖 denotes its position along that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Its first derivative is computed with finite differences at the first order as: 𝜕Φ𝑖 𝜕𝑥 = Φ𝑖+1 − Φ𝑖 ℎ + O(ℎ), (19) where ℎ is the size of the mesh cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Fourth-order Taylor expansion gives: 𝜕Φ𝑖 𝜕𝑥 = 8Φ𝑖+1 − Φ𝑖−1 12ℎ − Φ𝑖+2 − Φ𝑖−2 12ℎ + O(ℎ4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (20) This has a smaller error of order O(ℎ4), so it achieves higher pre- cision than (19) with the little cost of knowing the potential value at the second-nearest cell, that implies a negligible communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Another improvement with respect to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 of Gevolution, that follows an implementation of Gadget-4, was the application of cor- recting filters to the density in Fourier space to compensate for cloud- in-cell (CIC) interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Indeed, as discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' in Springel (2005) or Sefusatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (2016), CIC interpolation at some finite or- der leads to some loss of power that can be compensated for in Fourier space using suitable kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This was applied both to the compu- tation of the density and to the computation of energy-momentum tensor components in the relativistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Lastly, to make the Newtonian PM scheme equivalent to that of MNRAS 000, 1–14 (2022) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Gadget-4 we changed the form of the discrete Laplacian operator in the Poisson equation solver from its original form ∇2 → −4𝑁2 𝐿2 � sin2 𝜋𝑘𝑥 𝑁 + sin2 𝜋𝑘𝑦 𝑁 + sin2 𝜋𝑘𝑧 𝑁 � , (21) described in Adamek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (2016), equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5), to the form used in Gadget-4: ∇2 → −4𝜋2 𝐿2 � 𝑘2 𝑥 + 𝑘2 𝑦 + 𝑘2 𝑧 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 Calling Libgevolution from Gadget-4 The implementation of Libgevolution in Gadget-4 was performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We created a new PM class with a similar interface as the original one in Gadget-4, so that it is initialized and executed with the same functions as Gadget-4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' init_periodic() and pmforce_periodic().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A new class relativistic_pm was imple- mented within an gadget::gevolution_api namespace, avoiding to use the wider gadget namespace to make a clear distinction of purpose between the original Gadget-4 code and our additional fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This relativistic_pm class acts much like a mediator taking information in and out of gadget simulation particles, processing the correct units conversion and calling the methods on gevolution namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figure 2 shows a diagram that summarizes the contents of this PM class, its relation with Gadget-4’s resources and the entry points for gevolution’s api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' relativistic_pm consists of: A variable of type simparticle_handler that acts as a wrapper for providing particle information from Gadget-4’s simparticles global variable and writing back the data produced by gevolution’s PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A variable of type latfield_handler that takes care of cor- rectly initializing LATfield global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Indeed, while Gadget-4 can run with any number of MPI processes, LATfield has limitations that depend on the number of grid points in the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' latfield_handler also takes care of creating a sub-communicator from Gadget-4’s MPI global communicator that satisfies the constraints set by LATfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A variable of type gevolution::cosmology that contains the parameters for the background evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A container of type gevolution::Particles_gevolution that holds particle information, stored according to their location on the PM grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Variables of type gevolution::relativistic_pm and gevolution::newtonian_pm that perform the actual PM compu- tations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' construct the sources, either density or the components of the energy-momentum tensor, compute the gravitational potential or the metric perturbation fields and the forces that act upon the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The methods pm_init_periodic and pmforce_periodic, for initialization and execution of the PM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 Kick and drift operators In order to keep the Hamiltonian character of the equations of motion in Gadget-4, we have to describe the state of each particle through its position and momentum, not velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Following a leap-frog scheme, the momentum should be updated with a kick operation using the full relativistic Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, velocities in Gadget-4 are to be interpreted as momenta (per unit mass) of non-relativistic particles in the Newtonian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Then we redefine the Gadget-4 kick and drift operators assuming non-relativistic matter, 𝑝 ≪ 𝑚𝑐𝑎, and further neglecting the very small contribution coming from 𝜒: 𝑑𝑥𝑖 𝑑𝜏 = 𝑝𝑖 𝑚𝑎 (1 + 3Φ) + 𝑐𝐵𝑖 , (23) 𝑑𝑝𝑖 𝑑𝜏 = − 𝑐𝑝𝑛𝐵𝑛|𝑖 − Φ,𝑖𝑚𝑐2𝑎 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (24) The right hand side of (24) is what we call force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 Adding long-range and short-range forces To combine the forces computed with the relativistic PM and Gadget-4’s Newtonian Tree we have extended the idea of the TreePM coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' From equation (13) one obtains that the force acting on a particle in a TreePM scheme consists of two terms: �𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree Newton] + 𝐿𝑟𝑎 [ �𝐹PM Newton].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (25) The first term is the force computed using the Tree on which an exponential high-pass filter 𝑆𝑟𝑎 is applied, leaving short-wavelength modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The second term corresponds to the PM force on which the complementary low-pass filter 𝐿𝑟𝑎 is applied to leave long- wavelength modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The symbols 𝑆𝑎 and 𝐿𝑎 formally denote these linear operators: 𝑆𝑟𝑎 [ 𝑓 ](�𝑟) = 1 𝑁 ∑︁ �𝑘 ˜𝑓�𝑘 (1 − exp(−𝑘2𝑟𝑎2)) exp(−𝑖�𝑘 · �𝑟) , (26) and 𝐿𝑟𝑎 [ 𝑓 ](�𝑟) = 1 𝑁 ∑︁ �𝑘 ˜𝑓�𝑘 exp(−𝑘2𝑟𝑎2) exp(−𝑖�𝑘 · �𝑟) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (27) The grid smoothing scale 𝑟𝑎 scales with the PM mesh size, and its value is optimized in Gadget-4, in a way that will be tested below, to minimize the impact of the two different treatments of the gravitational force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In order to account for the relativistic dynamics while preserving the match between tree and PM contributions that is valid in the New- tonian case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' we choose the following strategy: Gadget-4 calls both newtonian_pm and relativistic_pm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the Newtonian value of the force is added to the Tree force as in a standard Newtonian simula- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' while the difference between the Newtonian and the relativistic forces is added on top as a correction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' but filtered on a different scale 𝑟𝑏,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' that we call gr-smoothing scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (25) then becomes: �𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree Newton] + 𝐿𝑟𝑎 [ �𝐹PM Newton] + 𝐿𝑟𝑏 [ �𝐹PM GR − �𝐹PM Newton] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (28) The case 𝑟𝑎 = 𝑟𝑏 would correspond to simply adding the relativistic force to the Tree: �𝐹 = 𝑆𝑟𝑎 [ �𝐹Tree Newton] + 𝐿𝑟𝑏 [ �𝐹PM GR ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (29) However, while the size of 𝑟𝑎, that regulates the match between Newtonian Tree and PM forces, is very well tested within Gadget-4, the optimal value of 𝑟𝑏 is to be found;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' we will show in the next Section that using 𝑟𝑏 larger than 𝑟𝑎 allows us to achieve percent accuracy at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4 VALIDATION The GrGadget code has been validated by running it on a few real- izations of initial conditions, listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These were generated MNRAS 000, 1–14 (2022) GrGadget 7 gadget:: simparticles LATfield2:: parallel particle_handler read/write latfield_handler read/write sim::begrun1() sim::gravity_long_range_force() init_periodic() pmforce_periodic() execute execute gevolution::cosmology gevolution::Particles_gevolution gevolution::relativistic_pm gevolution::newtonian_pm gadget::gevolution_api::relativistic_pm:: Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Diagram of resource ownership and relations for Libgevolution integrated into Gadget-4’s workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Each solid box represent a memory resource (an instantiation of a variable type) while the dashed boxes indicate ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The newly developed code, represented in the right part of the diagram denoted with the namespace gadget::gevolution_api, consists in a class named relativistic_pm that owns a particle_handler object that reads and writes directly into gadget::simparticles, a latfield_handler that takes care of setting up and inspect the state of LATfield2::parallel, and some types defined in Libgevolution, that are defined in gevolution namespace, like cosmology, Particles_gevolution and relativistic_pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The methods sim::begrun1() and sim::gravity_long_range_force() in gadget:: interact with the relativistic_pm through their interface init_periodic() and pmforce_periodic().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' name 𝑁𝑝 (particles) 𝑁 (PM grid points) 𝐿 (box size) N64 643 64 1 Gpc/ℎ N256 2563 256 1 Gpc/ℎ high_res 5123 512 500 Mpc/ℎ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Cosmological simulation configurations used to validate GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' with Gadget-4’s ngenic code at 𝑧 = 19, starting from a linear power spectrum generated with CAMB13 and with cosmological parame- ters consistent with Planck 2018 result (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020): Ω𝑏ℎ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0223, Ω𝑐ℎ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='120, 𝐻0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 km s−1 Mpc−1, 𝐴𝑠 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='097 × 10−9 and 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 Gevolution and Gadget-4 original codes As already discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1, the newtonian_pm imple- mentation in V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 of Gevolution computes the Newtonian forces differently from those obtained with Gadget-4’s PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Before imple- menting Libgevolution as the PM engine of Gadget-4, we need to make the two algorithms work in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' To this aim, we have run a set of simulations with the configuration N64 (described in table 1) with a small number of particles 𝑁𝑝 = 643 to be able to compute forces using a straightforward particle-particle (PP) scheme, that can be taken as the true force that we are trying to approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The same initial conditions at 𝑧 = 19 have been fed to both Gadget-4 (with Tree either on or switched off to have a pure PM run) and Gevolution (in Newtonian mode) codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' At later times, 𝑧 = 8 and 𝑧 = 0, we have written snapshots of the forces that the simulation particles experience, separating the PM and the TreePM components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' we have then compared those to the true Newtonian force computed with the PP scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The data we have obtained are summarized in the plots shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We have binned particles according to the value of the true force, then for each bin we have computed the mean (colored lines) and standard deviation 13 https://camb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='info/ (shaded regions) of the difference between the force computed with approximate methods (PM or TreePM) and the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Forces are given in Gadget-4’s default units, which is actually acceleration, measured in units of 10𝐻0 km/s = ℎ km2 s−2 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The green line shows the PM result using the original Gevolution code (the true force is anyway computed with Gadget-4 and matched particle by particle) while the red line is obtained from a pure PM using Gadget-4’s original code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The black line gives the TreePM method precision, obtained using Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Looking at the red and green lines (and their shaded areas) we find two known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Firstly, the TreePM method produces far less bias and dispersion when estimating forces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' for instance, in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 3 the error is of the order14 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 ℎ km2 s−2 kpc−1, while in the right panel it is larger but barely visible when compared with the other curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Secondly, while the PM force has low bias but a much larger variance than the TreePM one at high redshift, at low redshift, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' at higher level of non-linearity, it underestimates the value of the Newtonian force as its magnitude increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This underestimation is due to the failure of PM in resolving interaction at scales smaller than the grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' When comparing Gevolution PM and true forces, we notice an S-shaped feature in the plot, much more visible at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As anticipated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1, this is mostly due to the first-order interpolation used to find the gradient of the potential in the code version that we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4 we show the matter power spectra15 obtained at 𝑧 = 0 from a set of larger simulations with the configuration N256 (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The red solid line shows the result obtained with the orig- inal Gadget-4 code with its TreePM method, while the red dotted line shows the results obtained by switching off the Tree so that the 14 This quantification is in code units, we can take this value as a reference for a high accuracy gravity solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 15 In this paper all particle power spectra were computed using PowerI4 code presented in Sefusatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Unless otherwise stated, all power spectra are computed up to the the Nyquist frequency of the PM mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 Force (direct summation) 2 1 0 1 2 Force (estimated) - Force (direct summation) Force Test (z=8) TreePM (Gadget) Gevolution PM Gadget PM 100 75 50 25 0 25 50 75 100 Force (direct summation) 100 50 0 50 100 Force (estimated) - Force (direct summation) Force Test (z=0) TreePM (Gadget) Gevolution PM Gadget PM Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Difference of gravity force with respect to the true PP value, binned according to the true force, for N64 initial conditions, at 𝑧 = 8 (left panel) and 𝑧 = 0 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Lines represent the mean value of force difference in the bins, with colours explained in the legend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the shaded regions give the standard deviation of the corresponding force difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' forces are computed using the PM alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The green lines show re- sults obtained with the latest develop version of Gevolution that implements higher order schemes for finite differences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the dotted line gives results obtained with GRADIENT_ORDER=1 and is identi- cal to the result obtained with V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 of Gevolution, the green solid line uses GRADIENT_ORDER=2, that corresponds to a second-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These power spectra show that the matter distribution in Gevolution using first-order gradients loses power in what seems to be a uniform trend for large-scale modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is a behaviour which is not inherent to the PM nature of the code, since that type of numerical approximation should predict very well the linear evolu- tion at large scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' indeed, the higher-order scheme recovers power on large scales to sub-percent accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Conversely, Gadget-4’s PM and TreePM agree very well at wavenumbers below 𝑘 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1ℎ/Mpc scale, The higher-order differentiation worsens the loss of power of Gevolution for high values of 𝑘, that is not present in Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This can be explained as a consequence of the particle-to-mesh sam- pling and mesh-to-particle interpolation described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As discussed there, Gadget-4’s PM corrects for these effects, result- ing in a power spectrum that degrades only at very high values of 𝑘 as we approach the Nyquist frequency, while producing a ∼ 2 percent overcorrection at 𝑘 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 ℎ/Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' After implementing the higher-order differentiation scheme, the correction for the loss of power discussed above and the change in the discrete Laplacian operator (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1), the results of native Gadget-4 and Libgevolution PMs become indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 Newtonian forces We have tested our implementation of the GrGadget code by run- ning a standard test in Gadget-4: we create an N-body configuration in which there is a single massive particle in the entire simulation box, while other massless test particles are placed at different dis- tances from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this setting the exact value of the force on each particle is known, hence one can compare the numerical results coming from the TreePM algorithm to the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The results are shown in figure 5, where each dot represents a test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The x-axis gives the distance to the massive particle that sources the gravitational field, in units of the PM resolution (𝐿/𝑁), while the y-axis gives the corresponding absolute value of the relative difference of the true and estimated forces acting on the test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The red and blue lines correspond to the mean value of force residuals, for particles binned into distance bins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the red line denotes the statistics obtained from a simulation using Gadget-4’s original TreePM implementation and the blue line was produced using GrGadget, in this case with the Newtonian gravity engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This figure shows that the accuracy with which the TreePM code reproduces the gravitational force is at worst at percent level on scales of a few mesh cells, corresponding to the scale where the PM and Tree contributions are matched, and gets very accurate in the limits where either the Tree (small scales) or the PM (large scales) dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Gadget-4’s and GrGadget’s Newtonian PMs show basically the same accuracy, even though their PM implementations are very dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 6 we show the matter power spectra of a set of N256 simulations (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this case we are comparing the matter clustering of GrGadget, in blue (with Newtonian forces for testing purposes), against Gadget-4, in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In agreement with the previous test of force differences, we find that both codes produce the same matter power spectrum up to floating point errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is verified both in the case of simulations computing forces using a pure PM and in the case of TreePM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 Relativistic simulations with GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We present here results obtained by running GrGadget with relativistic_pm, comparing them with the corresponding rela- MNRAS 000, 1–14 (2022) GrGadget 9 102 103 104 P(k)/(Mpc3h−3) camb Gevolution Newton PM Gevolution Newton PM (2nd order) Gadget4 PM Gadget4 TreePM 10−2 10−1 100 k/(hMpc−1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 (P(k) − P(k)Gadget4)/P(k)Gadget4 Matter power spectrum z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Matter power spectrum of N256 cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The lower panel shows residuals with respect to Gadget-4’s original code (in red), used as baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The black line shows the linear power spectrum obtained with CAMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Red lines show results obtained with Gadget-4, with the Tree part on (solid line) or switched off (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Green lines show results obtained with Gevolution in Newtonian configuration, with finite differences at first order (dotted line) or second order (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Forces due to a point source: the points are test particles located at different distances (in units of the mesh resolution 𝐿/𝑁 ) from the source and the lines represent the RMS of the difference between real and TreePM forces in different distance bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The red line corresponds to Gadget-4 original TreePM while the blue line was obtained with GrGadget in Newtonian mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As for the the grid smoothing scale, the default value was used: 𝑟𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='25𝐿/𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For this test we have used 𝑁 = 256 and 𝐿 = 1 Gpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 102 103 104 P(k)/(Mpc3h−3) camb GrGadget (Newton) PM GrGadget (Newton) TreePM Gadget4 PM Gadget4 TreePM 10−2 10−1 100 k/(hMpc−1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 (P(k) − P(k)Gadget4)/P(k)Gadget4 Matter power spectrum z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Matter power spectrum of four simulations starting from the same initial conditions high_res: blue lines give results for Gadget-4 original code, red lines give results for GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In both cases dotted lines refer to runs with PM-only, solid lines refer to runs with full TreePM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' tivistic version of Gevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We expect that the power spectrum of the matter density displays some relativistic features at large scales due to terms preceded by H in the field equation (4), while at small scales results should be compatible with Gadget-4’s Newtonian sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, the matter power spectrum shown here is not an observable quantity, so this comparison is just meant to give a first validation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A more thorough comparison of observ- ables reconstructed on the past light cone will be presented in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figure 7 shows the matter power spectra for a series of N256 sim- ulations (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In this case Gevolution and GrGadget are run in GR mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The parameter that regulates the scale of the rela- tivistic correction (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 28 and 29) is set to 𝑟𝑏 = 6 𝐿/𝑁 ≈ 23Mpc/ℎ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the relativistic corrections of the PM method are smoothed at a distances below 6 grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The plot shows that relativistic PM-only simulations, GrGadget (blue dotted line) and Gevolution (green lines) are compatible on large scales (𝑘 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='03ℎ/Mpc) up to a small percent-level difference that it is likely caused by the use of differ- ent orders for finite difference gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' indeed, going from first- to second-order differences (from dotted to solid green line) the power spectrum gets nearer to GrGadget’s fourth-order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The plot also confirms that our combination of Tree and PM forces in the relativis- tic weak field limit with GrGadget (blue solid line) reproduces the Newtonian non-linear features to sub-percent level at small scales, that is for 𝑘 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1ℎ/Mpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' here Gadget-4 (red solid line) is again our reference for the non-linear clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Being designed for the use of Fourier methods from the beginning, Libgevolution offers an interface for the computation of the power spectrum of the fields defined through the library’s interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Thus we can also extract and analyse the power spectra of the individual components of the metric perturbations from the relativistic sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figures 8 and 9 show the power spectra of the relativistic potentials, Φ, 𝐵𝑖 and 𝜒, for a high resolution configuration high_res (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These plots show a comparison of PM (blue lines) and TreePM (red lines) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The power spectrum of the gravita- MNRAS 000, 1–14 (2022) Force Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0200 Gadget TreePM GrGadget (Newton) TreePM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0000 100 101 distance*N/L10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 102 103 104 P(k)/(Mpc3h−3) camb GrGadget TreePM GrGadget PM Gevolution Gr PM Gevolution Gr PM (2nd order) Gadget4 PM Gadget4 TreePM 10−2 10−1 100 k/(hMpc−1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 (P(k) − P(k)Gadget4)/P(k)Gadget4 Matter power spectrum z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Matter power spectrum of Gadget-4, Gevolution and GrGadget runs, the last code being run in relativistic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The upper panel shows the absolute value and the lower panel the relative difference with respect to Gadget-4’s TreePM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The black line gives the linear matter power spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' red and blue lines give Gadget-4 and GrGadget results, with full TreePM forces (solid lines) or with the Tree switched off (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Green lines give Gevolution results, dotted line referring to first-order finite differences (GRADIENT_ORDER=1) and solid line referring to second-order calculation (GRADIENT_ORDER=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' tional potentials converge for both methods on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, below 1 Mpc/ℎ the PM-only simulation loses power with respect to the TreePM one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the differences can reach up to 40% as we approach the Nyquist frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This pattern is equally found for the scalar fields Φ and 𝜒, as well as for the individual components of 𝐵𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 8 helps to understand the reason behind this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Generally speaking, energy density, momentum density and their respective density current (the components of the Energy- Momentum tensor) are sources of the metric perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Even though those quantities, as fields, are found at discrete positions of space defined by the mesh, their values are computed by sampling the energy and momentum carried by the particle distribution, which contain information on the clustering due to the short range inter- actions (through the Tree) that goes well below the mesh resolution 𝐿/𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Therefore, TreePM simulations, having power on scales well smaller than the PM mesh, give a better representation of the source of metric perturbation, and thus allow to recover power at frequency modes right below Nyquist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 8 highlights the particular case of 𝑇00 (the matter density) as a source for Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' by comparing 𝑇0 0 with 𝑘2Φ, we are verifying the Poisson equation 𝑘2 ˜Φ ≈ ˜𝑇00 that is valid for wavelengths below the Hubble horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This confirms that the presence of small-scale clustering in the particle distribution prop- agates to the gravitational fields up to the maximum resolution that the PM allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The same thing is visible in the vector modes 𝐵𝑖 and in 𝜒 (Figure 9), where we also notice a small, few-percent mismatch on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These fields are known to give sub-percent effects on observables, so this difference, that is likely due to some degree of numerical mode coupling, is non considered as a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In figure 10 we show how the matter power spectrum obtained using GrGadget is affected by the choice of the gr-smoothing scale parameter 𝑟𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We have used an N256 box configuration to perform this test, and tested values of 𝑟𝑏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5, 3, 6 in units of 𝐿/𝑁 ≈ 4 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We find that large-scales power is independent of the value of 𝑟𝑏 parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' structures one scales below the PM resolution are resolved by the Tree algorithm, hence for 𝑘 > 𝑘Nyquist there is a convergence of all simulations to a common non-linear power spectrum tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' It is in the medium to small scales 𝑘Nyquist > 𝑘 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 Mpc−1ℎ that we notice differences in the power spectrum above the ∼ 1% (dashed grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For small values of 𝑟𝑏 (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 𝐿/𝑁), we obtain discrepancies in the power spectrum at 𝑘 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 Mpc−1ℎ that can be as large as 5 percent and indicate the limitations of our force summation scheme, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' A value of 𝑟𝑏 = 3 𝐿/𝑁 or possibly higher is needed to obtain a good compatibility of GrGadget and Gadget-4 for all modes greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 Mpc−1ℎ, where relativistic features in the matter clustering is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The last test we present here regards the convergence of the nu- merical results for increasing resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figure 11 shows the matter power spectrum obtained from running Gadget-4’s TreePM (red lines), GrGadget with PM-only (blue dotted lines) and GrGadget with TreePM (blue continuous line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These various code configu- rations were run with different combinations of the number of grid points per dimension 𝑁 = 256, 𝑁 = 512 and box length 𝐿 = 250, 500, 1000, 2000 Mpc/ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' the number of particles was fixed as 𝑁𝑝 = 𝑁3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In all cases we have set the PM smoothing scale to 𝑟𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 𝐿/𝑁 and the gr-smoothing scale to 𝑟𝑏 = 3 𝐿/𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' It can be observed with the finest resolution, in the top plots, that there is a matching between General Relativity and Newtonian dynamics in the small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Then as the mesh size becomes coarser, in the middle plots, some discrepancies in the power spectrum start to appear which become more evident for even coarser meshes, in the bottom plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This mismatch may be caused by 𝑟𝑏 = 3 𝐿/𝑁 moving towards larger scales, so that the assumption that PM forces are Newtonian on the small scales breaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Indeed, while with 𝐿/𝑁 = 1 ℎ−1 Mpc (𝑟𝑏 = 3 ℎ−1 Mpc) the scales where relativistic effects become evident in the matter power spec- trum and the scales where the pure PM prediction starts to deviate from TreePM are well separated, for larger 𝐿/𝑁 values the two scales get nearer, indicating that the assumption of pure Newtonian forces on the mesh scale may not be very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This conclusion is appar- ently at variance with the discussion of Figure 10, where a larger value of 𝑟𝑏 was preferred;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' however, that figure refers to 𝐿/𝑁 = 1 and is shown at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5, where clustering is a bit weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' swe thus recommend to work with mesh sizes of 𝐿/𝑁 ∼ 1 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 5 CONCLUSIONS We have constructed a relativistic TreePM code, that we call Gr- Gadget, where the large-scale contribution to the gravitational force is computed using the relativistic C++ PM library Libgevolution, based on Gevolution code, while gravity coming from small scales is computed by the Tree code of Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The code works under the assumption that, in the context of cosmological simulations, dark matter can be treated non-relativistically and then the equations of motion of tracer particles tend to the Newtonian limit at scales well below the Hubble horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Following the Gevolution approach, we use a weak field approximation of GR, where the perturbations of the space-time metric with respect to FLRW background are encoded as fields and simulated by the PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Comparing the matter power spec- trum from GrGadget simulations with that of original Gadget-4 and Gevolution codes, we conclude that the code produces consis- tent results as long as the PM cell size 𝐿/𝑁 is smaller than 2 Mpc/ℎ and the gr-smoothing parameter is 𝑟𝑏 ≈ 3 𝐿/𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) GrGadget 11 10 30 10 28 10 26 10 24 10 22 10 20 10 18 Pk TreePM PM 10 2 10 1 100 k/(h Mpc 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 (Pk Pref)/Pref 101 102 103 104 Pk k2 TreePM k2 PM T00 TreePM 10 2 10 1 100 k/(h Mpc 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 (Pk Pref)/Pref Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In the left plot: power spectrum of the metric perturbation Φ in a high_res simulation obtained with GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In the right plot: power spectrum of 𝑘2Φ and 𝑇 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' For modes well below the Hubble horizon and small perturbations it should be verified that 𝑘2 ˜Φ ≈ ˜𝑇 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 10 2 10 1 100 10 29 10 27 10 25 10 23 10 21 10 19 Pk B0 TreePM B0 PM 10 2 10 1 100 k/(h Mpc 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 (Pk Pref)/Pref 10 39 10 37 10 35 10 33 10 31 10 29 10 27 Pk TreePM PM 10 2 10 1 100 k/(h Mpc 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 (Pk Pref)/Pref Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In the left plot: power spectrum of the metric perturbation 𝐵𝑖 (the 𝑥 component) in a high_res simulation obtained with GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In the right plot: power spectrum of 𝜒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' With respect to the pure PM implementation of Gevolution, the predictive power of GrGadget gives an improvement even on the scales sampled by the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is due to the fact that the energy-momentum tensor, that sources the equations of the fields that represent the perturbations of the metric, is computed from a fully non-linear distribution of particles, with gravity being resolved down to a much smaller softening length and not down to the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This may be very useful, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', when assessing the possibility of detecting the frame-dragging effect of a rotating dark-matter halo, if not of a spiral galaxy (Bruni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Furthermore, this code is a development of the widely used Gadget-4 code, and because the PM sector of the code is called only by the computation of the gravity force, our code can be easily extended to simulations of galaxies or galaxy clusters by switching on the hydrodynamics, star formation and feedback sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' All the physics described by these sectors can safely be treated in the Newtownian limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' one should in principle MNRAS 000, 1–14 (2022) 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 102 103 104 P(k)/(Mpc3h−3) camb GrGadget TreePM (rb=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0) GrGadget TreePM (rb=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0) GrGadget TreePM (rb=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5) Gadget4 TreePM 10−2 10−1 100 k/(hMpc−1) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 (P(k) − P(k)Gadget4)/P(k)Gadget4 Nyquist freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Matter power spectrum z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='50 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Power spectrum of matter density for Gadget-4 and GrGadget, on a N256 simulation configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The upper panel shows the absolute value and the lower panel the relative difference with respect to Gadget-4’s TreePM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Different shades of blue indicate different values of the gr-smoothing scale parameter 𝑟𝑏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5, 3, 6 in units of 𝐿/𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The PM smoothing scale is 𝑟𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 𝐿/𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The power spectra in this plot are computed beyond the Nyquist frequency to show the convergence of the matter distribution correlations for distances below the grid resolution, the Tree regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' add thermal energy of gas particles to the energy-momentum tensor, but while this extension is straightforward, it is likely to provide a negligible contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This is, for our group, a further step in the construction of an ecosystem of simulation codes and post-processing tools for model- ing the evolution of structure in the Universe, with the aim of making predictions for precision cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Sub-percent accuracy in cos- mological predictions, that matches the smallness of the statistical error that will be obtained with forthcoming galaxy surveys men- tioned in the Introduction, can only be obtained taking into account relativistic effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Lepori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020), and we can foresee that a self-consistent treatment of these effects (to within the required accuracy) will soon become the standard in cosmological simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' These effects can also be added by post-processing Newtonian simulations, but a validation of these procedures requires validation against a more self-consistent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Conversely, a large com- munity is developing Gevolution in the direction of adding modi- fications of gravity, whose formulation is typically worked out in a general relativistic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' This line of development, coupled with a Newtonian treatment of modified gravity in the Tree code, would be precious in the formulation of tests of gravity, because relativistic effects may hide smoking-gun features of specific classes of modified gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' APPENDIX A: CODE SCALING The code we presented in this work is the merging of two codes whose behaviour in terms of run-time scaling is well-known and characterized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' since we did not modify the underlying algorithms, it is expected that the run-time scaling of our code follows that of the parent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, the Libgevolution’s PM is obviously different from Gadget-4’s, and we added the translation of particles data from the host code to the target relativistic PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Both this facts require that we establish the overall scaling of GrGadget in its fully-relativistic configuration and the overhead associated to both the relativistic PM and the interface between the two codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In figure A1 we show the fraction of time spent in the PM in both the original and relativistic configurations as a function of the grid cell size (see the caption for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The relativistic PM is an order of magnitude more expensive than the original Gadget-4’s Newtonian PM, although in absolute sense it is still either negligible or secondary in the simulation sets that have been tested (it reaches a maximum value of 16% at highest resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' in the 𝑁 = 512,𝐿 = 250 Mpc/ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' However, it scales with both the resolution and the grid number as the original Newtonian PM does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Figures A2 and A3 report the scaling of run time in strong and weak scaling tests respectively for the total run time, the tree time and the PM time (left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' middle and right panels in both figures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' see the captions for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' As inferred from A1, the run-time and hence its scaling, are dominated by the Gadget-4’s Tree section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Julian Adamek for many fruitful discussions on gevolu- tion, Volker Springel for his comments on an early draft, Francesca Lepori, Marco Bruni, Marco Baldi and Emilio Bellini for discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Simulations were performed with the HOTCAT system of INAF (Taffoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Bertocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' PM acknowledges partial support by a Fondo di Ricerca di Ateneo grant of University of Trieste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' DATA AVAILABILITY The simulation codes presented in this paper are publicly available on github in the following path: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='com/GrGadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' REFERENCES Abbott B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2016, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 116, 061102 Adamek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Durrer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Kunz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2014, Classical and Quantum Gravity, 31, 234006 Adamek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Daverio D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Durrer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Kunz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2016, Journal of Cosmology and Astroparticle Physics, 2016, 053 Adamek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Durrer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Kunz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2017, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2017, 004 Adamek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='12457 Alam S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2021, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2021, 050 Barnes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Hut P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 1986, Nature, 324, 446 Barrera-Hinojosa C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Li B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, Journal of Cosmology and Astroparticle Physics, 2020, 007 Barrera-Hinojosa C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Li B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Bruni M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', hua He J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, Vector modes in ΛCDM: the gravitomagnetic potential in dark matter haloes from rela- tivistic 𝑁 -body simulations (arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='08257) Bartelmann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Schneider P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2001, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 340, 291 Bertocco S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, in Pizzo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Deul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Mol J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', de Plaa J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Verkouter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 527, Astronomical Data Analysis Software and Systems XXIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 303 (arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05340) Borzyszkowski M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Bertacca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Porciani C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2017, MNRAS, 471, 3899 Bruni M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Thomas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Wands D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2014, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' D, 89, 044010 Capozziello S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', De Laurentis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2012, Annalen der Physik, 524, 545 Chisari N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Zaldarriaga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2011, Physical Review D, 83 MNRAS 000, 1–14 (2022) GrGadget 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 (P(k) P(k)ref)/P(k)ref N=256 L/N = 1 Mpc/h L/N = 1 Mpc/h L/N = 1 Mpc/h L/N = 1 Mpc/h L/N = 1 Mpc/h L/N = 1 Mpc/h N=512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 (P(k) P(k)ref)/P(k)ref L/N = 2 Mpc/h L/N = 2 Mpc/h L/N = 2 Mpc/h L/N = 2 Mpc/h L/N = 2 Mpc/h L/N = 2 Mpc/h Gadget4 TreePM GrGadget PM GrGadget TreePM 10 2 10 1 100 k/(Mpc 1h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='100 (P(k) P(k)ref)/P(k)ref 10 2 10 1 100 k/(Mpc 1h) L/N = 4 Mpc/h L/N = 4 Mpc/h L/N = 4 Mpc/h L/N = 4 Mpc/h L/N = 4 Mpc/h L/N = 4 Mpc/h Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Matter power spectrum from cosmological simulations at 𝑧 = 0 using GrGadget (the blue lines) and compared to Gadget-4 (the red line) at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The dotted line is obtained with a simulation in which only the PM is used to compute forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The plots show the relative difference with respect to the power spectrum obtained with Gadget-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The left column corresponds to simulations with 𝑁 = 256 grid points per dimension while for the right column 𝑁 = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The boxsize changes along the ranks so that for the top plots the resolution is the highest 𝐿/𝑁 ≈ 1 Mpc/ℎ, in the middle 𝐿/𝑁 ≈ 2 Mpc/ℎ and the bottom plots correspond to 𝐿/𝑁 ≈ 4 Mpc/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' In all cases 𝑟𝑎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='5 𝐿/𝑁 and 𝑟𝑏 = 3 𝐿/𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The grey dashed line indicate a 1% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' DESI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2016, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00036 Daverio D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Hindmarsh M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Bevis N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2015, Latfield2: A c++ library for classical lattice field theory (arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05610) Doré O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2014, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4872 Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2019, ApJ, 875, L1 Ivezić Ž.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2019, ApJ, 873, 111 Krause E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2017, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='09359 Laureijs R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2011, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3193 Lepori F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Adamek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Durrer R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Clarkson C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Coates L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, Monthly Notices of the Royal Astronomical Society, 497, 2078–2095 MNRAS 000, 1–14 (2022) 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Quintana-Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 L/N (Mpc/h) 10 2 10 1 time_pm / time_total GR (N=256) Newton (N=256) GR (N=512) Newton (N=512) Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The fraction of PM time to the total running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Relativistic runs are shown in blue while Newtonian runs are shown in red, whereas symbols distinguish the value of grid points per dimension 𝑁 (squares and circles for 𝑁 = 256 and 512 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We plot the time fraction on the 𝑦–axis (log scale) vs the mesh resolution 𝐿/𝑁 on the 𝑥–axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, A&A, 641, A6 Puchwein E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Baldi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2013, MNRAS, 436, 348 Sefusatti E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Crocce M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Scoccimarro R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Couchman H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2016, mnras, 460, 3624 Silvestri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Trodden M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2009, Reports on Progress in Physics, 72, 096901 Spergel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2015, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='03757 Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2005, Monthly Notices of the Royal Astronomical Society, 364, 1105 Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Zier O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Reinecke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2021, MNRAS, 506, 2871 Taffoni G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Becciani U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Garilli B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Maggio G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Pasian F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Umana G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Smareglia R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Vitello F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', 2020, in Pizzo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Deul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Mol J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', de Plaa J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', Verkouter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=', eds, Astronomical Society of the Pacific Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 527, Astronomical Data Analysis Software and Systems XXIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 307 (arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='01283) This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 1–14 (2022) GrGadget 15 25 50 75 100 125 150 175 200 # process 25 50 75 100 125 150 175 200 speedup x processes_pivot Strong Scalability (Total time) N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 ideal 25 50 75 100 125 150 175 200 # process 25 50 75 100 125 150 175 200 speedup x processes_pivot Strong Scalability (PM time) N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 ideal 25 50 75 100 125 150 175 200 # process 25 50 75 100 125 150 175 200 speedup x processes_pivot Strong Scalability (Tree time) N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=250 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=500 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=1000 N = 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' L=2000 ideal Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Strong-scaling test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We present the code scaling as the number 𝑃 of MPI tasks is increased while running the same simulation set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' All the results refer to GrGadget, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' to the configuration with fully-relativistic PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' On the 𝑥–axis 𝑃 increases from 24 to 192, by ×2 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' On the 𝑦–axis we report the speed-up (normalized so that the ideal speed-up for 𝑃 = 1 is 1) for the total running time, the time spent in the PM and the time spent in the Tree on the Left, Middle and Right panels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Note that the ideal behaviour (black dotted line) would result in a linear speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The PM data includes the translation of particles data from Gadget-4 to Libgevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We show the results for 𝑁 = 128, 256 and 512 (solid, dashed and dot–dashed lines respectively) for 4 different box sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' mass resolutions), 𝐿 = 250, 500, 1000 and 2000 Mpc/ℎ (circles, squares and stars respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' See the discussion in Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' 25 50 75 100 125 150 175 200 # process 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 time / time_pivot Weak scalability (Total time) N = 128->256, L = 250->500 N = 128->256, L = 500->1000 N = 128->256, L = 1000->2000 N = 256->512, L = 250->500 N = 256->512, L = 500->1000 N = 256->512, L = 1000->2000 ideal 25 50 75 100 125 150 175 200 # process 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='0 time / time_pivot Weak scalability (PM time) N = 128->256, L = 250->500 N = 128->256, L = 500->1000 N = 128->256, L = 1000->2000 N = 256->512, L = 250->500 N = 256->512, L = 500->1000 N = 256->512, L = 1000->2000 ideal 25 50 75 100 125 150 175 200 # process 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='35 time / time_pivot Weak scalability (Tree time) N = 128->256, L = 250->500 N = 128->256, L = 500->1000 N = 128->256, L = 1000->2000 N = 256->512, L = 250->500 N = 256->512, L = 500->1000 N = 256->512, L = 1000->2000 ideal Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Weak-scaling test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We present the code scaling as the number 𝑃 of MPI tasks is increased for a proportionally increasing problem, then keeping constant the particles–per–task occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' All the results refer to GrGadget, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' to the configuration with fully-relativistic PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' On the 𝑥–axis 𝑃 increases from 24 to 192 with only 2 test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' On the 𝑦–axis we report the speed-up for the total running time, the time spent in the PM and the time spent in the Tree on the Left, Middle and Right panels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Note that the ideal behaviour would result in a constant running time (horizontal dotted black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' The PM data includes the translation of particles data from Gadget-4 to Libgevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' We show the results for two cases: from 𝑁 = 128, to 𝑁 = 256 (solid lines with circles), and from 𝑁 = 256, to 𝑁 = 512 (dashed lines with squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' Each of the two cases has been run for three different box sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' mass resolutions): 𝐿 = 250 → 𝐿 = 500 Mpc/ℎ, 𝐿 = 500 → 𝐿 = 1000 Mpc/ℎ and 𝐿 = 1000 → 𝐿 = 2000 Mpc/ℎ (red, green and blue colors respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' See the discussion in Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdFKT4oBgHgl3EQfli6N/content/2301.11854v1.pdf'} diff --git a/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/2301.03849v1.pdf.txt b/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/2301.03849v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..90745544aa5d6940e1419338aa1c21457b4a68ad --- /dev/null +++ b/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/2301.03849v1.pdf.txt @@ -0,0 +1,3000 @@ +A UNIVERSAL FRAMEWORK FOR ENTANGLEMENT +DETECTION UNDER GROUP SYMMETRY +SANG-JUN PARK, YEONG-GWANG JUNG, JEONGEUN PARK, +AND SANG-GYUN YOUN +ABSTRACT. One of the most fundamental questions in quantum infor- +mation theory is PPT-entanglement of quantum states, which is an NP- +hard problem in general. In this paper, however, we prove that all PPT +(πA ⊗ πB)-invariant quantum states are separable if and only if all ex- +tremal unital positive (πA, πB)-covariant maps are decomposable where +πA, πB are unitary representations of a compact group and πA is irre- +ducible. Moreover, an extremal unital positive (πB, πA)-covariant map +L is decomposable if and only if L is completely positive or completely +copositive. We apply the results to prove that all PPT quantum channels +of the form +Φ(ρ) = aρ + bρT + cTr(ρ) +d +Idd + (1 − a − b − c)diag(ρ) +are entanglement-breaking, and that all A-BC PPT (U⊗U⊗U)-invariant +tripartite quantum states are A-BC separable. The former resolves some +open questions raised in [DFV08, KMS20], and the latter is a strong +contrast to the fact that there exist PPT-entangled (U ⊗U ⊗U)-invariant +tripartite Werner states [EW01]. +1. INTRODUCTION +Quantum entanglement is one of the most non-classical manifestations of +quantum formalism and is considered a key resource for quantum communi- +cation. Indeed, quantum entanglement plays crucial roles in the existence of +Bell correlations [Bel64, Wer89], quantum cryptography [Eke91, JSW+00, +TBZG00, NPW+00], superdense coding [BW92, MWKZ96], quantum tele- +portation [BBC+93, BPM+97], entanglement-assisted classical communi- +cation [BSST99], and computational supremacy for communication com- +plexity problems [Bra03, BvDHT99, C¯D13]. +The question of whether a given quantum state is entangled or separa- +ble is of fundamental importance in quantum information theory(QIT). It +turned out that this question is NP-hard in general [Gur03, Gha10], so it is +unnatural to expect an efficient general scheme to characterize quantum en- +tanglement. Nevertheless, there have been numerous efforts to characterize +separability in some subclasses of quantum states. For example, a quantum +1 +arXiv:2301.03849v1 [math-ph] 10 Jan 2023 + +2 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +state ρ ∈ D(HA ⊗ HB) of positive partial transpose (PPT) is separable if +dim(HA) × dim(HB) ≤ 6 [HHH96, Wor76b] and, moreover, PPT implies +separability in some bipartite or multipartite systems of low-rank [KCKL00, +HLVC00, EW01, C¯D13]. Classification of entanglement of GHZ states has +also been studied in various contexts [Kay11, G¨11, HK16a, HK16b]. Note +that any PPT entangled state is bound entangled, which is applicable to +perform nonclassical tasks [HHH99, VW02, Mas06] and to produce secure +cryptographic key [HHHO05, HHHO09, HPHH08]. +In this paper, we restrict our interests to the so-called invariant quantum +states in a general context of compact group symmetries. More precisely, +for unitary representations πA : G → B(HA) and πB : G → B(HB) of +a compact group G, a bipartite quantum state ρ ∈ D(HA ⊗ HB) is called +(πA ⊗ πB)-invariant if +(πA(x) ⊗ πB(x))ρ = ρ(πA(x) ⊗ πB(x)) +(1.1) +for all x ∈ G. Werner states and isotropic states are standard examples of in- +variant quantum states for fundamental unitary group symmetries, and their +separability was characterized in [Wer89] and [HH99], respectively. Sep- +arability of invariant quantum states has been studied extensively for vari- +ous group symmetries [EW01, UDUPR07, DPR07, KCL05, TG09, AN14, +SN21, CKK+21]. +The dual objects of invariant quantum states are the so-called (πA, πB)- +covariant quantum channels, which are completely positive trace-preserving +(CPTP) maps L : B(HA) → B(HB) satisfying +L(πA(x)XπA(x)T) = πB(x)L(X)πB(x)∗ +(1.2) +for all X ∈ B(HB) and x ∈ G. Indeed, for a linear map L : B(HA) → +B(HB) and its normalized Choi matrix CL = +1 +dA +� +i,j=1 eij ⊗ L(eij), the +given map L is (πA, πB)-covariant if and only if CL is (πA ⊗ πB)-invariant +(Corollary 2.4). +One of the key observations of this paper is that all PPT (πA ⊗ πB)- +invariant quantum states are separable if and only if all (πB, πA)-covariant +positive maps are decomposable. If πA is irreducible, then the following +three statements are equivalent (Corollary 3.8): +• All PPT (πA⊗πB)-invariant quantum states are separable (PPT=SEP). +• All positive (πB, πA)-covariant maps are decomposable (POS=DEC). +• All PPT (πA, πB)-covariant quantum channels are entanglement- +breaking (PPT=EB). +Moreover, it is enough to consider only extremal elements (Theorem 3.9) +and, in particular, an extremal positive unital (πB, πA)-covariant linear map +L is decomposable if and only if L is completely positive (CP) or com- +pletely copositive (CCP) (Theorem 3.11). + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +3 +Our framework focusing on decomposability of extremal positive unital +(πB, πA)-covariant maps is applicable to study PPT entanglement for con- +crete examples. In Section 4, we prove that EB property coincides with PPT +property, i.e. PPT=EB holds for any quantum channels of the form +Φ(ρ) = aρ + bρT + cTr(ρ) +d +Idd + (1 − a − b − c)diag(ρ). +(1.3) +where diag(X) = +d +� +i=1 +Xiieii for X = (Xij)1≤i,j≤d. An important observa- +tion is that the quantum channels of the form (1.3) are irreducibly covariant +channels with respect to the standard representation of the signed symmetric +group (or the hyperoctahedral group) Hd (Lemma 4.3). Furthermore, we +characterize all positive unital covariant maps of the form (1.3) (Theorem +4.5) and our main theorem (Theorem 3.9) allows us to focus only on eight +extremal positive unital covariant maps to detect EB property for quantum +channels of the form (1.3). Our results strengthen Section 5 and Section 6 +of [KMS20] to a larger class, and resolve some of open questions posed in +[KMS20] and [DFV08]. +In Section 5, we focus on the question of whether all PPT quantum states +are separable, i.e. PPT=SEP problem for some tripartite quantum states +with unitary group symmetries. In Section 5.1, we present explicit positive +non-decomposable covariant linear maps L : Md(C) → Md2(C) satisfying +L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗ +(1.4) +for all d×d unitary matrices U ∈ Ud and X ∈ Md(C). This result gives an- +other explanation of the fact PPT̸=SEP for tripartite Werner states [EW01], +which implies the existence of PPT entangled quantum states ρ ∈ Md3(C) +satisfying +(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) +(1.5) +for all U ∈ U(d). On the other hand, in Section 5.2, we show that a +strong contrast PPT=SEP holds for quantum orthogonally invariant quan- +tum states. More generally, we prove that any PPT tripartite quantum state +ρ ∈ Md3(C) satisfying +(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) +(1.6) +for all unitary matrices U ∈ U(d) is separable (Theorem 5.6). +2. PRELIMINARIES +2.1. Separability and PPT property. In this paper, we focus only on finite- +dimensional complex Hilbert spaces H = Cd, HA = CdA, HB = CdB, and +their direct sums and tensor products. Recall that a quantum state ρ ∈ B(H) +is a positive matrix with Tr(ρ) = 1 and the set of all quantum states in B(H) + +4 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +is denoted by D(H). A bipartite positive operator X ∈ B(HA⊗HB) is said +to be of positive partial transpose (PPT) if +(idA ⊗ TB)(X) ≥ 0 +(2.1) +where TB is the transpose map on B(HB), and X is called separable if +there exist families of positive operators (XA +i )n +i=1 and (XB +i )n +i=1 such that +X = +n +� +i=1 +XA +i ⊗ XB +i . +(2.2) +In particular, if ρ ∈ D(HA ⊗ HB) is a separable quantum state, then there +exists a probability distribution (pi)n +i=1 and a family of product quantum +states (ρA +i ⊗ ρB +i )n +i=1 such that +ρ = +n +� +i=1 +piρA +i ⊗ ρB +i . +(2.3) +It is clear that separability implies PPT property, but the converse is not +true in general. More precisely, all PPT quantum states in B(HA ⊗ HB) +are separable if and only if dA · dB ≤ 6 [Per96, HHH96, Wor76a, Cho82]. +Moreover, it is known that the separability question is NP-hard [Gur03, +Gha10]. +For v ∈ H, we define linear maps |v⟩ : C → H given by λ �→ λv and +⟨v| : H → C given by w �→ ⟨v|w⟩ where ⟨v|w⟩ is the inner product of +v, w ∈ H whose first variable is the anti-linear part. In particular, |Ω⟩ = +�d +i=1 +1 +√ +d|i⟩⊗|i⟩ ∈ H⊗H is called the maximally entangled Bell state where +{|1⟩, |2⟩, · · · , |d⟩} is the standard orthonormal basis of H. The matrix unit +|i⟩⟨j| and the product vector |i1⟩ ⊗ |i2⟩ ⊗ · · · ⊗ |ik⟩ are also denoted by eij +and |i1i2 · · · ik⟩ respectively. +The (normalized) Choi matrix of a linear map L : B(HA) → B(HB) is +defined by +CL = (idA ⊗ L)(|ΩA⟩⟨ΩA|) = (idA ⊗ L) +� +1 +dA +dA +� +i,j=1 +eij ⊗ eij +� +(2.4) += 1 +dA +dA +� +i,j=1 +eij ⊗ L(eij) ∈ B(HA ⊗ HB). +(2.5) +Recall that L is completely positive (CP) if and only if the Choi matrix CL +is positive, and L is trace-preserving (TP) if and only if (idA ⊗TrB)(CL) = +1 +dA +IdA. In particular, if Φ : B(HA) → B(HB) is a CPTP linear map, i.e. a +quantum channel in the Schr¨odinger’s picture, then the Choi matrix CΦ is a +quantum state in D(HA ⊗ HB). We call this channel-state duality. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +5 +Let L : B(HA) → B(HB) be a linear map. Then L is called completely +copositive (CCP) if TB ◦L is completely positive, L is called decomposable +if there exist a CP map L1 and a CCP map L2 such that L = L1 + L2, and +L is called PPT if L is both CP and CCP. Thus, L is PPT if and only if CL +is PPT. +Another important property of quantum channels is the entanglement- +breaking (EB) property. A quantum channel Φ : B(HA) → B(HB) is +called EB if the Choi matrix CΦ is a separable quantum state. Note that any +EB quantum channel is PPT, but the converse is not true in general. +2.2. Invariance and covariance. In this section, we introduce two impor- +tant objects to discuss conservation of symmetry, namely invariant opera- +tors and covariant linear maps. Let us suppose that G is a compact group +throughout this paper. A continuous function π : G → U(Hπ) is called a +(finite-dimensional) unitary representation of G if it is a group homomor- +phism, i.e., +π(xy) = π(x)π(y) +(2.6) +for all x, y ∈ G. In this case, an operator X ∈ B(Hπ) is called π-invariant +if +π(x)Xπ(x)∗ = X +(2.7) +for all x ∈ G. The set of all π-invariant operators, the set of all π-invariant +quantum states, and the set of π-invariant PPT quantum states in B(Hπ) are +denoted by Inv(π), InvQS(π), and InvPPTQS(π), respectively. A unitary +representation π : G → B(Hπ) is called irreducible if Inv(π) = C · IdHπ. +If π is irreducible, so is the contragredient representation π : G → U(Hπ) +of π which is defined by π(x) = π(x) for all x ∈ G. +For unitary representations πA : G → U(HA) and πB : G → U(HB), the +tensor representation πA ⊗ πB : G → U(HA ⊗ HB) is given by +(πA ⊗ πB)(x) = πA(x) ⊗ πB(x) +(2.8) +for all x ∈ G. The tensor representation πA ⊗ πB is not irreducible in +general, but admits the so-called irreducible decomposition, i.e. irreducible +representations σ1, σ2, · · · , σk of G such that +πA ⊗ πB ∼= σ1 ⊕ σ2 ⊕ · · · ⊕ σk. +(2.9) +Here, (σ1 ⊕ σ2 ⊕ · · · ⊕ σk)(x) is the block diagonal matrix of σ1(x), σ2(x), +· · · , σk(x) for all x ∈ G, and π ∼= π′ means that there exists a unitary V +such that π(x) = V π′(x)V ∗ for all x ∈ G. We say that the irreducible +decomposition (2.9) is multiplicity-free if σi ≇ σj for all i ̸= j. This prop- +erty was highlighted in [GBW21] in view of programmability of covariant + +6 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +quantum channels. More generally, we may rewrite (2.9) as +πA ⊗ πB ∼= +l +� +i=1 +σi ⊗ Idmi +(2.10) +meaning that σi ̸∼= σj for all i ̸= j and each irreducible representation σi +appears mi times in the irreducible decomposition (2.9). In this case, we +have the following identification of Inv(πA ⊗ πB) as ∗-algebras [GBW21, +Lemma 6]: +Inv(πA ⊗ πB) ∼= +l +� +i=1 +Idni ⊗ Mmi(C), +(2.11) +where ni = dim Hσi. +For unitary representations πA : G → B(HA) and πB : G → B(HB), a +linear map L : B(HA) → B(HB) is called (πA, πB)-covariant if +L(πA(x)Y πA(x)∗) = πB(x)L(Y )πB(x)∗ +(2.12) +for all x ∈ G and Y ∈ B(HA), and let us denote by Cov(πA, πB) the space +of all (πA, πB)-covariant linear maps. Some subclasses of Cov(πA, πB) in +our interest are as follows: +• CovPos(πA, πB) is the set of all (πA, πB)-covariant positive maps, +• CovPos1(πA, πB) is the set of all (πA, πB)-covariant positive unital +maps, +• CovPosTP(πA, πB) is the set of all (πA, πB)-covariant positive TP +maps, +• CovQC(πA, πB) is the set of all (πA, πB)-covariant CPTP maps, +• CovPPTQC(πA, πB) is the set of all (πA, πB)-covariant PPT quan- +tum channels. +2.3. Twirling operation. An averaging technique called the twirling op- +eration is a standard method to analyze invariant operators and covariant +linear maps. For a unitary representation π : G → U(H), we define a +twirling map Tπ : B(H) → Inv(π) by +Tπ(X) = +� +G +π(x)Xπ(x)∗dx +(2.13) +for all X ∈ B(H), where dx denotes the normalized Haar measure on G. +Then Tπ is unital CPTP, and its well-definedness, i.e. Tπ(X) ∈ Inv(π), +is thanks to the translation-invariance property of the Haar measure. Fur- +thermore, we have X ∈ Inv(π) if and only if Tπ(X) = X, which means +that Tπ is a projection (more precisely, a conditional expectation) onto the +∗-subalgebra Inv(π) of B(H). Note that for any finite dimensional von +Neumann algebra M ⊂ Md(C), there is a unique TP conditional expec- +tation of Md(C) onto M [BO08, Lemma 1.5.11]. For example, the map + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +7 +X ∈ Mn ⊗ Mn �→ +1 +nIdn ⊗ (Trn ⊗ idm)(X) is the unique TP conditional +expectation onto M = Idn ⊗ Mm. This observation allow us to get the +following explicit formula of the twirling map Tπ for the case M = Inv(π). +Proposition 2.1. Suppose that a unitary representation π : G → U(H) +has an irreducible decomposition of the form (2.10) with the identification +Inv(π) ∼= �l +i=1 Idni ⊗ Mmi(C) ⊆ B +��l +i=1 Hi +� +. Let Πi be the orthogonal +projection from H onto Hi = Cni ⊗ Cmi. Then the twirling Tπ(X) of +X ∈ B(H) is given by +Tπ(X) = +l +� +i=1 +1 +ni +Idni ⊗ +� +(Trni ⊗ idmi)(ΠiXΠi) +� +. +(2.14) +In particular, if the irreducible decomposition of π is multiplicity-free, i.e., +if mi ≡ 1 for all i = 1, 2, · · · , l, then +Tπ(X) = +l +� +i=1 +Tr(ΠiX) +ni +Πi. +(2.15) +For unitary representations πA : G → U(HA) and πB : G → U(HB), the +twirling TπA,πBL of L : B(HA) → B(HB) is defined by +(TπA,πBL)(X) = +� +G +πB(x)∗L(πA(x)XπA(x)∗)πB(x) dx +(2.16) +for all X ∈ B(HA). Then similarly, the twirling operation TπA,πB is a well- +defined projection from B(B(HA), B(HB)) onto Cov(πA, πB). +Let us collect some useful properties of the twirling operations for the +next section. +Proposition 2.2. For any unitary representations πA and πB of G, the +twirling map TπA⊗πB preserves separability and PPT property of bipartite +operators. Furthermore, the twirling operation TπA,πB preserves positivity, +CP, TP, CCP, PPT, decomposability, and EB property of linear maps. +Proof. It is straightforward from the definitions and closedness of the spaces +associated with each of the properties mentioned above. For example, the +set of all decomposable linear maps L : B(HA) → B(HB) is closed in +B(B(HA), B(HB)) with respect to the natural (Euclidean) topology. +□ +For a linear map L : B(HA) → B(HB), the adjoint map L∗ : B(HB) → +B(HA) of L is a linear map satisfying +Tr(L(X) Y ) = Tr(X L∗(Y )) +(2.17) +for all X ∈ B(HA) and Y ∈ B(HB). Recall that the adjoint operation +L �→ L∗ preserves positivity, CP, CCP, PPT, and decomposability. + +8 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +Proposition 2.3. Let π : G → U(H), πA : G → U(HA) and πB : G → +U(HB) be unitary representations of G. Then we have the following. +(1) Tr((TπX) Y ) = Tr(X(TπY )) for any X, Y ∈ B(H). +(2) TπA⊗πB◦(TA⊗idB) = (TA⊗idB)◦TπA⊗πB where TA is the transpose +on B(HA). +(3) (TπA,πBL)∗ = TπB,πA(L∗) for any linear map L : B(HA) → B(HB). +(4) The Choi matrix of TπA,πBL is given by TπA⊗πB (CL) for any linear +map L : B(HA) → B(HB). +Proof. (1) Since the Haar measure on the compact group G is invariant +under the inverse x �→ x−1, we have +Tr((TπX) Y ) = +� +G +Tr(π(x)Xπ(x−1)Y )dx +(2.18) += Tr +� +X +� +G +π(x−1)Y π(x)dx +� +(2.19) += Tr +� +X +� +G +π(x)Y π(x−1)dx +� += Tr(X(TπY )) +(2.20) +for any X, Y ∈ B(H). +(2) It suffices to show the equality for product operators X = P ⊗Q, and +the conclusion follows immediately from the observation +πA(x)P TπA(x)∗ = +� +πA(x)PπA(x)T�T +. +(2.21) +(3) For any X ∈ B(HA) and Y ∈ B(HB), we have +Tr(X (TπB,πAL∗) (Y )) +(2.22) += +� +G +Tr(XπA(x)∗L∗(πB(x)Y πB(x)∗)πA(x))dx +(2.23) += +� +G +Tr(πB(x)∗L(πA(x)XπA(x)∗)πB(x) Y )dx +(2.24) += Tr((TπA,πBL) (X) Y ), +(2.25) +which gives us the desired conclusion. +(4) First of all, note that +dA +� +i,j=1 +(πA(x)eijπA(x)t) ⊗ (πB(x)L(eij)πB(x)∗) +(2.26) += +dA +� +i,j=1 +eij ⊗ (πB(x)L(πA(x)∗eijπA(x))πB(x)∗). +(2.27) + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +9 +for each x ∈ G. Indeed, the LHS (2.26) can be understood as +dA(idA ⊗ (AdπB(x) ◦ L)) +� +(πA(x) ⊗ IdA)|ΩA⟩⟨ΩA|(πA(x)t ⊗ IdA) +� +, +(2.28) +and the RHS (2.27) can be understood as +dA(idA ⊗ (AdπB(x) ◦ L)) ((IdA ⊗ πA(x)∗)|ΩA⟩⟨ΩA|(IdA ⊗ πA(x))) +(2.29) +where AdV (Y ) = V Y V ∗. Moreover, the so-called ricochet property +(X ⊗ IdA)|ΩA⟩ = (IdA ⊗ Xt)|ΩA⟩, X ∈ B(HA), +(2.30) +implies (2.28) = (2.29). Finally, taking the Haar integral on both sides com- +pletes the proof. +□ +Combining Proposition 2.3 (2), (3), and (4) with the fact that both Inv(πA⊗ +πB) and Cov(πA, πB) are the images of the twirling projections, we obtain +the following useful properties. +Corollary 2.4. Let X ∈ B(HA ⊗ HB) be a bipartite operator and L : +B(HA) → B(HB) be a linear map. Then +(1) X ∈ Inv(πA ⊗ πB) if and only if (TA ⊗ id)(X) ∈ Inv(πA ⊗ πB). +(2) L ∈ Cov(πA, πB) if and only if L∗ ∈ Cov(πB, πA). +(3) L ∈ Cov(πA, πB) if and only if CL ∈ Inv(πA ⊗ πB). +Remark 2.5. The results in Corollary 2.4 have been noted in various con- +texts, [EW01, Lemma 6], [GBW21, Lemma 11], and [LY22, Proposition +5.1, Theorem 3.5] for examples. Moreover, extendibility to more general +contexts of compact quantum group symmetry was proved in [LY22]. +3. A FRAMEWORK TO CHARACTERIZE ENTANGLEMENT UNDER GROUP +SYMMETRY +Let us recall a result of Horodecki on the characterization of entangle- +ment [HHH96]: a bipartite quantum state ρ ∈ D(HA ⊗ HB) is separable if +and only if (idA ⊗ L)(ρ) ≥ 0 for all positive linear maps L : B(HB) → +B(HA). Indeed, by duality arguments, the authors showed that they are +also equivalent to seemingly a weaker condition ‘⟨ρ, L⟩ ≥ 0’. Here, the +dual pairing ⟨·, ·⟩ is defined by +⟨X, N⟩ = Tr((idA ⊗ N)(X) |ΩA⟩⟨ΩA|) = Tr(XCN ∗) +(3.1) +for any operator X ∈ B(HA ⊗HB) and linear map N : B(HB) → B(HA). +In other words, positive linear maps play a crucial role as detectors for the +bipartite entanglement, in the sense that there should exist a positive linear +map L such that (id ⊗ L)(ρ) is non-positive whenever ρ is entangled. + +10 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +One technical issue in this characterization is that verifying whether a +linear map is positive or not is computationally intractable [Las10, NZ16]. +However, one of the main purposes of this paper is to develop a universal +framework to characterize separability of invariant quantum states. The +key ideas is that, for ρ ∈ InvQS(πA ⊗ πB), it is enough to consider only +L ∈ CovPos(πB, πA) to investigate separability of ρ. Let us begin with a +simple and useful lemma. +Lemma 3.1. For any bipartite operator X ∈ B(HA ⊗ HB) and linear map +L : B(HB) → B(HA), we have +⟨TπA⊗πBX, L⟩ = ⟨X, TπB,πAL⟩. +(3.2) +Proof. Thanks to Proposition 2.3, we have +⟨TπA⊗πBX, L⟩ = Tr((TπA⊗πBX)CL∗) +(3.3) += Tr(X(TπA⊗πBCL∗)) = Tr(XCL∗) = ⟨X, L⟩, +where L = (TπA,πBL∗)∗ = TπB,πAL. +□ +Then Lemma 3.1 allows us to conclude that covariant positive linear +maps are enough to characterize separability of bipartite invariant quantum +states. We remark that the ideas of the following proof appeared already for +some specified symmetries [Kay11, G¨11, SN21]. +Theorem 3.2. Let πA : G → B(HA) and πB : G → B(HB) be unitary +representations, and let ρ ∈ InvQS(πA⊗πB). The following are equivalent. +(1) ρ is a separable quantum state. +(2) (idA ⊗ L)(ρ) ≥ 0 in B(HA ⊗ HA) for any (πB, πA)-covariant pos- +itive linear map L : B(HB) → B(HA). +(3) ⟨ρ, L⟩ ≥ 0 for any (πB, πA)-covariant positive linear map L : +B(HB) → B(HA). +Proof. Two directions (1) ⇒ (2) and (2) ⇒ (3) are clear. For the last +direction (3) ⇒ (1), let us show that ⟨ρ, L⟩ ≥ 0 for all positive linear +maps L : B(HB) → B(HA). Indeed, since ρ is πA ⊗ πB-invariant and +TπB,πAL ∈ CovPos(πB, πA), we can apply Lemma 3.1 to obtain +⟨ρ, L⟩ = ⟨TπA⊗πBρ, L⟩ = ⟨ρ, TπB,πAL⟩ ≥ 0. +(3.4) +□ +From now on, let us focus on the question of whether PPT property coin- +cides with separability, i.e. PPT=SEP problem for invariant quantum states. +Recall that the dual notions of PPT property and separability correspond +to decomposability and positivity respectively. Indeed, many duality argu- +ments [Kye23] carry over into our framework, and the PPT=SEP problem + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +11 +in InvQS(πA ⊗ πB) is equivalent to the question whether all positive maps +are decomposable, i.e. POS=DEC problem in Cov(πB, πA). +Proposition 3.3. Let L : B(HB) → B(HA) be (πB, πA)-covariant. Then +(1) L is positive if and only if (idA ⊗ L)(ρ) ≥ 0 for any separable +ρ ∈ InvQS(πA ⊗ πB). +(2) L is decomposable if and only if (idA ⊗ L)(ρ) ≥ 0 for any PPT +ρ ∈ InvQS(πA ⊗ πB). +Proof. (1) Suppose (idA⊗L)(ρ) ≥ 0 for any separable ρ ∈ InvQS(πA⊗πB). +Then for every separable state ρ ∈ D(HA ⊗ HB), we have +⟨ρ, L⟩ = ⟨ρ, TπB,πAL⟩ = ⟨TπA⊗πBρ, L⟩ ≥ 0 +(3.5) +by Lemma 3.1 and by the separability of TπA⊗πBρ. Now positivity of L +follows from [EK00, Theorem 3.1]. The converse direction is clear. +(2) It is enough to repeat the arguments from (1) based on the following +duality result [Sr82]: L is decomposable if and only if ⟨ρ, L⟩ ≥ 0 for every +PPT state ρ. +□ +Corollary 3.4. The following are equivalent: +(1) PPT=SEP in InvQS(πA ⊗ πB). +(2) POS=DEC in Cov(πB, πA). +Proof. ((1) ⇒ (2)) If L ∈ CovPos(πB, πA), then (idA ⊗ L)(ρ) ≥ 0 for +every separable (hence every PPT) state ρ ∈ InvQS(πA ⊗ πB). Thus, L is +decomposable by Proposition 3.3. +((2) ⇒ (1)) If ρ ∈ InvQS(πA ⊗πB) is a PPT state, then (idA ⊗L)(ρ) ≥ 0 +for every decomposable (hence every positive) linear map L ∈ Cov(πB, πA). +Thus, ρ is separable by Theorem 3.2. +□ +Note that PPT=SEP in InvQS(πA ⊗ πB), or equivalently POS=DEC in +Cov(πB, πA), implies that PPT property coincides with the entanglement- +breaking property, i.e. PPT=EB in CovQC(πA, πB). Moreover, we have +the following characterization of EB property for Φ ∈ CovQC(πA, πB) by +Corollary 2.4 (3). +Corollary 3.5. Let Φ ∈ CovQC(πA, πB). Then the following are equiva- +lent. +(1) Φ is entanglement-breaking. +(2) CL◦Φ = (id ⊗ L)(CΦ) ≥ 0 for any L ∈ CovPos(πB, πA). +(3) L ◦ Φ is completely positive for any L ∈ CovPos(πB, πA). +To summarize, we have +PPT=SEP in InvQS(πA ⊗ πB) ⇔ DEC=POS in Cov(πB, πA), + +12 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +and these conditions imply PPT=EB in CovQC(πA, πB). One might ask +whether all these three problems are equivalent, but one technical issue is +that CovQC(πA, πB) is not identified with InvQS(πA ⊗πB) in general. This +leads us to question whether the (reduced) channel-state duality +�C : CovQC(πA, πB) → InvQS(πA ⊗ πB) +(3.6) +is bijective. The channel-state duality �C is not surjective in general, but it +is known to be the case if πA is irreducible, as already noted in [GBW21, +Lemma 15]. Moreover, we prove that the converse is also true, i.e. the +channel-state duality �C is bijective if and only if πA is irreducible. Let us +start with the following lemma. +Lemma 3.6. Let πA : G → U(HA) and πB : G → U(HB) be unitary +representations of G and let L ∈ Cov(πA, πB). +(1) If πB is irreducible, then L(IdA) = c IdB for some constant c. +(2) If πA is irreducible, then there is a constant c such that Tr(L(X)) = +c Tr(X) for X ∈ B(HA). +Proof. +(1) From the irreducibility of πB and the identity +πB(x)L(IdA)πB(x)∗ = L(πA(x)πA(x)∗) = L(IdA), +(3.7) +we have L(IdA) ∈ Inv(πB) = C · IdB. +(2) The adjoint map L∗ is (πB, πA)-covariant by Corollary 2.4 (2), so +L∗(IdB) = c IdA for some c by (1). In this case, we have +Tr(L(X)) = Tr(L(X) IdB) = Tr(X L∗(IdB)) = c Tr(X) +(3.8) +for any X ∈ B(HA). +□ +Now, let us apply Lemma 3.6 (2) to prove that the channel-state duality +�C : CovQC(πA, πB) → InvQS(πA ⊗ πB) should be bijective if and only if +πA is irreducible. +Proposition 3.7. Let πA : G → U(HA) and πB : G → U(HB) be unitary +representations of G and let L ∈ Cov(πB, πA). Then the channel-state +duality +�C : CovQC(πA, πB) → InvQS(πA ⊗ πB) +(3.9) +is bijective if and only if πA is irreducible. +Proof. Let us prove the if part first. For any ρ ∈ InvQS(πA ⊗ πB) there +exists completely positive L ∈ Cov(πA, πB) such that CL = ρ by Corollary +2.4 (3). Moreover, L should be trace-preserving. Indeed, irreducibility of + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +13 +πA implies that there exists a constant c such that Tr(L(X)) = cTr(X) for +all X ∈ B(HA) by Lemma 3.6 (2), and we have +c = c +dA +dA +� +i=1 +Tr(eii) = 1 +dA +dA +� +i=1 +Tr(eii ⊗ L(eii)) = Tr(CL) = 1. +(3.10) +Conversely, if we assume that πA = π(1) +A ⊕ π(2) +A with HA = H(1) +A ⊕ H(2) +A +and if Π1 is the orthogonal projection from HA onto H(1) +A , then we can take +a CP non-TP map L : B(HA) → B(HB) given by +L(X) = +dA +dB · dim H(1) +A +Tr(Π1X)IdB +(3.11) +whose Choi matrix is +CL = +� +1 +dim H(1) +A +Π1 +� +⊗ +� 1 +dB +IdB +� +∈ InvQS(πA ⊗ πB). +(3.12) +□ +Corollary 3.8. Let πA : G → U(HA) and πB : G → U(HB) be unitary +representations of G and suppose that πA is irreducible. Then the following +are equivalent. +(1) PPT=SEP in InvQS(πA ⊗ πB). +(2) PPT=EB in CovQC(πA, πB). +(3) POS=DEC in CovPos1(πB, πA). +Proof. It is enough to note that Lemma 3.6 allows us to focus on a smaller +convex set CovQC(πA, πB) and CovPos1(πB, πA) rather than Cov(πA, πB) +and CovPos(πB, πA), respectively, in Corollary 3.4. +□ +Moreover, we prove that the extreme points of CovPos1(πB, πA) are enough +for the entanglement detection, which we propose as a universal machinery +to characterize entangled invariant quantum states with general compact +group symmetries. Let us denote by Ext(S) the set of all extreme points of +a convex set S. +Theorem 3.9. Let πA : G → B(HA) and πB : G → B(HB) be unitary +representations, and suppose that πA is irreducible. Let ρ ∈ InvQS(πA ⊗ +πB) and Φ ∈ CovQC(πA, πB) such that CΦ = ρ from Proposition 3.7. The +following are equivalent. +(1) ρ is separable. +(2) (id ⊗ L)(ρ) ≥ 0 for any L ∈ CovPos1(πB, πA). +(3) (id ⊗ L)(ρ) ≥ 0 for any L ∈ Ext (CovPos1(πB, πA)). +(4) Φ is entanglement-breaking. +(5) L ◦ Φ is completely positive for any L ∈ CovPos1(πB, πA). + +14 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +(6) L ◦ Φ is completely positive for any L ∈ Ext (CovPos1(πB, πA)). +Proof. The equivalences (1) ⇔ (4), (2) ⇔ (5), (3) ⇔ (6) and one-side impli- +cations (1) ⇒ (2) ⇒ (3) are clear. Moreover, the direction (2) ⇒ (1) follows +from Lemma 3.6 (1). For the proof of (3) ⇒ (2), note that CovPos1(πB, πA) +is a compact subset of +{L ∈ B(B(HB), B(HA)) : ∥L∥op ≤ 1} , +(3.13) +where ∥ · ∥op denotes the operator norm with respect to Hilbert-Schmidt +norms on B(HB) and B(HA), since the positivity of L implies ∥L∥op = +∥L(IdB)∥ = 1 [Pau02, Corollary 2.9]. Therefore, CosPos1(πB, πA) can be +written as a convex hull of its extreme points, which completes the proof. +□ +Remark 3.10. If πB is irreducible instead of irreducibility of πA, then Φ +is chosen to be a unital CP map up to constant, and CovPosTP(πB, πA) +replaces the role of CovPos1(πB, πA) in (2), (3), (5), (6). Note that compact- +ness of CovPosTP(πB, πA) comes from the identification with CovPos1(πA, πB) +(up to constant) via taking the adjoint operation. +Finally, we claim that the decomposability of the extremal elements in +CovPos1(πB, πA) is much easier to check thanks to the following theorem. +Theorem 3.11. Suppose that πA (resp. πB) is irreducible, and let L ∈ +Ext(CovPos1(πB, πA)) (resp. L ∈ Ext(CovPosTP(πB, πA)). Then L is de- +composable if and only if L is CP or CCP. +Proof. Let us focus only on the case where πA is irreducible since the other +case is analogous. If L is decomposable, then there exist a CP map L1 and +a CCP map L2 such that L = L1 + L2. By taking the twirling operation +TπB,πA, we have L = L′ +1 + L′ +2 where L′ +i = TπB,πA(Li) ∈ Cov(πB, πA). +Note that L′ +1 is CP, L′ +2 is CCP, and we can write L′ +i = λiL′′ +i for some +λi ≥ 0, λ1 + λ2 = 1, and L′′ +i ∈ CovPos1(πB, πA) by Lemma 3.6 (1). Then +extremality of L allows us to conclude that L = L′′ +1 or L = L′′ +2, which +proves the assertion. The other direction is immediate. +□ +To summarize, our strategy to study PPT=SEP and PPT=EB problems +consists of the following three independent steps, assuming πA is irreducible. +[Step 1] The first step is to characterize all elements in CovPos1(πB, πA) for +given specific unitary representations πA and πB. +[Step 2] The next step is to solve POS=DEC problem in CovPos1(πB, πA). +In particular, for extremal elements L ∈ Ext(CovPos1(πB, πA)), the +given L is decomposable if and only if L is CP or CCP. If POS=DEC +holds, then PPT=SEP problem in InvQS(πA ⊗ πB) and PPT=EB +problem in CovQC(πA, πB) has the affirmative answer. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +15 +[Step 3] If there exists a non-decomposable element in CovPos1(πB, πA), +then the last step is to find the following objects: +– Φ ∈ CovPPTQC(πA, πB) for which L ◦ Φ is non-CP, +– ρ ∈ InvPPTQS(πA ⊗ πB) for which (id ⊗ L)(ρ) ≱ 0. +4. PPT=EB HOLDS FOR (H, H)-COVARIANT QUANTUM CHANNELS +One of the main applications of the results in Section 3 is a complete char- +acterization of EB property for quantum channels Φ : Md(C) → Md(C) of +the form +Φ(X) = aTr(X) +d +Idd + bX + cXT + (1 − a − b − c) diag(X). +(4.1) +The main result of this section is as follows. +Theorem 4.1. Let Φ be a quantum channel of the form (4.1). Then Φ is +entanglement-breaking if and only if Φ is PPT. +Remark 4.2. +(1) The quantum channels of the form (4.1) include two +important one-parameter families of quantum channels, namely de- +polarizing channels +∆b(X) = (1 − b)Tr(X) +d +Idd + bX +(4.2) +with − +1 +d2−1 ≤ b ≤ 1, and transpose depolarizing channels +Λc(X) = (1 − c)Tr(X) +d +Idd + cXT +(4.3) +with − +1 +d−1 ≤ c ≤ +1 +d+1. It was already known that PPT=EB for these +channels from various perspectives [Wer89, HH99, Wat18, SN21], +and our Theorem 4.1 covers these classes. +(2) Moreover, Theorem 4.1 gives an affirmative answer to PPT=EB +problem for the so-called generalized Werner-Holevo quantum chan- +nels, which was once conjectured to be false [DFV08]. See Appen- +dix A for more details on the generalized Werner-Holevo channels. +A starting point for a proof of Theorem 4.1 is to observe that any quantum +channel of the form (4.1) is covariant with respect to the signed symmetric +group Hd. One of the equivalent ways to realize the signed symmetric group +is to define Hd as a subgroup of the orthogonal group Od generated by +permutation matrices and diagonal orthogonal matrices. In other words, +every element in Hd is written as an orthogonal matrix +d +� +i=1 +si|σ(i)⟩⟨i| for +s1, s2, . . . , sn ∈ {±1} and σ ∈ Sd. We define Inv(H ⊗ H) and Cov(H, H) + +16 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +with respect to the fundamental representation H ∈ Hd �→ H ∈ Od, which +is irreducible as proved below. +Lemma 4.3. The fundamental representation H ∈ Hd �→ H ∈ Od is +irreducible. +Proof. The identity +HXHT = +d +� +i,j=1 +sisjXij|σ(i)⟩⟨σ(j)| = +d +� +i,j=1 +sσ−1(i)sσ−1(j)Xσ−1(i)σ−1(j)|i⟩⟨j| +(4.4) +and the invariance property HXHT = X for all H ∈ Hd tell us that +sσ(i)sσ(j)Xσ(i)σ(j) = Xij +(4.5) +for all s1, . . . , sd ∈ {±1} and σ ∈ Sd. This implies that Xii ≡ X11 for all +1 ≤ i ≤ d and Xij = 0 for all i ̸= j, i.e., X = X11 Idd ∈ C · Idd. +□ +Let us denote by W the space of linear maps spanned by the following +four unital TP maps ψ0, ψ1, ψ2, ψ3 : Md(C) → Md(C), where +� +� +� +� +� +� +� +ψ0(X) = Tr(X) +d +Idd, +ψ1(X) = X, +ψ2(X) = XT, +ψ3(X) = diag(X) = �d +i=1 Xii|i⟩⟨i|. +(4.6) +It is straightforward to check ψi ∈ Cov(H, H) for i = 0, . . . , 3, so we have +W ⊆ Cov(H, H). To prove Cov(H, H) = W, let us note the fact that +any L ∈ Cov(H, H) satisfies the so-called diagonal orthogonal covariance +(DOC) property, i.e. +L(ZXZT) = ZL(X)ZT +(4.7) +for all X ∈ Md(C) and diagonal orthogonal matrices Z. This class of +channels has been analyzed recently in [SN21, SN22, SDN22]. In partic- +ular, it is shown that any DOC map L can be parameterized by a triple +(A, B, C) ∈ Md(C)3 satisfying diag(A) = diag(B) = diag(C) such that +L(X) = diag(A|diag X⟩) + �B ⊙ X + �C ⊙ XT, +(4.8) +where |diag Y ⟩ = �d +i=1 Yii|i⟩, �Y = Y − diag(Y ), and ⊙ denotes the Schur +product (or Hadamard product) between matrices. In this case, let us denote +by L = LA,B,C. +Proposition 4.4. The space Cov(H, H) is spanned by the four unital TP +positive maps ψ0, ψ1, ψ2, and ψ3 from (4.6). + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +17 +Proof. We already know W ⊆ Cov(H, H), and let us pick an arbitrary +L ∈ Cov(H, H). Since L is DOC, there exists (A, B, C) ∈ Md(C)3 such +that L = LA,B,C of the form (4.8). Note that L further satisfies +L(PσXP T +σ ) = PσL(X)P T +σ +(4.9) +for all X ∈ Md(C) and σ ∈ Sd. Here, Pσ = �d +i=1 |σ(i)⟩⟨i| is the permuta- +tion matrix associated with σ. +Let us take X = eij. If i = j, then (4.9) implies +d +� +k=1 +Akσ(i)|k⟩⟨k| = +d +� +k=1 +Aki|σ(k)⟩⟨σ(k)|, +(4.10) +which means that Aik = Aσ(i)σ(k) for all 1 ≤ i, k ≤ d and σ ∈ Sd. There- +fore, Aii ≡ A11 for all i and Aik ≡ A12 for all i ̸= k. On the other hand, if +i ̸= j, then (4.9) becomes +Bσ(i)σ(j)|σ(i)⟩⟨σ(j)| + Cσ(j)σ(i)|σ(j)⟩⟨σ(i)| +(4.11) += Bij|σ(i)⟩⟨σ(j)| + Cji|σ(j)⟩⟨σ(i)|, +(4.12) +which gives Bij ≡ B12 and Cij ≡ C12 for all i ̸= j. Consequently, the +formula (4.8) now gives +L = dA12ψ0 +B12ψ1 +C12ψ2 +(A11 −A12 −B12 −C12)ψ3 ∈ W, (4.13) +which in turn shows Cov(H, H) ⊆ W. +□ +From now, let us denote (H, H)-covariant unital (and TP) maps by +ψa,b,c = aψ0 + bψ1 + cψ2 + (1 − a − b − c)ψ3 +(4.14) +for simplicity, where ψ0, . . . , ψ3 are from (4.6). Note that ψa,b,c can be +understood as a DOC map LA,B,C under the correspondence +� +� +� +� +� +A = a +dJd + (1 − a)Idd, +B = b(Jd − Idd) + ( a +d + (1 − a))Idd, +C = c(Jd − Idd) + ( a +d + (1 − a))Idd, +(4.15) +where Jd = �d +i,j=1 eij. According to [SN21, Section 6], LA,B,C is CPTP if +and only if +� +� +� +� +� +Aij ≥ 0, �d +k=1 Akj = 1 for all i, j +B ≥ 0, +Cij = Cji, |Cij|2 ≤ AijAji for all i, j. +(4.16) + +18 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +In terms of the parameters a, b, c, the map ψa,b,c is CPTP if and only if +� +� +� +0 ≤ a ≤ +d +d−1, +a +d − +1 +d−1 ≤ b ≤ 1 − d−1 +d a, +− a +d ≤ c ≤ a +d. +(4.17) +Note that the set of (a, b, c) ∈ R3 satisfying (4.17) is a tetrahedral depicted +in Figure 1. +FIGURE 1. The region of CovQC(H, H) +In particular, there are exactly four extremal (H, H)-covariant quantum +channels corresponding to the four vertices given by +� +� +� +� +� +� +� +� +� +Ψ1 = ψ1, +Ψ2 = +d +d−1ψ0 + +1 +d−1ψ2 − +2 +d−1ψ3, +Ψ3 = − +1 +d−1ψ1 + +d +d−1ψ3, +Ψ4 = +d +d−1ψ0 − +1 +d−1ψ1. +(4.18) +whose Choi matrices are (up to normalization) four mutually orthogonal +projections. On the other hand, it is easy to see that +Td ◦ ψa,b,c = ψa,b,c ◦ Td = ψa,c,b, a, b, c ∈ C. +(4.19) +Therefore, ψa,b,c is a PPT quantum channel if and only if both ψa,b,c and +ψa,c,b are CPTP. Let us denote by CovPPTQC(H, H) the set of all PPT +quantum channels ψa,b,c. Then CovPPTQC(H, H) can be realized as a con- +vex set in R3 as in the following Figure 2. +If d ≥ 3, the eight vertices of the polytope CovPPTQC(H, H) are explic- +itly given by ψvi (i = 1, . . . , 8), where +• v0 = (0, 0, 0), + +亚4 +亚2 +1 +1 +- +- +- +1 +- +- +- +- +1 +1 +亚ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +19 +FIGURE 2. The region of CovPPTQC(H, H) +• v1 = +� +d +2(d−1), +1 +2(d−1), − +1 +2(d−1) +� +, v2 = +� +d +2(d−1), − +1 +2(d−1), +1 +2(d−1) +� +, +v3 = +� +d +2(d−1), − +1 +2(d−1), − +1 +2(d−1) +� +, +• v4 = +� +1, 1 +d, 1 +d +� +, v5 = +� +1, 1 +d, − +1 +d(d−1) +� +, v6 = +� +1, − +1 +d(d−1), 1 +d +� +, +• v7 = +� +d +d−1, 0, 0 +� +. +[Step 1+Step 2] One of the main steps in this section is to characterize +all elements in CovPos1(H, H). Indeed, CovPos1(H, H) is given as follows +with eight extreme points (see Figure 3). +FIGURE 3. The region of CovPos1(H, H) + +- +1 +1 +V5 +V6 +. +1 +- +- +1 +V3 +1 +V2 +1 +- +1 +- +1 +4 +1 +1 +1 +- +- +1 +- +o +W亚 +亚2 +亚3 +I3 0 Td +ioTd +2 +亚20 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +Theorem 4.5. Let d ≥ 3. Then the convex set CovPos1(H, H) has exactly +8 extreme points +Ψ1, Ψ2, Ψ3, Ψ4, +Ψ1 ◦ Td, Ψ2 ◦ Td, Ψ3 ◦ Td, Ψ4 ◦ Td, +(4.20) +where Ψ1, . . . , Ψ4 are given by (4.18). In particular, all positive (H, H)- +covariant maps are decomposable. +Proof. Since Ψ1, . . . , Ψ4 are CP and Ψ1◦Td, . . . , Ψ4◦Td are CCP, the convex +hull Vd of these 8 maps is obviously contained in CovPos1(H, H). To show +the reverse inclusion CovPos1(H, H) ⊆ Vd, we observe that the set +Vd := +� +(a, b, c) ∈ R3 : ψa,b,c ∈ Vd +� +⊂ R3 +(4.21) +is the convex hull of 8 points +� (0, 1, 0) , +� +d +d−1, 0, +1 +d−1 +� +, +� +0, − +1 +d−1, 0 +� +, +� +d +d−1, 0, − +1 +d−1 +� +, +(0, 0, 1) , +� +d +d−1, +1 +d−1, 0 +� +, +� +0, 0, − +1 +d−1 +� +, +� +d +d−1, − +1 +d−1, 0 +� +, +(4.22) +which are got from (4.18) and (4.19). Therefore, Vd can be understood as +the region of (a, b, c) ∈ R3 satisfying the following inequalities: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(1) +0 ≤ a ≤ +d +d−1, +(2) +d−2 +d a + b + c ≤ 1, +(3) +d−2 +d a + |b − c| ≤ 1, +(4) +b + c ≥ − +1 +d−1, +(5) +b − (d − 1) c ≤ 1, +(6) +c − (d − 1) b ≤ 1. +(4.23) +Now if ψa,b,c /∈ Vd (which is equivalent to (a, b, c) /∈ Vd, and hence violates +at least one of the inequalities (1) - (6) in (4.23)), we can choose a unit +vector ξ ∈ Cd such that ψa,b,c(|ξ⟩⟨ξ|) is not positive semidefinite as in Table +1. This shows CovPos1(H, H) ⊆ Vd. +TABLE 1. Non-positivity outside Vd +(a, b, c) violates (1) +|ξ⟩ = |1⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 +(a, b, c) violates (2) +|ξ⟩ = +1 +√ +2(|1⟩ + |2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 +(a, b, c) violates (3) +|ξ⟩ = +1 +√ +2(|1⟩ + i|2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 +(a, b, c) violates (4) +|ξ⟩ = +1 +√ +d +d +� +k=1 +|k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 +(a, b, c) violates (5) or (6) +|ξ⟩ = +1 +√ +d +d +� +k=1 +e +2πik +d |k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 +□ + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +21 +Proof of Theorem 4.1. Now, the conclusion is straightforward from Corol- +lary 3.8 and Theorem 4.5. +□ +Remark 4.6. +(1) Note that Theorem 4.5 gives a complete characteri- +zation of all positive linear maps ψ spanned by ψ0, ψ1, ψ2, ψ3. This +strengthens the results in Section 5 of [KMS20] focusing on positive +linear maps spanned only by ψ0, ψ1, ψ3 without ψ2. +(2) Theorem 4.5 tells us not only POS=DEC, but also explicit decom- +positions of our positive covariant maps into sums of CP and CCP +maps. Note that this was one of the open questions raised in Section +6.c of [KMS20]. We refer to Appendix B for more details. +5. PPT=SEP PROBLEMS IN TRIPARTITE SYSTEMS WITH UNITARY +GROUP SYMMETRIES +Recall that a tripartite quantum state ρ ∈ D(HA ⊗ HB ⊗ HC) is called +A-BC separable (resp. A-BC PPT) if ρ is separable (resp. PPT) in the +situation where B(HA ⊗ HB ⊗ HC) is understood as the bipartite system +B(HA) ⊗ B(HB ⊗ HC). Furthermore, C-AB or B-AC separability (resp. +PPT) is defined similarly. We will focus on the situation where HA = HB = +HC = Cd, and let us denote by +� +� +� +XTA = (Td ⊗ idd2)(X), +XTB = (idd ⊗ Td ⊗ idd)(X), +XTC = (idd2 ⊗ Td)(X), +(5.1) +the three partial transposes of X ∈ B(HA ⊗ HB ⊗ HC) = Md3(C). +The main purpose of this section is to apply our results in Section 3 as +new sources to study PPT=SEP problems, equivalently POS=DEC prob- +lems for some tripartite invariant quantum states. In Section 5.1, we exhibit +positive non-decomposable covariant maps L : Md(C) → Md2(C) satisfy- +ing +L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗ +(5.2) +for all unitary matrices U ∈ Ud and X ∈ Md(C). This result is parallel +to the fact PPT̸=SEP for tripartite Werner states [EW01], i.e. tripartite +quantum states ρ ∈ Md3(C) satisfying +(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) +(5.3) +for all unitary matrices U ∈ U(d). +On the other hand, in Section 5.2, we show that a strong contrast PPT=SEP +holds for quantum orthogonally invariant quantum states. More generally, +we prove that PPT=SEP holds for any tripartite quantum states ρ ∈ Md3(C) +satisfying +(U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) +(5.4) + +22 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +for all unitary matrices U ∈ U(d). +5.1. Tripartite Werner states. Let πA, πBC be unitary representations of +the unitary group Ud given by πA(U) = U and πBC(U) = U ⊗ U. Then +the elements in InvQS(πA ⊗ πBC) are called tripartite Werner states. Let +us write Inv(U ⊗3) = Inv(πA ⊗ πBC) and Cov(U, UU) = Cov(πA, πBC) +for simplicity. The application of Schur-Weyl duality [EW01] or von Neu- +mann’s bicommutant theorem [Wat18, Theorem 7.15] implies that the space +Inv(U ⊗3) is spanned by six unitary operators {Vσ : σ ∈ S3}. Here, Vσ : +(Cd)⊗3 → (Cd)⊗3 is determined by Vσ(ξ1 ⊗ ξ2 ⊗ ξ3) = ξσ−1(1) ⊗ ξσ−1(2) ⊗ +ξσ−1(3) for any ξ1, ξ2, ξ3 ∈ Cd and σ ∈ S3, or equivalently, +Vσ = +d +� +j1,j2,j3=1 +|j1j2j3⟩⟨jσ(1)jσ(2)jσ(3)|. +(5.5) +Recall that A-BC PPT property and separability of ρ ∈ InvQS(U ⊗3) +were already characterized in [EW01], and it was shown that PPT=SEP if +and only if d = 2. Therefore, a direct application of Corollary 3.4 gives us +the following result. +Theorem 5.1. All positive (UU, U)-covariant maps are decomposable if +and only if d = 2. By taking the adjoint operation L �→ L∗, the same +conclusion holds for positive (U, UU)-covariant maps. +In the remaining of this section, we will assume d ≥ 3 and exhibit posi- +tive non-decomposable (U, UU)-covariant maps. +[Step 1] First of all, let us characterize all elements in CovPos(U, UU). +Note that Corollary 2.4 (3) implies that the space Cov(U, UU) is spanned +by the following six linear maps Lσ whose unnormalized Choi matrices are +the operators Vσ ∈ Inv(U ⊗3) in (5.5): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Le(X) = (Tr X) · Idd ⊗ Idd, +L(12)(X) = XT ⊗ Idd, +L(13)(X) = Idd ⊗ XT, +L(23)(X) = (Tr X) · �d +j2,j3=1 |j3j2⟩⟨j2j3|, +L(123)(X) = �d +j1,j2,j3=1 Xj1j2|j2j3⟩⟨j3j1|, +L(132)(X) = �d +j1,j2,j3=1 Xj1j3|j2j3⟩⟨j1j2|. +(5.6) +Lemma 5.2. Let L = � +σ∈S3 aσLσ ∈ Cov(U, UU). Then L is positive if +and only if +� +� +� +� +� +� +� +(1) +ae, a(12), a(13), a(23) ∈ R and a(132) = a(123), +(2) +ae ≥ max +� +−a(12), −a(13), |a(23)| +� +, +(3) +ae + a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0, +(4) +� +ae + a(12) +� � +ae + a(13) +� +≥ +��a(23) + a(123) +��2 . +(5.7) + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +23 +Proof. Since every unit vector ξ ∈ Cd can be written as |ξ⟩ = U|1⟩ for +some U ∈ Ud, the (U, UU)-covariance property implies that L is positive if +and only if L(e11) ≥ 0. Moreover, L(e11) has a matrix decomposition +L(e11) ∼= (ae + a(12) + a(13) + a(23) + a(123) + a(132))1 ⊕ (ae + a(23))Idd−1 +⊕ +� +� +d +� +j=2 +� ae + a(12) +a(23) + a(123) +a(23) + a(132) +ae + a(13) +�� +� ⊕ +� +� +� +2≤i C2+s2, then we can choose s′ > |s| such that AB = C2+(s′)2. +In this case, x(0) +± = (ae, A, B, C, r, ±s′) ∈ P, and x is a (nontrivial) convex +combination of x(0) ++ and x(0) +− . Thus, x is not extremal in P. +From now on, we assume AB = C2 + s2 (i.e., s = ± +√ +AB − C2) and +divide the condition (3) of (D.3) into the following cases. +[Case 1] A + B + C + r ≥ ae > |C − r|. Then for sufficiently small +δ > 0, +x(1) +± = +� +ae ∓ 2(A + B + C) +d − 2 +δ, A ± Aδ, B ± Bδ, C ± Cδ, r ∓ d(A + B + C) +d − 2 +δ, s ± sδ +� +(D.9) +are elements of P, and x = (x(1) ++ + x(1) +− )/2. Therefore, x /∈ Ext(P). +[Case 2] A + B + C + r > ae = |C − r| > 0. Here we consider only the +case C > r since the other case C < r can be argued similarly. Then for +sufficiently small δ > 0, +x(2) +± = (ae ± (C − k)δ, A ± Aδ, B ± Bδ, C ± Cδ, r ± kδ, s ± sδ) +(D.10) + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +35 +are elements of P, where k ∈ R satisfies +d(d − 2)(C − k) + d(A + B + C) − (d − 2)k = 0 +(D.11) +so that the condition (4) of (D.3) holds for x(2) +± . Since x = (x(2) ++ + x(2) +− )/2, +it is not extremal. +[Case 3] A + B + C + r ≥ ae = |C − r| = 0, so C = r. We claim that +x = (0, A, B, C, C, s) ∈ Ext(P) corresponding to Type II′. Suppose that x +is a convex combination of x(3) +± = (a±, A±, B±, C±, r±, s±) ∈ P. Then the +condition ae = 0 and a± ≥ 0 imply a± = 0, which again forces |C±−r±| = +0. Therefore, Lemma D.1 implies that x(3) +± = (0, A±, B±, C±, C±, s±) = +λ±x for some λ± ≥ 0. Now the TP condition (D.3) (4) implies λ± = 1, so +x = x(3) +± . +[Case 4] A + B + C + r = ae = C − r ≥ 0. Then r = − A+B +2 +and +x = ( A+B+2C +2 +, A, B, C, − A+B +2 , s). Here we claim that x ∈ Ext(P) which +corresponds to Type III′ (note that A + B ≥ 2 +√ +AB = 2 +√ +C2 + s2 ≥ 2|C|, +so r = − A+B +2 +conversely implies C ≥ r). If x is a convex combination +of x(4) +± += (a±, A±, B±, C±, r±, s±) ∈ P, then the condition (D.3) (3) for +x(4) +± implies A± + B± + C± + r± = a± = |C± − r±|. We may assume +A+ ≥ A ≥ A− without loss of generality, so Lemma D.1 implies +x(4) ++ = +�A + B + 2C +2 ++ δ′, A + Aδ, B + Bδ, C + Cδ, −A + B +2 ++ δ′′, s + sδ +� +(D.12) +for some δ ≥ 0, δ′, δ′′ ∈ R, and δ′ = (A + B + C)δ + δ′′ from A+ + B+ + +C+ + r+ = a+. Now for the case a+ = r+ − C+ ≥ 0, we have +0 ≤ A + B + 2C = −(A + B + 2C)δ ≤ 0. +(D.13) +However, this says A+B = −2C and x = (0, A, B, C, C, s), which can be +absorbed into Case 3. For the case a+ = C+ − r+, we have δ′′ = − A+B +2 δ +and δ′ = A+B+2C +2 +δ. However, then the TP condition (D.3) (4) implies +� +d(d − 2)A + B + 2C +2 ++ d(A + B + C) + (d − 2)A + B +2 +� +δ = 0, +(D.14) +which is possible only if δ = 0. Therefore, x = x(4) ++ = x(4) +− . +[Case 5] A+B +C +r = ae = r−C ≥ 0. Then A+B = −2C, and the +previous inequality A + B ≥ 2 +√ +AB ≥ 2|C| implies A = B = −C ≥ 0 +and s = 0. Thus, x = (r − C, −C, −C, C, r, 0) with C ≤ 0 and r ≥ C. +Then our problem is divided into the following three subcases. + +36 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +• If C < 0 and r > C, then x /∈ Ext(P) since x = (x(5) ++ + x(5) +− )/2, +where +x(5) +± = +� +r − C ∓ +2 +d − 2δ, −C ± δ, −C ± δ, C ∓ δ, r ∓ +d +d − 2δ, 0 +� +∈ P +(D.15) +for sufficiently small δ > 0. +• If r = C, then x = (0, −C, −C, C, C, 0) is extremal since it can be +absorbed into Case 3. +• If C = 0, then x = r(1, 0, 0, 0, 1, 0) is indeed extremal (corre- +sponding to Type I′) since the point (A, B, C, s) = (0, 0, 0, 0) is an +extreme point of S in Lemma D.1 and since r is uniquely deter- +mined by the TP condition (D.3) (4). +□ +Now we shall prove Lemma 5.10 using similar arguments. Let Q0 be the +set of all tuples (ae, a(12), a(13), a(23), r, s) satisfying (5.25) and (D.1), and +then consider a linear isomorphism +β : (ae, a(12), a(13), a(23), r, s) �→ (A, B, C, p, q, s) +(D.16) +of R6, where +� +A = � +σ∈S3 aσ, B = +ae +d−1 + a(23), C = a(23) + r, +p = ae + a(12), q = ae + a(13). +Then +Q = β(Q0) becomes the set of all tuples (A, B, C, p, q, s) ∈ R6 satisfying +� +� +� +� +� +� +� +(1) +A, B, p, q ≥ 0, +(2) +AB ≥ C2 + s2, +(3) +A + B − 2C ≤ p + q, +(4) +(−d2 + d + 1)A − (d − 1)2B + 2d(d − 1)C + (d2 − 1)(p + q) = 1. +(D.17) +Proof of Lemma 5.10. It is sufficient to show that the extreme points y = +(A, B, C, p, q, s) of Q are classified into the following four types up to nor- +malizing constants: for A, B ≥ 0 and AB ≥ C2, +Type I′ +(0, 0, 0, 1, 0, 0), +Type II′ +(0, 0, 0, 0, 1, 0), +Type III′ +(A, B, C, A + B − 2C, 0, ± +√ +AB − C2), +Type IV′ +(A, B, C, 0, A + B − 2C, ± +√ +AB − C2). +As in the proof of Lemma 5.4, we may assume AB = C2+s2. Furthermore, +we may assume p = 0 or q = 0 since y is a convex combination of y(0) +± ∈ Q, +where y(0) ++ = (A, B, C, p + q, 0, s) and y(0) +− = (A, B, C, 0, p + q, s). Let us +first assume q = 0, and divide the condition (3) of (D.17) into the following +three cases. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +37 +[Case 1] (A, B) ̸= (0, 0) and A + B − 2C < p. Then for sufficiently +small δ > 0, +y(1) +± = (A ± Aδ, B ± Bδ, C ± Cδ, p ± δ′, 0, s ± sδ) ∈ Q +(D.18) +where δ′ ∈ R satisfies +� +(−d2 + d + 1)A − (d − 1)2B + 2d(d − 1)C +� +δ+(d2−1)δ′ = 0, (D.19) +so that the condition (4) of (D.17) holds for y(1) +± . Since y = (y(1) ++ + y(1) +− )/2 +and y(1) ++ ̸= y(1) +− , we have y /∈ Ext(Q). +[Case 2] A = B = 0 (hence C = s = 0). Then y = p(0, 0, 0, 1, 0, 0) is +extremal in Q (corresponding to Type I′) since (A, B, C, s) = (0, 0, 0, 0) is +an extreme point of S in Lemma D.1 and since p is uniquely determined by +(D.17) (4). +[Case 3] A+B −2C = p. In this case, we claim that y = (A, B, C, A+ +B − 2C, 0, s) ∈ Ext(Q), which corresponds to Type III′. Indeed, if y is +a convex combination of y(2) +± += (A±, B±, C±, p±, q±, s±) ∈ Q, then the +conditions q = 0 and q± ≥ 0 imply q± = 0. Moreover, the conditions +A + B − 2C = p and A± + B± − 2C± ≤ p± imply A± + B± − 2C± = p±. +Now applying Lemma D.1, we can write +y(2) ++ = (A(1 + δ), B(1 + δ), C(1 + δ), (A + B − 2C)(1 + δ), 0, s(1 + δ)) +(D.20) +for some δ ∈ R. On the other hand, the TP condition (D.17) (4) for y(2) ++ +gives +(dA + 2(d − 1)B − 2(d − 1)C) δ = 0. +(D.21) +However, +dA + 2(d − 1)B = A + (d − 1)B + (d − 1)(A + B) ≥ 2(d − 1)C (D.22) +since A + B ≥ 2C, and the equality above holds only if A = B = C = +p = s = 0 which is impossible. Therefore, (D.21) holds only if δ = 0, and +hence we have y = y(2) ++ = y(2) +− . +Finally, we can proceed analogously when p = 0 and get the tuples of +Type II′ and Type IV′ as extreme points of Q. +□ +APPENDIX E. PROOF OF THEOREM 5.6 WHEN d = 2 +When d = 2, we have an additional relation +Ve − V(12) − V(13) − V(23) + V(123) + V(132) = 0. +(E.1) +Therefore, {Vσ}σ∈S3 is no longer linearly independent, and both the spaces +Inv(U ⊗3) = span {Vσ : σ ∈ S3} and Inv(U ⊗U ⊗U) = span {Tσ : σ ∈ S3} +are 5-dimensional. In particular, we have Inv(O⊗3 ++ ) = Inv(U ⊗ U ⊗ U) in +this case. + +38 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +We can write a general element in Cov(U, UU) as M = � +σ∈S3\{e} aσMσ. +Then M is positive if and only if +� +� +� +a(12), a(13), a(23) ≥ 0 and a(132) = a(123), +a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0, +(a(12) + a(13) + a(23) + a(123) + a(132))a(23) ≥ |a(23) + a(123)|2, +(E.2) +by following the same proof in Lemma 5.8. Now let us write (r, s) = +(Re(a(123)), Im(a(123))) for convenience and consider a linear isomorphism +˜β : (a(12), a(13), a(23), r, s) �→ (a(12), A, B, C, s), +(E.3) +of R5, where A = a(12) + a(13) + a(23) + 2r, B = a(23), and C = a(23) + r. +Then the set �Q = +� +˜β(a(12), a(13), a(23), r, s) : M ∈ CovPosTP(U, UU) +� +is +equal to the set of tuples (a(12), A, B, C, s) ∈ R5 satisfying +� +� +� +� +� +� +� +(1) +A, B ≥ 0, +(2) +AB ≥ C2 + s2, +(3) +0 ≤ a(12) ≤ A + B − 2C, +(4) +A + B − C = 1 +2. +(E.4) +In order to find the extreme points y = (a(12), A, B, C, s) of �Q, note that +we still have AB = C2 + s2 as in the proof of Lemma 5.10. Moreover, +we have a(12) = 0 or A + B − 2C since y is a convex combination of +y+ = (A + B − 2C, A, B, C, s) and y− = (0, A, B, C, s). Therefore, we +can list all possible extreme points of �Q in the following two types: +Type I′ +(A + B − 2C, A, B, C, ± +√ +AB − C2), +Type II′ +(0, A, B, C, ± +√ +AB − C2), +for A, B ≥ 0, AB ≥ C2, and A + B − C = 1 +2. Moreover, any extreme +point of Type I′ corresponds to a tuple +(a(12), a(13), a(23), r, s) = (A+B −2C, 0, B, C −B, ± +√ +AB − C2), (E.5) +so the associated linear map M = � +σ∈S3\{e} aσMσ is CP by Lemma 5.9 +(note that Lemma 5.9 (1) still gives a sufficient condition for M to be CP +when d = 2). Similarly, any extreme point of Type II′ corresponds a CCP +map. In other words, every element in Ext(CovPosTP(U, UU)) is CP or +CCP, thus POS=DEC holds in CovPos1(UU, U). This completes the proof +of Theorem 5.6 by Theorem 3.9. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +39 +APPENDIX F. QUANTUM ORTHOGONAL SYMMETRY +Within the framework of compact quantum groups, the orthogonal group +Od is understood as the space C(Od) of continuous functions on Od en- +dowed with the co-multiplication ∆ : C(Od) → C(Od × Od) given by +(∆f)(x, y) = f(xy) +(F.1) +for all x, y ∈ Od and f ∈ C(Od). Moreover, there exists a family of +continuous functions (πij)1≤i,j≤d generating C(Od) and +∆(πij) = +d +� +k=1 +πik ⊗ πkj ∈ C(Od) ⊗min C(Od) ∼= C(Od × Od) +(F.2) +for all 1 ≤ i, j ≤ d, where ⊗min means the minimal tensor product of +C∗-algebras. +The free orthogonal quantum group O+ +d is a liberation of Od in the sense +that the space C(O+ +d ) of ‘non-commutative’ continuous functions on O+ +d +is the universal unital C∗-algebra generated by d2 self-adjoint operators uij +satisfying that u = +d +� +i,j=1 +eij ⊗ uij is a unitary, i.e. u∗u = uu∗ = Idd ⊗ 1 +in Md(C) ⊗ C(O+ +d ). The quantum group structure is encoded in the unital +∗-homomorphism ∆ : C(O+ +d ) → C(O+ +d ) ⊗min C(O+ +d ) determined by +∆(uij) = +d +� +k=1 +uik ⊗ ukj. +Then u = +d +� +i,j=1 +eij ⊗ uij is the standard unitary representation of O+ +d satis- +fying uc = +d +� +i,j=1 +eij ⊗ u∗ +ij = u in the sense of [Wor87, Ban96]. The 3-fold +tensor representation of u is defined by +u ⊤ u ⊤ u = +d +� +i1,j1,i2,j2,i3,j3=1 +ei1j1 ⊗ ei2j2 ⊗ ei3j3 ⊗ ui1j1ui2j2ui3j3. +(F.3) +Then the space Inv(O⊗3 ++ ) in Section 5.2 is understood as the space Inv(u ⊤ u ⊤ u) +of operators X ∈ Md(C)⊗3 satisfying +(u ⊤ u ⊤ u) · (X ⊗ 1) = (X ⊗ 1) · (u ⊤ u ⊤ u) +(F.4) +in view of [LY22]. To sketch a proof of this fact, we can observe that the +5 operators Tσ (σ ∈ S3\ {(13)}) in (5.20) are linearly independent, and the + +40 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +operators Tσ satisfy (F.4) using the identity +(u ⊤ u)(|Ωd⟩ ⊗ 1) = |Ωd⟩ ⊗ 1. +(F.5) +Thus, Inv(O⊗3 ++ ) ⊆ Inv(u ⊤ u ⊤ u). Moreover, the space Inv(u ⊤ u ⊤ u) +should be of dimension five thanks to the representation theory of O+ +d (see +Corollary 6.4.12 and Corollary 5.3.5 of [Tim08]). Hence, we have Inv(O⊗3 ++ ) = +Inv(u ⊤ u ⊤ u). +REFERENCES +[AN14] +Muneerah Al Nuwairan. The extreme points of SU(2)-irreducibly covariant +channels. Internat. J. Math., 25(6):1450048, 30, 2014. +[Ban96] +Teodor Banica. Th´eorie des repr´esentations du groupe quantique compact +libre O(n). C. R. Acad. Sci. Paris S´er. I Math., 322(3):241–244, 1996. +[BBC+93] +Charles H. Bennett, Gilles Brassard, Claude Cr´epeau, Richard Jozsa, Asher +Peres, and William K. Wootters. Teleporting an unknown quantum state +via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., +70(13):1895–1899, 1993. +[Bel64] +J. S. Bell. On the Einstein Podolsky Rosen paradox. Phys. Phys. Fiz., +1(3):195–200, 1964. +[BO08] +Nathanial P. Brown and Narutaka Ozawa. C∗-algebras and finite- +dimensional approximations, volume 88 of Graduate Studies in Mathemat- +ics. American Mathematical Society, Providence, RI, 2008. +[BPM+97] +Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald We- +infurter, and Anton Zeilinger. Experimental quantum teleportation. Nature, +390(6660):575–579, 1997. +[Bra03] +Gilles Brassard. Quantum communication complexity. volume 33, pages +1593–1616. 2003. Special issue dedicated to David Mermin, Part II. +[BSST99] +Charles H. Bennett, Peter W. Shor, John A. Smolin, and Ashish V. Thapliyal. +Entanglement-assisted classical capacity of noisy quantum channels. Phys. +Rev. Lett., 83:3081–3084, Oct 1999. +[BvDHT99] Harry Buhrman, Wim van Dam, Peter Høyer, and Alain Tapp. Multiparty +quantum communication complexity. Phys. Rev. A, 60:2737–2741, Oct 1999. +[BW92] +Charles H. Bennett and Stephen J. Wiesner. Communication via one- and +two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., +69(20):2881–2884, 1992. +[C¯D13] +Lin Chen and Dragomir ˇZ ¯Dokovi´c. Separability problem for multipartite +states of rank at most 4. J. Phys. A, 46(27):275304, 24, 2013. +[Cho82] +M.D. Choi. Positive linear-maps. In in Operator Algebras and Applica- +tions (Kingston, 1980), Proc.Sympos.Pure.Math., volume 38, pages 583– +590. Amer.Math.Soc., 1982. +[CKK+21] +Euijung Chang, Jaeyoung Kim, Hyesun Kwak, Hun Hee Lee, and Sang- +Gyun Youn. Irreducibly su(2)-covariant quantum channels of low rank. to +appear in Rev. Math. Phys., arXiv preprint arXiv:2105.00709, 2021. +[DFV08] +B. Dierckx, M. Fannes, and C. Vandenplas. Additivity of the Renyi entropy +of order 2 for positive-partial-transpose-inducing channels. Phys. Rev. A (3), +77(6):060302, 4, 2008. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +41 +[DPR07] +A. R. Usha Devi, R. Prabhu, and A. K. Rajagopal. Characterizing multipar- +ticle entanglement in symmetric n-qubit states via negativity of covariance +matrices. Phys. Rev. Lett., 98:060501, Feb 2007. +[EK00] +Myoung-Hoe Eom and Seung-Hyeok Kye. Duality for positive linear maps +in matrix algebras. Math. Scand., 86(1):130–142, 2000. +[Eke91] +Artur K. Ekert. Quantum cryptography based on Bell’s theorem. Phys. Rev. +Lett., 67(6):661–663, 1991. +[EW01] +T. Eggeling and R. F. Werner. Separability properties of tripartite states with +U � U � U symmetry. Phys. Rev. A, 63:042111, Mar 2001. +[G¨11] +Otfried G¨uhne. Entanglement criteria and full separability of multi-qubit +quantum states. Phys. Lett. A, 375(3):406–410, 2011. +[GBW21] +Martina Gschwendtner, Andreas Bluhm, and Andreas Winter. Programma- +bility of covariant quantum channels. Quant., 5(6):488, 2021. +[Gha10] +Sevag Gharibian. Strong NP-hardness of the quantum separability problem. +Quantum Inf. Comput., 10(3-4):343–360, 2010. +[Gur03] +Leonid Gurvits. Classical deterministic complexity of Edmond’s problem +and quantum entanglement. In Proceedings of the Thirty-Fifth Annual ACM +Symposium on Theory of Computing, pages 10–19. ACM, New York, 2003. +[Has18] +Anna-Lena K. Hashagen. Symmetry Methods in Quantum Information The- +ory. 2018. Thesis (Ph.D.)–Technische Universit¨at M¨unchen Lehrstuhl f¨ur +mathematische Physik. +[HH99] +Michał Horodecki and Paweł Horodecki. Reduction criterion of separability +and limits for a class of distillation protocols. Phys. Rev. A, 59:4206–4216, +Jun 1999. +[HHH96] +MichałHorodecki, PawełHorodecki, and Ryszard Horodecki. Separability of +mixed states: necessary and sufficient conditions. Phys. Lett. A, 223(1-2):1– +8, 1996. +[HHH99] +Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. Bound entan- +glement can be activated. Phys. Rev. Lett., 82:1056–1059, Feb 1999. +[HHHO05] +Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Jonathan Op- +penheim. Secure key from bound entanglement. Phys. Rev. Lett., 94:160502, +Apr 2005. +[HHHO09] +Karol Horodecki, MichałHorodecki, PawełHorodecki, and Jonathan Oppen- +heim. General paradigm for distilling classical key from quantum states. +IEEE Trans. Inform. Theory, 55(4):1898–1929, 2009. +[HK16a] +Kil-Chan Ha and Seung-Hyeok Kye. Construction of three-qubit gen- +uine entanglement with bipartite positive partial transposes. Phys. Rev. A, +93:032315, Mar 2016. +[HK16b] +Kyung Hoon Han and Seung-Hyeok Kye. Construction of multi-qubit opti- +mal genuine entanglement witnesses. J. Phys. A, 49(17):17503, 16, 2016. +[HLVC00] +Paweł Horodecki, Maciej Lewenstein, Guifr´e Vidal, and Ignacio Cirac. Op- +erational criterion and constructive checks for the separability of low-rank +density matrices. Phys. Rev. A, 62:032310, Aug 2000. +[HPHH08] +Karol Horodecki, Łukasz Pankowski, Michał Horodecki, and Paweł +Horodecki. Low-dimensional bound entanglement with one-way distillable +cryptographic key. IEEE Trans. Inform. Theory, 54(6):2621–2625, 2008. + +42 +S.-J. PARK, Y.-G. JUNG, J. PARK, AND S.-G. YOUN +[JSW+00] +Thomas Jennewein, Christoph Simon, Gregor Weihs, Harald Weinfurter, and +Anton Zeilinger. Quantum cryptography with entangled photons. Phys. Rev. +Lett., 84:4729–4732, May 2000. +[Kay11] +Alastair Kay. Optimal detection of entanglement in greenberger-horne- +zeilinger states. Phys. Rev. A, 83:020303, Feb 2011. +[KCKL00] +B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein. Separability in 2 × N +composite quantum systems. Phys. Rev. A (3), 61(6):062302, 10, 2000. +[KCL05] +J. K. Korbicz, J. I. Cirac, and M. Lewenstein. Spin squeezing inequalities +and entanglement of n qubit states. Phys. Rev. Lett., 95:120502, Sep 2005. +[KMS20] +Piotr Kopszak, Marek Mozrzymas, and MichałStudzi´nski. Positive maps +from irreducibly covariant operators. J. Phys. A, 53(39):395306, 33, 2020. +[Kye23] +Seung-Hyeok Kye. Compositions and tensor products of linear maps be- +tween matrix algebras. Linear Algebra Appl., 658:283–309, 2023. +[Las10] +Jean Bernard Lasserre. Moments, positive polynomials and their applica- +tions, volume 1 of Imperial College Press Optimization Series. Imperial Col- +lege Press, London, 2010. +[LY22] +Hun Hee Lee and Sang-Gyun Youn. Quantum channels with quantum group +symmetry. Comm. Math. Phys., 389(3):1303–1329, 2022. +[Mas06] +Llu´ıs Masanes. All bipartite entangled states are useful for information pro- +cessing. Phys. Rev. Lett., 96:150501, Apr 2006. +[Mic07] +S. Michalakis. Multiplicativity of the maximal output 2-norm for depolarized +Werner-Holevo channels. J. Math. Phys., 48(12):122102, 5, 2007. +[MRS15] +Marek Mozrzymas, +Adam Rutkowski, +and MichałStudzi´nski. Using +non-positive maps to characterize entanglement witnesses. J. Phys. A, +48(39):395302, 11, 2015. +[MWKZ96] Klaus Mattle, Harald Weinfurter, Paul G. Kwiat, and Anton Zeilinger. Dense +coding in experimental quantum communication. Phys. Rev. Lett., 76:4656– +4659, Jun 1996. +[NPW+00] +D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund, and P. G. Kwiat. +Entangled state quantum cryptography: Eavesdropping on the ekert protocol. +Phys. Rev. Lett., 84:4733–4736, May 2000. +[NS21] +Ion Nechita and Satvik Singh. A graphical calculus for integration over ran- +dom diagonal unitary matrices. Linear Algebra and its Applications, 613:46– +86, 2021. +[NZ16] +Jiawang Nie and Xinzhen Zhang. Positive maps and separable matrices. +SIAM J. Optim., 26(2):1236–1256, 2016. +[Pau02] +Vern Paulsen. Completely bounded maps and operator algebras, volume 78 +of Cambridge Studies in Advanced Mathematics. Cambridge University +Press, Cambridge, 2002. +[Per96] +Asher Peres. Separability criterion for density matrices. Phys. Rev. Lett., +77(8):1413–1415, 1996. +[SDN22] +Satvik Singh, Nilanjana Datta, and Ion Nechita. Ergodic theory of diagonal +orthogonal covariant quantum channels, 2022. +[SN21] +Satvik Singh and Ion Nechita. Diagonal unitary and orthogonal symmetries +in quantum theory. Quantum, 5:519, August 2021. +[SN22] +Satvik Singh and Ion Nechita. The PPT2 Conjecture Holds for All Choi-Type +Maps. Annales Henri Poincare, 23(9):3311–3329, 2022. + +ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES +43 +[Sr82] +Erling Stø rmer. Decomposable positive maps on C∗-algebras. Proc. Amer. +Math. Soc., 86(3):402–404, 1982. +[TBZG00] +W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. Quantum cryptography using +entangled photons in energy-time bell states. Phys. Rev. Lett., 84:4737–4740, +May 2000. +[TG09] +G´eza T´oth and Otfried G¨uhne. Entanglement and permutational symmetry. +Phys. Rev. Lett., 102:170503, May 2009. +[Tim08] +Thomas Timmermann. An Invitation to Quantum Groups and Duality. Euro- +pean Mathematical Society, 2008. +[UDUPR07] A. R. Usha Devi, M. S. Uma, R. Prabhu, and A. K. Rajagopal. Constraints on +the uncertainties of entangled symmetric qubits. Phys. Lett. A, 364(3-4):203– +207, 2007. +[VW02] +Karl Gerd H. Vollbrecht and Michael M. Wolf. Activating distillation with +an infinitesimal amount of bound entanglement. Phys. Rev. Lett., 88:247901, +May 2002. +[Wan95] +Shuzhou Wang. Free products of compact quantum groups. Comm. Math. +Phys., 167(3):671–692, 1995. +[Wat18] +John Watrous. The Theory of Quantum Information. Cambridge University +Press, 2018. +[Wer89] +Reinhard F. Werner. Quantum states with einstein-podolsky-rosen correla- +tions admitting a hidden-variable model. Phys. Rev. A, 40:4277–4281, Oct +1989. +[Wor76a] +S. L. Woronowicz. Nonextendible positive maps. Comm. Math. Phys., +51(3):243–282, 1976. +[Wor76b] +S. L. Woronowicz. Positive maps of low dimensional matrix algebras. Rep. +Math. Phys., 10(2):165–183, 1976. +[Wor87] +S. L. Woronowicz. Compact matrix pseudogroups. Comm. Math. Phys., +111(4):613–665, 1987. +SANG-JUN PARK, DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL +UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF KOREA +Email address: psj05071@snu.ac.kr +YEONG-GWANG JUNG, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NA- +TIONAL UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF +KOREA +Email address: wollow21@snu.ac.kr +JEONGEUN PARK, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NATIONAL +UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF KOREA +Email address: pju0013@snu.ac.kr +SANG-GYUN YOUN, DEPARTMENT OF MATHEMATICS EDUCATION, SEOUL NA- +TIONAL UNIVERSITY, GWANAK-RO 1, GWANAK-GU, SEOUL 08826, REPUBLIC OF +KOREA +Email address: s.youn@snu.ac.kr + diff --git a/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/load_file.txt b/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ccea25594d63c624d190e240737b7b84f6d8e55 --- /dev/null +++ b/V9E2T4oBgHgl3EQfYAcW/content/tmp_files/load_file.txt @@ -0,0 +1,1643 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf,len=1642 +page_content='A UNIVERSAL FRAMEWORK FOR ENTANGLEMENT DETECTION UNDER GROUP SYMMETRY SANG-JUN PARK, YEONG-GWANG JUNG, JEONGEUN PARK, AND SANG-GYUN YOUN ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' One of the most fundamental questions in quantum infor- mation theory is PPT-entanglement of quantum states, which is an NP- hard problem in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this paper, however, we prove that all PPT (πA ⊗ πB)-invariant quantum states are separable if and only if all ex- tremal unital positive (πA, πB)-covariant maps are decomposable where πA, πB are unitary representations of a compact group and πA is irre- ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, an extremal unital positive (πB, πA)-covariant map L is decomposable if and only if L is completely positive or completely copositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We apply the results to prove that all PPT quantum channels of the form Φ(ρ) = aρ + bρT + cTr(ρ) d Idd + (1 − a − b − c)diag(ρ) are entanglement-breaking, and that all A-BC PPT (U⊗U⊗U)-invariant tripartite quantum states are A-BC separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The former resolves some open questions raised in [DFV08, KMS20], and the latter is a strong contrast to the fact that there exist PPT-entangled (U ⊗U ⊗U)-invariant tripartite Werner states [EW01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' INTRODUCTION Quantum entanglement is one of the most non-classical manifestations of quantum formalism and is considered a key resource for quantum communi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, quantum entanglement plays crucial roles in the existence of Bell correlations [Bel64, Wer89], quantum cryptography [Eke91, JSW+00, TBZG00, NPW+00], superdense coding [BW92, MWKZ96], quantum tele- portation [BBC+93, BPM+97], entanglement-assisted classical communi- cation [BSST99], and computational supremacy for communication com- plexity problems [Bra03, BvDHT99, C¯D13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The question of whether a given quantum state is entangled or separa- ble is of fundamental importance in quantum information theory(QIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' It turned out that this question is NP-hard in general [Gur03, Gha10], so it is unnatural to expect an efficient general scheme to characterize quantum en- tanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Nevertheless, there have been numerous efforts to characterize separability in some subclasses of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For example, a quantum 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='03849v1 [math-ph] 10 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN state ρ ∈ D(HA ⊗ HB) of positive partial transpose (PPT) is separable if dim(HA) × dim(HB) ≤ 6 [HHH96, Wor76b] and, moreover, PPT implies separability in some bipartite or multipartite systems of low-rank [KCKL00, HLVC00, EW01, C¯D13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Classification of entanglement of GHZ states has also been studied in various contexts [Kay11, G¨11, HK16a, HK16b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that any PPT entangled state is bound entangled, which is applicable to perform nonclassical tasks [HHH99, VW02, Mas06] and to produce secure cryptographic key [HHHO05, HHHO09, HPHH08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this paper, we restrict our interests to the so-called invariant quantum states in a general context of compact group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' More precisely, for unitary representations πA : G → B(HA) and πB : G → B(HB) of a compact group G, a bipartite quantum state ρ ∈ D(HA ⊗ HB) is called (πA ⊗ πB)-invariant if (πA(x) ⊗ πB(x))ρ = ρ(πA(x) ⊗ πB(x)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Werner states and isotropic states are standard examples of in- variant quantum states for fundamental unitary group symmetries, and their separability was characterized in [Wer89] and [HH99], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Sep- arability of invariant quantum states has been studied extensively for vari- ous group symmetries [EW01, UDUPR07, DPR07, KCL05, TG09, AN14, SN21, CKK+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The dual objects of invariant quantum states are the so-called (πA, πB)- covariant quantum channels, which are completely positive trace-preserving (CPTP) maps L : B(HA) → B(HB) satisfying L(πA(x)XπA(x)T) = πB(x)L(X)πB(x)∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2) for all X ∈ B(HB) and x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, for a linear map L : B(HA) → B(HB) and its normalized Choi matrix CL = 1 dA � i,j=1 eij ⊗ L(eij), the given map L is (πA, πB)-covariant if and only if CL is (πA ⊗ πB)-invariant (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' One of the key observations of this paper is that all PPT (πA ⊗ πB)- invariant quantum states are separable if and only if all (πB, πA)-covariant positive maps are decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If πA is irreducible, then the following three statements are equivalent (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8): All PPT (πA⊗πB)-invariant quantum states are separable (PPT=SEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' All positive (πB, πA)-covariant maps are decomposable (POS=DEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' All PPT (πA, πB)-covariant quantum channels are entanglement- breaking (PPT=EB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, it is enough to consider only extremal elements (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) and, in particular, an extremal positive unital (πB, πA)-covariant linear map L is decomposable if and only if L is completely positive (CP) or com- pletely copositive (CCP) (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 3 Our framework focusing on decomposability of extremal positive unital (πB, πA)-covariant maps is applicable to study PPT entanglement for con- crete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In Section 4, we prove that EB property coincides with PPT property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=EB holds for any quantum channels of the form Φ(ρ) = aρ + bρT + cTr(ρ) d Idd + (1 − a − b − c)diag(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) where diag(X) = d � i=1 Xiieii for X = (Xij)1≤i,j≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' An important observa- tion is that the quantum channels of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) are irreducibly covariant channels with respect to the standard representation of the signed symmetric group (or the hyperoctahedral group) Hd (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Furthermore, we characterize all positive unital covariant maps of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) and our main theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) allows us to focus only on eight extremal positive unital covariant maps to detect EB property for quantum channels of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Our results strengthen Section 5 and Section 6 of [KMS20] to a larger class, and resolve some of open questions posed in [KMS20] and [DFV08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In Section 5, we focus on the question of whether all PPT quantum states are separable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=SEP problem for some tripartite quantum states with unitary group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1, we present explicit positive non-decomposable covariant linear maps L : Md(C) → Md2(C) satisfying L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4) for all d×d unitary matrices U ∈ Ud and X ∈ Md(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This result gives an- other explanation of the fact PPT̸=SEP for tripartite Werner states [EW01], which implies the existence of PPT entangled quantum states ρ ∈ Md3(C) satisfying (U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) for all U ∈ U(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' On the other hand, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2, we show that a strong contrast PPT=SEP holds for quantum orthogonally invariant quan- tum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' More generally, we prove that any PPT tripartite quantum state ρ ∈ Md3(C) satisfying (U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6) for all unitary matrices U ∈ U(d) is separable (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PRELIMINARIES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Separability and PPT property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this paper, we focus only on finite- dimensional complex Hilbert spaces H = Cd, HA = CdA, HB = CdB, and their direct sums and tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Recall that a quantum state ρ ∈ B(H) is a positive matrix with Tr(ρ) = 1 and the set of all quantum states in B(H) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN is denoted by D(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A bipartite positive operator X ∈ B(HA⊗HB) is said to be of positive partial transpose (PPT) if (idA ⊗ TB)(X) ≥ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) where TB is the transpose map on B(HB), and X is called separable if there exist families of positive operators (XA i )n i=1 and (XB i )n i=1 such that X = n � i=1 XA i ⊗ XB i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2) In particular, if ρ ∈ D(HA ⊗ HB) is a separable quantum state, then there exists a probability distribution (pi)n i=1 and a family of product quantum states (ρA i ⊗ ρB i )n i=1 such that ρ = n � i=1 piρA i ⊗ ρB i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) It is clear that separability implies PPT property, but the converse is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' More precisely, all PPT quantum states in B(HA ⊗ HB) are separable if and only if dA · dB ≤ 6 [Per96, HHH96, Wor76a, Cho82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, it is known that the separability question is NP-hard [Gur03, Gha10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For v ∈ H, we define linear maps |v⟩ : C → H given by λ �→ λv and ⟨v| : H → C given by w �→ ⟨v|w⟩ where ⟨v|w⟩ is the inner product of v, w ∈ H whose first variable is the anti-linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In particular, |Ω⟩ = �d i=1 1 √ d|i⟩⊗|i⟩ ∈ H⊗H is called the maximally entangled Bell state where {|1⟩, |2⟩, · · · , |d⟩} is the standard orthonormal basis of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The matrix unit |i⟩⟨j| and the product vector |i1⟩ ⊗ |i2⟩ ⊗ · · · ⊗ |ik⟩ are also denoted by eij and |i1i2 · · · ik⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The (normalized) Choi matrix of a linear map L : B(HA) → B(HB) is defined by CL = (idA ⊗ L)(|ΩA⟩⟨ΩA|) = (idA ⊗ L) � 1 dA dA � i,j=1 eij ⊗ eij � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4) = 1 dA dA � i,j=1 eij ⊗ L(eij) ∈ B(HA ⊗ HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) Recall that L is completely positive (CP) if and only if the Choi matrix CL is positive, and L is trace-preserving (TP) if and only if (idA ⊗TrB)(CL) = 1 dA IdA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In particular, if Φ : B(HA) → B(HB) is a CPTP linear map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' a quantum channel in the Schr¨odinger’s picture, then the Choi matrix CΦ is a quantum state in D(HA ⊗ HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We call this channel-state duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 5 Let L : B(HA) → B(HB) be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then L is called completely copositive (CCP) if TB ◦L is completely positive, L is called decomposable if there exist a CP map L1 and a CCP map L2 such that L = L1 + L2, and L is called PPT if L is both CP and CCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Thus, L is PPT if and only if CL is PPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Another important property of quantum channels is the entanglement- breaking (EB) property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A quantum channel Φ : B(HA) → B(HB) is called EB if the Choi matrix CΦ is a separable quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that any EB quantum channel is PPT, but the converse is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Invariance and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this section, we introduce two impor- tant objects to discuss conservation of symmetry, namely invariant opera- tors and covariant linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us suppose that G is a compact group throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A continuous function π : G → U(Hπ) is called a (finite-dimensional) unitary representation of G if it is a group homomor- phism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=', π(xy) = π(x)π(y) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6) for all x, y ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this case, an operator X ∈ B(Hπ) is called π-invariant if π(x)Xπ(x)∗ = X (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The set of all π-invariant operators, the set of all π-invariant quantum states, and the set of π-invariant PPT quantum states in B(Hπ) are denoted by Inv(π), InvQS(π), and InvPPTQS(π), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A unitary representation π : G → B(Hπ) is called irreducible if Inv(π) = C · IdHπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If π is irreducible, so is the contragredient representation π : G → U(Hπ) of π which is defined by π(x) = π(x) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For unitary representations πA : G → U(HA) and πB : G → U(HB), the tensor representation πA ⊗ πB : G → U(HA ⊗ HB) is given by (πA ⊗ πB)(x) = πA(x) ⊗ πB(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The tensor representation πA ⊗ πB is not irreducible in general, but admits the so-called irreducible decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' irreducible representations σ1, σ2, · · · , σk of G such that πA ⊗ πB ∼= σ1 ⊕ σ2 ⊕ · · · ⊕ σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) Here, (σ1 ⊕ σ2 ⊕ · · · ⊕ σk)(x) is the block diagonal matrix of σ1(x), σ2(x), · · , σk(x) for all x ∈ G, and π ∼= π′ means that there exists a unitary V such that π(x) = V π′(x)V ∗ for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We say that the irreducible decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) is multiplicity-free if σi ≇ σj for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This prop- erty was highlighted in [GBW21] in view of programmability of covariant 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' More generally, we may rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) as πA ⊗ πB ∼= l � i=1 σi ⊗ Idmi (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='10) meaning that σi ̸∼= σj for all i ̸= j and each irreducible representation σi appears mi times in the irreducible decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this case, we have the following identification of Inv(πA ⊗ πB) as ∗-algebras [GBW21, Lemma 6]: Inv(πA ⊗ πB) ∼= l � i=1 Idni ⊗ Mmi(C), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11) where ni = dim Hσi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For unitary representations πA : G → B(HA) and πB : G → B(HB), a linear map L : B(HA) → B(HB) is called (πA, πB)-covariant if L(πA(x)Y πA(x)∗) = πB(x)L(Y )πB(x)∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='12) for all x ∈ G and Y ∈ B(HA), and let us denote by Cov(πA, πB) the space of all (πA, πB)-covariant linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Some subclasses of Cov(πA, πB) in our interest are as follows: CovPos(πA, πB) is the set of all (πA, πB)-covariant positive maps, CovPos1(πA, πB) is the set of all (πA, πB)-covariant positive unital maps, CovPosTP(πA, πB) is the set of all (πA, πB)-covariant positive TP maps, CovQC(πA, πB) is the set of all (πA, πB)-covariant CPTP maps, CovPPTQC(πA, πB) is the set of all (πA, πB)-covariant PPT quan- tum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Twirling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' An averaging technique called the twirling op- eration is a standard method to analyze invariant operators and covariant linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For a unitary representation π : G → U(H), we define a twirling map Tπ : B(H) → Inv(π) by Tπ(X) = � G π(x)Xπ(x)∗dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='13) for all X ∈ B(H), where dx denotes the normalized Haar measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then Tπ is unital CPTP, and its well-definedness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Tπ(X) ∈ Inv(π), is thanks to the translation-invariance property of the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Fur- thermore, we have X ∈ Inv(π) if and only if Tπ(X) = X, which means that Tπ is a projection (more precisely, a conditional expectation) onto the ∗-subalgebra Inv(π) of B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that for any finite dimensional von Neumann algebra M ⊂ Md(C), there is a unique TP conditional expec- tation of Md(C) onto M [BO08, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For example, the map ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 7 X ∈ Mn ⊗ Mn �→ 1 nIdn ⊗ (Trn ⊗ idm)(X) is the unique TP conditional expectation onto M = Idn ⊗ Mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This observation allow us to get the following explicit formula of the twirling map Tπ for the case M = Inv(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Suppose that a unitary representation π : G → U(H) has an irreducible decomposition of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='10) with the identification Inv(π) ∼= �l i=1 Idni ⊗ Mmi(C) ⊆ B ��l i=1 Hi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let Πi be the orthogonal projection from H onto Hi = Cni ⊗ Cmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the twirling Tπ(X) of X ∈ B(H) is given by Tπ(X) = l � i=1 1 ni Idni ⊗ � (Trni ⊗ idmi)(ΠiXΠi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='14) In particular, if the irreducible decomposition of π is multiplicity-free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=', if mi ≡ 1 for all i = 1, 2, · · · , l, then Tπ(X) = l � i=1 Tr(ΠiX) ni Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='15) For unitary representations πA : G → U(HA) and πB : G → U(HB), the twirling TπA,πBL of L : B(HA) → B(HB) is defined by (TπA,πBL)(X) = � G πB(x)∗L(πA(x)XπA(x)∗)πB(x) dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='16) for all X ∈ B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then similarly, the twirling operation TπA,πB is a well- defined projection from B(B(HA), B(HB)) onto Cov(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us collect some useful properties of the twirling operations for the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For any unitary representations πA and πB of G, the twirling map TπA⊗πB preserves separability and PPT property of bipartite operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Furthermore, the twirling operation TπA,πB preserves positivity, CP, TP, CCP, PPT, decomposability, and EB property of linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' It is straightforward from the definitions and closedness of the spaces associated with each of the properties mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For example, the set of all decomposable linear maps L : B(HA) → B(HB) is closed in B(B(HA), B(HB)) with respect to the natural (Euclidean) topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ For a linear map L : B(HA) → B(HB), the adjoint map L∗ : B(HB) → B(HA) of L is a linear map satisfying Tr(L(X) Y ) = Tr(X L∗(Y )) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='17) for all X ∈ B(HA) and Y ∈ B(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Recall that the adjoint operation L �→ L∗ preserves positivity, CP, CCP, PPT, and decomposability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let π : G → U(H), πA : G → U(HA) and πB : G → U(HB) be unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) Tr((TπX) Y ) = Tr(X(TπY )) for any X, Y ∈ B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) TπA⊗πB◦(TA⊗idB) = (TA⊗idB)◦TπA⊗πB where TA is the transpose on B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) (TπA,πBL)∗ = TπB,πA(L∗) for any linear map L : B(HA) → B(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4) The Choi matrix of TπA,πBL is given by TπA⊗πB (CL) for any linear map L : B(HA) → B(HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) Since the Haar measure on the compact group G is invariant under the inverse x �→ x−1, we have Tr((TπX) Y ) = � G Tr(π(x)Xπ(x−1)Y )dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='18) = Tr � X � G π(x−1)Y π(x)dx � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='19) = Tr � X � G π(x)Y π(x−1)dx � = Tr(X(TπY )) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='20) for any X, Y ∈ B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) It suffices to show the equality for product operators X = P ⊗Q, and the conclusion follows immediately from the observation πA(x)P TπA(x)∗ = � πA(x)PπA(x)T�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='21) (3) For any X ∈ B(HA) and Y ∈ B(HB), we have Tr(X (TπB,πAL∗) (Y )) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='22) = � G Tr(XπA(x)∗L∗(πB(x)Y πB(x)∗)πA(x))dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='23) = � G Tr(πB(x)∗L(πA(x)XπA(x)∗)πB(x) Y )dx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='24) = Tr((TπA,πBL) (X) Y ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='25) which gives us the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4) First of all, note that dA � i,j=1 (πA(x)eijπA(x)t) ⊗ (πB(x)L(eij)πB(x)∗) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='26) = dA � i,j=1 eij ⊗ (πB(x)L(πA(x)∗eijπA(x))πB(x)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='27) ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 9 for each x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, the LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='26) can be understood as dA(idA ⊗ (AdπB(x) ◦ L)) � (πA(x) ⊗ IdA)|ΩA⟩⟨ΩA|(πA(x)t ⊗ IdA) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='28) and the RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='27) can be understood as dA(idA ⊗ (AdπB(x) ◦ L)) ((IdA ⊗ πA(x)∗)|ΩA⟩⟨ΩA|(IdA ⊗ πA(x))) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='29) where AdV (Y ) = V Y V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, the so-called ricochet property (X ⊗ IdA)|ΩA⟩ = (IdA ⊗ Xt)|ΩA⟩, X ∈ B(HA), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='30) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='28) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Finally, taking the Haar integral on both sides com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Combining Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3 (2), (3), and (4) with the fact that both Inv(πA⊗ πB) and Cov(πA, πB) are the images of the twirling projections, we obtain the following useful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let X ∈ B(HA ⊗ HB) be a bipartite operator and L : B(HA) → B(HB) be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then (1) X ∈ Inv(πA ⊗ πB) if and only if (TA ⊗ id)(X) ∈ Inv(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) L ∈ Cov(πA, πB) if and only if L∗ ∈ Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) L ∈ Cov(πA, πB) if and only if CL ∈ Inv(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The results in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 have been noted in various con- texts, [EW01, Lemma 6], [GBW21, Lemma 11], and [LY22, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5] for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, extendibility to more general contexts of compact quantum group symmetry was proved in [LY22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A FRAMEWORK TO CHARACTERIZE ENTANGLEMENT UNDER GROUP SYMMETRY Let us recall a result of Horodecki on the characterization of entangle- ment [HHH96]: a bipartite quantum state ρ ∈ D(HA ⊗ HB) is separable if and only if (idA ⊗ L)(ρ) ≥ 0 for all positive linear maps L : B(HB) → B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, by duality arguments, the authors showed that they are also equivalent to seemingly a weaker condition ‘⟨ρ, L⟩ ≥ 0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Here, the dual pairing ⟨·, ·⟩ is defined by ⟨X, N⟩ = Tr((idA ⊗ N)(X) |ΩA⟩⟨ΩA|) = Tr(XCN ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) for any operator X ∈ B(HA ⊗HB) and linear map N : B(HB) → B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In other words, positive linear maps play a crucial role as detectors for the bipartite entanglement, in the sense that there should exist a positive linear map L such that (id ⊗ L)(ρ) is non-positive whenever ρ is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN One technical issue in this characterization is that verifying whether a linear map is positive or not is computationally intractable [Las10, NZ16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' However, one of the main purposes of this paper is to develop a universal framework to characterize separability of invariant quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The key ideas is that, for ρ ∈ InvQS(πA ⊗ πB), it is enough to consider only L ∈ CovPos(πB, πA) to investigate separability of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us begin with a simple and useful lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For any bipartite operator X ∈ B(HA ⊗ HB) and linear map L : B(HB) → B(HA), we have ⟨TπA⊗πBX, L⟩ = ⟨X, TπB,πAL⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Thanks to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3, we have ⟨TπA⊗πBX, L⟩ = Tr((TπA⊗πBX)CL∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) = Tr(X(TπA⊗πBCL∗)) = Tr(XCL∗) = ⟨X, L⟩, where L = (TπA,πBL∗)∗ = TπB,πAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 allows us to conclude that covariant positive linear maps are enough to characterize separability of bipartite invariant quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We remark that the ideas of the following proof appeared already for some specified symmetries [Kay11, G¨11, SN21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA : G → B(HA) and πB : G → B(HB) be unitary representations, and let ρ ∈ InvQS(πA⊗πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) ρ is a separable quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) (idA ⊗ L)(ρ) ≥ 0 in B(HA ⊗ HA) for any (πB, πA)-covariant pos- itive linear map L : B(HB) → B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) ⟨ρ, L⟩ ≥ 0 for any (πB, πA)-covariant positive linear map L : B(HB) → B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Two directions (1) ⇒ (2) and (2) ⇒ (3) are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For the last direction (3) ⇒ (1), let us show that ⟨ρ, L⟩ ≥ 0 for all positive linear maps L : B(HB) → B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, since ρ is πA ⊗ πB-invariant and TπB,πAL ∈ CovPos(πB, πA), we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 to obtain ⟨ρ, L⟩ = ⟨TπA⊗πBρ, L⟩ = ⟨ρ, TπB,πAL⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4) □ From now on, let us focus on the question of whether PPT property coin- cides with separability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=SEP problem for invariant quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Recall that the dual notions of PPT property and separability correspond to decomposability and positivity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, many duality argu- ments [Kye23] carry over into our framework, and the PPT=SEP problem ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 11 in InvQS(πA ⊗ πB) is equivalent to the question whether all positive maps are decomposable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' POS=DEC problem in Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let L : B(HB) → B(HA) be (πB, πA)-covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then (1) L is positive if and only if (idA ⊗ L)(ρ) ≥ 0 for any separable ρ ∈ InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) L is decomposable if and only if (idA ⊗ L)(ρ) ≥ 0 for any PPT ρ ∈ InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) Suppose (idA⊗L)(ρ) ≥ 0 for any separable ρ ∈ InvQS(πA⊗πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then for every separable state ρ ∈ D(HA ⊗ HB), we have ⟨ρ, L⟩ = ⟨ρ, TπB,πAL⟩ = ⟨TπA⊗πBρ, L⟩ ≥ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 and by the separability of TπA⊗πBρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Now positivity of L follows from [EK00, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The converse direction is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) It is enough to repeat the arguments from (1) based on the following duality result [Sr82]: L is decomposable if and only if ⟨ρ, L⟩ ≥ 0 for every PPT state ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The following are equivalent: (1) PPT=SEP in InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) POS=DEC in Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ((1) ⇒ (2)) If L ∈ CovPos(πB, πA), then (idA ⊗ L)(ρ) ≥ 0 for every separable (hence every PPT) state ρ ∈ InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Thus, L is decomposable by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ((2) ⇒ (1)) If ρ ∈ InvQS(πA ⊗πB) is a PPT state, then (idA ⊗L)(ρ) ≥ 0 for every decomposable (hence every positive) linear map L ∈ Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Thus, ρ is separable by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Note that PPT=SEP in InvQS(πA ⊗ πB), or equivalently POS=DEC in Cov(πB, πA), implies that PPT property coincides with the entanglement- breaking property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=EB in CovQC(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, we have the following characterization of EB property for Φ ∈ CovQC(πA, πB) by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let Φ ∈ CovQC(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the following are equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) Φ is entanglement-breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) CL◦Φ = (id ⊗ L)(CΦ) ≥ 0 for any L ∈ CovPos(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) L ◦ Φ is completely positive for any L ∈ CovPos(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' To summarize, we have PPT=SEP in InvQS(πA ⊗ πB) ⇔ DEC=POS in Cov(πB, πA), 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN and these conditions imply PPT=EB in CovQC(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' One might ask whether all these three problems are equivalent, but one technical issue is that CovQC(πA, πB) is not identified with InvQS(πA ⊗πB) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This leads us to question whether the (reduced) channel-state duality �C : CovQC(πA, πB) → InvQS(πA ⊗ πB) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The channel-state duality �C is not surjective in general, but it is known to be the case if πA is irreducible, as already noted in [GBW21, Lemma 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, we prove that the converse is also true, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' the channel-state duality �C is bijective if and only if πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us start with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA : G → U(HA) and πB : G → U(HB) be unitary representations of G and let L ∈ Cov(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) If πB is irreducible, then L(IdA) = c IdB for some constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) If πA is irreducible, then there is a constant c such that Tr(L(X)) = c Tr(X) for X ∈ B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) From the irreducibility of πB and the identity πB(x)L(IdA)πB(x)∗ = L(πA(x)πA(x)∗) = L(IdA), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7) we have L(IdA) ∈ Inv(πB) = C · IdB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) The adjoint map L∗ is (πB, πA)-covariant by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 (2), so L∗(IdB) = c IdA for some c by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this case, we have Tr(L(X)) = Tr(L(X) IdB) = Tr(X L∗(IdB)) = c Tr(X) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8) for any X ∈ B(HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Now, let us apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6 (2) to prove that the channel-state duality �C : CovQC(πA, πB) → InvQS(πA ⊗ πB) should be bijective if and only if πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA : G → U(HA) and πB : G → U(HB) be unitary representations of G and let L ∈ Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the channel-state duality �C : CovQC(πA, πB) → InvQS(πA ⊗ πB) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) is bijective if and only if πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us prove the if part first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For any ρ ∈ InvQS(πA ⊗ πB) there exists completely positive L ∈ Cov(πA, πB) such that CL = ρ by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, L should be trace-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, irreducibility of ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 13 πA implies that there exists a constant c such that Tr(L(X)) = cTr(X) for all X ∈ B(HA) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6 (2), and we have c = c dA dA � i=1 Tr(eii) = 1 dA dA � i=1 Tr(eii ⊗ L(eii)) = Tr(CL) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='10) Conversely, if we assume that πA = π(1) A ⊕ π(2) A with HA = H(1) A ⊕ H(2) A and if Π1 is the orthogonal projection from HA onto H(1) A , then we can take a CP non-TP map L : B(HA) → B(HB) given by L(X) = dA dB · dim H(1) A Tr(Π1X)IdB (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11) whose Choi matrix is CL = � 1 dim H(1) A Π1 � ⊗ � 1 dB IdB � ∈ InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='12) □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA : G → U(HA) and πB : G → U(HB) be unitary representations of G and suppose that πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) PPT=SEP in InvQS(πA ⊗ πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) PPT=EB in CovQC(πA, πB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) POS=DEC in CovPos1(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' It is enough to note that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6 allows us to focus on a smaller convex set CovQC(πA, πB) and CovPos1(πB, πA) rather than Cov(πA, πB) and CovPos(πB, πA), respectively, in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Moreover, we prove that the extreme points of CovPos1(πB, πA) are enough for the entanglement detection, which we propose as a universal machinery to characterize entangled invariant quantum states with general compact group symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us denote by Ext(S) the set of all extreme points of a convex set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA : G → B(HA) and πB : G → B(HB) be unitary representations, and suppose that πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let ρ ∈ InvQS(πA ⊗ πB) and Φ ∈ CovQC(πA, πB) such that CΦ = ρ from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) ρ is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) (id ⊗ L)(ρ) ≥ 0 for any L ∈ CovPos1(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (3) (id ⊗ L)(ρ) ≥ 0 for any L ∈ Ext (CovPos1(πB, πA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4) Φ is entanglement-breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (5) L ◦ Φ is completely positive for any L ∈ CovPos1(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN (6) L ◦ Φ is completely positive for any L ∈ Ext (CovPos1(πB, πA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The equivalences (1) ⇔ (4), (2) ⇔ (5), (3) ⇔ (6) and one-side impli- cations (1) ⇒ (2) ⇒ (3) are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, the direction (2) ⇒ (1) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' For the proof of (3) ⇒ (2), note that CovPos1(πB, πA) is a compact subset of {L ∈ B(B(HB), B(HA)) : ∥L∥op ≤ 1} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='13) where ∥ · ∥op denotes the operator norm with respect to Hilbert-Schmidt norms on B(HB) and B(HA), since the positivity of L implies ∥L∥op = ∥L(IdB)∥ = 1 [Pau02, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Therefore, CosPos1(πB, πA) can be written as a convex hull of its extreme points, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If πB is irreducible instead of irreducibility of πA, then Φ is chosen to be a unital CP map up to constant, and CovPosTP(πB, πA) replaces the role of CovPos1(πB, πA) in (2), (3), (5), (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that compact- ness of CovPosTP(πB, πA) comes from the identification with CovPos1(πA, πB) (up to constant) via taking the adjoint operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Finally, we claim that the decomposability of the extremal elements in CovPos1(πB, πA) is much easier to check thanks to the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Suppose that πA (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' πB) is irreducible, and let L ∈ Ext(CovPos1(πB, πA)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' L ∈ Ext(CovPosTP(πB, πA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then L is de- composable if and only if L is CP or CCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us focus only on the case where πA is irreducible since the other case is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If L is decomposable, then there exist a CP map L1 and a CCP map L2 such that L = L1 + L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' By taking the twirling operation TπB,πA, we have L = L′ 1 + L′ 2 where L′ i = TπB,πA(Li) ∈ Cov(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that L′ 1 is CP, L′ 2 is CCP, and we can write L′ i = λiL′′ i for some λi ≥ 0, λ1 + λ2 = 1, and L′′ i ∈ CovPos1(πB, πA) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then extremality of L allows us to conclude that L = L′′ 1 or L = L′′ 2, which proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The other direction is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ To summarize, our strategy to study PPT=SEP and PPT=EB problems consists of the following three independent steps, assuming πA is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' [Step 1] The first step is to characterize all elements in CovPos1(πB, πA) for given specific unitary representations πA and πB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' [Step 2] The next step is to solve POS=DEC problem in CovPos1(πB, πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In particular, for extremal elements L ∈ Ext(CovPos1(πB, πA)), the given L is decomposable if and only if L is CP or CCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If POS=DEC holds, then PPT=SEP problem in InvQS(πA ⊗ πB) and PPT=EB problem in CovQC(πA, πB) has the affirmative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 15 [Step 3] If there exists a non-decomposable element in CovPos1(πB, πA), then the last step is to find the following objects: – Φ ∈ CovPPTQC(πA, πB) for which L ◦ Φ is non-CP, – ρ ∈ InvPPTQS(πA ⊗ πB) for which (id ⊗ L)(ρ) ≱ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=EB HOLDS FOR (H, H)-COVARIANT QUANTUM CHANNELS One of the main applications of the results in Section 3 is a complete char- acterization of EB property for quantum channels Φ : Md(C) → Md(C) of the form Φ(X) = aTr(X) d Idd + bX + cXT + (1 − a − b − c) diag(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) The main result of this section is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let Φ be a quantum channel of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then Φ is entanglement-breaking if and only if Φ is PPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) The quantum channels of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) include two important one-parameter families of quantum channels, namely de- polarizing channels ∆b(X) = (1 − b)Tr(X) d Idd + bX (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2) with − 1 d2−1 ≤ b ≤ 1, and transpose depolarizing channels Λc(X) = (1 − c)Tr(X) d Idd + cXT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) with − 1 d−1 ≤ c ≤ 1 d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' It was already known that PPT=EB for these channels from various perspectives [Wer89, HH99, Wat18, SN21], and our Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 covers these classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) Moreover, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 gives an affirmative answer to PPT=EB problem for the so-called generalized Werner-Holevo quantum chan- nels, which was once conjectured to be false [DFV08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' See Appen- dix A for more details on the generalized Werner-Holevo channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A starting point for a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1 is to observe that any quantum channel of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) is covariant with respect to the signed symmetric group Hd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' One of the equivalent ways to realize the signed symmetric group is to define Hd as a subgroup of the orthogonal group Od generated by permutation matrices and diagonal orthogonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In other words, every element in Hd is written as an orthogonal matrix d � i=1 si|σ(i)⟩⟨i| for s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , sn ∈ {±1} and σ ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We define Inv(H ⊗ H) and Cov(H, H) 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN with respect to the fundamental representation H ∈ Hd �→ H ∈ Od, which is irreducible as proved below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The fundamental representation H ∈ Hd �→ H ∈ Od is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The identity HXHT = d � i,j=1 sisjXij|σ(i)⟩⟨σ(j)| = d � i,j=1 sσ−1(i)sσ−1(j)Xσ−1(i)σ−1(j)|i⟩⟨j| (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4) and the invariance property HXHT = X for all H ∈ Hd tell us that sσ(i)sσ(j)Xσ(i)σ(j) = Xij (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) for all s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , sd ∈ {±1} and σ ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This implies that Xii ≡ X11 for all 1 ≤ i ≤ d and Xij = 0 for all i ̸= j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=', X = X11 Idd ∈ C · Idd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Let us denote by W the space of linear maps spanned by the following four unital TP maps ψ0, ψ1, ψ2, ψ3 : Md(C) → Md(C), where � � � � � � � ψ0(X) = Tr(X) d Idd, ψ1(X) = X, ψ2(X) = XT, ψ3(X) = diag(X) = �d i=1 Xii|i⟩⟨i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6) It is straightforward to check ψi ∈ Cov(H, H) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , 3, so we have W ⊆ Cov(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' To prove Cov(H, H) = W, let us note the fact that any L ∈ Cov(H, H) satisfies the so-called diagonal orthogonal covariance (DOC) property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' L(ZXZT) = ZL(X)ZT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7) for all X ∈ Md(C) and diagonal orthogonal matrices Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This class of channels has been analyzed recently in [SN21, SN22, SDN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In partic- ular, it is shown that any DOC map L can be parameterized by a triple (A, B, C) ∈ Md(C)3 satisfying diag(A) = diag(B) = diag(C) such that L(X) = diag(A|diag X⟩) + �B ⊙ X + �C ⊙ XT, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8) where |diag Y ⟩ = �d i=1 Yii|i⟩, �Y = Y − diag(Y ), and ⊙ denotes the Schur product (or Hadamard product) between matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In this case, let us denote by L = LA,B,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The space Cov(H, H) is spanned by the four unital TP positive maps ψ0, ψ1, ψ2, and ψ3 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We already know W ⊆ Cov(H, H), and let us pick an arbitrary L ∈ Cov(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Since L is DOC, there exists (A, B, C) ∈ Md(C)3 such that L = LA,B,C of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that L further satisfies L(PσXP T σ ) = PσL(X)P T σ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) for all X ∈ Md(C) and σ ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Here, Pσ = �d i=1 |σ(i)⟩⟨i| is the permuta- tion matrix associated with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us take X = eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If i = j, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) implies d � k=1 Akσ(i)|k⟩⟨k| = d � k=1 Aki|σ(k)⟩⟨σ(k)|, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='10) which means that Aik = Aσ(i)σ(k) for all 1 ≤ i, k ≤ d and σ ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' There- fore, Aii ≡ A11 for all i and Aik ≡ A12 for all i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' On the other hand, if i ̸= j, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='9) becomes Bσ(i)σ(j)|σ(i)⟩⟨σ(j)| + Cσ(j)σ(i)|σ(j)⟩⟨σ(i)| (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='11) = Bij|σ(i)⟩⟨σ(j)| + Cji|σ(j)⟩⟨σ(i)|, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='12) which gives Bij ≡ B12 and Cij ≡ C12 for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Consequently, the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8) now gives L = dA12ψ0 +B12ψ1 +C12ψ2 +(A11 −A12 −B12 −C12)ψ3 ∈ W, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='13) which in turn shows Cov(H, H) ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ From now, let us denote (H, H)-covariant unital (and TP) maps by ψa,b,c = aψ0 + bψ1 + cψ2 + (1 − a − b − c)ψ3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='14) for simplicity, where ψ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , ψ3 are from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that ψa,b,c can be understood as a DOC map LA,B,C under the correspondence � � � � � A = a dJd + (1 − a)Idd, B = b(Jd − Idd) + ( a d + (1 − a))Idd, C = c(Jd − Idd) + ( a d + (1 − a))Idd, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='15) where Jd = �d i,j=1 eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' According to [SN21, Section 6], LA,B,C is CPTP if and only if � � � � � Aij ≥ 0, �d k=1 Akj = 1 for all i, j B ≥ 0, Cij = Cji, |Cij|2 ≤ AijAji for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='16) 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN In terms of the parameters a, b, c, the map ψa,b,c is CPTP if and only if � � � 0 ≤ a ≤ d d−1, a d − 1 d−1 ≤ b ≤ 1 − d−1 d a, − a d ≤ c ≤ a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='17) Note that the set of (a, b, c) ∈ R3 satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='17) is a tetrahedral depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The region of CovQC(H, H) In particular, there are exactly four extremal (H, H)-covariant quantum channels corresponding to the four vertices given by � � � � � � � � � Ψ1 = ψ1, Ψ2 = d d−1ψ0 + 1 d−1ψ2 − 2 d−1ψ3, Ψ3 = − 1 d−1ψ1 + d d−1ψ3, Ψ4 = d d−1ψ0 − 1 d−1ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='18) whose Choi matrices are (up to normalization) four mutually orthogonal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' On the other hand, it is easy to see that Td ◦ ψa,b,c = ψa,b,c ◦ Td = ψa,c,b, a, b, c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='19) Therefore, ψa,b,c is a PPT quantum channel if and only if both ψa,b,c and ψa,c,b are CPTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us denote by CovPPTQC(H, H) the set of all PPT quantum channels ψa,b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then CovPPTQC(H, H) can be realized as a con- vex set in R3 as in the following Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' If d ≥ 3, the eight vertices of the polytope CovPPTQC(H, H) are explic- itly given by ψvi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , 8), where v0 = (0, 0, 0), 亚4 亚2 1 1 1 1 1 亚ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 19 FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The region of CovPPTQC(H, H) v1 = � d 2(d−1), 1 2(d−1), − 1 2(d−1) � , v2 = � d 2(d−1), − 1 2(d−1), 1 2(d−1) � , v3 = � d 2(d−1), − 1 2(d−1), − 1 2(d−1) � , v4 = � 1, 1 d, 1 d � , v5 = � 1, 1 d, − 1 d(d−1) � , v6 = � 1, − 1 d(d−1), 1 d � , v7 = � d d−1, 0, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' [Step 1+Step 2] One of the main steps in this section is to characterize all elements in CovPos1(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Indeed, CovPos1(H, H) is given as follows with eight extreme points (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The region of CovPos1(H, H) 1 1 V5 V6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 1 1 V3 1 V2 1 1 1 4 1 1 1 1 o W亚 亚2 亚3 I3 0 Td ioTd 2 亚20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the convex set CovPos1(H, H) has exactly 8 extreme points Ψ1, Ψ2, Ψ3, Ψ4, Ψ1 ◦ Td, Ψ2 ◦ Td, Ψ3 ◦ Td, Ψ4 ◦ Td, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='20) where Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , Ψ4 are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In particular, all positive (H, H)- covariant maps are decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Since Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , Ψ4 are CP and Ψ1◦Td, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' , Ψ4◦Td are CCP, the convex hull Vd of these 8 maps is obviously contained in CovPos1(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' To show the reverse inclusion CovPos1(H, H) ⊆ Vd, we observe that the set Vd := � (a, b, c) ∈ R3 : ψa,b,c ∈ Vd � ⊂ R3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='21) is the convex hull of 8 points � (0, 1, 0) , � d d−1, 0, 1 d−1 � , � 0, − 1 d−1, 0 � , � d d−1, 0, − 1 d−1 � , (0, 0, 1) , � d d−1, 1 d−1, 0 � , � 0, 0, − 1 d−1 � , � d d−1, − 1 d−1, 0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='22) which are got from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Therefore, Vd can be understood as the region of (a, b, c) ∈ R3 satisfying the following inequalities: � � � � � � � � � � � � � � � (1) 0 ≤ a ≤ d d−1, (2) d−2 d a + b + c ≤ 1, (3) d−2 d a + |b − c| ≤ 1, (4) b + c ≥ − 1 d−1, (5) b − (d − 1) c ≤ 1, (6) c − (d − 1) b ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='23) Now if ψa,b,c /∈ Vd (which is equivalent to (a, b, c) /∈ Vd, and hence violates at least one of the inequalities (1) - (6) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='23)), we can choose a unit vector ξ ∈ Cd such that ψa,b,c(|ξ⟩⟨ξ|) is not positive semidefinite as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This shows CovPos1(H, H) ⊆ Vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Non-positivity outside Vd (a, b, c) violates (1) |ξ⟩ = |1⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 (a, b, c) violates (2) |ξ⟩ = 1 √ 2(|1⟩ + |2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 (a, b, c) violates (3) |ξ⟩ = 1 √ 2(|1⟩ + i|2⟩) =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 (a, b, c) violates (4) |ξ⟩ = 1 √ d d � k=1 |k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 (a, b, c) violates (5) or (6) |ξ⟩ = 1 √ d d � k=1 e 2πik d |k⟩ =⇒ ψa,b,c(|ξ⟩⟨ξ|) ≱ 0 □ ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 21 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Now, the conclusion is straightforward from Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='8 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (1) Note that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5 gives a complete characteri- zation of all positive linear maps ψ spanned by ψ0, ψ1, ψ2, ψ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This strengthens the results in Section 5 of [KMS20] focusing on positive linear maps spanned only by ψ0, ψ1, ψ3 without ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (2) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5 tells us not only POS=DEC, but also explicit decom- positions of our positive covariant maps into sums of CP and CCP maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that this was one of the open questions raised in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='c of [KMS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We refer to Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT=SEP PROBLEMS IN TRIPARTITE SYSTEMS WITH UNITARY GROUP SYMMETRIES Recall that a tripartite quantum state ρ ∈ D(HA ⊗ HB ⊗ HC) is called A-BC separable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' A-BC PPT) if ρ is separable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT) in the situation where B(HA ⊗ HB ⊗ HC) is understood as the bipartite system B(HA) ⊗ B(HB ⊗ HC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Furthermore, C-AB or B-AC separability (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PPT) is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' We will focus on the situation where HA = HB = HC = Cd, and let us denote by � � � XTA = (Td ⊗ idd2)(X), XTB = (idd ⊗ Td ⊗ idd)(X), XTC = (idd2 ⊗ Td)(X), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1) the three partial transposes of X ∈ B(HA ⊗ HB ⊗ HC) = Md3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The main purpose of this section is to apply our results in Section 3 as new sources to study PPT=SEP problems, equivalently POS=DEC prob- lems for some tripartite invariant quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1, we exhibit positive non-decomposable covariant maps L : Md(C) → Md2(C) satisfy- ing L(UXU T) = (U ⊗ U)L(X)(U ⊗ U)∗ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2) for all unitary matrices U ∈ Ud and X ∈ Md(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' This result is parallel to the fact PPT̸=SEP for tripartite Werner states [EW01], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' tripartite quantum states ρ ∈ Md3(C) satisfying (U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='3) for all unitary matrices U ∈ U(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' On the other hand, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2, we show that a strong contrast PPT=SEP holds for quantum orthogonally invariant quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' More generally, we prove that PPT=SEP holds for any tripartite quantum states ρ ∈ Md3(C) satisfying (U ⊗ U ⊗ U)ρ = ρ(U ⊗ U ⊗ U) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4) 22 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' JUNG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' PARK, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' YOUN for all unitary matrices U ∈ U(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Tripartite Werner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let πA, πBC be unitary representations of the unitary group Ud given by πA(U) = U and πBC(U) = U ⊗ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then the elements in InvQS(πA ⊗ πBC) are called tripartite Werner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let us write Inv(U ⊗3) = Inv(πA ⊗ πBC) and Cov(U, UU) = Cov(πA, πBC) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' The application of Schur-Weyl duality [EW01] or von Neu- mann’s bicommutant theorem [Wat18, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='15] implies that the space Inv(U ⊗3) is spanned by six unitary operators {Vσ : σ ∈ S3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Here, Vσ : (Cd)⊗3 → (Cd)⊗3 is determined by Vσ(ξ1 ⊗ ξ2 ⊗ ξ3) = ξσ−1(1) ⊗ ξσ−1(2) ⊗ ξσ−1(3) for any ξ1, ξ2, ξ3 ∈ Cd and σ ∈ S3, or equivalently, Vσ = d � j1,j2,j3=1 |j1j2j3⟩⟨jσ(1)jσ(2)jσ(3)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5) Recall that A-BC PPT property and separability of ρ ∈ InvQS(U ⊗3) were already characterized in [EW01], and it was shown that PPT=SEP if and only if d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Therefore, a direct application of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 gives us the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' All positive (UU, U)-covariant maps are decomposable if and only if d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' By taking the adjoint operation L �→ L∗, the same conclusion holds for positive (U, UU)-covariant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' In the remaining of this section, we will assume d ≥ 3 and exhibit posi- tive non-decomposable (U, UU)-covariant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' [Step 1] First of all, let us characterize all elements in CovPos(U, UU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Note that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='4 (3) implies that the space Cov(U, UU) is spanned by the following six linear maps Lσ whose unnormalized Choi matrices are the operators Vσ ∈ Inv(U ⊗3) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='5): � � � � � � � � � � � � � � � Le(X) = (Tr X) · Idd ⊗ Idd, L(12)(X) = XT ⊗ Idd, L(13)(X) = Idd ⊗ XT, L(23)(X) = (Tr X) · �d j2,j3=1 |j3j2⟩⟨j2j3|, L(123)(X) = �d j1,j2,j3=1 Xj1j2|j2j3⟩⟨j3j1|, L(132)(X) = �d j1,j2,j3=1 Xj1j3|j2j3⟩⟨j1j2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='6) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Let L = � σ∈S3 aσLσ ∈ Cov(U, UU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Then L is positive if and only if � � � � � � � (1) ae, a(12), a(13), a(23) ∈ R and a(132) = a(123), (2) ae ≥ max � −a(12), −a(13), |a(23)| � , (3) ae + a(12) + a(13) + a(23) + a(123) + a(132) ≥ 0, (4) � ae + a(12) � � ae + a(13) � ≥ ��a(23) + a(123) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content='7) ENTANGLEMENT DETECTION OF INVARIANT QUANTUM STATES 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Since every unit vector ξ ∈ Cd can be written as |ξ⟩ = U|1⟩ for some U ∈ Ud, the (U, UU)-covariance property implies that L is positive if and only if L(e11) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E2T4oBgHgl3EQfYAcW/content/2301.03849v1.pdf'} +page_content=' Moreover, L(e11) has a matrix decomposition L(e11) ∼= (ae + a(12) + a(13) + a(23) + a(123) + a(132))1 ⊕ (ae + a(23))Idd−1 ⊕ � � d � j=2 � ae + a(12) a(23) + a(123) a(23) + a(132) ae + a(13) �� � ⊕ � � � 2≤i 0 +~ 5 nm +0.25 +Inlet +[Norm.] +Diamagnetic +Microfluidic +0.5 +Device +Contrast [ +Paramagnetic +Near-Surface +0.5 +1.5 +2.5 +NV-Ensemble +3.5 +Outlet +Pure Water +NaCI (aq.) +MnCl2 (ag.) +Laser (532 nm) +Diamond +0 +1 +2 +4 +7 +8 +9 +Fluorescence (635 to 800 nm) +Sweep Time t [ms]the relaxation curves depicting the contrast as a function of sweep time t (see Supplementary +Note 1 for fitting details). +We perform the measurements by filling the microfluidic channel and covering the di- +amond surface either with pure water or aqueous electrolyte solutions (see Methods for +detail). Figure 1b depicts the T1 relaxation curves when water and solutions of diamagnetic +NaCl or paramagnetic MnCl2 cover the diamond surface. Paramagnetic MnCl2 (1 µM) on +the diamond leads to a T1 time reduction of 0.47 ± 0.19 with respect to water, which is in +accordance with other studies and can be ascribed to the strong dipole-dipole interaction of +the NV-center with the paramagnetic species.25,32 In contrast, when we repeat the same ex- +periment with diamagnetic NaCl (500 mM) solution, we observe an extension of the T1 time +by a factor of 2.04 ± 0.45 compared to water. Experiments supporting this observation are +also conducted with other diamagnetic salt solutions (mono-, di- and trivalent) and reveal +similar results (see Supplementary Note 2). Therefore, we choose NaCl as a representative +of a standard diamagnetic electrolyte for the following measurements in our work and expect +comparable results for other diamagnetic salt solutions. +Moreover, by tuning the magnetic field B0 and thereby the NV-center’s Larmor fre- +quency NV0,-1 (i.e., the ms = 0 → ms = −1 transition frequency) we are able to map the +spectral noise density. We probe water/NaCl (500 mM) solution with NV0,-1 frequencies from +131 MHz to 2.87 GHz and observe a similar effect over the entire frequency range (see Supple- +mentary Note 2). Consequently, the extension of the T1 time of near-surface NV-ensembles +with exposure to diamagnetic electrolyte solutions is an effect that covers a broad range of +(high) frequencies (i.e., from ∼ hundreds of MHz to GHz). +Additionally, in order to exclude an impact of the solvent’s physical properties (i.e., +polarity) on our experiments,19 we choose typical organic solvents whose dielectric constants +(κ) and chemical structure differ significantly from water (κ = 8040) and probe them with +relaxometry (see Supplementary Note 3). Since the T1 time remains unaffected, we conclude +that the herein described effect is not induced by the physical properties of water, but by +6 + +the diamagnetic electrolyte. +Sensitivity of T1 Relaxometry on Electrolytes +In order to obtain information about the sensitivity of the NV-relaxometry protocol +to para- and diamagnetic electrolyte solutions, we perform additional measurement series +where the electrolyte concentration is increased stepwise by one order of magnitude (from +10−5 to 10−2 mM in the case of paramagnetic MnCl2 and from 10−4 to 103 mM in the case +of diamagnetic NaCl). +Paramagnetic MnCl2 shows a stepwise T1 decrease in micromolar concentrations reaching +a decline of up to 86 ± 10% for a 10 µM solution with respect to water covering the diamond +(see Figure 2a and Supplementary Note 4). +Note that a further concentration increase +(> 10 µM) is not measurable, as it leads to a collapse of the T1 time. In contrast, diamagnetic +NaCl shows a slight T1 increase compared to water, which then fluctuates moderately from +micromolar to lower millimolar concentrations. Importantly, a significant and gradual T1 +increase is measurable from 10 mM to 500 mM NaCl solution, where the effect saturates at +81 ± 11% with respect to water (see Figure 2b and Supplementary Note 4). +The decrease of the T1 time with paramagnetic species (e.g., MnCl2) is expected and well +studied.17,25,26,30 Here, high frequency (∼ GHz) noise originates from magnetic dipole-dipole +interactions of the NV-center’s electronic spin and the sample’s electronic spin (“spin-flips”), +resulting in a decline of the T1 time if unpaired electrons are near the sensor. On the other +hand, for diamagnetic ions (e.g., NaCl) these interactions are absent as only paired electrons +without a (net) magnetic moment are present. +Surprisingly, here we observe a gradual +extension of the T1 time with increasing millimolar concentrations of diamagnetic NaCl +solution. +Importantly, both sensitivity regimes (∼ nano- to micromolar for paramagnetic and ∼ +millimolar for diamagnetic species) match the typical physiological41,43 or (for diamagnetic +electrolytes) electrochemical concentrations,44 opening up sensing applications in cell biology +7 + +Figure 2: NV-relaxometry with increasing concentrations of para- and diamag- +netic electrolyte solutions. a) Paramagnetic MnCl2 shows a stepwise T1 time decrease +for concentrations in the micromolar regime until the effect reaches a maximum measur- +able decline of 86 ± 10% for 10 µM solutions with respect to water covering the diamond. +b) In contrast to that, diamagnetic NaCl (right) shows a slight increase of the T1 time +compared to water, which then fluctuates moderately from the micromolar to the lower +millimolar regime. For concentrations ≥ 10 mM the T1 time increases gradually along with +the NaCl concentration until the effect saturates to 81 ± 11% for NaCl (500 mM) solutions. +Shaded areas indicate typical physiological concentration regimes for para- and diamag- +netic ions.41–43 Experiments are performed at fNV = 1.88 GHz. +or electrochemistry. +8 + +InCl2 (aq.) +Conc. +Flips +Conc.Reversibility, Passivation and NV-Center Depth +Because of the surprising observation, that the T1 time increases with diamagnetic elec- +trolyte solutions compared to water covering the diamond, the next experiments concentrate +on the mechanism behind this effect. Therefore, we probe the reversibility and passivation +of the effect along with the sensor’s response in dependence of its implantation depth. First, +we evaluate if the extension of the T1 relaxation time is a reversible process by exposing an +oxygen-terminated diamond alternatingly to water and NaCl (500 mM) solution. Thereby, +we show that the T1 relaxation time is altered from “short” in case of water exposure to +“long” when NaCl solution covers the surface (see Figure 3a in green). Alternating between +water and electrolyte solution demonstrates a 1.83±0.35 fold increase of the T1 time with elec- +trolyte exposure on the oxygen-terminated diamond. In a next step, we examine if the effect +is specific to the diamond surface termination. Therefore, the formerly oxygen-terminated +diamond is coated with an aluminium oxide (Al2O3) thin film (thickness ∼ 1 nm)13 prepared +by Atomic Layer Deposition (ALD). The aluminium oxide thin film ensures a controllable +and uniform surface termination with hydroxyl groups. We repeat the previous experiment, +but this time the T1 relaxation time remains unaffected by the NaCl solution (see Figure 3a +in black). +Additionally, we investigate if the extent of the electrolyte’s effect is dependent on the +depth of the embedded NV-center ensemble. Therefore, we prepare two diamonds with 15N +implantation energies of 2.5 and 4 keV with the tri-acid clean procedure described before +and probe them with NV-relaxometry. Near-surface NV-centers implanted with an energy +of 2.5 keV are mainly distributed within a depth of ∼ 5 nm below the surface, while ensembles +created with 4 keV 15N are located about ∼ 12 nm beneath the surface.37 Figure 3b shows a +significantly larger effect of the electrolyte on the relaxation time of the shallow implanted +NV-diamond with respect to the deeper one, although a T1 time extension is still detectable +in the latter case (see also Supplementary Note 5). Importantly, while the effect with the +∼ 1 nm thick aluminium oxide layer is completely passivated, a T1 time increase can still be +9 + +Figure 3: NV-relaxometry experiments with different surface terminations and +NV-center ensemble implantation depths. a) An oxygen-terminated diamond sur- +face is alternatingly exposed to water and NaCl (500 mM) solution. The T1 time increases +with electrolyte exposure by a factor of 1.83 ± 0.35 with respect to water. Importantly, +this behavior can be altered by either the presence or absence of water or NaCl solution. +After coating the same diamond with an aluminium oxide (Al2O3) thin film (thickness +∼ 1 nm) the T1 time is unaffected by the NaCl. b) T1 relaxation curves of water and NaCl +(500 mM) solution covering the diamond surface. Diamonds were implanted with 15N at +an energy of 2.5 keV (top) and 4 keV (bottom) resulting in different NV-center ensemble +depths (dNV). While on the shallower implanted diamond the T1 time increases by a factor +of around two with exposure to NaCl solution, on the deeper implanted diamond the effect +is strongly reduced. Experiments are performed at fNV = 1.88 GHz. +10 + +2.2 +1.8 +1.4 +1.0 +dv ~ 5 nm +Oxygen-Terminated Diamond +OHO OH @ OH e +~ 1 nm Al203 +dnv ~ 5 nm +dnv ~ 5 nm +dnv ~ 10-15 nmobserved with the oxygen-termination when the sensor is ∼ 5 to 10 nm further away from +the electrolyte. Note that the same measurements conducted with LiCl (500 mM) solution +lead to similar results (see Supplementary Note 5). +From these experiments we conclude that the extension of the T1 relaxation time is a +reversible and interfacial process which is dependent on the distance of the sensor to the +sample. +Additionally, NV-charge state alterations or changes in NV-dephasing and NV-coherence +are not observed in our experiments (see Supplementary Note 6). +Probing Magnetic and Electric Noise Contributions +In the next set of experiments we investigate if the T1 relaxation time increase originates +from a reduction of electric and/or magnetic field noise. Since we are dealing with electrolytes +dissolved in water, it is particularly interesting to explore the influence that charged ions +and their randomly fluctuating electric fields might have on the T1 relaxation time of the +near-surface NV-ensembles. Whereas the typically used relaxometry experiments use single +quantum (SQ) transitions to probe magnetic field noise (as in the experiments from the +previous sections), double quantum (DQ) transitions are influenced by electric field noise.45 +For these transitions, the full NV-center ground state (S = 1) is considered, where an +additional relaxation pathway between ms = −1 ↔ ms = +1 with ∆ms = 2 (see Figure 4a) +becomes accessible, the DQ transition. Regarding the NV-center as a “qutrit” rather than a +qubit allows to probe the effect of the diamagnetic electrolyte solution on both electric and +magnetic field noise at the same time. +SQ and DQ relaxometry measurements on the system water/NaCl (500 mM) solution +reveal that the diamagnetic electrolyte has an effect on both relaxation channels, i.e., it +reduces magnetic as well as electric field fluctuations (see Figure 4b). Here, two different T1 +times can be defined: T1,SQ for the relaxation time in the SQ and T1,DQ in the DQ channel. +An extension of T1,SQ by a factor of 1.52 ± 0.16 and a 2.84 ± 0.31 increase of T1,DQ can +11 + +Figure 4: Single quantum (SQ) and double quantum (DQ) relaxation experi- +ments. a) Energy level scheme of the NV-center ground state transitions. SQ transitions +(∆ms = ±1) with relaxation rates Ω are susceptible to magnetic noise. The DQ relax- +ation (∆ms = ±2) with relaxation rate γ is magnetically forbidden but susceptible to +electric noise.45 b) Top: DQ pulse sequence. Bottom: SQ and DQ relaxation curves of +water and NaCl (500 mM) solution covering the diamond. Experiments are performed at +B0 = 15 G, where the NV0,-1 transition is at fNV = 2.83 GHz (corresponding to a DQ tran- +sition frequency of 80 MHz). T1,SQ increases by a factor of 1.52±0.16 and T1,DQ by a factor +of 2.84 ± 0.31 +when the diamond is exposed to NaCl solution. +12 + +SQ Relaxatior +ating +Spin +bles +~MHz +Q ~GHzbe measured in presence of NaCl solution with respect to water (see also Supplementary +Note 7). Thus, diamagnetic electrolytes reduce both – electric and magnetic – noise at the +diamond surface. Importantly, when we repeat the same experiments with paramagnetic +MnCl2, only T1,SQ reduces by 80%, whereas the DQ transition remains unaffected compared +to water (see Supplementary Note 7). This indicates an exclusive impact of the paramagnetic +electrolyte on magnetic field noise and could provide a possible pathway to distinguish para- +from diamagnetic ions in solution. +Additionally, we can exclude an influence of diamagnetic NaCl solution on the static +electric field environment by performing zero field ESR Measurements (see Supplementary +Note 7). +Influence of Electrolytes on Surface Dark Spins: DEER Experi- +ments +So far, we have focused on the direct influence of electrolytes on the NV-centers. How- +ever, the diamond as the NV-center’s host material provides various surface dark spins, e.g. +dangling bonds, whose response to the electrolytes is probed in the next set of experiments. +Intrinsic T1 times of these surface dark spins are often long (∼ a few microseconds)46 which +allows us to probe them with NV-based double electron electron resonance (DEER) spec- +troscopy.16,47 Figure 5a shows the pulse sequence of a typical DEER experiment: A spin-echo +is performed on the NV-center’s electronic spin (MWNV spin-echo), at the same time, in the +second free precession time of the echo, an additional microwave pulse (MWDEER) is ap- +plied to drive the target electronic spins. Sweeping the MW frequency (f DEER) flips the +surface dark spins when their Larmor frequency is matched and causes a dip in the DEER +signal (see Figure 5b). As shown in Figure 5b, a clear dip in the DEER spectrum appears +when the diamond interface is covered with NaCl (500 mM) solution. +The resonance at +f DEER = 0.887 GHz corresponds to ge ∼ 2 spins and is typically assigned to dangling bonds +at the diamond surface.50 Interestingly, the dip is drastically reduced when the experiment +13 + +Figure 5: NV-DEER experiments probing the response of surface dark spins +to electrolyte exposure. a) Pulse sequence of the double electron electron resonance +(DEER) experiment. Sweeping the microwave frequency of the MWDEER pulse (f DEER) +while applying a spin-echo experiment on the NV-center (MWNV spin-echo) allows for the +detection of electronic (surface dark) spins coupled to the NV-centers.48 b) DEER ex- +periment with water and NaCl (500 mM) solution covering the diamond surface. A pro- +nounced dip in the DEER spectrum appears at around 0.887 GHz (where ge ∼ 2) when the +diamond is exposed to NaCl solution. The dip gets drastically reduced when water cov- +ers the diamond surface. c) Top: Pulse sequence of the DEER-Rabi experiment. Bottom: +When the pulse duration of the microwave drive (MWDEER) at the surface dark spins’ +resonance frequency is swept during the spin-echo, DEER-Rabi oscillations can be ob- +served.49 While the πds-pulse lengths remain equal when water or NaCl (500 mM) solution +cover the diamond surface (πds ∼ 24 ns), the latter causes a by a factor of around three +more pronounced Rabi amplitude. d) Top: Pulse sequence of the DEER-T1 experiment. +Bottom: Varying the correlation time (tcorr) between two subsequent DEER segments,46 +shows a surface dark spin relaxation with T1,ds = 7.20±1.10 µs in case of NaCl exposure. In +contrast, no significant relaxation decay can be observed when water covers the diamond +surface. DEER experiments are performed at fNV = 1.98 GHz. +14 + +MWDEER +Surface +Dark Spins ++ +dNvis repeated with water. +Once the resonance condition for the ge ∼ 2 spins is found, coherent control of the +surface dark spin state can be demonstrated. Figure 5c shows a DEER-Rabi experiment +on the surface dark spins. Sweeping the microwave pulse duration (MWDEER) during the +spin-echo causes oscillations of the defect’s spin state.49 While we determine equal πds pulse +lengths (πds ∼ 24 ns) for water and the electrolyte, the former leads to an about three +times increased DEER-Rabi amplitude with respect to the latter. +In the next step, we +probe the surface dark spin relaxation time (T1,ds). Figure 5d depicts a pulse sequence from +Sushkov et al.,46 where the T1 time of the surface dark spins can be measured by correlating +two subsequent DEER segments and varying the correlation time tcorr. Interestingly, we +measure a relaxation time T1,ds of 7.20 ± 1.10 µs for the surface dark spins exposed to the +NaCl solution. In contrast, a clear relaxation decay cannot be observed in case of pure water +covering the diamond. Although we cannot exclude that the surface dark spins vanish, the +more likely case is that the dark spins act as reporter spins46 experiencing the same effect of +the electrolyte solution as the NV-center: “fast” relaxation in the case of water and “slow” +relaxation when diamagnetic electrolyte solutions cover the diamond, which is also expressed +in the increased DEER signals (see Figure 5b-d). Accordingly, we enhance the sensitivity of +our sensor by the proximity of the reporter spins to the electrolyte solutions.46,51 +Theoretical Modeling +Our experimental results show an influence of diamagnetic electrolyte solutions on near- +surface spin defects in diamond where electric as well as magnetic noise is suppressed resulting +in an increase of the T1 relaxation time. To further study this surprising effect computational +modeling is used. Here, as a working assumption, we focus on charge fluctuations within the +diamond lattice. We note, that further processes such as proton hopping at the interface or +water and ion dynamics within the electric double layer can also play a role but have not +15 + +been treated herein. +To this end, we model an interface between a slab of diamond and a thin layer of water +subsequently enriched by Na+ and Cl- ions (see Methods for detail). Then, we probe the +interfacial structure and vacuum level shifts (VLS) based on the configurations obtained +from the ab initio molecular dynamics (MD)(see Figure 6a). The calculated alignment of +the electronic levels of water and the model diamond surface is shown in Figure S10. Here, +a mismatch of the chemical potentials (defined as a center of the band gap) promotes an +electron leakage from the diamond surface towards the water. The resulting redistribution +of charges leads to the development of an electric field, that further rearranges the charged +solvated Na+ and Cl- ions. The large positive VLS of 1.1 eV (see Figure S10a) causes the +ions to rearrange with the direction of the field, facilitating the effect of band bending. By +adding a carboxyl group, we observe a stabilization of the downwards band bending relative +to the case of the model diamond surface in water. However, in both cases, we obtain a +broad distribution of surface dipoles, owing to the complexity of ion dynamics within the +Stern layer. By contrast, a sharp distribution of the dipole moments is observed between a +dissociated carboxyl group and a solvated Na+ ion nearby. This stable configuration gives +rise to a large VLS of ∼ -1.9 eV. This value is further used to trace the evolution of the +electrostatic potential at the microscopic level. More specifically, we set it as a boundary +condition for solving the Poisson equation to access the modifications of the potential inside +a semi-infinite diamond slab. As shown in Figure 6b, we observe that the interfacial region +of ∼ 40 nm is affected by the respective readjustments of the charges, resulting in a rapid +decay of the potential near the interface and a slow saturation towards the bulk. +To establish a relation between the band bending and the noise reduction, we consider the +electric and magnetic fluctuations caused by a pair of active defects around the NV-centers. +The charge transfer process leads to a continuous change in the charge/spin state of the +nearby defects, which can affect the relaxation and coherence time of the NV-center when +the rate approaches the timescale of the quantum sensing experiment. In the Marcus theory, +16 + +Figure 6: Ab initio MD simulations of the diamond/electrolyte interface. a) +Representative snapshot of the diamond/electrolyte interface from the ab initio MD simu- +lations. Color code: C (grey), O (red), H (white), Na+ (yellow), Cl- (green). b) Variations +of the electrostatic potential in a semi-infinite diamond due the arrangement of NaCl at +the interface. Inset: Electrostatic potential (∆Vmax) for a pair of defects in 3 nm distance +to each other surrounding a ∼ 5 nm deep NV-center. c) Schematic representation of a con- +tinuous charge hopping between two defects around the NV-center. HAB is the transfer +integral, ∆G0 is calculated as ∆Vmax/2 from b). d) Calculated rate constants for pairs of +substitutional nitrogen and vacancy defects as a function of distance between the defects. +e) Rate constants for the forward and backward electron transfer between a pair of va- +cancy defects as a function of bias due to the interfacial band bending. Vertical lines refer +to an estimated difference in the effects by replacing the electrolyte with pure water, con- +sidering a pair of defects and a NV-center in a configuration from the inset in b). +17 + +a) +b) +c) +forward +Electrostatic potential [eV] +0 +A +B +-0.5 +-1.25 +V- +NV +↑ HAB +-1 +vo +AV +-1.5 +-1.5 +-1.75 +△G0 +3 +4 +5 +6 +7 +-2 +back +0 +10 +20 30 +40 +50 60 +Distance [nm] +d) +10 12 +10 10 +108 +_s] - +108 +INON+ +H20 +NaCI +107 +106 +forward +104 +back +10 6 +0 +5 10 15 20 25 30 +0 +0.05 +0.1 +0.15 +△G° [eV] +Distance [A]such fluctuations are described as a sequence of thermally activated hopping events, whilst +the rate constants are determined from the distance-dependent coupling parameters and +the required structural reorganizations (see Figure 6c). The dipoles at the diamond/solvent +interface affect this equilibrium by altering the onsite Gibbs energy term with a contribution +from the electrostatic potential (∆V ). As shown in Figures 6c and 6e, the band bending +accelerates a forward charge transfer process (until reaching the Marcus inverted region), but +at the same time, the magnitude of the back charge transfer (BCT) rate drops exponentially. +Hence, regardless of the defect type, large interfacial band bending can lead to a dynamical +trapping of the charges around a site with the lower Gibbs energy. For a numerical validation, +we focus on the electron fluctuations between a pair of carbon vacancies as well as on the hole +fluctuations between two substitutional nitrogen defects. We note that carbon vacancies are +often generated by the implantation and irradiation techniques used to create the NV-centers, +and substitutional N defects are present around NV-centers to stabilize the negative charge +state of the NV-center. After determining the relevant parameters in the periodic supercells +(see Methods for detail), we first analyze the possible contribution of each charge transfer +reaction to the electrical noise. To this end, we compare the fluctuation rates, calculated for +both defect pairs in a bulk-like environment (∆V = 0). As shown in Figure 6d, the charge +fluctuation rates for a pair of nitrogen defects is remarkably smaller than for a vacancy pair. +This difference is attributed to a formation of an energetically unfavorable N0 configuration, +that hinders the charge fluctuations by large structural reorganizations (λreorg = 1.89 eV). +Hence, substitutional N0 pairs are unlikely to be the origin of electric and magnetic noise, +affecting the T1 time due to charge fluctuation. By contrast, owing to a rather modest λreorg +of 0.28 eV, a vacancy pair gives rise to noise in a broad frequency range, where the respective +rate constants are controlled by the separation between the active sites. +The relevant distance between a pair of vacancies is readily obtained from an exponential +fit of the rate constants in Figure 6d. More specifically, we find the charge fluctuation rates, +which would be relevant for an influence on the T1 time (∼ 0.1 GHz) at a separation of ∼ +18 + +3 nm. The molecular dynamics simulations performed by F´avaro de Oliveira et al. show +that such a high local density of vacancies around the region of a NV-center is achieved +during the nitrogen implantation due to the cascade process of the “kick-out” mechanism.52 +Using the variations of electrostatic potential from Figure 6b for the implantation depths of +5 and 12 nm, we calculate a contribution to the onsite Gibbs energies by 0.125 and 0.075 eV, +respectively (see Methods for details). +As shown in Figure 6e, the BCT rate at 12 nm +reduces by a factor of ∼ 7 relative to the case of ∆V = 0. Moreover, in agreement with +our experimental results, the faster changes in the potential at 5 nm enhances the reduction +factor to ∼ 19 for the shallower NV-centers. Relative to pure water, the BCT rates decrease +by factors of ∼ 10 and 3.5 for the depths of 5 nm and 12 nm, respectively. Furthermore, the +proposed mechanism is consistent with the experimental results on dark spins (see Figure 5). +Interestingly, our calculations point to an even larger decrease of the BCT rate at smaller +distances from the interface which should translate to an even larger sensitivity. +Given +the favorable downwards band bending, these results call for a further optimization of the +implantation parameters as well as the surface structure to fully exploit the extension of the +T1 time by diamagnetic electrolyte solutions. +Conclusion and Outlook +We report on the effect of diamagnetic electrolyte solutions on highly dense near-surface +spin defects in oxygen-terminated diamonds. Surprisingly, we observe that diamagnetic ions +increase the T1 relaxation time of NV-centers. We demonstrate that this effect is reversible, +surface sensitive and responsive to millimolar concentrations. We find that also interfacial +spin defects are sensitive to diamagnetic species, anticipating their possible use as reporter +spins for future optimization. Furthermore, we investigate the underlying mechanism by +single and double quantum NV-relaxometry experiments in combination with ab initio sim- +ulations. We propose that ions at the interface stabilize charge fluctuations between pairs of +19 + +carbon vacancies and alike deep defects, surrounding the NV-centers. This reduces magnetic +as well as electric noise at the diamond interface by a dynamical trapping of mobile electrons +to a site with lower Gibbs energy. These findings encourage further simulations and experi- +ments (e.g, on other NV-diamond systems, such as nanodiamonds or single NV-centers) to +elaborate on a comprehensive understanding of the complex processes at the solid/liquid +interface. +We would like to emphasize that the sensitivities of relaxometry to para- and diamagnetic +electrolyte solutions both represent scientifically relevant concentration regimes. Paramag- +netic species in the physiological environment, e.g., reactive oxygen species (ROS) or trace +metals, e.g., manganese can typically be found in ∼ nanomolar to micromolar concentra- +tions fitting the highly sensitive feedback of NV-relaxometry (see Figure 2a).41,53 However, +diamagnetic ion concentrations are typically orders of magnitude higher in the cytoplasm +(∼ millimolar)42,43 or in electrochemistry (∼ 0.1 to 1 molar)44 which fit very well the NV- +center’s response reported in our work (see Figure 2b). Importantly, these two effects may +counteract if both species are present. A possible pathway to differentiate between these +two could be recording single and double quantum experiments, which are only affected by +diamagnetic species in the latter case (see Figure 4 and Supplementary Note 7). Therefore, +we propose to probe both relaxation times in future relaxometry studies. We envision ap- +plications ranging from probing electrochemical interfaces54 to nanoscale ion sensing in cells +or neuroscience, where changes in the membrane potential occur as a result of concentration +gradients of diamagnetic ions.55–57 +Methods +Sample Preparation +Two 2 × 2 × 0.5 mm electronic grade diamond samples (natural 13C abundance, Element +Six) were implanted with 15N at an energy of 2.5 keV or 4 keV, an off-axis tilt of 7° and a +20 + +fluence of 2×1012 cm−2 by Innovion and annealed according to Bucher et al.18 Before exper- +iments are conducted, the diamonds are cleaned with a tri-acid cleaning protocol according +to Brown et al.:38 Samples are boiled in equal parts of sulfuric, nitric and perchloric acid at +a temperature of 280 °C for two hours. This cleaning procedure is also applied before the +deposition of aluminium oxide (Al2O3) on the diamond. +Preparation of Electrolyte Solutions +For the measurements where pure water is used, deionized water with a resistivity of +18.2 MΩ·cm at 25 °C (Merck Millipore) is utilized. +Sodium chloride (NaCl, Merck 106404) is prepared in a 1 M stock solution, where NaCl is +dissolved in deionized water. Before the experiments, NaCl is diluted from the stock solution +to obtain 500, 250, 100, 50, 10 and 1 mM concentrated solutions. The other salt solutions +used within this work are prepared in the same manner. +Atomic Layer Deposition (ALD) +The 2.5 keV 15N implanted diamond is coated with an aluminium oxide (Al2O3) thin +film by ALD according to Liu et al.13 The deposition includes 10 cycles of alternated sample +exposure to trimethyl aluminium (TMA) and H2O. This procedure results in a film thickness +of ∼ 1 nm and ensures surface termination with hydroxyl groups by exposing the diamond +to a remote oxygen plasma within the ALD system.13,58 The Al2O3 layer can be removed +from the diamond surface by soaking the sample overnight in 5% NaOH solution. +Experimental Setup +The quantum sensing setup is based on a modified version of the experiment described +in Bucher et al.18 Before experiments are performed, the diamond is glued to a thin glass +cover slide (48393026, VWR) together with a microfluidic device that encloses the diamond +21 + +edges and covers its surface, such that a volume of ∼ 0.60 µL of the sample liquid can be +applied in a controllable way. On the other side of the cover slide a 6 mm diameter glass +hemisphere (TECHSPEC® N-BK7 Half-Ball Lenses, Edmund Optics) is glued, in order to +improve the fluorescence light collection efficiency. The glass cover slide is then fixed on a +30 mm cage plate (CP4S, Thorlabs). This whole assembly is then positioned between two +permanent magnets, that are rotated and tilted in order to align the B0 field with one of +the four possible NV-center orientations. The distance between the two magnets can be +adjusted in order to correspond to the working magnetic field strengths B0 (in this work: +15, 316, 352 and 978 G). Initialization of the NV-ensemble is realized with a 532 nm laser +(Verdi G5, Coherent) with a power of ∼ 250 mW (CW) after the AOM. The laser light is +focused on the diamond by a Plano-Convex Lens (LA 1986-A-M, Thorlabs) in a total internal +reflection geometry. Laser pulses are regulated by an acousto-optic modulator (Gooch and +Housego, model 3260-220) with pulse durations of 5 µs. Photoluminescence (PL) is collected +and focused on a large area photodiode (OE-300-SI-10, Femto Messtechnik GmbH, Berlin, +Germany) by two condenser lenses (ACL25416U-B, Thorlabs). The excitation light is filtered +by a long-pass optical filter (Edge Basic 647 Long Wave Pass, Semrock) placed between +the bottom condenser lens and the photodiode. The output voltage of the photodiode is +digitized with a data acquisition unit (USB-6229 DAQ, National Instruments). A 500 MHz +PulseBlaster card (ESR-Pro-II, Spincore) is utilized to trigger and to time the microwave +and light pulses used for quantum control of the NV-centers. The microwave frequencies +are produced by a signal source (SynthHD, Windfreak Technologies, LLC.). +Microwave +phase control is obtained by a combination of a phase-shifting splitter (ZX10Q-2-27-S+, +Mini-Circuits), two switches (ZASWA-2-50dRA+, Mini-Circuits) and a combiner (ZX10-2- +42-S+, Mini-Circuits). The amplified microwave pulses (ZHL-16W-43-S+, Mini-Circuits) +are delivered by a homebuilt microwave loop on top of the microfluidic chip. The electron +spin resonance (ESR) frequency is used to determine the magnetic field strength B0 as well +as the NV0,-1 resonance frequency f NV. +22 + +T1 Relaxometry Experiments (Single and Double Quantum) +Single quantum (SQ) relaxometry experiments: To obtain a signal-to-noise ratio (SNR) +as shown in Figure 1b the sequence is repeated 5,000 times for every data point. +Each +experiment consists of 31 data points measured in a logarithmic increasing sweep time t to +guarantee more sampling points at short times t. Note that these parameters are also used +for double quantum (DQ) relaxometry. For normalization and noise cancellation, the second +half of the sequence contains a MW π0,-1-pulse, where the subscripts 0 and -1 indicate the +initialization of the spin state from ms = 0 to ms = −1.18 The spectra are then plotted +as the measurement result of the first half divided by the result of the second half of the +sequence. +Double quantum (DQ) relaxometry experiments: For a detailed discussion of the DQ +relaxation and pulse sequence, the reader is referred to Myers et al.45 In short, the DQ pulse +sequence (see inset of Figure 4b) consists of two consecutive measurements where MW π- +pulses are used to control spin state initialization and readout. In both halves of the sequence +the NV-center is initialized in ms = −1. After a sweep time t the spin state population of +either ms = −1 (in the first part) or ms = +1 (in the second part) is read out. Dividing the +second by the first part yields a population ratio of the two states. +Sensitivity of T1 Relaxometry on Electrolytes +Experiments to determine the sensitivity of T1 relaxometry measurements on para- and +diamagnetic electrolytes are conducted for MnCl2 and NaCl solutions using the SQ relax- +ometry pulse sequence. Probing each concentration results in a relaxation curve of which +the T1 time is determined. The T1 time is then normalized to the one of water covering +the diamond. Before probing any electrolyte concentration, we wash the microfluidic device +with water to ensure equal starting conditions, i.e. a constant T1 time for water covering +the diamond. We perform each series three times resulting in a mean T1 value for each +concentration (see also Supplementary Note 4). Figure 2 in the main text shows the mean +23 + +(normalized) T1 time along with the standard deviation. +DEER Measurements +DEER spectra (see Figure 5b) are recorded by performing a spin-echo sequence on the +NV-center spins with a free evolution time of 1 µs. The duration of the MW-pulse (MWDEER) +applied to the surface dark spins is set to 200 ns and the driving frequency (f DEER) is swept +over 90 MHz (from fDEER = 0.84 to 0.93 GHz). To obtain a SNR as shown in Figure 5b +the sequence is repeated 10,000 times for every data point. Each experiment consists of +67 data points in equally separated time steps and this whole experiment is repeated four +times. Referencing for noise cancellation is achieved by alternating the last MW-pulse of the +spin-echo sequence from π/2 to 3/2π. +Once the resonance condition for ge = 2 is found, DEER-Rabi experiments on the surface +dark spins are performed by sweeping the MW-pulse duration (MWDEER) during the NV +spin-echo (see Figure 5c) as described above. The sequence is repeated 10,000 times for +every data point. +Each experiment consists of 101 equally spaced data points and this +whole experiment is repeated ten times. To account for MW (MWDEER) noise, the same +procedure is repeated 20 MHz off the resonance condition. The outcome of both on- and +off-resonant measurements are subtracted resulting in the spectra shown in Figure 5c. After +that, measurements of the surface dark spin population relaxation are carried out according +to Sushkov et al. with a πds-pulse length of 24 ns.46 The sequence shown in Figure 5d is +repeated 10,000 times for every data point. Each experiment consists of 21 data points in +equally separated time steps. This whole experiment is then repeated 50 times. Background +subtraction is achieved by performing the experiment in the same procedure without the +additional MW drive (MWDEER). Subtracting the outcome of both MW-on and MW-off +measurements then yields the spectra shown in Figure 5d. +24 + +Simulation of the Diamond/Water Interface +In our simulations, we use a slab of a model diamond surface with hydrogen, hydroxyl, +and ether surface terminations (see Figure S10d). It is a symmetric (100) surface of ∼ 1.4 nm +with a 2 × 1 surface reconstruction pattern, exhibiting a positive electron affinity and no +surface states inside the band gap.59 The water layer on top of the diamond (thickness +∼ 2 nm) was constructed as follows. First, we equilibrate 74 water molecules with the clas- +sical molecular dynamics (MD) for 5 ns in a simulation box of commensurate lateral size +with the diamond slab. These calculations are done with the GROMACS software in the +canonical NVT ensemble,60 using the GROMOS 54A7 force field.61 After that, we superim- +pose the water box and the diamond surface and allow for an additional equilibration step +of 10 ps with the ab initio MD, as implemented in the VASP package.62 We also incorporate +∼ 1.9 nm of vacuum together with a dipole correction scheme to eliminate the interaction +with the periodic images. This yields the simulation supercell of 1.0097 × 1.0097 × 5.3 nm3, +which is further used in the ab initio MD calculations. Ab initio calculations are performed +using the PBE functional63 in conjunction with the D2 dispersion correction, using a pro- +jector augmented wave method with the kinetic energy cutoff of 370 eV. We note that the +PBE functional provides semi-quantitative results for the electronic structure but is able to +accurately yield the trends in the change of the electronic structure upon different surface +terminations and environments of diamond. Further, we note that we focus on the difference +in the electrostatic environment due to the interaction of the electrolyte with the surface +groups, assuming no change in the microstructure of the carbon layer. +Charge Transfer Rates +We calculate the charge transfer rate with an expression from the Marcus theory,64 given +as: +kCT = 2π +ℏ |HAB|2 +1 +√4πλkBT exp +� +−(λ + ∆G)2 +4πλkBT +� +25 + +where HAB is the transfer integral, λ the reorganization energy, ∆G the Gibbs energy dif- +ference due to an external field, kB the Boltzmann constant, ℏ the reduced Planck constant, +and T the temperature. The reorganization energy is determined for a single defect (either +a carbon vacancy or a substitutional nitrogen) in a 1000-carbon supercell. For computing +the transfer integrals as a function of distance, we use diamond supercells of different sizes, +varying between 64 and 1000-carbon atoms. The reorganization energies are calculated by +the four-point scheme, while the transfer integrals are estimated at a high symmetry con- +figuration as 1/4 of the bandwidth along the Γ-X direction. The contribution to the Gibbs +energy is computed by solving the one dimensional Poisson equation given the experimental +depth of the NV center.65 Noteworthy, the effect from the band bending is governed by the +orientation of defect pairs relative to the direction of the electric field. At a reference depth +of the NV-center, the maximum strength, corresponding to a change in the electrostatic +potential (∆Vmax), is reached in a parallel configuration, whilst the effect is quenched to- +wards the orthogonal arrangement. Considering a uniform distribution of the defects in our +samples, we compute an expectation value of ∆V as ∆Vmax/2. +Acknowledgement +This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) - 412351169 within the Emmy Noether program. R.R. acknowledges support +from the DFG Walter Benjamin Programme (Project RI 3319/1-1). D.B.B. acknowledges +support from the DFG under Germany’s Excellence Strategy—EXC 2089/1—390776260 and +the EXC-2111 390814868. M.S.B. acknowledges support from the DFG through the Munich +Center of Quantum Science and Technology (MCQST, EXC-2111) and by BMBF (epiNV, +13N15702). A.G. acknowledges the Hungarian NKFIH grant No. KKP129866 of the National +Excellence Program of Quantum-coherent materials project, the support for the Quantum +Information National Laboratory from the Ministry of Culture and Innovation of Hungary +26 + +(NKFIH grant No. 2022-2.1.1-NL-2022-00004), the EU EIC Pathfinder project ”QuMicro” +(grant No. 101046911) and the EU QuantERA for the project MAESTRO. We acknowledge +KIF¨U for awarding us access to computational resources based in Hungary. +Author Contributions +D.B.B., R.R. and F.A.F.-M. discovered the effect of diamagnetic electrolytes on the +relaxation of NV-centers. D.B.B., R.R. and F.A.F.-M. designed the experiments. D.B.B. +supervised the study. F.A.F.-M. performed the experiments and was supported by M.R.S. for +NV-relaxometry. F.A.F.-M., R.R. and D.B.B. analyzed the data. R.D.A. built the quantum +sensing setup and designed the microfluidic device. A.G. and A.P. incorporated theoretical +modeling and simulations. M.S.B. and L.M.T. helped with the charge state experiments. +All authors discussed the results and contributed to the writing of the manuscript. +Additional Information +Competing interests: The authors declare no competing interests. +Data availability +The data supporting our findings are available within the paper and the Supplementary +Information. Additional relevant data are available from the corresponding author upon +reasonable request. +Code availability +The codes used for data acquisition and processing are available from the corresponding +author upon reasonable request. +27 + +Supplementary Information +Supplementary Note 1: Fitting of T1 Single Quantum (SQ), Double +Quantum (DQ) and Surface Dark Spin Relaxation Curves +Recorded single quantum (SQ) and double quantum (DQ) relaxation curves are fitted +with a biexponential function as the T1 decay exhibited two components according to prior +work:25,27,28,32,66 +C(t) = A · exp(− 1 +T1a +· t) + (1 − A) · exp(− 1 +T1b +· t) +where C is the contrast, A is the amplitude and T1a >> T1b. For completeness, relaxation +times in the tables are given by both time constants. In agreement with prior work,25,27,32 +values of T1 in the main text are only considering the longer component T1a. However, both +time constants are longer in all cases where diamagnetic electrolytes are measured with NV- +relaxometry and compared to water (see Table S1). Errors and errorbars from SQ and DQ +relaxation curves shown in tables or figures are standard deviations from the biexponential fit +function or in case of the sensitivity experiments (see Figure 2) the standard deviation from +three consecutive measurements. T1 time constants in the tables are given to three significant +digits. In case of the T1,ds relaxation measurements (see Figure 5d), the relaxation curve is +fitted to a single exponential decay: C(t) = A · exp(− +1 +T1,ds · t).46 +Supplementary Note 2: T1 Time Constants of Measured Electrolytes +and T1 Time Magnetic Field Dependence for Pure Water/NaCl +(500 mM) +Table S1 and Figure S1 show the T1 time constants (T1a and T1b) and T1 relaxation +curves of the measured electrolyte solutions in this work. Experiments are conducted with +the relaxometry pulse sequence according to the main text. +28 + +Figure S1: T1 relaxation curves of water and a-f) diamagnetic electrolyte +(500 mM) solutions as well as g) and h) paramagnetic electrolyte (1 µM) so- +lutions covering the diamond surface. Experiments are performed at f NV = +1.88 GHz. +29 + +Table S1: T1 time constants (T1a and T1b) of water and measured diamagnetic +electrolyte solutions (500 mM) as well as paramagnetic electrolyte solutions +(1 µM) covering the diamond surface. Experiments are performed at fNV = 1.88 GHz +. +Electrolyte [c = 500 mM] +T1a [µs] +T1b [µs] +Water +920 ± 170 +140 ± 60.0 +CsF +1510 ± 250 +200 ± 60.0 +KCl +1720 ± 300 +270 ± 90.0 +KNO3 +1360 ± 220 +240 ± 50.0 +LiCl +2940 ± 720 +930 ± 40.0 +NaCl +1920 ± 200 +170 ± 20.0 +CaCl2 +2920 ± 260 +460 ± 190 +MgSO4 +3600 ± 160 +310 ± 100 +AlCl3 +2070 ± 370 +360 ± 190 +Electrolyte [c = 1 µM] +T1a [µs] +T1b [µs] +MnCl2 +430 ± 160 +140 ± 50.0 +Gd(NO3)3 +250 ± 15.0 +21 ± 6.00 +Further measurements of water/NaCl (500 mM) solution are performed in different mag- +netic fields B0 (978, 352, 15 and 0 G), i.e., different resonance frequencies of the NV-center’s +ms = 0 → ms = −1 transition (f NV = 0.131, 1.88, 2.83 and 2.87 GHz). Figure S2 shows the +T1a time constants depending on f NV (see Supplementary Note 1 for details). In Table S2 +both time constants (T1a and T1b) are listed. +Figure S2: T1a time constants for water and NaCl (500 mM) solution and their +dependence on the NV0,-1 resonance frequency f NV. +30 + +Table S2: T1 time constants (T1a and T1b) for water and NaCl (500 mM) solution +covering the diamond surface depending on the NV0,-1 resonance frequency +f NV. +T1a [µs] +T1b [µs] +f NV = 0.131 GHz +Water +1810±460 +400±60.0 +NaCl 500 mM +2300±340 +670±130 +f NV = 1.88 GHz +Water +940±180 +130±20.0 +NaCl 500 mM +1990±200 +170±20.0 +f NV = 2.83 GHz +Water +2660±110 +510±80.0 +NaCl 500 mM +3970±400 +1210±410 +f NV = 2.87 GHz +Water +3460±630 +930±190 +NaCl 500 mM +4830±740 +1830±510 +Supplementary Note 3: NV-Relaxometry Experiments with Differ- +ent Organic Solvents +Following measurements are performed in order to investigate the impact of the solvent’s +physical properties on NV-relaxometry experiments. Therefore, we choose organic solvents +with dielectric constants (κ) which differ significantly from the properties of water.40 The +diamond is covered three times alternatingly with water and the organic solvent. T1 times +of the solvents are then normalized to the T1 time of water. Figure S3 shows that the T1 +time remains unaffected by the solvent. +31 + +Figure S3: NV-relaxometry experiments showing the impact of the solvent’s di- +electric constant (κ). Experiments are performed at f NV = 1.88 GHz. +Supplementary Note 4: NV-Relaxometry Measurement Series of +Para- and Diamagnetic Electrolyte Solutions in Increasing Con- +centrations +Figure S4: NV-relaxometry measurement series with increasing concentrations +of a) MnCl2 and b) NaCl solutions. Data points are T1 times normalized to the +T1 time of water for each series. Solid lines connect the mean values of three +consecutive performed series. Experiments are performed at f NV = 1.88 GHz. +The NV-relaxometry measurement series with paramagnetic MnCl2 and diamagnetic +NaCl solutions are performed in order to determine the sensitivity of the protocol to increas- +32 + +ing electrolyte solutions in each case. Experiments are conducted using the SQ relaxometry +pulse sequence (see Methods for detail). We perform each series three times resulting in a +mean value for each concentration (see color codes in Figure S4). Figure 2 in the main text +shows the mean (normalized) T1 time along with the standard deviation. +Paramagnetic MnCl2 solutions decrease the T1 time in ∼ nano- to micromolar concen- +trations with respect to water. In contrast to that, diamagnetic NaCl solutions increase the +T1 time in ∼ millimolar concentrations. +Supplementary Note 5: NV-Depth Dependence Measurements with +Water/LiCl (500 mM) +Figure S5: T1 relaxation curves of water and LiCl 500 mM solution covering +the diamond surface. Diamonds were implanted with 15N at an energy of a) +2.5 keV and b) 4 keV. Experiments are performed at f NV = 1.88 GHz. +NV-relaxometry with water/LiCl (500 mM) covering the diamond is performed in order +to investigate the impact of another diamagnetic electrolyte and to support the experiments +with NaCl (500 mM) solution using differently deep NV-center ensembles (implanted with +2.5 keV and 4 keV, see Figure 3b). Figure S5 and Table S3 show similar results for both +NaCl and LiCl (500 mM) solution. +33 + +Table S3: T1 time constants (T1a and T1b) of water, NaCl (500 mM) and LiCl +(500 mM) solution on the diamond surface depending on the nitrogen implan- +tation energy. Experiments are performed at f NV = 1.88 GHz. +Implantation energy [keV] +T1a [µs] +T1b [µs] +2.5 +Water +940 ± 180 +130 ± 20.0 +NaCl 500 mM +1920 ± 200 +170 ± 20.0 +Water +660 ± 180 +200 ± 50.0 +LiCl 500 mM +2940 ± 720 +930 ± 40.0 +4 +Water +1090 ± 190 +190 ± 30.0 +NaCl 500 mM +1270 ± 180 +220 ± 40.0 +Water +2750 ± 340 +380 ± 80.0 +LiCl 500 mM +2880 ± 270 +440 ± 90.0 +Supplementary Note 6: NV-Charge State, Coherence and Dephas- +ing Measurements +We observe an increase of T1 by diamagnetic electrolyte solutions. However, it is known +that NV-charge state alteration (i.e., NV0 ↔ NV–) can influence the outcome of NV- +relaxometry measurements.67,68 For that reason, we perform NV-Rabi experiments with +water/NaCl (500 mM) solution (see Figure S6a). Any change in the NV-Rabi contrast indi- +cates an alteration of the NV-center’s charge state. For instance, an ionization of NV– would +increase the proportion of NV0, thereby raising the background fluorescence and lowering +the contrast. The NV-Rabi experiments show no difference in the outcome between water +and the electrolyte implying a constant charge state distribution during the measurement. +Secondly, to supplement the NV-Rabi experiments, we conduct NV-relaxometry with dis- +tinct optical readout of the NV0 and NV– charge states and with three different laser powers +(see Figure S6b and Figure S6c). Possible ionization of NV– in the dark or recombination +processes would be visible as an alteration in the readout signal of the NV0 charge state +(see Figure S6b).67,68 These measurements are carried out using the first half of the relax- +ometry pulse sequence (i.e., without a π-pulse) and with two different optical filters. The +647 nm long pass filter predominantly reads out the fluorescence from the NV– state and the +600 ± 40 band pass filter mostly reads out the fluorescence from the NV0 state.69 While a T1 +34 + +Figure S6: Pulse sequences and spectra of NV-charge state measurements. a) +NV-Rabi experiments, b) NV-charge state measurements with selective read- +out of the NV0 or the NV– state and c) T1 relaxation curves using three differ- +ent laser powers. +35 + +fluorescence decay curve can be extracted from the measurements with the long pass filter, +no decisive change in the NV0 state is visible using the band pass filter. Probable impact of +the laser power on the NV–/NV0 ratio and a subsequent change in the T1 relaxation curves +is probed with relaxometry experiments using laser powers of 25, 50 and 100 µW µm−2 (see +Figure S6c). Both NV-Rabi and NV-charge state experiments do not show an impact on +NV-charge state alteration on the relevant timescales of the relaxometry measurements we +conduct herein. +Figure S7: Pulse sequences and spectra of Ramsey and T2 Hahn-echo mea- +surements. a) Ramsey oscillations performed at a 4 MHz detuned NV0,-1 reso- +nance frequency f NV and b) T2 Hahn-echo experiments with water and NaCl +(500 mM) solution covering the diamond surface at f NV = 1.88 GHz. +Additionally, we perform Ramsey (NV-dephasing) and T2 (NV-coherence) Hahn-echo +experiments, whose outcome is typically affected by changes in the low frequency components +of the noise (see Figure S7a and S7b).30 Both experiments show no difference in the outcome +for water or NaCl (500 mM) solution. However, we note that probable changes in this noise +frequency regime might not be observable with the high-dense NV-center ensemble we use in +this work, since the surrounding spin-bath (e.g. P1-centers or other paramagnetic impurities) +is limiting the NV-dephasing and NV-coherence in this case.37,50 +36 + +Supplementary Note 7: T1 Time Constants for Single Quantum and +Double Quantum Experiments at B0 = 15 G and Zero Field ESR +Measurements +Figure S8: a) SQ and b) DQ relaxation curves of water and MnCl2 (100 µM) so- +lution covering the diamond. Experiments are performed at B0 = 15 G, where +the NV0,-1 transition is at fNV = 2.83 GHz (corresponding to a DQ transi- +tion frequency of 80 MHz). T1,SQ decreases by 80%, whereas T1,DQ remains un- +changed compared to water when MnCl2 solution covers the diamond. +Single and double quantum T1 experiments of water/NaCl (500 mM) and water/MnCl2 +(100 µM) solution covering the diamond surface are performed in order to elucidate the effect +of the electrolyte on magnetic and electric field noise. In the case of the NaCl solution, both +T1 time constants (T 1a,SQ and T 1a,DQ) increase compared to water, indicating a reduction of +both magnetic and electric field noise (see also Table S4). Importantly, MnCl2 only reduces +the T1 time for the SQ relaxation, whereas the DQ transition remains unaffected compared to +37 + +Table S4: T1 time constants (T 1a,SQ and T 1a,DQ) for water/NaCl (500 mM) and +water/MnCl2 (100 µM) solution covering the diamond surface. Experiments +are performed at B0 = 15 G +. +T 1a,SQ [µs] +T 1a,DQ [µs] +Water +2600 ± 280 +440 ± 24.0 +NaCl 500 mM +3970 ± 400 +1250 ± 120 +Water +2000 ± 340 +410 ± 71 +MnCl2 100 µM +390 ± 71.0 +390 ± 78.0 +water (see also Table S4). This indicates an exclusive impact of the paramagnetic electrolyte +on magnetic field noise. However, we note that probing MnCl2 in higher (> 100 µM) concen- +trations would lead to a collapse of the NV-center’s T1 time (see also Figure 2). Therefore, a +final statement on the impact of higher concentrated paramagnetic electrolyte solutions on +the DQ (as well as the SQ) relaxation cannot be made. +Additionally, we investigate the static electric field environment of the NV-center, i.e., +charges within the diamond and adjacent to the NV-center (e.g., N+ and NV–).70 Therefore, +we measure ESR at zero magnetic field (here the earth’s magnetic field ∼ 0.5 G), because +any difference in the static electric field in the proximity of the NV-center with respect to +water or the electrolyte solution covering the surface would induce a shifting and/or splitting +of the ms = ±1 states apparent in the ESR spectra.70 Figure S9 shows no significant change +of the ESR resonance lines for the exposure of water or electrolyte solution, indicating that +static electric fields do not contribute. +38 + +Figure S9: ESR experiments at zero magnetic field with water and NaCl +(500 mM) solution covering the diamond surface. +39 + +Supplementary Note 8: Results of DFT-PBE Ab Initio Molecular +Dynamics Simulations +Figure S10: a) Band alignment of the water layer and the model diamond sur- +face. b) Distribution of interfacial dipoles, sampled from the MD trajectories +for three different compositions of the interface and solvent. c) Average elec- +trostatic potentials and vacuum level shifts (VLS) computed for the configu- +rations corresponding to the middle of the distributions in b). Vertical lines +show the parts of the simulation box, spanned by diamond (C), water or aque- +ous NaCl solution and vacuum. d) Structures of the model diamond surface +before and after adding a COOH group. +40 + +a) +Band alignment +b) +600 +model +-0.86 eV +Intensity [arb.units] +model+COOH +CBM +500 +model+CoOo'Na +LUMO +-1.94 eV +400 +300 +200 +VBM +-5.20 eV +100 +HOMO +-6.32 eV +Diamond +0 +-2 +-1 +0 +c) +Water +1 +2 +Dipole moment [at.units] +10 +Electrostatic potential [eV] +model+ Na*cl +model +COOH +model+Coo'Nat +5 +士 +VLS=1.1 +VLS=-0.6 +VLS=-1.9 +0 +-5 +-10 +-15 +c +H20 +-20 +NaCI +-25 +WW +-10 +0 +10 20 30 +40 +50 +-10 +10 20 30 +4050 +10 +0 +10 +20 30 40 +50 +Z-coordinate [A] +Z-coordinate [A] +Z-coordinate [A] +d) +model +model+COOHReferences +(1) Acosta, V. M.; Bauch, E.; Ledbetter, M. P.; Waxman, A.; Bouchard, L.-S.; Budker, D. +Temperature Dependence of the Nitrogen-Vacancy Magnetic Resonance in Diamond. +Physical Review Letters 2010, 104, 070801. +(2) Iv´ady, V.; Simon, T.; Maze, J. R.; Abrikosov, I. A.; Gali, A. Pressure and temperature +dependence of the zero-field splitting in the ground state of NV centers in diamond: A +first-principles study. Physical Review B 2014, 90, 235205. +(3) Teissier, J.; Barfuss, A.; Appel, P.; Neu, E.; Maletinsky, P. Strain Coupling of a +Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator. Physical Review +Letters 2014, 113, 020503. +(4) Dolde, F.; Fedder, H.; Doherty, M. W.; N¨obauer, T.; Rempp, F.; Balasubramanian, G.; +Wolf, T.; Reinhard, F.; Hollenberg, L. C. L.; Jelezko, F.; Wrachtrup, J. Electric-field +sensing using single diamond spins. Nature Physics 2011, 7, 459–463. +(5) Balasubramanian, G.; Chan, I. Y.; Kolesov, R.; Al-Hmoud, M.; Tisler, J.; Shin, C.; +Kim, C.; Wojcik, A.; Hemmer, P. R.; Krueger, A.; Hanke, T.; Leitenstorfer, A.; Brats- +chitsch, R.; Jelezko, F.; Wrachtrup, J. Nanoscale imaging magnetometry with diamond +spins under ambient conditions. Nature 2008, 455, 648–651. +(6) Maze, J. R.; Stanwix, P. L.; Hodges, J. S.; Hong, S.; Taylor, J. M.; Cappellaro, P.; +Jiang, L.; Dutt, M. V. G.; Togan, E.; Zibrov, A. S.; Yacoby, A.; Walsworth, R. L.; +Lukin, M. D. Nanoscale magnetic sensing with an individual electronic spin in diamond. +Nature 2008, 455, 644–647. +(7) Glenn, D. R.; Lee, K.; Park, H.; Weissleder, R.; Yacoby, A.; Lukin, M. D.; Lee, H.; +Walsworth, R. L.; Connolly, C. B. Single-cell magnetic imaging using a quantum dia- +mond microscope. Nature Methods 2015, 12, 736–738. +41 + +(8) Lovchinsky, I.; Sushkov, A. O.; Urbach, E.; de Leon, N. P.; Choi, S.; De Greve, K.; +Evans, R.; +Gertner, R.; +Bersin, E.; +Muller, C.; +McGuinness, L.; +Jelezko, F.; +Walsworth, R. L.; Park, H.; Lukin, M. D. Nuclear magnetic resonance detection and +spectroscopy of single proteins using quantum logic. Science 2016, 351, 836–841. +(9) M¨uller, C.; Kong, X.; Cai, J.-M.; Melentijevi´c, K.; Stacey, A.; Markham, M.; +Twitchen, D.; Isoya, J.; Pezzagna, S.; Meijer, J.; Du, J. F.; Plenio, M. B.; Nayde- +nov, B.; McGuinness, L. P.; Jelezko, F. Nuclear magnetic resonance spectroscopy with +single spin sensitivity. Nature Communications 2014, 5, 4703. +(10) Sushkov, A. O.; Lovchinsky, I.; Chisholm, N.; Walsworth, R. L.; Park, H.; Lukin, M. D. +Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters. +Physical Review Letters 2014, 113, 197601. +(11) Bucher, D. B. Principles of nano-and microscale NMR-spectroscopy with NV-diamond +sensors. eMagRes 2019, 8, 363–370. +(12) Lovchinsky, I.; Sanchez-Yamagishi, J. D.; Urbach, E. K.; Choi, S.; Fang, S.; Ander- +sen, T. I.; Watanabe, K.; Taniguchi, T.; Bylinskii, A.; Kaxiras, E.; Kim, P.; Park, H.; +Lukin, M. D. Magnetic resonance spectroscopy of an atomically thin material using a +single-spin qubit. Science 2017, 355, 503–507. +(13) Liu, K. S.; Henning, A.; Heindl, M. W.; Allert, R. D.; Bartl, J. D.; Sharp, I. D.; +Rizzato, R.; Bucher, D. B. Surface NMR using quantum sensors in diamond. Proceedings +of the National Academy of Sciences 2022, 119, e2111607119. +(14) Allert, R. D.; Briegel, K. D.; Bucher, D. B. Advances in nano- and microscale NMR +spectroscopy using diamond quantum sensors. Chemical Communications 2022, 58, +8165–8181. +(15) Allert, R. D.; Bruckmaier, F.; Neuling, N. R.; Freire-Moschovitis, F. A.; Liu, K. S.; +42 + +Schrepel, C.; Sch¨atzle, P.; Knittel, P.; Hermans, M.; Bucher, D. B. Microfluidic quantum +sensing platform for lab-on-a-chip applications. Lab on a Chip 2022, 22, 4831–4840. +(16) Shi, F.; Zhang, Q.; Wang, P.; Sun, H.; Wang, J.; Rong, X.; Chen, M.; Ju, C.; Rein- +hard, F.; Chen, H.; Wrachtrup, J.; Wang, J.; Du, J. Single-protein spin resonance +spectroscopy under ambient conditions. Science 2015, 347, 1135–1138. +(17) Schirhagl, R.; Chang, K.; Loretz, M.; Degen, C. L. Nitrogen-Vacancy Centers in Dia- +mond: Nanoscale Sensors for Physics and Biology. Annual Review of Physical Chemistry +2014, 65, 83–105. +(18) Bucher, D. B.; Aude Craik, D. P.; Backlund, M. P.; Turner, M. J.; Ben Dor, O.; +Glenn, D. R.; Walsworth, R. L. Quantum diamond spectrometer for nanoscale NMR +and ESR spectroscopy. Nature Protocols 2019, 14, 2707–2747. +(19) Kim, M.; Mamin, H. J.; Sherwood, M. H.; Ohno, K.; Awschalom, D. D.; Rugar, D. +Decoherence of Near-Surface Nitrogen-Vacancy Centers Due to Electric Field Noise. +Physical Review Letters 2015, 115, 087602. +(20) Stacey, A.; Dontschuk, N.; Chou, J.; Broadway, D. A.; Schenk, A. K.; Sear, M. J.; +Tetienne, J.; Hoffman, A.; Prawer, S.; Pakes, C. I.; Tadich, A.; de Leon, N. P.; Gali, A.; +Hollenberg, L. C. L. Evidence for Primal sp 2 Defects at the Diamond Surface: Can- +didates for Electron Trapping and Noise Sources. Advanced Materials Interfaces 2019, +6, 1801449. +(21) Sangtawesin, S. et al. Origins of Diamond Surface Noise Probed by Correlating Single- +Spin Measurements with Surface Spectroscopy. Physical Review X 2019, 9, 031052. +(22) Romach, Y.; M¨uller, C.; Unden, T.; Rogers, L. J.; Isoda, T.; Itoh, K. M.; Markham, M.; +Stacey, A.; Meijer, J.; Pezzagna, S.; Naydenov, B.; McGuinness, L. P.; Bar-Gill, N.; +Jelezko, F. Spectroscopy of Surface-Induced Noise Using Shallow Spins in Diamond. +Physical Review Letters 2015, 114, 017601. +43 + +(23) Chrostoski, P.; Sadeghpour, H. R.; Santamore, D. H. Electric Noise Spectra of a Near- +Surface Nitrogen-Vacancy Center in Diamond with a Protective Layer. Physical Review +Applied 2018, 10, 064056. +(24) Degen, C. L.; Reinhard, F.; Cappellaro, P. Quantum sensing. Reviews of Modern +Physics 2017, 89, 035002. +(25) Steinert, S.; Ziem, F.; Hall, L. T.; Zappe, A.; Schweikert, M.; G¨otz, N.; Aird, A.; Bala- +subramanian, G.; Hollenberg, L.; Wrachtrup, J. Magnetic spin imaging under ambient +conditions with sub-cellular resolution. Nature Communications 2013, 4, 1607. +(26) Mzyk, A.; Sigaeva, A.; Schirhagl, R. Relaxometry with Nitrogen Vacancy (NV) Centers +in Diamond. Accounts of Chemical Research 2022, 55, 3572–3580. +(27) Perona Mart´ınez, F.; Nusantara, A. C.; Chipaux, M.; Padamati, S. K.; Schirhagl, R. +Nanodiamond Relaxometry-Based Detection of Free-Radical Species When Produced in +Chemical Reactions in Biologically Relevant Conditions. ACS Sensors 2020, 5, 3862– +3869. +(28) Li, R.; Vedelaar, T.; Mzyk, A.; Morita, A.; Padamati, S. K.; Schirhagl, R. Following +Polymer Degradation with Nanodiamond Magnetometry. ACS Sensors 2022, 7, 123– +130. +(29) Jarmola, A.; Acosta, V. M.; Jensen, K.; Chemerisov, S.; Budker, D. Temperature- and +Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensem- +bles in Diamond. Physical Review Letters 2012, 108, 197601. +(30) Rosskopf, T.; Dussaux, A.; Ohashi, K.; Loretz, M.; Schirhagl, R.; Watanabe, H.; +Shikata, S.; Itoh, K. M.; Degen, C. L. Investigation of Surface Magnetic Noise by +Shallow Spins in Diamond. Physical Review Letters 2014, 112, 147602. +44 + +(31) Nie, L.; Nusantara, A. C.; Damle, V. G.; Baranov, M. V.; Chipaux, M.; Reyes-San- +Martin, C.; Hamoh, T.; Epperla, C. P.; Guricova, M.; Cigler, P.; van den Bogaart, G.; +Schirhagl, R. Quantum Sensing of Free Radicals in Primary Human Dendritic Cells. +Nano Letters 2022, 22, 1818–1825. +(32) Ziem, F. C.; G¨otz, N. S.; Zappe, A.; Steinert, S.; Wrachtrup, J. Highly Sensitive Detec- +tion of Physiological Spins in a Microfluidic Device. Nano Letters 2013, 13, 4093–4098. +(33) Ermakova, A.; Pramanik, G.; Cai, J.-M.; Algara-Siller, G.; Kaiser, U.; Weil, T.; +Tzeng, Y.-K.; Chang, H. C.; McGuinness, L. P.; Plenio, M. B.; Naydenov, B.; Jelezko, F. +Detection of a Few Metallo-Protein Molecules Using Color Centers in Nanodiamonds. +Nano Letters 2013, 13, 3305–3309. +(34) Fujisaku, T.; Tanabe, R.; Onoda, S.; Kubota, R.; Segawa, T. F.; So, F. T.-K.; +Ohshima, T.; Hamachi, I.; Shirakawa, M.; Igarashi, R. pH Nanosensor Using Electronic +Spins in Diamond. ACS Nano 2019, 13, 11726–11732. +(35) Simpson, D. A.; Ryan, R. G.; Hall, L. T.; Panchenko, E.; Drew, S. C.; Petrou, S.; +Donnelly, P. S.; Mulvaney, P.; Hollenberg, L. C. L. Electron paramagnetic resonance +microscopy using spins in diamond under ambient conditions. Nature Communications +2017, 8, 458. +(36) Pham, L. M.; DeVience, S. J.; Casola, F.; Lovchinsky, I.; Sushkov, A. O.; Bersin, E.; +Lee, J.; Urbach, E.; Cappellaro, P.; Park, H.; Yacoby, A.; Lukin, M.; Walsworth, R. L. +NMR technique for determining the depth of shallow nitrogen-vacancy centers in dia- +mond. Physical Review B 2016, 93, 045425. +(37) Henshaw, J.; Kehayias, P.; Saleh Ziabari, M.; Titze, M.; Morissette, E.; Watanabe, K.; +Taniguchi, T.; Li, J. I. A.; Acosta, V. M.; Bielejec, E. S.; Lilly, M. P.; Mounce, A. M. +Nanoscale solid-state nuclear quadrupole resonance spectroscopy using depth-optimized +nitrogen-vacancy ensembles in diamond. Applied Physics Letters 2022, 120, 174002. +45 + +(38) Brown, K. J.; Chartier, E.; Sweet, E. M.; Hopper, D. A.; Bassett, L. C. Cleaning +diamond surfaces using boiling acid treatment in a standard laboratory chemical hood. +Journal of Chemical Health & Safety 2019, 26, 40–44. +(39) Li, C.; Zhang, X.; Oliveira, E. F.; Puthirath, A. B.; Neupane, M. R.; Weil, J. D.; +Birdwell, A. G.; Ivanov, T. G.; Kong, S.; Gray, T.; Kannan, H.; Biswas, A.; Vajtai, R.; +Galvao, D. S.; Ajayan, P. M. Systematic comparison of various oxidation treatments +on diamond surface. Carbon 2021, 182, 725–734. +(40) Seyferth, D. Organic Solvents. Physical Properties and Methods of Purification. Fourth +Edition. Volume 11 of Weissberger’s ”Techniques of Chemistry”. Organometallics 1987, +6, 1375–1376. +(41) Flora, S. J. Biomarkers in Toxicology; Elsevier, 2014; pp 485–519. +(42) Flowers, T. J.; Munns, R.; Colmer, T. D. Sodium chloride toxicity and the cellular +basis of salt tolerance in halophytes. Annals of Botany 2015, 115, 419–431. +(43) Ladenson, J. H.; Apple, F. S.; Aguanno, J. J.; Koch, D. D. Sodium measurements in +multiple myeloma: two techniques compared. Clinical Chemistry 1982, 28, 2383–2386. +(44) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, +2nd ed.; Wiley, 2001. +(45) Myers, B. A.; Ariyaratne, A.; Jayich, A. C. B. Double-Quantum Spin-Relaxation Limits +to Coherence of Near-Surface Nitrogen-Vacancy Centers. Physical Review Letters 2017, +118, 197201. +(46) Sushkov, A. O.; Lovchinsky, I.; Chisholm, N.; Walsworth, R. L.; Park, H.; Lukin, M. D. +Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters. +Physical Review Letters 2014, 113, 197601. +46 + +(47) Schlipf, L.; Oeckinghaus, T.; Xu, K.; Dasari, D. B. R.; Zappe, A.; de Oliveira, F. F.; +Kern, B.; Azarkh, M.; Drescher, M.; Ternes, M.; Kern, K.; Wrachtrup, J.; Finkler, A. A +molecular quantum spin network controlled by a single qubit. Science Advances 2017, +3. +(48) Mamin, H. J.; Sherwood, M. H.; Rugar, D. Detecting external electron spins using +nitrogen-vacancy centers. Physical Review B 2012, 86, 195422. +(49) Bluvstein, D.; Zhang, Z.; McLellan, C. A.; Williams, N. R.; Jayich, A. C. B. Extending +the Quantum Coherence of a Near-Surface Qubit by Coherently Driving the Paramag- +netic Surface Environment. Physical Review Letters 2019, 123, 146804. +(50) Barry, J. F.; Schloss, J. M.; Bauch, E.; Turner, M. J.; Hart, C. A.; Pham, L. M.; +Walsworth, R. L. Sensitivity optimization for NV-diamond magnetometry. Reviews of +Modern Physics 2020, 92, 015004. +(51) Zhang, Z.; Joos, M.; Bluvstein, D.; Lyu, Y.; Jayich, A. C. B. Reporter-spin-assisted T1 +relaxometry. 2022, arXiv:2208.11470 [quant–ph]. +(52) F´avaro de Oliveira, F.; Antonov, D.; Wang, Y.; Neumann, P.; Momenzadeh, S. A.; +H¨außermann, T.; Pasquarelli, A.; Denisenko, A.; Wrachtrup, J. Tailoring spin defects +in diamond by lattice charging. Nature Communications 2017, 8, 15409. +(53) Sies, H.; Jones, D. P. Reactive oxygen species (ROS) as pleiotropic physiological sig- +nalling agents. Nature Reviews Molecular Cell Biology 2020, 21, 363–383. +(54) Favaro, M.; Jeong, B.; Ross, P. N.; Yano, J.; Hussain, Z.; Liu, Z.; Crumlin, E. J. Unrav- +elling the electrochemical double layer by direct probing of the solid/liquid interface. +Nature Communications 2016, 7, 12695. +(55) Kaufmann, S.; Simpson, D. A.; Hall, L. T.; Perunicic, V.; Senn, P.; Steinert, S.; +McGuinness, L. P.; +Johnson, B. C.; +Ohshima, T.; +Caruso, F.; +Wrachtrup, J.; +47 + +Scholten, R. E.; Mulvaney, P.; Hollenberg, L. Detection of atomic spin labels in a lipid +bilayer using a single-spin nanodiamond probe. Proceedings of the National Academy +of Sciences 2013, 110, 10894–10898. +(56) Hall, L. T.; Hill, C. D.; Cole, J. H.; St¨adler, B.; Caruso, F.; Mulvaney, P.; Wrachtrup, J.; +Hollenberg, L. C. L. Monitoring ion-channel function in real time through quantum +decoherence. Proceedings of the National Academy of Sciences 2010, 107, 18777–18782. +(57) Szatm´ari, D.; S´ark´any, P.; Kocsis, B.; Nagy, T.; Miseta, A.; Bark´o, S.; Longauer, B.; +Robinson, R. C.; Nyitrai, M. Intracellular ion concentrations and cation-dependent +remodelling of bacterial MreB assemblies. Scientific Reports 2020, 10, 12002. +(58) Henning, A.; Bartl, J. D.; Zeidler, A.; Qian, S.; Bienek, O.; Jiang, C.; Paulus, C.; +Rieger, B.; Stutzmann, M.; Sharp, I. D. Aluminum Oxide at the Monolayer Limit via +Oxidant-Free Plasma-Assisted Atomic Layer Deposition on GaN. Advanced Functional +Materials 2021, 31, 2101441. +(59) Kaviani, M.; De´ak, P.; Aradi, B.; Frauenheim, T.; Chou, J.-P.; Gali, A. Proper Sur- +face Termination for Luminescent Near-Surface NV Centers in Diamond. Nano Letters +2014, 14, 4772–4777. +(60) Berendsen, H.; van der Spoel, D.; van Drunen, R. GROMACS: A message-passing par- +allel molecular dynamics implementation. Computer Physics Communications 1995, +91, 43–56. +(61) Schmid, N.; Eichenberger, A. P.; Choutko, A.; Riniker, S.; Winger, M.; Mark, A. E.; +van Gunsteren, W. F. Definition and testing of the GROMOS force-field versions 54A7 +and 54B7. European Biophysics Journal 2011, 40, 843–856. +(62) Kresse, G.; Furthm¨uller, J. Efficient iterative schemes for ab initio total-energy calcu- +lations using a plane-wave basis set. Physical Review B 1996, 54, 11169–11186. +48 + +(63) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made +Simple. Physical Review Letters 1996, 77, 3865–3868. +(64) Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron +Transfer. I. The Journal of Chemical Physics 1956, 24, 966–978. +(65) Broadway, D. A.; Dontschuk, N.; Tsai, A.; Lillie, S. E.; Lew, C. T.-K.; McCallum, J. C.; +Johnson, B. C.; Doherty, M. W.; Stacey, A.; Hollenberg, L. C. L.; Tetienne, J.-P. Spatial +mapping of band bending in semiconductor devices using in situ quantum sensors. +Nature Electronics 2018, 1, 502–507. +(66) Rioux, J. A.; Levesque, I. R.; Rutt, B. K. Biexponential longitudinal relaxation in white +matter: Characterization and impact on T1 mapping with IR-FSE and MP2RAGE. +Magnetic Resonance in Medicine 2016, 75, 2265–2277. +(67) Bluvstein, D.; Zhang, Z.; Jayich, A. C. B. Identifying and Mitigating Charge Insta- +bilities in Shallow Diamond Nitrogen-Vacancy Centers. Physical Review Letters 2019, +122, 076101. +(68) Dhomkar, S.; Jayakumar, H.; Zangara, P. R.; Meriles, C. A. Charge Dynamics in near- +Surface, Variable-Density Ensembles of Nitrogen-Vacancy Centers in Diamond. Nano +Letters 2018, 18, 4046–4052. +(69) Doherty, M. W.; Manson, N. B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollen- +berg, L. C. The nitrogen-vacancy colour centre in diamond. Physics Reports 2013, +528, 1–45. +(70) Mittiga, T.; Hsieh, S.; Zu, C.; Kobrin, B.; Machado, F.; Bhattacharyya, P.; Rui, N. Z.; +Jarmola, A.; Choi, S.; Budker, D.; Yao, N. Y. Imaging the Local Charge Environment +of Nitrogen-Vacancy Centers in Diamond. Physical Review Letters 2018, 121, 246402. +49 + diff --git a/WdE4T4oBgHgl3EQfNAwx/content/tmp_files/load_file.txt b/WdE4T4oBgHgl3EQfNAwx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34cb099dec2957080964f3fc1b3ff14ee2116bde --- /dev/null +++ b/WdE4T4oBgHgl3EQfNAwx/content/tmp_files/load_file.txt @@ -0,0 +1,1948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf,len=1947 +page_content='Sensing Diamagnetic Electrolytes with Spin Defects in Diamond Fabian A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Freire-Moschovitis,† Roberto Rizzato,† Anton Pershin,‡ Moritz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schepp,† Robin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Allert,† Lina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Todenhagen,¶ Martin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Brandt,¶ ´Ad´am Gali,‡,§ and Dominik B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bucher∗,† †TUM School of Natural Sciences, Department of Chemistry, Technical University of Munich, Lichtenbergstraße 4, 85748 Garching bei M¨unchen, Germany ‡Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, PO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Box 49, Budapest H-1525, Hungary ¶Walter Schottky Institut and Physik-Department, Technical University of Munich, Am Coulombwall 4, 85748 Garching bei M¨unchen, Germany §Department of Atomic Physics, Institute of Physics, Budapest University of Technology and Economics, M˝uegyetem rakpart 3, Budapest H-1111, Hungary E-mail: dominik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='bucher@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='de Abstract Quantum sensing with spin defects in diamond, such as the nitrogen vacancy (NV) center, enables the detection of various chemical species on the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Molecules or ions with unpaired electronic spins are typically probed by their influence on the NV- center’s spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Whereas it is well-known that paramagnetic ions reduce the NV-center’s relaxation time (T1), here we report on the opposite effect for diamagnetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We demonstrate that millimolar concentrations of aqueous diamagnetic elec- trolyte solutions increase the T1 time of near-surface NV-center ensembles compared 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='04952v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='app-ph] 12 Jan 2023 to pure water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To elucidate the underlying mechanism of this surprising effect, single and double quantum NV experiments are performed, which indicate a reduction of magnetic and electric noise in the presence of diamagnetic electrolytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In combination with ab initio simulations, we propose that a change in the interfacial band bending due to the formation of an electric double layer leads to a stabilization of fluctuating charges at the interface of an oxygen-terminated diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This work not only helps to understand noise sources in quantum systems but also broadens the application space of quantum sensors towards electrolyte sensing in cell biology, neuroscience and electrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Introduction Nitrogen vacancy (NV) centers in diamond offer a broad platform for quantum sensing ap- plications ranging from the measurement of basic physical properties, such as temperature,1 pressure,2 strain,3 electric4 and magnetic fields5,6 down to single cells,7 single molecules,8 or even single nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='9,10 This unprecedented sensitivity is achieved due to the atomic size of the qubit enabling its location only a few nanometer away from the diamond interface (see Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='11 Near surface NV-centers (≤ 20 nm below the surface) can be used to sense nuclear magnetic resonance (NMR)8,12–15 and electron spin resonance (ESR) signals16 from nanoscale chemical and biological samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The NV-center translates these magnetic field fluctuations directly to an optical signal, detected by a change in the fluorescence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Together with optical spin state initialization with green laser light and coherent spin state manipulation with microwave pulses, these key features make the NV-center a unique tool for (bio)chemical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='11,15,17,18 NV-centers are also perceptive to electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='4,19 Even a single elementary charge located ∼ 150 nm away from the quantum sensor produces a strong enough static electric field to affect an NV-center’s ODMR (optically detected magnetic resonance) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='4 This high sensitivity of NV-centers to magnetic or electric fields marks a narrow ridge: On the 2 one hand, it allows for single nuclear spin or elementary charge detection, on the other hand it makes the qubit prone to magnetic or electronic spin noise in its close environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This is particularly relevant for NV-centers close to the surface, where interfacial processes and defects cause additional noise and reduce the performance of the NV-quantum sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='20–23 Due to the NV-center’s susceptibility to a broad range of frequencies (from DC to GHz),24 noise can be measured by applying different sensing protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 High frequency noise (∼ GHz) can be probed by the longitudinal spin-lattice relaxation time (T1) with a protocol that is usually referred to as NV-relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='17,18,24–26 The T1 time defines the time constant of the spin-state-dependent fluorescence decay from the magnetic sublevel ms = 0 (bright state) to thermal equilibrium (mixed state) and is on the order of a few milliseconds for near- surface NV-ensembles in bulk diamonds,11 or on the order of a few hundred microseconds for nanodiamonds (depending on their size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='17 Generally, any magnetic noise overlapping with the NV-center’s Larmor frequency (on the order of the zero-field splitting D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='87 GHz) will decrease the T1 relaxation time (see Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18,25,27,28 Relaxometry has been successfully applied to map high frequency magnetic noise originating from inside the diamond (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' paramagnetic impurities29), at the diamond interface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' dangling bonds20,30) or from samples on top of the diamond (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' organic radicals27,31 or paramagnetic ions, such as Mn2+,32 Fe3+,33 or Gd3+ 25,28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Furthermore, nanoscale NV-relaxometry has also been used to determine the pH value34 or to monitor chemical reactions in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='35 While the increase of the relaxation rate due to paramagnetic species has been studied extensively, herein we detect the opposite effect for diamagnetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' When near-surface NV-centers in oxygen-terminated diamonds are exposed to aqueous diamagnetic electrolytes, we observe a systematic extension of the T1 relaxation time compared to pure (deionized) water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We show that this unexpected effect is proportional to the electrolyte concentration, reversible and dependent on the NV-center’s implantation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In order to shed light on the underlying sensing mechanism, we perform single and double quantum relaxometry experiments which indicate a reduction of electric as well as magnetic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Furthermore, 3 double electron electron resonance (DEER) spectroscopy experiments show a similar effect on surface dark spins, which possibly act as surface reporter spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In combination with theoretical methods including ab initio simulations of the diamond/electrolyte interface, we propose that diamagnetic ions alter the interfacial band bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This leads to a stabilization of fluctuating charges at the interface and to the increase of the T1 relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Results T1 Relaxometry on Electrolytes with Near-Surface NV-Center En- sembles In this study we use near-surface high dense NV-center ensembles (implanted with 15N at an energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV and a fluence of 2 × 1012 cm−2), distributed ∼ 5 nm underneath the diamond surface,13,36,37 to investigate the effect of aqueous electrolyte solutions on the spin- lattice relaxation time T1 probed by NV-relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Before experiments are conducted, we prepare the diamond surface with a tri-acid clean procedure according to Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='38 This procedure not only ensures to remove non-diamond carbon material from the interface but also creates an oxygen-terminated surface comprised of mixed carbon oxide species including hydroxyl groups, ethers, ketones, aldehydes and carboxylic acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='39 We position the diamond in a microfluidic device that guarantees controllable in- and output of the applied liquids, prevents sample evaporation and provides a constant and defined volume for following measurements (see Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='15 Importantly, the microfluidic device avoids a direct contact between the liquid and the microwave delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The NV-relaxometry protocol is depicted in Figure 1b and essentially consists of two 532 nm laser pulses of 5 µs duration for optical spin state initialization and readout separated by a sweep time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A subsequent measurement with a π0,-1-pulse (where the subscripts 0 and 1 indicate transitions between ms = 0 ↔ ms = −1) at the start is used for normalization and noise cancellation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 The T1 time can be extracted from the (bi)exponential fit of 4 Figure 1: Scheme of NV-relaxometry experiments with aqueous electrolyte so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Top (left): The NV-center in the diamond crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A nitrogen atom (blue) replaces a carbon atom (black) in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Together with an adjacent vacancy an NV-center is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The orbitals (petrol) indicate four possible NV-center orienta- tions within the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Top (right): Scheme of an oxygen-terminated diamond surface with an ensemble of near-surface NV-centers (dNV ∼ 5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The oxygen termi- nation consists of hydroxyl groups, ethers, ketones, aldehydes and carboxylic acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We probe pure water, paramagnetic (purple) and diamagnetic (yellow) ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bottom: A mi- crofluidic device placed on top of the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In- and outlet allow for adding and remov- ing aqueous electrolyte solutions or pure water from the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' b) Top (left): NV-relaxometry pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Two laser pulses for spin state initialization and read- out are separated by a sweep time (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A consecutive measurement with a π-pulse at the start is used for noise cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 Top (right): Energy levels of the NV-center’s elec- tronic ground state (S = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The zero-field splitting (D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='87 GHz) separates the ms = 0 (bright) and ms = ±1 states (dark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A bias magnetic field B0 splits the degen- erate ms = ±1 states according to the Zeeman effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bottom: T1 relaxation curves of pure water and solutions of NaCl (500 mM) and MnCl2 (1 µM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' While paramagnetic MnCl2 re- duces the T1 relaxation time with respect to pure water, diamagnetic NaCl extends the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at fNV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 5 a) b) Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Read ms = +1 Laser t ms = -1 GHz 元0,1 MW ms = 0 元0,1 OH Bo = 0 Bo > 0 ~ 5 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='25 Inlet [Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='] Diamagnetic Microfluidic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 Device Contrast [ Paramagnetic Near-Surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 NV-Ensemble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 Outlet Pure Water NaCI (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=') MnCl2 (ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=') Laser (532 nm) Diamond 0 1 2 4 7 8 9 Fluorescence (635 to 800 nm) Sweep Time t [ms]the relaxation curves depicting the contrast as a function of sweep time t (see Supplementary Note 1 for fitting details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We perform the measurements by filling the microfluidic channel and covering the di- amond surface either with pure water or aqueous electrolyte solutions (see Methods for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure 1b depicts the T1 relaxation curves when water and solutions of diamagnetic NaCl or paramagnetic MnCl2 cover the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Paramagnetic MnCl2 (1 µM) on the diamond leads to a T1 time reduction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='19 with respect to water, which is in accordance with other studies and can be ascribed to the strong dipole-dipole interaction of the NV-center with the paramagnetic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='25,32 In contrast, when we repeat the same ex- periment with diamagnetic NaCl (500 mM) solution, we observe an extension of the T1 time by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='45 compared to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments supporting this observation are also conducted with other diamagnetic salt solutions (mono-, di- and trivalent) and reveal similar results (see Supplementary Note 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, we choose NaCl as a representative of a standard diamagnetic electrolyte for the following measurements in our work and expect comparable results for other diamagnetic salt solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Moreover, by tuning the magnetic field B0 and thereby the NV-center’s Larmor fre- quency NV0,-1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', the ms = 0 → ms = −1 transition frequency) we are able to map the spectral noise density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We probe water/NaCl (500 mM) solution with NV0,-1 frequencies from 131 MHz to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='87 GHz and observe a similar effect over the entire frequency range (see Supple- mentary Note 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Consequently, the extension of the T1 time of near-surface NV-ensembles with exposure to diamagnetic electrolyte solutions is an effect that covers a broad range of (high) frequencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', from ∼ hundreds of MHz to GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, in order to exclude an impact of the solvent’s physical properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', polarity) on our experiments,19 we choose typical organic solvents whose dielectric constants (κ) and chemical structure differ significantly from water (κ = 8040) and probe them with relaxometry (see Supplementary Note 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Since the T1 time remains unaffected, we conclude that the herein described effect is not induced by the physical properties of water, but by 6 the diamagnetic electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sensitivity of T1 Relaxometry on Electrolytes In order to obtain information about the sensitivity of the NV-relaxometry protocol to para- and diamagnetic electrolyte solutions, we perform additional measurement series where the electrolyte concentration is increased stepwise by one order of magnitude (from 10−5 to 10−2 mM in the case of paramagnetic MnCl2 and from 10−4 to 103 mM in the case of diamagnetic NaCl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Paramagnetic MnCl2 shows a stepwise T1 decrease in micromolar concentrations reaching a decline of up to 86 ± 10% for a 10 µM solution with respect to water covering the diamond (see Figure 2a and Supplementary Note 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Note that a further concentration increase (> 10 µM) is not measurable, as it leads to a collapse of the T1 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In contrast, diamagnetic NaCl shows a slight T1 increase compared to water, which then fluctuates moderately from micromolar to lower millimolar concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, a significant and gradual T1 increase is measurable from 10 mM to 500 mM NaCl solution, where the effect saturates at 81 ± 11% with respect to water (see Figure 2b and Supplementary Note 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The decrease of the T1 time with paramagnetic species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', MnCl2) is expected and well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='17,25,26,30 Here, high frequency (∼ GHz) noise originates from magnetic dipole-dipole interactions of the NV-center’s electronic spin and the sample’s electronic spin (“spin-flips”), resulting in a decline of the T1 time if unpaired electrons are near the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' On the other hand, for diamagnetic ions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', NaCl) these interactions are absent as only paired electrons without a (net) magnetic moment are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Surprisingly, here we observe a gradual extension of the T1 time with increasing millimolar concentrations of diamagnetic NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, both sensitivity regimes (∼ nano- to micromolar for paramagnetic and ∼ millimolar for diamagnetic species) match the typical physiological41,43 or (for diamagnetic electrolytes) electrochemical concentrations,44 opening up sensing applications in cell biology 7 Figure 2: NV-relaxometry with increasing concentrations of para- and diamag- netic electrolyte solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Paramagnetic MnCl2 shows a stepwise T1 time decrease for concentrations in the micromolar regime until the effect reaches a maximum measur- able decline of 86 ± 10% for 10 µM solutions with respect to water covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' b) In contrast to that, diamagnetic NaCl (right) shows a slight increase of the T1 time compared to water, which then fluctuates moderately from the micromolar to the lower millimolar regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For concentrations ≥ 10 mM the T1 time increases gradually along with the NaCl concentration until the effect saturates to 81 ± 11% for NaCl (500 mM) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Shaded areas indicate typical physiological concentration regimes for para- and diamag- netic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='41–43 Experiments are performed at fNV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' or electrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 8 InCl2 (aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=') Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Flips Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='Reversibility, Passivation and NV-Center Depth Because of the surprising observation, that the T1 time increases with diamagnetic elec- trolyte solutions compared to water covering the diamond, the next experiments concentrate on the mechanism behind this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, we probe the reversibility and passivation of the effect along with the sensor’s response in dependence of its implantation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' First, we evaluate if the extension of the T1 relaxation time is a reversible process by exposing an oxygen-terminated diamond alternatingly to water and NaCl (500 mM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Thereby, we show that the T1 relaxation time is altered from “short” in case of water exposure to “long” when NaCl solution covers the surface (see Figure 3a in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Alternating between water and electrolyte solution demonstrates a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='35 fold increase of the T1 time with elec- trolyte exposure on the oxygen-terminated diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In a next step, we examine if the effect is specific to the diamond surface termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, the formerly oxygen-terminated diamond is coated with an aluminium oxide (Al2O3) thin film (thickness ∼ 1 nm)13 prepared by Atomic Layer Deposition (ALD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The aluminium oxide thin film ensures a controllable and uniform surface termination with hydroxyl groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We repeat the previous experiment, but this time the T1 relaxation time remains unaffected by the NaCl solution (see Figure 3a in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, we investigate if the extent of the electrolyte’s effect is dependent on the depth of the embedded NV-center ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, we prepare two diamonds with 15N implantation energies of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 and 4 keV with the tri-acid clean procedure described before and probe them with NV-relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Near-surface NV-centers implanted with an energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV are mainly distributed within a depth of ∼ 5 nm below the surface, while ensembles created with 4 keV 15N are located about ∼ 12 nm beneath the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='37 Figure 3b shows a significantly larger effect of the electrolyte on the relaxation time of the shallow implanted NV-diamond with respect to the deeper one, although a T1 time extension is still detectable in the latter case (see also Supplementary Note 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, while the effect with the ∼ 1 nm thick aluminium oxide layer is completely passivated, a T1 time increase can still be 9 Figure 3: NV-relaxometry experiments with different surface terminations and NV-center ensemble implantation depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) An oxygen-terminated diamond sur- face is alternatingly exposed to water and NaCl (500 mM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The T1 time increases with electrolyte exposure by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='35 with respect to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, this behavior can be altered by either the presence or absence of water or NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' After coating the same diamond with an aluminium oxide (Al2O3) thin film (thickness ∼ 1 nm) the T1 time is unaffected by the NaCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' b) T1 relaxation curves of water and NaCl (500 mM) solution covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Diamonds were implanted with 15N at an energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV (top) and 4 keV (bottom) resulting in different NV-center ensemble depths (dNV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' While on the shallower implanted diamond the T1 time increases by a factor of around two with exposure to NaCl solution, on the deeper implanted diamond the effect is strongly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at fNV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 dv ~ 5 nm Oxygen-Terminated Diamond OHO OH @ OH e ~ 1 nm Al203 dnv ~ 5 nm dnv ~ 5 nm dnv ~ 10-15 nmobserved with the oxygen-termination when the sensor is ∼ 5 to 10 nm further away from the electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Note that the same measurements conducted with LiCl (500 mM) solution lead to similar results (see Supplementary Note 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' From these experiments we conclude that the extension of the T1 relaxation time is a reversible and interfacial process which is dependent on the distance of the sensor to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, NV-charge state alterations or changes in NV-dephasing and NV-coherence are not observed in our experiments (see Supplementary Note 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Probing Magnetic and Electric Noise Contributions In the next set of experiments we investigate if the T1 relaxation time increase originates from a reduction of electric and/or magnetic field noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Since we are dealing with electrolytes dissolved in water, it is particularly interesting to explore the influence that charged ions and their randomly fluctuating electric fields might have on the T1 relaxation time of the near-surface NV-ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Whereas the typically used relaxometry experiments use single quantum (SQ) transitions to probe magnetic field noise (as in the experiments from the previous sections), double quantum (DQ) transitions are influenced by electric field noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='45 For these transitions, the full NV-center ground state (S = 1) is considered, where an additional relaxation pathway between ms = −1 ↔ ms = +1 with ∆ms = 2 (see Figure 4a) becomes accessible, the DQ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Regarding the NV-center as a “qutrit” rather than a qubit allows to probe the effect of the diamagnetic electrolyte solution on both electric and magnetic field noise at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' SQ and DQ relaxometry measurements on the system water/NaCl (500 mM) solution reveal that the diamagnetic electrolyte has an effect on both relaxation channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', it reduces magnetic as well as electric field fluctuations (see Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Here, two different T1 times can be defined: T1,SQ for the relaxation time in the SQ and T1,DQ in the DQ channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' An extension of T1,SQ by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='16 and a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='31 increase of T1,DQ can 11 Figure 4: Single quantum (SQ) and double quantum (DQ) relaxation experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Energy level scheme of the NV-center ground state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' SQ transitions (∆ms = ±1) with relaxation rates Ω are susceptible to magnetic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The DQ relax- ation (∆ms = ±2) with relaxation rate γ is magnetically forbidden but susceptible to electric noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='45 b) Top: DQ pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bottom: SQ and DQ relaxation curves of water and NaCl (500 mM) solution covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at B0 = 15 G, where the NV0,-1 transition is at fNV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83 GHz (corresponding to a DQ tran- sition frequency of 80 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T1,SQ increases by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='16 and T1,DQ by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='31 when the diamond is exposed to NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 12 SQ Relaxatior ating Spin bles ~MHz Q ~GHzbe measured in presence of NaCl solution with respect to water (see also Supplementary Note 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Thus, diamagnetic electrolytes reduce both – electric and magnetic – noise at the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, when we repeat the same experiments with paramagnetic MnCl2, only T1,SQ reduces by 80%, whereas the DQ transition remains unaffected compared to water (see Supplementary Note 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This indicates an exclusive impact of the paramagnetic electrolyte on magnetic field noise and could provide a possible pathway to distinguish para- from diamagnetic ions in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, we can exclude an influence of diamagnetic NaCl solution on the static electric field environment by performing zero field ESR Measurements (see Supplementary Note 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Influence of Electrolytes on Surface Dark Spins: DEER Experi- ments So far, we have focused on the direct influence of electrolytes on the NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' How- ever, the diamond as the NV-center’s host material provides various surface dark spins, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' dangling bonds, whose response to the electrolytes is probed in the next set of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Intrinsic T1 times of these surface dark spins are often long (∼ a few microseconds)46 which allows us to probe them with NV-based double electron electron resonance (DEER) spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='16,47 Figure 5a shows the pulse sequence of a typical DEER experiment: A spin-echo is performed on the NV-center’s electronic spin (MWNV spin-echo), at the same time, in the second free precession time of the echo, an additional microwave pulse (MWDEER) is ap- plied to drive the target electronic spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sweeping the MW frequency (f DEER) flips the surface dark spins when their Larmor frequency is matched and causes a dip in the DEER signal (see Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' As shown in Figure 5b, a clear dip in the DEER spectrum appears when the diamond interface is covered with NaCl (500 mM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The resonance at f DEER = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='887 GHz corresponds to ge ∼ 2 spins and is typically assigned to dangling bonds at the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='50 Interestingly, the dip is drastically reduced when the experiment 13 Figure 5: NV-DEER experiments probing the response of surface dark spins to electrolyte exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Pulse sequence of the double electron electron resonance (DEER) experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sweeping the microwave frequency of the MWDEER pulse (f DEER) while applying a spin-echo experiment on the NV-center (MWNV spin-echo) allows for the detection of electronic (surface dark) spins coupled to the NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='48 b) DEER ex- periment with water and NaCl (500 mM) solution covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A pro- nounced dip in the DEER spectrum appears at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='887 GHz (where ge ∼ 2) when the diamond is exposed to NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The dip gets drastically reduced when water cov- ers the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' c) Top: Pulse sequence of the DEER-Rabi experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bottom: When the pulse duration of the microwave drive (MWDEER) at the surface dark spins’ resonance frequency is swept during the spin-echo, DEER-Rabi oscillations can be ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='49 While the πds-pulse lengths remain equal when water or NaCl (500 mM) solution cover the diamond surface (πds ∼ 24 ns), the latter causes a by a factor of around three more pronounced Rabi amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' d) Top: Pulse sequence of the DEER-T1 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bottom: Varying the correlation time (tcorr) between two subsequent DEER segments,46 shows a surface dark spin relaxation with T1,ds = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='10 µs in case of NaCl exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In contrast, no significant relaxation decay can be observed when water covers the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' DEER experiments are performed at fNV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='98 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 14 MWDEER Surface Dark Spins + dNvis repeated with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Once the resonance condition for the ge ∼ 2 spins is found, coherent control of the surface dark spin state can be demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure 5c shows a DEER-Rabi experiment on the surface dark spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sweeping the microwave pulse duration (MWDEER) during the spin-echo causes oscillations of the defect’s spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='49 While we determine equal πds pulse lengths (πds ∼ 24 ns) for water and the electrolyte, the former leads to an about three times increased DEER-Rabi amplitude with respect to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In the next step, we probe the surface dark spin relaxation time (T1,ds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure 5d depicts a pulse sequence from Sushkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=',46 where the T1 time of the surface dark spins can be measured by correlating two subsequent DEER segments and varying the correlation time tcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Interestingly, we measure a relaxation time T1,ds of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='20 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='10 µs for the surface dark spins exposed to the NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In contrast, a clear relaxation decay cannot be observed in case of pure water covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Although we cannot exclude that the surface dark spins vanish, the more likely case is that the dark spins act as reporter spins46 experiencing the same effect of the electrolyte solution as the NV-center: “fast” relaxation in the case of water and “slow” relaxation when diamagnetic electrolyte solutions cover the diamond, which is also expressed in the increased DEER signals (see Figure 5b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Accordingly, we enhance the sensitivity of our sensor by the proximity of the reporter spins to the electrolyte solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='46,51 Theoretical Modeling Our experimental results show an influence of diamagnetic electrolyte solutions on near- surface spin defects in diamond where electric as well as magnetic noise is suppressed resulting in an increase of the T1 relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To further study this surprising effect computational modeling is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Here, as a working assumption, we focus on charge fluctuations within the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We note, that further processes such as proton hopping at the interface or water and ion dynamics within the electric double layer can also play a role but have not 15 been treated herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To this end, we model an interface between a slab of diamond and a thin layer of water subsequently enriched by Na+ and Cl- ions (see Methods for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Then, we probe the interfacial structure and vacuum level shifts (VLS) based on the configurations obtained from the ab initio molecular dynamics (MD)(see Figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The calculated alignment of the electronic levels of water and the model diamond surface is shown in Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Here, a mismatch of the chemical potentials (defined as a center of the band gap) promotes an electron leakage from the diamond surface towards the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The resulting redistribution of charges leads to the development of an electric field, that further rearranges the charged solvated Na+ and Cl- ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The large positive VLS of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1 eV (see Figure S10a) causes the ions to rearrange with the direction of the field, facilitating the effect of band bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' By adding a carboxyl group, we observe a stabilization of the downwards band bending relative to the case of the model diamond surface in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' However, in both cases, we obtain a broad distribution of surface dipoles, owing to the complexity of ion dynamics within the Stern layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' By contrast, a sharp distribution of the dipole moments is observed between a dissociated carboxyl group and a solvated Na+ ion nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This stable configuration gives rise to a large VLS of ∼ -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='9 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This value is further used to trace the evolution of the electrostatic potential at the microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' More specifically, we set it as a boundary condition for solving the Poisson equation to access the modifications of the potential inside a semi-infinite diamond slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' As shown in Figure 6b, we observe that the interfacial region of ∼ 40 nm is affected by the respective readjustments of the charges, resulting in a rapid decay of the potential near the interface and a slow saturation towards the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To establish a relation between the band bending and the noise reduction, we consider the electric and magnetic fluctuations caused by a pair of active defects around the NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The charge transfer process leads to a continuous change in the charge/spin state of the nearby defects, which can affect the relaxation and coherence time of the NV-center when the rate approaches the timescale of the quantum sensing experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In the Marcus theory, 16 Figure 6: Ab initio MD simulations of the diamond/electrolyte interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Representative snapshot of the diamond/electrolyte interface from the ab initio MD simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Color code: C (grey), O (red), H (white), Na+ (yellow), Cl- (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' b) Variations of the electrostatic potential in a semi-infinite diamond due the arrangement of NaCl at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Inset: Electrostatic potential (∆Vmax) for a pair of defects in 3 nm distance to each other surrounding a ∼ 5 nm deep NV-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' c) Schematic representation of a con- tinuous charge hopping between two defects around the NV-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' HAB is the transfer integral, ∆G0 is calculated as ∆Vmax/2 from b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' d) Calculated rate constants for pairs of substitutional nitrogen and vacancy defects as a function of distance between the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' e) Rate constants for the forward and backward electron transfer between a pair of va- cancy defects as a function of bias due to the interfacial band bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Vertical lines refer to an estimated difference in the effects by replacing the electrolyte with pure water, con- sidering a pair of defects and a NV-center in a configuration from the inset in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 17 a) b) c) forward Electrostatic potential [eV] 0 A B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='25 V- NV ↑ HAB 1 vo AV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='75 △G0 3 4 5 6 7 2 back 0 10 20 30 40 50 60 Distance [nm] d) 10 12 10 10 108 _s] - 108 INON+ H20 NaCI 107 106 forward 104 back 10 6 0 5 10 15 20 25 30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='15 △G° [eV] Distance [A]such fluctuations are described as a sequence of thermally activated hopping events, whilst the rate constants are determined from the distance-dependent coupling parameters and the required structural reorganizations (see Figure 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The dipoles at the diamond/solvent interface affect this equilibrium by altering the onsite Gibbs energy term with a contribution from the electrostatic potential (∆V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' As shown in Figures 6c and 6e, the band bending accelerates a forward charge transfer process (until reaching the Marcus inverted region), but at the same time, the magnitude of the back charge transfer (BCT) rate drops exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hence, regardless of the defect type, large interfacial band bending can lead to a dynamical trapping of the charges around a site with the lower Gibbs energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For a numerical validation, we focus on the electron fluctuations between a pair of carbon vacancies as well as on the hole fluctuations between two substitutional nitrogen defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We note that carbon vacancies are often generated by the implantation and irradiation techniques used to create the NV-centers, and substitutional N defects are present around NV-centers to stabilize the negative charge state of the NV-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' After determining the relevant parameters in the periodic supercells (see Methods for detail), we first analyze the possible contribution of each charge transfer reaction to the electrical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To this end, we compare the fluctuation rates, calculated for both defect pairs in a bulk-like environment (∆V = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' As shown in Figure 6d, the charge fluctuation rates for a pair of nitrogen defects is remarkably smaller than for a vacancy pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This difference is attributed to a formation of an energetically unfavorable N0 configuration, that hinders the charge fluctuations by large structural reorganizations (λreorg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='89 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hence, substitutional N0 pairs are unlikely to be the origin of electric and magnetic noise, affecting the T1 time due to charge fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' By contrast, owing to a rather modest λreorg of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='28 eV, a vacancy pair gives rise to noise in a broad frequency range, where the respective rate constants are controlled by the separation between the active sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The relevant distance between a pair of vacancies is readily obtained from an exponential fit of the rate constants in Figure 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' More specifically, we find the charge fluctuation rates, which would be relevant for an influence on the T1 time (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1 GHz) at a separation of ∼ 18 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The molecular dynamics simulations performed by F´avaro de Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' show that such a high local density of vacancies around the region of a NV-center is achieved during the nitrogen implantation due to the cascade process of the “kick-out” mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='52 Using the variations of electrostatic potential from Figure 6b for the implantation depths of 5 and 12 nm, we calculate a contribution to the onsite Gibbs energies by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='125 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='075 eV, respectively (see Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' As shown in Figure 6e, the BCT rate at 12 nm reduces by a factor of ∼ 7 relative to the case of ∆V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Moreover, in agreement with our experimental results, the faster changes in the potential at 5 nm enhances the reduction factor to ∼ 19 for the shallower NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Relative to pure water, the BCT rates decrease by factors of ∼ 10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 for the depths of 5 nm and 12 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Furthermore, the proposed mechanism is consistent with the experimental results on dark spins (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Interestingly, our calculations point to an even larger decrease of the BCT rate at smaller distances from the interface which should translate to an even larger sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Given the favorable downwards band bending, these results call for a further optimization of the implantation parameters as well as the surface structure to fully exploit the extension of the T1 time by diamagnetic electrolyte solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Conclusion and Outlook We report on the effect of diamagnetic electrolyte solutions on highly dense near-surface spin defects in oxygen-terminated diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Surprisingly, we observe that diamagnetic ions increase the T1 relaxation time of NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We demonstrate that this effect is reversible, surface sensitive and responsive to millimolar concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We find that also interfacial spin defects are sensitive to diamagnetic species, anticipating their possible use as reporter spins for future optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Furthermore, we investigate the underlying mechanism by single and double quantum NV-relaxometry experiments in combination with ab initio sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We propose that ions at the interface stabilize charge fluctuations between pairs of 19 carbon vacancies and alike deep defects, surrounding the NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This reduces magnetic as well as electric noise at the diamond interface by a dynamical trapping of mobile electrons to a site with lower Gibbs energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' These findings encourage further simulations and experi- ments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g, on other NV-diamond systems, such as nanodiamonds or single NV-centers) to elaborate on a comprehensive understanding of the complex processes at the solid/liquid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We would like to emphasize that the sensitivities of relaxometry to para- and diamagnetic electrolyte solutions both represent scientifically relevant concentration regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Paramag- netic species in the physiological environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', reactive oxygen species (ROS) or trace metals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', manganese can typically be found in ∼ nanomolar to micromolar concentra- tions fitting the highly sensitive feedback of NV-relaxometry (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='41,53 However, diamagnetic ion concentrations are typically orders of magnitude higher in the cytoplasm (∼ millimolar)42,43 or in electrochemistry (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1 to 1 molar)44 which fit very well the NV- center’s response reported in our work (see Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, these two effects may counteract if both species are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A possible pathway to differentiate between these two could be recording single and double quantum experiments, which are only affected by diamagnetic species in the latter case (see Figure 4 and Supplementary Note 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, we propose to probe both relaxation times in future relaxometry studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We envision ap- plications ranging from probing electrochemical interfaces54 to nanoscale ion sensing in cells or neuroscience, where changes in the membrane potential occur as a result of concentration gradients of diamagnetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='55–57 Methods Sample Preparation Two 2 × 2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 mm electronic grade diamond samples (natural 13C abundance, Element Six) were implanted with 15N at an energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV or 4 keV, an off-axis tilt of 7° and a 20 fluence of 2×1012 cm−2 by Innovion and annealed according to Bucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 Before exper- iments are conducted, the diamonds are cleaned with a tri-acid cleaning protocol according to Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' :38 Samples are boiled in equal parts of sulfuric, nitric and perchloric acid at a temperature of 280 °C for two hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This cleaning procedure is also applied before the deposition of aluminium oxide (Al2O3) on the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Preparation of Electrolyte Solutions For the measurements where pure water is used, deionized water with a resistivity of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='2 MΩ·cm at 25 °C (Merck Millipore) is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sodium chloride (NaCl, Merck 106404) is prepared in a 1 M stock solution, where NaCl is dissolved in deionized water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Before the experiments, NaCl is diluted from the stock solution to obtain 500, 250, 100, 50, 10 and 1 mM concentrated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The other salt solutions used within this work are prepared in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Atomic Layer Deposition (ALD) The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV 15N implanted diamond is coated with an aluminium oxide (Al2O3) thin film by ALD according to Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='13 The deposition includes 10 cycles of alternated sample exposure to trimethyl aluminium (TMA) and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This procedure results in a film thickness of ∼ 1 nm and ensures surface termination with hydroxyl groups by exposing the diamond to a remote oxygen plasma within the ALD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='13,58 The Al2O3 layer can be removed from the diamond surface by soaking the sample overnight in 5% NaOH solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experimental Setup The quantum sensing setup is based on a modified version of the experiment described in Bucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 Before experiments are performed, the diamond is glued to a thin glass cover slide (48393026, VWR) together with a microfluidic device that encloses the diamond 21 edges and covers its surface, such that a volume of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='60 µL of the sample liquid can be applied in a controllable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' On the other side of the cover slide a 6 mm diameter glass hemisphere (TECHSPEC® N-BK7 Half-Ball Lenses, Edmund Optics) is glued, in order to improve the fluorescence light collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The glass cover slide is then fixed on a 30 mm cage plate (CP4S, Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This whole assembly is then positioned between two permanent magnets, that are rotated and tilted in order to align the B0 field with one of the four possible NV-center orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The distance between the two magnets can be adjusted in order to correspond to the working magnetic field strengths B0 (in this work: 15, 316, 352 and 978 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Initialization of the NV-ensemble is realized with a 532 nm laser (Verdi G5, Coherent) with a power of ∼ 250 mW (CW) after the AOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The laser light is focused on the diamond by a Plano-Convex Lens (LA 1986-A-M, Thorlabs) in a total internal reflection geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Laser pulses are regulated by an acousto-optic modulator (Gooch and Housego, model 3260-220) with pulse durations of 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Photoluminescence (PL) is collected and focused on a large area photodiode (OE-300-SI-10, Femto Messtechnik GmbH, Berlin, Germany) by two condenser lenses (ACL25416U-B, Thorlabs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The excitation light is filtered by a long-pass optical filter (Edge Basic 647 Long Wave Pass, Semrock) placed between the bottom condenser lens and the photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The output voltage of the photodiode is digitized with a data acquisition unit (USB-6229 DAQ, National Instruments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A 500 MHz PulseBlaster card (ESR-Pro-II, Spincore) is utilized to trigger and to time the microwave and light pulses used for quantum control of the NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The microwave frequencies are produced by a signal source (SynthHD, Windfreak Technologies, LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Microwave phase control is obtained by a combination of a phase-shifting splitter (ZX10Q-2-27-S+, Mini-Circuits), two switches (ZASWA-2-50dRA+, Mini-Circuits) and a combiner (ZX10-2- 42-S+, Mini-Circuits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The amplified microwave pulses (ZHL-16W-43-S+, Mini-Circuits) are delivered by a homebuilt microwave loop on top of the microfluidic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The electron spin resonance (ESR) frequency is used to determine the magnetic field strength B0 as well as the NV0,-1 resonance frequency f NV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 22 T1 Relaxometry Experiments (Single and Double Quantum) Single quantum (SQ) relaxometry experiments: To obtain a signal-to-noise ratio (SNR) as shown in Figure 1b the sequence is repeated 5,000 times for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Each experiment consists of 31 data points measured in a logarithmic increasing sweep time t to guarantee more sampling points at short times t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Note that these parameters are also used for double quantum (DQ) relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For normalization and noise cancellation, the second half of the sequence contains a MW π0,-1-pulse, where the subscripts 0 and -1 indicate the initialization of the spin state from ms = 0 to ms = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='18 The spectra are then plotted as the measurement result of the first half divided by the result of the second half of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Double quantum (DQ) relaxometry experiments: For a detailed discussion of the DQ relaxation and pulse sequence, the reader is referred to Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='45 In short, the DQ pulse sequence (see inset of Figure 4b) consists of two consecutive measurements where MW π- pulses are used to control spin state initialization and readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In both halves of the sequence the NV-center is initialized in ms = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' After a sweep time t the spin state population of either ms = −1 (in the first part) or ms = +1 (in the second part) is read out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dividing the second by the first part yields a population ratio of the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sensitivity of T1 Relaxometry on Electrolytes Experiments to determine the sensitivity of T1 relaxometry measurements on para- and diamagnetic electrolytes are conducted for MnCl2 and NaCl solutions using the SQ relax- ometry pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Probing each concentration results in a relaxation curve of which the T1 time is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The T1 time is then normalized to the one of water covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Before probing any electrolyte concentration, we wash the microfluidic device with water to ensure equal starting conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a constant T1 time for water covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We perform each series three times resulting in a mean T1 value for each concentration (see also Supplementary Note 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure 2 in the main text shows the mean 23 (normalized) T1 time along with the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' DEER Measurements DEER spectra (see Figure 5b) are recorded by performing a spin-echo sequence on the NV-center spins with a free evolution time of 1 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The duration of the MW-pulse (MWDEER) applied to the surface dark spins is set to 200 ns and the driving frequency (f DEER) is swept over 90 MHz (from fDEER = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='84 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='93 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To obtain a SNR as shown in Figure 5b the sequence is repeated 10,000 times for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Each experiment consists of 67 data points in equally separated time steps and this whole experiment is repeated four times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Referencing for noise cancellation is achieved by alternating the last MW-pulse of the spin-echo sequence from π/2 to 3/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Once the resonance condition for ge = 2 is found, DEER-Rabi experiments on the surface dark spins are performed by sweeping the MW-pulse duration (MWDEER) during the NV spin-echo (see Figure 5c) as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The sequence is repeated 10,000 times for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Each experiment consists of 101 equally spaced data points and this whole experiment is repeated ten times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' To account for MW (MWDEER) noise, the same procedure is repeated 20 MHz off the resonance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The outcome of both on- and off-resonant measurements are subtracted resulting in the spectra shown in Figure 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' After that, measurements of the surface dark spin population relaxation are carried out according to Sushkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' with a πds-pulse length of 24 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='46 The sequence shown in Figure 5d is repeated 10,000 times for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Each experiment consists of 21 data points in equally separated time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This whole experiment is then repeated 50 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Background subtraction is achieved by performing the experiment in the same procedure without the additional MW drive (MWDEER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Subtracting the outcome of both MW-on and MW-off measurements then yields the spectra shown in Figure 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 24 Simulation of the Diamond/Water Interface In our simulations, we use a slab of a model diamond surface with hydrogen, hydroxyl, and ether surface terminations (see Figure S10d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' It is a symmetric (100) surface of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='4 nm with a 2 × 1 surface reconstruction pattern, exhibiting a positive electron affinity and no surface states inside the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='59 The water layer on top of the diamond (thickness ∼ 2 nm) was constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' First, we equilibrate 74 water molecules with the clas- sical molecular dynamics (MD) for 5 ns in a simulation box of commensurate lateral size with the diamond slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' These calculations are done with the GROMACS software in the canonical NVT ensemble,60 using the GROMOS 54A7 force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='61 After that, we superim- pose the water box and the diamond surface and allow for an additional equilibration step of 10 ps with the ab initio MD, as implemented in the VASP package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='62 We also incorporate ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='9 nm of vacuum together with a dipole correction scheme to eliminate the interaction with the periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This yields the simulation supercell of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0097 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0097 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='3 nm3, which is further used in the ab initio MD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ab initio calculations are performed using the PBE functional63 in conjunction with the D2 dispersion correction, using a pro- jector augmented wave method with the kinetic energy cutoff of 370 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We note that the PBE functional provides semi-quantitative results for the electronic structure but is able to accurately yield the trends in the change of the electronic structure upon different surface terminations and environments of diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Further, we note that we focus on the difference in the electrostatic environment due to the interaction of the electrolyte with the surface groups, assuming no change in the microstructure of the carbon layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Charge Transfer Rates We calculate the charge transfer rate with an expression from the Marcus theory,64 given as: kCT = 2π ℏ |HAB|2 1 √4πλkBT exp � −(λ + ∆G)2 4πλkBT � 25 where HAB is the transfer integral, λ the reorganization energy, ∆G the Gibbs energy dif- ference due to an external field, kB the Boltzmann constant, ℏ the reduced Planck constant, and T the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The reorganization energy is determined for a single defect (either a carbon vacancy or a substitutional nitrogen) in a 1000-carbon supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For computing the transfer integrals as a function of distance, we use diamond supercells of different sizes, varying between 64 and 1000-carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The reorganization energies are calculated by the four-point scheme, while the transfer integrals are estimated at a high symmetry con- figuration as 1/4 of the bandwidth along the Γ-X direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The contribution to the Gibbs energy is computed by solving the one dimensional Poisson equation given the experimental depth of the NV center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='65 Noteworthy, the effect from the band bending is governed by the orientation of defect pairs relative to the direction of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' At a reference depth of the NV-center, the maximum strength, corresponding to a change in the electrostatic potential (∆Vmax), is reached in a parallel configuration, whilst the effect is quenched to- wards the orthogonal arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Considering a uniform distribution of the defects in our samples, we compute an expectation value of ∆V as ∆Vmax/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Acknowledgement This study was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 412351169 within the Emmy Noether program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' acknowledges support from the DFG Walter Benjamin Programme (Project RI 3319/1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' acknowledges support from the DFG under Germany’s Excellence Strategy—EXC 2089/1—390776260 and the EXC-2111 390814868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' acknowledges support from the DFG through the Munich Center of Quantum Science and Technology (MCQST, EXC-2111) and by BMBF (epiNV, 13N15702).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' acknowledges the Hungarian NKFIH grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' KKP129866 of the National Excellence Program of Quantum-coherent materials project, the support for the Quantum Information National Laboratory from the Ministry of Culture and Innovation of Hungary 26 (NKFIH grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 2022-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1-NL-2022-00004), the EU EIC Pathfinder project ”QuMicro” (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 101046911) and the EU QuantERA for the project MAESTRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We acknowledge KIF¨U for awarding us access to computational resources based in Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Author Contributions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' discovered the effect of diamagnetic electrolytes on the relaxation of NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' designed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' supervised the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' performed the experiments and was supported by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' for NV-relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' built the quantum sensing setup and designed the microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' incorporated theoretical modeling and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' helped with the charge state experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' All authors discussed the results and contributed to the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additional Information Competing interests: The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Data availability The data supporting our findings are available within the paper and the Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additional relevant data are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Code availability The codes used for data acquisition and processing are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 27 Supplementary Information Supplementary Note 1: Fitting of T1 Single Quantum (SQ), Double Quantum (DQ) and Surface Dark Spin Relaxation Curves Recorded single quantum (SQ) and double quantum (DQ) relaxation curves are fitted with a biexponential function as the T1 decay exhibited two components according to prior work:25,27,28,32,66 C(t) = A · exp(− 1 T1a t) + (1 − A) · exp(− 1 T1b t) where C is the contrast, A is the amplitude and T1a >> T1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For completeness, relaxation times in the tables are given by both time constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In agreement with prior work,25,27,32 values of T1 in the main text are only considering the longer component T1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' However, both time constants are longer in all cases where diamagnetic electrolytes are measured with NV- relaxometry and compared to water (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Errors and errorbars from SQ and DQ relaxation curves shown in tables or figures are standard deviations from the biexponential fit function or in case of the sensitivity experiments (see Figure 2) the standard deviation from three consecutive measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T1 time constants in the tables are given to three significant digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In case of the T1,ds relaxation measurements (see Figure 5d), the relaxation curve is fitted to a single exponential decay: C(t) = A · exp(− 1 T1,ds · t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='46 Supplementary Note 2: T1 Time Constants of Measured Electrolytes and T1 Time Magnetic Field Dependence for Pure Water/NaCl (500 mM) Table S1 and Figure S1 show the T1 time constants (T1a and T1b) and T1 relaxation curves of the measured electrolyte solutions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are conducted with the relaxometry pulse sequence according to the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 28 Figure S1: T1 relaxation curves of water and a-f) diamagnetic electrolyte (500 mM) solutions as well as g) and h) paramagnetic electrolyte (1 µM) so- lutions covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 29 Table S1: T1 time constants (T1a and T1b) of water and measured diamagnetic electrolyte solutions (500 mM) as well as paramagnetic electrolyte solutions (1 µM) covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at fNV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Electrolyte [c = 500 mM] T1a [µs] T1b [µs] Water 920 ± 170 140 ± 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 CsF 1510 ± 250 200 ± 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 KCl 1720 ± 300 270 ± 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 KNO3 1360 ± 220 240 ± 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 LiCl 2940 ± 720 930 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 1920 ± 200 170 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 CaCl2 2920 ± 260 460 ± 190 MgSO4 3600 ± 160 310 ± 100 AlCl3 2070 ± 370 360 ± 190 Electrolyte [c = 1 µM] T1a [µs] T1b [µs] MnCl2 430 ± 160 140 ± 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 Gd(NO3)3 250 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 21 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='00 Further measurements of water/NaCl (500 mM) solution are performed in different mag- netic fields B0 (978, 352, 15 and 0 G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', different resonance frequencies of the NV-center’s ms = 0 → ms = −1 transition (f NV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='131, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='87 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure S2 shows the T1a time constants depending on f NV (see Supplementary Note 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In Table S2 both time constants (T1a and T1b) are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure S2: T1a time constants for water and NaCl (500 mM) solution and their dependence on the NV0,-1 resonance frequency f NV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 30 Table S2: T1 time constants (T1a and T1b) for water and NaCl (500 mM) solution covering the diamond surface depending on the NV0,-1 resonance frequency f NV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T1a [µs] T1b [µs] f NV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='131 GHz Water 1810±460 400±60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 2300±340 670±130 f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz Water 940±180 130±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 1990±200 170±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 f NV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83 GHz Water 2660±110 510±80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 3970±400 1210±410 f NV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='87 GHz Water 3460±630 930±190 NaCl 500 mM 4830±740 1830±510 Supplementary Note 3: NV-Relaxometry Experiments with Differ- ent Organic Solvents Following measurements are performed in order to investigate the impact of the solvent’s physical properties on NV-relaxometry experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, we choose organic solvents with dielectric constants (κ) which differ significantly from the properties of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='40 The diamond is covered three times alternatingly with water and the organic solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T1 times of the solvents are then normalized to the T1 time of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure S3 shows that the T1 time remains unaffected by the solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 31 Figure S3: NV-relaxometry experiments showing the impact of the solvent’s di- electric constant (κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Supplementary Note 4: NV-Relaxometry Measurement Series of Para- and Diamagnetic Electrolyte Solutions in Increasing Con- centrations Figure S4: NV-relaxometry measurement series with increasing concentrations of a) MnCl2 and b) NaCl solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Data points are T1 times normalized to the T1 time of water for each series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Solid lines connect the mean values of three consecutive performed series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The NV-relaxometry measurement series with paramagnetic MnCl2 and diamagnetic NaCl solutions are performed in order to determine the sensitivity of the protocol to increas- 32 ing electrolyte solutions in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are conducted using the SQ relaxometry pulse sequence (see Methods for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' We perform each series three times resulting in a mean value for each concentration (see color codes in Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure 2 in the main text shows the mean (normalized) T1 time along with the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Paramagnetic MnCl2 solutions decrease the T1 time in ∼ nano- to micromolar concen- trations with respect to water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In contrast to that, diamagnetic NaCl solutions increase the T1 time in ∼ millimolar concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Supplementary Note 5: NV-Depth Dependence Measurements with Water/LiCl (500 mM) Figure S5: T1 relaxation curves of water and LiCl 500 mM solution covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Diamonds were implanted with 15N at an energy of a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV and b) 4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' NV-relaxometry with water/LiCl (500 mM) covering the diamond is performed in order to investigate the impact of another diamagnetic electrolyte and to support the experiments with NaCl (500 mM) solution using differently deep NV-center ensembles (implanted with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 keV and 4 keV, see Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure S5 and Table S3 show similar results for both NaCl and LiCl (500 mM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 33 Table S3: T1 time constants (T1a and T1b) of water, NaCl (500 mM) and LiCl (500 mM) solution on the diamond surface depending on the nitrogen implan- tation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Implantation energy [keV] T1a [µs] T1b [µs] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 Water 940 ± 180 130 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 1920 ± 200 170 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 Water 660 ± 180 200 ± 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 LiCl 500 mM 2940 ± 720 930 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 4 Water 1090 ± 190 190 ± 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 1270 ± 180 220 ± 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 Water 2750 ± 340 380 ± 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 LiCl 500 mM 2880 ± 270 440 ± 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 Supplementary Note 6: NV-Charge State, Coherence and Dephas- ing Measurements We observe an increase of T1 by diamagnetic electrolyte solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' However, it is known that NV-charge state alteration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', NV0 ↔ NV–) can influence the outcome of NV- relaxometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='67,68 For that reason, we perform NV-Rabi experiments with water/NaCl (500 mM) solution (see Figure S6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Any change in the NV-Rabi contrast indi- cates an alteration of the NV-center’s charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' For instance, an ionization of NV– would increase the proportion of NV0, thereby raising the background fluorescence and lowering the contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The NV-Rabi experiments show no difference in the outcome between water and the electrolyte implying a constant charge state distribution during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Secondly, to supplement the NV-Rabi experiments, we conduct NV-relaxometry with dis- tinct optical readout of the NV0 and NV– charge states and with three different laser powers (see Figure S6b and Figure S6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Possible ionization of NV– in the dark or recombination processes would be visible as an alteration in the readout signal of the NV0 charge state (see Figure S6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='67,68 These measurements are carried out using the first half of the relax- ometry pulse sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', without a π-pulse) and with two different optical filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The 647 nm long pass filter predominantly reads out the fluorescence from the NV– state and the 600 ± 40 band pass filter mostly reads out the fluorescence from the NV0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='69 While a T1 34 Figure S6: Pulse sequences and spectra of NV-charge state measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) NV-Rabi experiments, b) NV-charge state measurements with selective read- out of the NV0 or the NV– state and c) T1 relaxation curves using three differ- ent laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 35 fluorescence decay curve can be extracted from the measurements with the long pass filter, no decisive change in the NV0 state is visible using the band pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Probable impact of the laser power on the NV–/NV0 ratio and a subsequent change in the T1 relaxation curves is probed with relaxometry experiments using laser powers of 25, 50 and 100 µW µm−2 (see Figure S6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Both NV-Rabi and NV-charge state experiments do not show an impact on NV-charge state alteration on the relevant timescales of the relaxometry measurements we conduct herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Figure S7: Pulse sequences and spectra of Ramsey and T2 Hahn-echo mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' a) Ramsey oscillations performed at a 4 MHz detuned NV0,-1 reso- nance frequency f NV and b) T2 Hahn-echo experiments with water and NaCl (500 mM) solution covering the diamond surface at f NV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='88 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, we perform Ramsey (NV-dephasing) and T2 (NV-coherence) Hahn-echo experiments, whose outcome is typically affected by changes in the low frequency components of the noise (see Figure S7a and S7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='30 Both experiments show no difference in the outcome for water or NaCl (500 mM) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' However, we note that probable changes in this noise frequency regime might not be observable with the high-dense NV-center ensemble we use in this work, since the surrounding spin-bath (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P1-centers or other paramagnetic impurities) is limiting the NV-dephasing and NV-coherence in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='37,50 36 Supplementary Note 7: T1 Time Constants for Single Quantum and Double Quantum Experiments at B0 = 15 G and Zero Field ESR Measurements Figure S8: a) SQ and b) DQ relaxation curves of water and MnCl2 (100 µM) so- lution covering the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at B0 = 15 G, where the NV0,-1 transition is at fNV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='83 GHz (corresponding to a DQ transi- tion frequency of 80 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T1,SQ decreases by 80%, whereas T1,DQ remains un- changed compared to water when MnCl2 solution covers the diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Single and double quantum T1 experiments of water/NaCl (500 mM) and water/MnCl2 (100 µM) solution covering the diamond surface are performed in order to elucidate the effect of the electrolyte on magnetic and electric field noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' In the case of the NaCl solution, both T1 time constants (T 1a,SQ and T 1a,DQ) increase compared to water, indicating a reduction of both magnetic and electric field noise (see also Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Importantly, MnCl2 only reduces the T1 time for the SQ relaxation, whereas the DQ transition remains unaffected compared to 37 Table S4: T1 time constants (T 1a,SQ and T 1a,DQ) for water/NaCl (500 mM) and water/MnCl2 (100 µM) solution covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Experiments are performed at B0 = 15 G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T 1a,SQ [µs] T 1a,DQ [µs] Water 2600 ± 280 440 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 NaCl 500 mM 3970 ± 400 1250 ± 120 Water 2000 ± 340 410 ± 71 MnCl2 100 µM 390 ± 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 390 ± 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='0 water (see also Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' This indicates an exclusive impact of the paramagnetic electrolyte on magnetic field noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' However, we note that probing MnCl2 in higher (> 100 µM) concen- trations would lead to a collapse of the NV-center’s T1 time (see also Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Therefore, a final statement on the impact of higher concentrated paramagnetic electrolyte solutions on the DQ (as well as the SQ) relaxation cannot be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Additionally, we investigate the static electric field environment of the NV-center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', charges within the diamond and adjacent to the NV-center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=', N+ and NV–).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='70 Therefore, we measure ESR at zero magnetic field (here the earth’s magnetic field ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='5 G), because any difference in the static electric field in the proximity of the NV-center with respect to water or the electrolyte solution covering the surface would induce a shifting and/or splitting of the ms = ±1 states apparent in the ESR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='70 Figure S9 shows no significant change of the ESR resonance lines for the exposure of water or electrolyte solution, indicating that static electric fields do not contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 38 Figure S9: ESR experiments at zero magnetic field with water and NaCl (500 mM) solution covering the diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 39 Supplementary Note 8: Results of DFT-PBE Ab Initio Molecular Dynamics Simulations Figure S10: a) Band alignment of the water layer and the model diamond sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' b) Distribution of interfacial dipoles, sampled from the MD trajectories for three different compositions of the interface and solvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' c) Average elec- trostatic potentials and vacuum level shifts (VLS) computed for the configu- rations corresponding to the middle of the distributions in b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Vertical lines show the parts of the simulation box, spanned by diamond (C), water or aque- ous NaCl solution and vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' d) Structures of the model diamond surface before and after adding a COOH group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 40 a) Band alignment b) 600 model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='86 eV Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content="units] model+COOH CBM 500 model+CoOo'Na LUMO 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='94 eV 400 300 200 VBM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='20 eV 100 HOMO 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='32 eV Diamond 0 2 1 0 c) Water 1 2 Dipole moment [at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content="units] 10 Electrostatic potential [eV] model+ Na*cl model +COOH model+Coo'Nat 5 士 VLS=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='1 VLS=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='6 VLS=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='9 0 5 10 15 c H20 20 NaCI 25 WW 10 0 10 20 30 40 50 10 10 20 30 4050 10 0 10 20 30 40 50 Z-coordinate [A] Z-coordinate [A] Z-coordinate [A] d) model model+COOHReferences (1) Acosta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bauch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ledbetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Waxman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bouchard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Budker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Temperature Dependence of the Nitrogen-Vacancy Magnetic Resonance in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2010, 104, 070801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (2) Iv´ady, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Simon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Maze, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Abrikosov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Gali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pressure and temperature dependence of the zero-field splitting in the ground state of NV centers in diamond: A first-principles study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review B 2014, 90, 235205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (3) Teissier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Barfuss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Appel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Neu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Maletinsky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Strain Coupling of a Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2014, 113, 020503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (4) Dolde, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Fedder, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Doherty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' N¨obauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rempp, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Balasubramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wolf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reinhard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Electric-field sensing using single diamond spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Physics 2011, 7, 459–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (5) Balasubramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kolesov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Al-Hmoud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tisler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Shin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wojcik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hemmer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Krueger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hanke, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Leitenstorfer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Brats- chitsch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nanoscale imaging magnetometry with diamond spins under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature 2008, 455, 648–651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (6) Maze, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Stanwix, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hodges, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cappellaro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dutt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Togan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zibrov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Yacoby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nanoscale magnetic sensing with an individual electronic spin in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature 2008, 455, 644–647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (7) Glenn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Weissleder, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Yacoby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Connolly, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Single-cell magnetic imaging using a quantum dia- mond microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Methods 2015, 12, 736–738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 41 (8) Lovchinsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sushkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Urbach, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' de Leon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' De Greve, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Evans, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Gertner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bersin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Muller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McGuinness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nuclear magnetic resonance detection and spectroscopy of single proteins using quantum logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Science 2016, 351, 836–841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (9) M¨uller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Melentijevi´c, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Stacey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Markham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Twitchen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Isoya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pezzagna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Meijer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Plenio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nayde- nov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McGuinness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nuclear magnetic resonance spectroscopy with single spin sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Communications 2014, 5, 4703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (10) Sushkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lovchinsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chisholm, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2014, 113, 197601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (11) Bucher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Principles of nano-and microscale NMR-spectroscopy with NV-diamond sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' eMagRes 2019, 8, 363–370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (12) Lovchinsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sanchez-Yamagishi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Urbach, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Fang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ander- sen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bylinskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kaxiras, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Magnetic resonance spectroscopy of an atomically thin material using a single-spin qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Science 2017, 355, 503–507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (13) Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Henning, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Heindl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Allert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bartl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sharp, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rizzato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bucher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Surface NMR using quantum sensors in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 2022, 119, e2111607119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (14) Allert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Briegel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bucher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Advances in nano- and microscale NMR spectroscopy using diamond quantum sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chemical Communications 2022, 58, 8165–8181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (15) Allert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bruckmaier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Neuling, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Freire-Moschovitis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 42 Schrepel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sch¨atzle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Knittel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hermans, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bucher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Microfluidic quantum sensing platform for lab-on-a-chip applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lab on a Chip 2022, 22, 4831–4840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (16) Shi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ju, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rein- hard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Single-protein spin resonance spectroscopy under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Science 2015, 347, 1135–1138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (17) Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Loretz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Degen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nitrogen-Vacancy Centers in Dia- mond: Nanoscale Sensors for Physics and Biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Annual Review of Physical Chemistry 2014, 65, 83–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (18) Bucher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Aude Craik, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Backlund, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Turner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ben Dor, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Glenn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Quantum diamond spectrometer for nanoscale NMR and ESR spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Protocols 2019, 14, 2707–2747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (19) Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mamin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sherwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ohno, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Awschalom, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rugar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Decoherence of Near-Surface Nitrogen-Vacancy Centers Due to Electric Field Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2015, 115, 087602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (20) Stacey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dontschuk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Broadway, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schenk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sear, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tetienne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hoffman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Prawer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pakes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tadich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' de Leon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Gali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Evidence for Primal sp 2 Defects at the Diamond Surface: Can- didates for Electron Trapping and Noise Sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Advanced Materials Interfaces 2019, 6, 1801449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (21) Sangtawesin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Origins of Diamond Surface Noise Probed by Correlating Single- Spin Measurements with Surface Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review X 2019, 9, 031052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (22) Romach, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M¨uller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Unden, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rogers, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Isoda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Itoh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Markham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Stacey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Meijer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pezzagna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Naydenov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McGuinness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bar-Gill, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Spectroscopy of Surface-Induced Noise Using Shallow Spins in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2015, 114, 017601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 43 (23) Chrostoski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sadeghpour, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Santamore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Electric Noise Spectra of a Near- Surface Nitrogen-Vacancy Center in Diamond with a Protective Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Applied 2018, 10, 064056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (24) Degen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reinhard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cappellaro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Quantum sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reviews of Modern Physics 2017, 89, 035002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (25) Steinert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ziem, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hall, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zappe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schweikert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G¨otz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Aird, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bala- subramanian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Magnetic spin imaging under ambient conditions with sub-cellular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Communications 2013, 4, 1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (26) Mzyk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sigaeva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Relaxometry with Nitrogen Vacancy (NV) Centers in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Accounts of Chemical Research 2022, 55, 3572–3580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (27) Perona Mart´ınez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nusantara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chipaux, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Padamati, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nanodiamond Relaxometry-Based Detection of Free-Radical Species When Produced in Chemical Reactions in Biologically Relevant Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' ACS Sensors 2020, 5, 3862– 3869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (28) Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Vedelaar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mzyk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Morita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Padamati, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Following Polymer Degradation with Nanodiamond Magnetometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' ACS Sensors 2022, 7, 123– 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (29) Jarmola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Acosta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jensen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chemerisov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Budker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensem- bles in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2012, 108, 197601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (30) Rosskopf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dussaux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ohashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Loretz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Watanabe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Shikata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Itoh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Degen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Investigation of Surface Magnetic Noise by Shallow Spins in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2014, 112, 147602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 44 (31) Nie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nusantara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Damle, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Baranov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chipaux, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reyes-San- Martin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hamoh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Epperla, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Guricova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cigler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' van den Bogaart, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schirhagl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Quantum Sensing of Free Radicals in Primary Human Dendritic Cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nano Letters 2022, 22, 1818–1825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (32) Ziem, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G¨otz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zappe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Steinert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Highly Sensitive Detec- tion of Physiological Spins in a Microfluidic Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nano Letters 2013, 13, 4093–4098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (33) Ermakova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pramanik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Weil, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tzeng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McGuinness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Plenio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Naydenov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Detection of a Few Metallo-Protein Molecules Using Color Centers in Nanodiamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nano Letters 2013, 13, 3305–3309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (34) Fujisaku, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tanabe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Onoda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kubota, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ohshima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hamachi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Shirakawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Igarashi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' pH Nanosensor Using Electronic Spins in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' ACS Nano 2019, 13, 11726–11732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (35) Simpson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ryan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hall, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Panchenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Drew, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Petrou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Donnelly, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mulvaney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Electron paramagnetic resonance microscopy using spins in diamond under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Communications 2017, 8, 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (36) Pham, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' DeVience, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Casola, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lovchinsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sushkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bersin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Urbach, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cappellaro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Yacoby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' NMR technique for determining the depth of shallow nitrogen-vacancy centers in dia- mond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review B 2016, 93, 045425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (37) Henshaw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kehayias, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Saleh Ziabari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Titze, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Morissette, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Watanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Acosta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bielejec, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lilly, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mounce, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nanoscale solid-state nuclear quadrupole resonance spectroscopy using depth-optimized nitrogen-vacancy ensembles in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Applied Physics Letters 2022, 120, 174002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 45 (38) Brown, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chartier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sweet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hopper, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bassett, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cleaning diamond surfaces using boiling acid treatment in a standard laboratory chemical hood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Journal of Chemical Health & Safety 2019, 26, 40–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (39) Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Oliveira, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Puthirath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Neupane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Weil, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Birdwell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ivanov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Gray, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kannan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Biswas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Vajtai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Galvao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ajayan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Systematic comparison of various oxidation treatments on diamond surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Carbon 2021, 182, 725–734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (40) Seyferth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Organic Solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Properties and Methods of Purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Fourth Edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Volume 11 of Weissberger’s ”Techniques of Chemistry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Organometallics 1987, 6, 1375–1376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (41) Flora, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Biomarkers in Toxicology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Elsevier, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' pp 485–519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (42) Flowers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Munns, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Colmer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sodium chloride toxicity and the cellular basis of salt tolerance in halophytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Annals of Botany 2015, 115, 419–431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (43) Ladenson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Apple, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Aguanno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Koch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sodium measurements in multiple myeloma: two techniques compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Clinical Chemistry 1982, 28, 2383–2386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (44) Bard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Faulkner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Electrochemical Methods: Fundamentals and Applications, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wiley, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (45) Myers, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ariyaratne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jayich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Double-Quantum Spin-Relaxation Limits to Coherence of Near-Surface Nitrogen-Vacancy Centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2017, 118, 197201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (46) Sushkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lovchinsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chisholm, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lukin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Magnetic Resonance Detection of Individual Proton Spins Using Quantum Reporters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2014, 113, 197601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 46 (47) Schlipf, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Oeckinghaus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Xu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dasari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zappe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' de Oliveira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kern, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Azarkh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Drescher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ternes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kern, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Finkler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A molecular quantum spin network controlled by a single qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Science Advances 2017, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (48) Mamin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sherwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rugar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Detecting external electron spins using nitrogen-vacancy centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review B 2012, 86, 195422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (49) Bluvstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McLellan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Williams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jayich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Extending the Quantum Coherence of a Near-Surface Qubit by Coherently Driving the Paramag- netic Surface Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2019, 123, 146804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (50) Barry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Schloss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bauch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Turner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pham, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Walsworth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sensitivity optimization for NV-diamond magnetometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reviews of Modern Physics 2020, 92, 015004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (51) Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Joos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bluvstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lyu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jayich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reporter-spin-assisted T1 relaxometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 2022, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='11470 [quant–ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (52) F´avaro de Oliveira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Antonov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Neumann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Momenzadeh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H¨außermann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Pasquarelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Denisenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tailoring spin defects in diamond by lattice charging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Communications 2017, 8, 15409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (53) Sies, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Reactive oxygen species (ROS) as pleiotropic physiological sig- nalling agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Reviews Molecular Cell Biology 2020, 21, 363–383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (54) Favaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jeong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ross, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Yano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hussain, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Crumlin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Unrav- elling the electrochemical double layer by direct probing of the solid/liquid interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Communications 2016, 7, 12695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (55) Kaufmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Simpson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hall, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Perunicic, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Senn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Steinert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McGuinness, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Johnson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ohshima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Caruso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 47 Scholten, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mulvaney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Detection of atomic spin labels in a lipid bilayer using a single-spin nanodiamond probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 2013, 110, 10894–10898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (56) Hall, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hill, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Cole, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' St¨adler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Caruso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mulvaney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Monitoring ion-channel function in real time through quantum decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 2010, 107, 18777–18782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (57) Szatm´ari, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' S´ark´any, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kocsis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nagy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Miseta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bark´o, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Longauer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Robinson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nyitrai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Intracellular ion concentrations and cation-dependent remodelling of bacterial MreB assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Scientific Reports 2020, 10, 12002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (58) Henning, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bartl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zeidler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Qian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bienek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Paulus, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rieger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Stutzmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Sharp, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Aluminum Oxide at the Monolayer Limit via Oxidant-Free Plasma-Assisted Atomic Layer Deposition on GaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Advanced Functional Materials 2021, 31, 2101441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (59) Kaviani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' De´ak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Aradi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Frauenheim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Chou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Gali, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Proper Sur- face Termination for Luminescent Near-Surface NV Centers in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nano Letters 2014, 14, 4772–4777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (60) Berendsen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' van der Spoel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' van Drunen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' GROMACS: A message-passing par- allel molecular dynamics implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Computer Physics Communications 1995, 91, 43–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (61) Schmid, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Eichenberger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Choutko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Riniker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Winger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Mark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' van Gunsteren, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Definition and testing of the GROMOS force-field versions 54A7 and 54B7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' European Biophysics Journal 2011, 40, 843–856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (62) Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Furthm¨uller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Efficient iterative schemes for ab initio total-energy calcu- lations using a plane-wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review B 1996, 54, 11169–11186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 48 (63) Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Generalized Gradient Approximation Made Simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 1996, 77, 3865–3868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (64) Marcus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The Journal of Chemical Physics 1956, 24, 966–978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (65) Broadway, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Dontschuk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tsai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lillie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Lew, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' McCallum, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Johnson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Doherty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Stacey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollenberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Tetienne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Spatial mapping of band bending in semiconductor devices using in situ quantum sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nature Electronics 2018, 1, 502–507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (66) Rioux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Levesque, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rutt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Biexponential longitudinal relaxation in white matter: Characterization and impact on T1 mapping with IR-FSE and MP2RAGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Magnetic Resonance in Medicine 2016, 75, 2265–2277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (67) Bluvstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jayich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Identifying and Mitigating Charge Insta- bilities in Shallow Diamond Nitrogen-Vacancy Centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2019, 122, 076101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (68) Dhomkar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jayakumar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zangara, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Meriles, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Charge Dynamics in near- Surface, Variable-Density Ensembles of Nitrogen-Vacancy Centers in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Nano Letters 2018, 18, 4046–4052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (69) Doherty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Manson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Delaney, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jelezko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Wrachtrup, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hollen- berg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' The nitrogen-vacancy colour centre in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physics Reports 2013, 528, 1–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' (70) Mittiga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Hsieh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Zu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Kobrin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Machado, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Bhattacharyya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Rui, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Jarmola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Budker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Yao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Imaging the Local Charge Environment of Nitrogen-Vacancy Centers in Diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' Physical Review Letters 2018, 121, 246402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} +page_content=' 49' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE4T4oBgHgl3EQfNAwx/content/2301.04952v1.pdf'} diff --git a/YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss b/YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2a63e41636e7d9c8d61b732ff13798da0be15956 --- /dev/null +++ b/YNFQT4oBgHgl3EQfdzaO/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6b216a73a369bdf832b9621c567a83e4a6115a2568d6a8d1d8245870406ab55 +size 2949165 diff --git a/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/2301.00169v1.pdf.txt b/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/2301.00169v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..14542a7ce04c663f93457ff83290e6990a4e0af9 --- /dev/null +++ b/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/2301.00169v1.pdf.txt @@ -0,0 +1,2171 @@ +Generative Graph Neural Networks for Link Prediction +Xingping Xiana, Tao Wua,∗, Xiaoke Mab, Shaojie Qiaoc, Yabin Shaod, Chao Wange, Lin Yuana, Yu Wua +aSchool of Cybersecurity and Information Law, Chongqing University of Posts and Telecommunications, Chongqing, China. +bSchool of Computer Science and Technology, XiDian University, XiAn, China. +cSchool of Software Engineering, Chengdu University of Information Technology, Chengdu, China. +dSchool of Science, Chongqing University of Posts and Telecommunications, Chongqing, China. +eSchool of Computer and Information Science, Chongqing Normal University, Chongqing, China. +Abstract +Inferring missing links or detecting spurious ones based on observed graphs, known as link prediction, is a long-standing challenge +in graph data analysis. With the recent advances in deep learning, graph neural networks have been used for link prediction and +have achieved state-of-the-art performance. Nevertheless, existing methods developed for this purpose are typically discriminative, +computing features of local subgraphs around two neighboring nodes and predicting potential links between them from the perspec- +tive of subgraph classification. In this formalism, the selection of enclosing subgraphs and heuristic structural features for subgraph +classification significantly affects the performance of the methods. To overcome this limitation, this paper proposes a novel and rad- +ically different link prediction algorithm based on the network reconstruction theory, called GraphLP. Instead of sampling positive +and negative links and heuristically computing the features of their enclosing subgraphs, GraphLP utilizes the feature learning abil- +ity of deep-learning models to automatically extract the structural patterns of graphs for link prediction under the assumption that +real-world graphs are not locally isolated. Moreover, GraphLP explores high-order connectivity patterns to utilize the hierarchical +organizational structures of graphs for link prediction. Our experimental results on all common benchmark datasets from different +applications demonstrate that the proposed method consistently outperforms other state-of-the-art methods. Unlike the discriminative +neural network models used for link prediction, GraphLP is generative, which provides a new paradigm for neural-network-based link +prediction. The code is available at https://github.com/star4455/GraphLP. +Keywords: Graph Machine Learning, Graph Neural Networks, Link Prediction, Structural Patterns, Network Reconstruction. +1. Introduction +Graphs provide an elegant representation for characterizing +entities and their interrelations in complex systems. Given that +real-world graphs can usually only be partially observed and +are often noisy, link prediction aimed at inferring missing and +spurious links based on observed graphs is a paradigmatic and +fundamental problem across many scientific domains, including +knowledge graph completion [52], experimental design in bio- +logical networks [6], fake account detection in online social net- +works [24], and product recommendation on e-commerce web- +sites [31]. +To address the link prediction problem, numerous heuristic +methods have been proposed, including local indices such as +Common Neighbors (CN) [2], and Resource Allocation (RA) +[61], global indices such as Katz [25], and SimRank [23], and +quasi-local indices such as the Local Path Index (LP) [61]. How- +ever, heuristic methods have a strong assumption on when two +nodes are likely to be linked in real-world graphs and lack uni- +versal applicability to diverse areas [8]. Subsequently, statistical +learning-based algorithms have been proposed to obtain ground- +breaking results, such as maximum likelihood-based hierarchi- +cal structure model [13], stochastic block model [21], matrix +factorization-based link prediction method [37], Linear Opti- +mization (LO) link prediction method [38], and Low Frobenius +norm-based Link Prediction (LFLP) method [53]. With the pro- +posal of network representation learning, various network em- +∗Corresponding author. +Email addresses: xxp0213@gmail.com (Xingping Xian), +wutaoadeny@gmail.com (Tao Wu) +bedding algorithms have been put forth so that the likelihood of +a non-observed links can be estimated based on the proximity +of nodes in low-dimensional vector space, including LINE [44], +Node2Vec [20], and DNGR [10]. +Recently, driven by the dramatic advances in deep learning +techniques, neural networks have gradually been used to solve +the link prediction problem. [56] trained a fully-connected neu- +ral network on the enclosing subgraphs of target links for link +prediction, wherein a Weisfeiler-Lehman (WL) algorithm-based +graph labeling mechanism was proposed to encode subgraphs. +Based on the enclosing subgraphs extracted around links, [57] +trained a Graph Neural Network (GNN) for link prediction to +achieve a performance comparable to that of heuristic meth- +ods. Along this line of research, [36] encoded subgraphs into +random-walk transition probabilities and then computed fea- +tures using these probabilities to classify positive and negative +links. +Although these subgraph classification-based methods +have achieved state-of-the-art link prediction performance, the +prediction results are found to be considerably affected by the +extraction process of the k-hop enclosing subgraphs and the +graph structure features for them. For example, in representa- +tion learning on graphs [55], the range of enclosing subgraphs +strongly depends on the graph structure, and the effective range +should be different for subgraphs with varying properties. +Typically, from the perspective of subgraph classification, link +prediction methods treat subgraphs in real-world graphs inde- +pendently and equivalently. That is, the global structural in- +formation of real-world graphs is totally neglected during this +process. +However, extensive empirical analyses indicate that +real-world graphs are not locally isolated but globally relevant +Preprint submitted to Journal of LATEX Templates +January 3, 2023 +arXiv:2301.00169v1 [cs.SI] 31 Dec 2022 + +Figure 1: An illustrative example depicting the global and high-order organizations of real-world graphs. (a) Gene network for C. +elegans [12]. (b) Representative hierarchical star-like structure [45]. (c) Representative hierarchical modular organization [39]. (b) and +(c) depict the representative structural patterns of real-world graphs such as (a). +[35, 51]; here, nodes and edges naturally portray different struc- +tural roles and contribute differently to the global organization of +real-world graphs [53, 54]. Moreover, subgraph classification- +based link prediction methods assume that real-world graphs ex- +hibit low-order connectivity patterns and can be captured at the +level of individual nodes and edges. However, empirical studies +have discovered that real-world graphs exhibit high-order orga- +nizations at the level of small subgraphs, which are recursively +grouped into a hierarchical structure [7, 3]. An illustrative ex- +ample of the global and high-order organization in real-world +graphs is depicted in Figure 1. Hence, two challenges need to be +addressed for link prediction: (i) how to learn good representa- +tion preserving both local and global graph structural features? +and (ii) how to characterize and utilize hierarchical structure pat- +terns? +To address these challenges, instead of predicting poten- +tial links through subgraph classification, this study designs a +novel generative and multi-order GNN for link prediction, called +GraphLP. Evidently, real-world graphs share some global prop- +erties, such as low-rank and sparsity, that can be used to pro- +vide guidance for graph learning. +Hence, motivated by the +network reconstruction theory [21], GraphLP defines a self- +representation model-based collaborative inference operation to +refine the observed graphs globally, which assumes that the orig- +inal graph can be reconstructed utilizing the correlation between +subgraph patterns. +Assuming that the paths between a pair +of nodes provide evidence for the existence of potential links, +GraphLP extracts the local structural information via a high- +order connectivity operation on the observed graphs. Thus, ev- +ery neural network layer obtains the connectivity of node pairs +within two-hop neighborhood, and a neural network with multi- +ple connectivity layers captures the degree of connectivity be- +tween node pairs with various path lengths. +Meanwhile, the +weighted adjacency matrices generated by the connectivity op- +eration in every neural network layer reflect the multi-order con- +nectivity pattern in the graphs. Further, the hierarchical organi- +zational structure of real-world graphs is explored by applying +a collaborative inference operation. The contributions of this +study can be summarized as follows: +• Generative framework. +Rather than subgraph classi- +fication based discriminative schemes, a novel network +reconstruction-based generative GNN is proposed for link +prediction, which provides a new paradigm for the applica- +tion of neural networks in link prediction problem. +• End-to-end learning. Instead of designing heuristic graph +structural features for subgraph representation, local and +global structural patterns are extracted and fused in an end- +to-end fashion for link prediction. +• Algorithm. A novel collaborative inference operation and +high-order connectivity computation mechanism are devel- +oped to characterize the structural patterns in real-world +graphs at different scales. +• Experiment. Extensive experiments on real-world datasets +from different areas reveal that the proposed method, +GraphLP, achieves promising performance and consistently +outperforms other state-of-the-art methods. +Paper Organization. The rest of this work is organized as +follows. Section 2 discusses related studies. Section 3 presents +the problem definitions and describes the preliminaries. Section +4 describes the proposed method. Section 5 presents the experi- +mental results, and finally, Section 6 presents the conclusion and +discussion. +2. Related Work +GNNs and link prediction task have been extensively investi- +gated in recent years. A brief review of related studies is pro- +vided in this section. +2.1. Graph Neural Networks +Owing to their potential in modeling the complex structures +of non-Euclidean graphs, GNNs have achieved state-of-the-art +performance on almost all graph-based tasks, such as node clas- +sification, graph classification, link prediction. Based on differ- +ent theories and perspectives, a plethora of different GNNs have +2 + +(a) +(b) +(c) +8.80been proposed over the years. Generally, GNNs can be divided +into two categories: spectral-based and spatial-based methods. +Of these, spectral-based GNNs are types of GNNs that design +graph convolution operators in the spectral domain using Fourier +transform. The involved convolution operation is defined as fol- +lows: +f1 ∗ f2 = U[(UT f1) ⊙ (UT f2)], +(1) +where ⊙ denotes an element-wise product. The spectral filter is +defined as g = UT f1, and the node signal X can be processed as +follows: +Z = U[g(Λ) ⊙ (UTX)] = Ug(Λ)UTX. +(2) +where U denotes a matrix of eigenvectors of the normalized +Laplacian graph L = I − D− 1 +2 AD− 1 +2 = UΛUT [11]. Assum- +ing that feature representation of node should be affected only +by its k-hop neighborhood, [16] proposed a Chebyshev poly- +nomial based k-localized convolution and developed a convolu- +tional neural network, ChebNet, which eliminated the need to +compute the eigenvectors of the Laplacian. Subsequently, [50] +simplified the Chebshev polynomial filter using its first-order +approximation and proposed the popular spectral-based method +called Graph Convolutional Networks (GCNs). Notably, spatial- +based GNNs define graph convolution operator based on graph +topology wherein the feature vectors of node’s neighbors are ag- +gregated via a permutation-invariant function. Specifically, [22] +proposed a GraphSAGE approach that sampled fixed size neigh- +borhood nodes and used max pooling, mean pooling, and LSTM +pooling scheme to aggregate neighbor information. Considering +the different weights of node’s neighbors, [46] proposed a Graph +Attention Network (GAT) algorithm to calculate attention coeffi- +cient and then aggregated the neighborhood information. Other +related models include PATCHY-SAN [34], DCNN [4], and fur- +ther details on GNNs can be found in the review [60]. +2.2. Neural Networks based Link Prediction +Following heuristic methods, matrix completion-based meth- +ods and network embedding-based methods, neural networks +have been gradually applied to link prediction problem and +have achieved state-of-the-art results. +Specifically, [56] pro- +posed a link prediction method called Weisfeiler-Lehman Neural +Machine (WLNM), which labeled nodes using the Weisfeiler- +Lehman algorithm and encoded subgraphs to construct a feed- +forward neural network-based classification model. Next, from +the perspective of subgraph classification, [57] proposed a novel +GNN-based link prediction framework, SEAL, to learn subgraph +structures and node features from local enclosing subgraphs. +Along this line, to directly leverage the topology features of lo- +cal subgraphs, [36] proposed a new random-walk-based pool- +ing scheme, WalkPool, and built features for subgraph classifi- +cation. Moreover, [18] proposed a neural network-based link +prediction method with only one-hop neighborhood informa- +tion, which demonstrated almost equivalent performance to the +WLNM and SEAL. Instead of subgraph classification, [8] con- +verted the original graph into a corresponding line graph and +solved the node classification problem for link prediction. To +perform link prediction for general directed or undirected com- +plex networks, [48] represented the adjacency matrices of net- +works as binary images and developed a generative adversarial +networks (GANs)-based method. In addition, because existing +GNN-based methods do not scale appropriately to large graphs, +[30] extracted sparse enclosing subgraphs based on multiple ran- +dom walks and presented a scalable link prediction solution, +called ScaLed. To reduce the time required to determine the +distances between two nodes, [27] defined an anchor-based dis- +tance and proposed a new distance-enhanced GNN method for +link prediction. +Among all existing methods for link prediction, the work clos- +est to the one condidered in this study is the GANs-based method +[48]. However, this method predicts potential links via image +processing within the GANs framework, whereas the proposed +method conducts link prediction via GNNs-based network re- +construction. +2.3. Network Structure Analysis +Real-world graphs, also known as complex networks, are ab- +stract representation of complex systems and have been exten- +sively studied in the field of network science. Consequently, +numerous studies have revealed that complex networks exhibit +rich and diverse connectivity patterns. [32] augmented that the +organization of real networks usually embodies both regularities +and irregularities, where the former can be modeled and decides +the extent to which the formation of a network can be explained. +Notably, link predictability reflects the structural regularities in +real-world networks and denotes the inherent difficulty of link +prediction. [53] proposed a self-representation network model- +based method, called NetSRE, for measuring and regulating link +predictability of networks. [54] proposed a deep linear coding- +based link prediction adversarial attack method by disturbing +the underlying structural pattern of networks, which proved that +links play global structural roles in network organization. More- +over, [7] suggested that high-order connectivity patterns are es- +sential for understanding the fundamental structures of networks +and developed a framework that identified clusters of network +motifs. [41] claimed that hierarchical structure plays an impor- +tant role in complex systems. To prove the existence of hierar- +chical organization, an unsupervised method for extracting the +hierarchical organization of complex networks was introduced +and validated. +Although real-world graphs exhibit various structural pat- +terns, most existing neural networks-based link prediction meth- +ods simply assume that they are flattened and locally isolated, +and these methods judge the existence of links explicitly based +only on local enclosing subgraphs. With the exception of local +structural features, this study focuses on integrating global and +hierarchical structural patterns into neural networks for link pre- +diction. +3. Problem Definition and Preliminaries +3.1. Problem Definition +Notations. +Let G = (V, E) denote an undirected and un- +weighted graph, where V = {v1, · · · , vN} denotes the set of nodes +and E = {e1, · · · , eM} denotes the set of edges. The adjacency +matrix of graph G is denoted as A ∈ {0, 1}N×N, where Ai j = 1 if +nodes i and j are connected and Ai j = 0 otherwise. Each edge +e can be represented as a node pair (u, v, ), where u, v ∈ V. Let +N (u) denote the neighbors of node u, N (u) = {v|(u, v) ∈ E}. +Link Prediction. Given an observed graph Go = (V, Eo) that +corresponds to the original graph G = (V, E), link prediction +aims to infer the presence or absence of an edge between a +pair of target nodes based on Go, thereby generating a recovered +graph G∗ to approximate the original graph G. In particular, the +prediction problem involves identifying a function that generates +3 + +a likelihood score for a pair of nodes (u, v) � E to infer the miss- +ing link (u, v), or to produce a likelihood score for an existing +edge (u, v) ∈ E to identify spurious links. Thus, the link predic- +tion problem can be formulated as suv = f(u, v, A|θ), where θ +denotes the parameter of link prediction model. In this work, Em +and Es denote the identified missing and spurious links, respec- +tively. +Note that data augmentation is a set of techniques that in- +creases the amount and diversity of data by creating reasonable +virtual data points from existing data, such that better machine +learning models can be constructed based on them. According +to [59], this study considers graph data augmentation and adopts +a random mapping mechanism to produce augmented graph set +D based on the observed graph Go = (V, Eo). Specifically, the +set of all possible edges in the graph Go is denoted as Ω, the ex- +isting edge set is denoted as Eo, and the non-existing edge set is +denoted as Enon = Ω − Eo. Thus, the candidate sets for random +mapping are defined as follows: +Ec +del = Eo, +Ec +add = Enon. +(3) +Thereafter, samples are randomly produced from the candidate +sets to obtain the edge sets Edel and Eadd. Finally, a new aug- +mented graph is generated by modifying the graph Go based on +Edel and Eadd: +G′ = (V, (E ∪ Eadd)\Edel). +(4) +Each input graph can be viewed as an instance for link pre- +diction, owing to the generative learning scheme of the models +considered in this work. Thus, the dataset containing a series of +augmented graphs can be denoted as D = {Gi|i = 1, ..., t} and +split to yield disjoint training and validation sets. These can be +denoted as Dtrain and Dval respectively, wherein the missing and +spurious links of the validation set are guaranteed not to appear +in the training set. The observed graph Go used to generate the +augmented graphs is defined as test set Dtest. +3.2. Graph Convolutional Networks +GCNs are a class of neural networks designed to general- +ize traditional convolution operator for non-euclidean graph- +structured data. In essence, GCNs aim to learn new feature rep- +resentations of nodes in graphs by exploiting their structural in- +formation. Let adjacency matrix A ∈ {0, 1}N×N denote the struc- +tural information of the graph G, and X ∈ RN×F denote the fea- +ture matrix of all graph nodes. Mathematically, using the output +of the l-th layer as the input for the next layer, each neural net- +work layer can be formulated as a nonlinear function: +H(l+1) = f(H(l), A) +(5) +where H(l) corresponds to the feature matrix of the l-th layer, and +H(0) = X is the input feature matrix of the first layer. Specific +GCNs models differ only in the manner in which the nonlinear +function f(·) is instantiated. A simple example of f(·) is as fol- +lows: +f(H(l), A) = σ(AH(l)W(l)) +(6) +where σ(·) denotes a nonlinear activation function, such as +a Rectified Linear Unit (ReLU), and W(l) denotes a trainable +weight matrix for the l-th layer. With this propagation rule, the +neighbour’s features are aggregated to represent each node at +every layer, and the features become increasingly abstract by +stacking layers on top of each other. However, there exist two +limitations: the propagation rule simply aggregates the features +Table 1: Notations and meanings. +Notations +Descriptions +G +Original graph +Go +Observed graph +A +Adjacency matrix of graph +Em +Missing links +Es +Spurious links +D +Dataset that contains augmented graphs +H(l) +Feature matrix of l-th neural network layer +W(l) +Trainable weight matrix for the l-th layer +|| · ||0 +ℓ0−norm +|| · ||2,1 +ℓ2,1−norm +of neighboring nodes but not the node itself, and the multiplica- +tion with A expected to change the scale of the feature vectors. +That is, the nodes with a high degree will have a larger value, +and the nodes with a low degree may have smaller values. To +address the problems, a new propagation function, f(·), is pre- +sented as follows: +f(H(l), A) = σ( ˆD− 1 +2 ˆA ˆD− 1 +2 H(l)W(l)) +(7) +where ˆA is obtained by adding an identity matrix I to the adja- +cency matrix ˆA = A + I, ˆD denotes the diagonal node degree +matrix of ˆA, and ˆD− 1 +2 ˆA ˆD− 1 +2 denotes symmetric normalization. +3.3. Low-rank and Sparse Modeling +Traditionally, Principal component analysis (PCA) was pro- +posed to determine a low-dimensional representation of data +while retaining as much information as possible. However, the +PCA is particularly effective when dealing with Gaussian noise, +which is independent and identically distributed with respect to +the original data. Hence, the Robust Principal Component Anal- +ysis (RPCA) [9] has been proposed to eliminate the effect of +erratic noise (outliers). PCA and RPCA methods implicitly as- +sume that the underlying data structure is a single low-rank sub- +space; however, real-world data may be drawn from a union of +multiple subspaces, and therefore, modeling may be inaccurate. +To this end, Low-Rank Representation (LRR) [28] has been pro- +posed. +Considering the correlation between the connectivity patterns +of nodes in real-world graphs, the adjacency matrix of the graphs +should be low-rank. In other words, the rows or columns of the +adjacency matrix must not be linearly independent. Thus, as- +suming that hidden non-zero entries representing missing links +can be recovered according to the adjacency matrix, [37] pro- +posed an RPCA-based link prediction method, which is formu- +lated as the following optimization problem: +min +X∗,E rank(X∗) + γ||E||0 s.t., A = X∗ + E +(8) +where rank(X∗) denotes the rank of matrix X∗, || · ||0 is the +ℓ0−norm, and γ denots the balancing parameter. The method +searches for X∗ with a low rank as low as possible and E as +sparse as possible from A. Moreover, by representing a net- +work structure with as few representative subgraphs as possible, +[53] proposed an LRR-based link prediction method, wherein +networks could be modeled via a low-rank and sparse represen- +4 + +Figure 2: Demonstration of our link prediction method, GraphLP. (a) Link prediction method, GraphLP. The original graph is per- +turbed using a random mapping mechanism to obtain the observed graph; after this, the observed graph is further perturbed to generate +augmented graphs. These augmented graphs are fed into GraphLP to learn the model using the observed graph as the label. Subse- +quently, the learned model is used to infer the original graph based on the observed graph. (b) Self-representation-based collaborative +inference. Based on the structural regularity of graphs, the original graph can be reconstructed by utilizing the correlation between +subgraph patterns. (c) Example of high-order connectivity. In addition to the 1-hop neighborhood, multi-hop connectivity influences +the existence of links. The right graph represents the two-hop connectivity of the graph on the left, and the red dotted lines in the left +graph provide an example of the two-hop connectivity path of node 2. +tation, as follows: +min +Z,E rank(Z) + α||Z||0 + β||E||2,1 s.t., A = AZ + E +(9) +where Z denotes the representation matrix reflecting the organi- +zation principle of the network, and || · ||2,1 is the ℓ2,1−norm. +The notations used in this study are listed in Table 1. +4. The Proposed Method +This section presents the proposed link prediction method, +GraphLP. As depicted in Figure 2, the framework of GraphLP +consists of three main components: +• Collaborative inference operation. There exist certain sim- +ilarities between the connection patterns of individuals in +a complex system such that the perturbed structure of real- +world graphs can be recovered globally based on the corre- +lation between subgraph patterns (Section 4.1). +• High-order connectivity computation. The existence of a +link between any two target nodes is intended to be primar- +ily determined by the connectivity degree between nodes, +i.e., the number of paths and their length. Thus, the like- +lihood of a link can be estimated locally by computing the +connectivity (Section 4.2). +• Pattern fusion operation. In addition to the first-order adja- +cency matrix, the connection patterns of nodes in the high- +order adjacency matrix are also considered to be correlated, +and the high-order connectivity can be reconstructed based +on the collaborative inference. Thus, the graph topology +can be estimated by fusing the k-order (k ≥ 1) adjacency +matrix (Section 4.3). +4.1. Collaborative Inference Operation +[32] suggested that link formation in real-world graphs is usu- +ally driven by both regular and irregular factors, and the for- +mer can be explained based on the mixture of multiple mech- +anisms, such as homophily, triadic closure, preferential attach- +ment. Meanwhile, assuming that high-dimensional data are a +mixture of simple data and are drawn from a union of multiple +low-dimensional linear subspaces, the LRR has been proposed +to represent the data A = [a1, a2, ..., aN] as a linear combination +of the basis in a ”dictionary” D = [d1, d2, ..., dM]: +min +Z rank(Z) s.t., A = DZ, +(10) +5 + +(a) Link Prediction Method GraphLP +Target Graph for +A* = AZ* +(D 2AD 2H()W() +Original Graph Observed Graph +Model Testing +Collaborative +Collaborative +Connectivity +Connectivity +Connectivity +High-order +Inference +Inference +Inference +Concat +Augmented Graph +Target Graph for + Process of Model Training + Flatten of Adjacency Matrix +Model Training +-→ Process of Link Prediction +(c) High-order Connectivity +(b) Self-representation based Collaborative Inference +2-order Connectivity Path +A +E +A +* +Z ++Thus, the optimal representation matrix Z∗ uncovers the under- +lying subspaces in the data. By using each subspace to model +a homogeneous subset of the data, multiple subspaces in LRR +can capture heterogeneous structures within the data. There- +fore, considering the above ideas, the regular structure of real- +world graphs can be described appropriately by the LRR model, +wherein the generation mechanisms of graph organization essen- +tially corresponds to subspaces and the low rankness constraint +captures the global correlation in graphs. +Meanwhile, based +on the generation mechanisms of graph organization, individual +nodes may have similar connection patterns, and substructures +that follow the same generation mechanism can be represented +by each other, as depicted in Figure 2(b). Therefore, by using +the adjacency matrix A as the dictionary, the real-world graph +can be represented by itself, as follows: +min +Z rank(Z) s.t., A = AZ. +(11) +In addition to their regular structure, real-world graphs also +contain irregular components. Thus, we let matrix E denote such +irregular connections; then, the proposed self-representation +model can be modified as A = AZ + E. According to the LRR, +data are considered to be ”sample specific”, and the ℓ21−norm +is adopted to constrain the matrix E, i.e., ||E||2,1. However, al- +though the proposed method can be used to model real-world +graphs, the low-rank model and ℓ21−norm constraints are usu- +ally solved using Alternating direction method (ADM), which +requires a large number of iterations and has high complexity. +Therefore, a reasonable strategy is to relax the constraints with +Frobenius norm: +min +Z ||Z||2 +F + λ||A − AZ||2 +F s.t. , A = AZ + E +(12) +Let L = ||Z||2 +F + λ||A − AZ||2 +F denote the partial derivative of L +with respect to Z is ∂L/∂Z = 2Z + λ(−2ATA + 2ATAZ). By +setting ∂L/∂Z = 0, the optimal representation Z∗ can be obtained +as follows: +Z∗ = λ(λATA+I)−1ATA. +(13) +where I denotes the identity matrix. Thus, in the case that the +clean data is sufficient enough to represent the graph’s struc- +tural patterns and the irregular connections are properly char- +acterized, the structure perturbations can be inferred using AZ∗. +Hence, the collaborative inference operation is defined as fol- +lows: +CI(A) = λA(λATA+I)−1ATA +(14) +4.2. High-order Connectivity Computation +According to local similarity indices for link prediction, the +more the number of paths two nodes possess, the greater the +similarity between them. Specifically, two nodes with a high +mutual connectivity are more likely to generate a link between +them. Thus, n-hop-based (n ≥ 2) paths must be explored to char- +acterize the local structural features for link prediction. Using a +deep learning framework, the n-hop computation can be decom- +posed into two-hop operations on each neural layer. Hence, a +high-order connectivity computation calculates the two hop con- +nectivity of graph nodes in each layer, and the mutual connec- +tivity of two nodes can be estimated by stacking the high-order +connectivity computation mechanism. Assuming that the integer +powers of the adjacency matrix characterizes the mutual connec- +tivity of graph nodes, that is, [An]ij denotes the number of paths +Figure 3: Illustration of high-order connectivity computation. +with length n connecting nodes i and j, the high-order connectiv- +ity computation in each neural layer can be defined based on the +idea of the second power of adjacency matrix A. From the per- +spective of graph convolution networks, high-order connectivity +computation can be defined as +HCCA(A) = ˆD− 1 +2 ˆA ˆD− 1 +2 CI(A), +(15) +where the weighted adjacency matrix generated by the proposed +collaborative inference operation is viewed as the features of +graph nodes. Figure 3 illustrates a high-order connectivity com- +putation. As presented in Equation (15), the global and local +structural features can be captured for link prediction at the level +of individual nodes and edges. Thus, the nonlinear propagation +function can be defined as follows: +H(l+1) = ˆD− 1 +2 ˆA ˆD− 1 +2 CI(H(l))W(l). +(16) +Thus, the hierarchical structure of real-world graphs can be char- +acterized by executing the nonlinear propagation function itera- +tively, in which HCCA(H(l)) = ˆD− 1 +2 ˆA ˆD− 1 +2 CI(H(l)) represents +the high-order connectivity of graph nodes, as depicted in Figure +2(c), and CI(H(l+1)) denotes the collaborative inference. +4.3. Pattern Fusion Operation +To estimate the likelihood of potential links, the output of the +(l − 1)-th layer, i.e., H(l), is fed as the input of the l-th layer. +Based on CI(H(l)) and HCCA(H(l)), the shallow layers extract +the low-order global and local structure features, while the deep +layers extract the high-order global and local structure features. +Meanwhile, the effective range that the local structure features +drawn from increases as the model depth increases. Therefore, +the structure features in different range at various order, i.e., +HCCA(H(l)) and CI(H(l)), 0 ≤ l ≤ L, all contribute to the +inference of potential links, although the exact extent of their +contribution depends on the graph data. +To overcome the issues mentioned above, in addition to being +used as the inputs of the next layer, the outputs of neural net- +work layers are mapped to skip a block of several layers based +on residual connections, as illustrated in Figure 2(a). Next, all +outputs are concatenated and used as the input of a two-layer +Multi-layer Perceptron (MLP), which is defined as: +O= MLP(concat(CI(H(l)), HCCA(H(l)))),0 ≤ l ≤ L. +(17) +where O is a vector containing the probabilities of links between +all possible node pairs, and missing and spurious links can be +inferred based on it. +6 + +Central node +1-hop neighbor nodes +2-hop neighbor nodes +Node features representing link +weights to 2-hop neighbors +Computing connectivity of +central node to 2-hop neighbors +Adjacency relations of central node +Weighted links produced by C(A)4.4. Model Training +To train the proposed model, augmented graphs generated +based on the observed graph are used as training data, and the +adjacency matrix of the observed graph is flatten as its labels Y, +where Yi∗N+j denotes the existence of the link between nodes i +and j. Correspondingly, O represents the prediction results ob- +tained by the proposed model for all possible links. Here, the +Binary Cross-Entropy (BCE) is used as the loss function: +L = − 1 +N2 +N2 +� +i=1 +Yi log(Oi) + (1 − Yi) log(1 − Oi). +(18) +The learned model is then deployed on the observed graph to +reconstruct the original graph. The training process of GraphLP +is outlined in Algorithm 1. +Algorithm 1 Training Process of GraphLP +Input: Training set Dtrain, validation set Dval, and test set Dtest, +number of neural network layers L. +Output: The well-trained model GraphLP. +1: while not convergence do +2: +for 0 ≤ l ≤ L do +3: +Conduct collaborative inference operation using (14); +4: +Compute high-order connectivity using (16); +5: +end for +6: +Fuse the outputs based on MLP using (17); +7: +Update the model by minimizing the loss function (18); +8: end while +4.5. Model Analysis +(1) Generalized local similarity indices. The high-order con- +nectivity computation HCCA(H(l)) in every neural network +layer is essentially the second power of the adjacency matrix, +and it obtains the connectivity of node pairs within two-hop +neighborhood. As the model depth increases, the connectivity +of node pairs in a wider range is considered. Thus, GraphLP can +degenerate to S i j = A2 + αA3 + βA4 + · · · when collaborative +inference and deep learning mechanism are abolished. +(2) Connection to WalkPool. WalkPool [36] first generates +node representations based on GNN and encodes them into edge +weights of the extracted enclosing subgraphs; following this, it +uses the edge weights to compute the transition probabilities of +random walk. +Next, the method calculates a list of features +based on the transition probabilities to classify the subgraphs. +However, for an enclosing subgraph G = (V, E), its variants +G+ = (V, E ∪ {i, j}) and G− = (V, E\{i, j}) are used as positive +and negative samples, respectively. In essence, this method dis- +criminates only those subgraphs that differ by a single edge and +is not suitable for practical link prediction scenarios. In contrast, +GraphLP can predict any potential links based on graph structure +features. +(3) Connection to LFLP. The LFLP [53] constructs an ad- +jacency matrix based on a self-representation model and then +combines it with the observed network to identify missing and +spurious links. The collaborative inference operation CI(H(l)) +of our work is similar to that in the LFLP with respect to model- +ing the global structure of graphs; however, the difference is that +only low-order global structural features are considered in LFLP, +whereas multi-order global and local structural features are char- +acterized based on the deep-learning framework in GraphLP. +5. Experiments +Further, extensive experiments are conducted on real-world +graphs to evaluate the performance of the proposed method +GraphLP: (1) Compare GraphLP with state-of-the-art meth- +ods; (2) Compare GraphLP with traditional baseline methods; +(3) Model architecture analysis; (4) Model sensitivity analysis. +Here, Area Under Curve (AUC) and Average Precision (AP) are +adopted to evaluate the performance of the methods. Further- +more, Precision is used to verify the superiority of GraphLP over +traditional link prediction methods. Based on the link prediction +results O, the scores are sorted in descending and ascending or- +ders, and following this, their top-L links are taken as the pre- +dicted missing and spurious links. Note that Precision is defined +by calculating the ratio of accurately discovered links to the total +number of links in the probe set: +Precision = T/R +(19) +where T is the number of accurately identified links, and R is +the total number of links in the probe set. +5.1. Experimental Settings +5.1.1. Experimental Datasets +Herein, seven widely used graph datasets are used for link +prediction. (1) USAir [40]. This is the transportation network +of the United States, including 332 airports as nodes and 2,126 +airlines as edges, connecting the United States worldwide. The +average node degree is 12.81. (2) C.ele [49]. This is a neu- +ral network of C. elegans, with 297 neurons representing nodes +and 2,148 synaptic connections representing edges. The average +node degree is 14.46. (3) PB [1]. This dataset is a network of +hyperlinks between weblogs on US political blogs, with 1,222 +blogs on US politics as nodes and 16,714 hyperlinks between +blogs as edges. The average node degree is 27.36. (4) NS [33]. +This is an undirected co-authorship network with 1,589 nodes +and 2,742 edges, where the nodes denote the scientists engaged +in network science research, and the edges denote two scientists +have co-authored a publication. The average node degree is 3.45. +(5) Yeast [47]. This represents a protein-protein interaction net- +work formed in yeast with 2,375 proteins as nodes and 11,693 +protein-protein interactions as edges. The average node degree +is 9.85. (6) E.coli [58]. This is a pairwise reaction network of +metabolites with 1,805 nodes and 14,660 edges. The average +node degree is 12.55. (7) Router [43]. It is a snapshot of the In- +ternet structure at the level of autonomous systems, with 5,022 +nodes and 6,258 edges, in which the nodes represent routers and +the edges represent the data transmission between routers. The +average node degree is 2.49. The properties of the datasets are +listed in Table 2. +To extensively validate the performance of the proposed +method, 90% and 50% of the links of the original graph are se- +lected randomly to first construct the observed graphs. There- +after, based on the observed graph Go, 10% nonexisting links +are add randomly as spurious links, and 10% existing links are +removed randomly as missing links, denoted as Edel and Eadd +respectively, to generate the augmented graph set D = {Gi|i = +1, ..., t}. Following this, 90% and 10% graphs are randomly se- +lect from D as the training and validation set, respectively, and +the observed graph Go is used as the test set. +7 + +Table 2: Summary of the datasets. ACC is the average clustering +coefficient, and AD is the average node degree. +Dataset USAir +NS +PB +Yeast +C.ele +Router +E.coli +Node +332 +1589 +1222 +2375 +297 +5022 +1805 +Edges +2126 +2742 +16714 11693 +2148 +6258 +14660 +ACC +0.625 +0.638 +0.320 +0.306 +0.292 +0.012 +0.516 +AD +12.81 +3.45 +27.36 +9.85 +14.46 +2.49 +12.55 +5.1.2. Comparison Methods +The proposed method was compared with six state-of-the-art +deep learning-based link prediction methods, including: +(1) Weisfeiler-Lehman graph kernel (WLK) [42] is a fast fea- +ture extraction scheme based on the WL test for graph isomor- +phism, which maps the original graph to a graph sequence and +adds the pair-wise similarities between the graphs. +(2) Weifeiler-Lehmam Neural Machine (WLNM) [56] is a +subgraph classification-based link prediction method that lever- +age deep learning to automatically learn topological features +from enclosing subgraphs. +(3) Node2Vec [20] is a network embedding method that en- +codes proximity information into low-dimensional vectors. The +node features and low-dimensional vectors are then fed into the +MLP for link prediction. +(4) LINE [44] learns network embeddings that preserve the +first-order and second-order proximity, and the resulting low- +dimensional vectors are used for link prediction. +(5) SEAL [57] extracts the enclosing subgraphs of positive +and negative links and marks different roles of their nodes. The +method then trains a GNN based on the node information matrix +to classify subgraphs for link prediction. +(6) WalkPool (WP) [36] is a subgraph classification-based +link prediction method. It encodes node feature and graph topol- +ogy into the transition probabilities of random walk, and follow- +ing this, a list of features is computed to classify subgraphs. +5.1.3. Parameter Settings +GraphLP is implemented on a Pytorch platform with a +NVIDIA GeForce RTX GPU and optimized using Adam op- +timizer. All models are implemented using Python 3.6. The +learning rate is set to 0.0012 for the NS dataset and 0.0005 for +the other graphs. For all the datasets, the weight decay is set to +0.0. The number of epochs on the E.coli and Yeast dataset is +300, whereas it was 200 on the other datasets. Dropout is ap- +plied to the MLP, and the dropout rate is set to 0.5 on Router +and 0.2 on the others. The trade-off parameter λ is set to 0.13, +and the number of neural network layers in the GraphLP is set +to three. The detailed hyperparameter settings for the model are +listed in Table 3. +Table 3: Hyperparameter setting for the proposed method. +Name +Value +optimizer +Adam +loss function +binary cross entropy +learning rate +NS=0.0012, others=0.0005 +weight decay +0.0 +epochs +Ecoli, Yeast=300, others=200 +dropout +Router=0.5, others=0.2 +λ +0.13 +number of network layers +3 +5.2. Experimental Result +For 90% of the observed links, the results about the AUC +and AP with standard deviations are presented in Table 4 and +5, which indicate that GraphLP significantly outperforms other +state-of-the-art algorithms in terms of both AUC and AP, with +exception of the NS and Router datasets. The results demon- +strate that the learning of local and global graph structure en- +tirely characterizes the underlying structural patterns; thus, the +missing links and spurious links can be better identified. Ta- +ble 4 indicates that GraphLP significantly improves the AUC +on the PB, C.ele, and Router datasets, with approximately 4%, +7%, and 3% performance improvement, respectively, compared +to the WP algorithm. In addition, the proposed method still per- +forms better than other state-of-the-art methods on the USAir, +Yeast and NS datasets. Moreover, the results for the AP pre- +sented in Table 5 also indicate that GraphLP outperforms state- +of-the-art methods on most of datasets, and GraphLP achieves a +maximum performance enhancement of approximately 9% com- +pared to the best performing graph neural network method WP. +For 50% of the observed links, the results also demonstrate +that the proposed model achieves remarkable performance com- +pared to the methods, as described in Table 6 and 7. The results +illustrate that, as the amount of structure perturbation increases, +GraphLP can still appropriately learn the real graph structure, +thus recovering the original graph effectively. Therefore, the +values of the AUC and AP decreased to a lower extent. Fur- +thermore, by comparing Table 4 with Table 6 and Table 5 with +Table 7, we can infer that the AUC and AP values drop faster +for the other state-of-the-art methods than those for GraphLP, +which demonstrates that GraphLP can better capture the under- +lying structural patterns to demonstrate better performance. +5.3. Compared with Traditional Link Prediction Methods +To further verify the proposed method, the precision of +GraphLP and traditional link prediction methods are calculated +based on the following datasets: (1) Macaque [15], cortical net- +works of the macaque monkey; (2) Mangwet [5], the food web +in Mangrove Estuary during the wet season; (3) Jazz [19], a +collaboration network of jazz musicians; (4) Metabolic [17], a +metabolic network of C.elegans; (5) USAir, (6) C.ele, (7) E.coli +and (8) Yeast. Here, six representative traditional link prediction +methods are selected for comparison. +(1) The CN [29] metric is among the most widely used meth- +ods for link prediction problem. It assumes that two nodes will +be more easily connected if they share more common neighbors; +(2) The RA [61] metric is inspired by the physical processes +involved in resource allocation, which suppresses the contribu- +tion of high-degree common neighbors; +(3) The LP [61] index measures the structural similarity of +node pairs within three-hops; +(4) The Non-negative Matrix Factorization (NMF) [26] model +is used for structure prediction by learning the latent features of +real-world graphs; +(5) The Robust Principal Component Analysis (RPCA) [37] +represents a real-world graph based on the sparsity and low rank +property of its adjacency matrix and infers potential links based +on matrix completion. +(6) The LFLP [53] uses a self-representation model to re- +construct the original graph based on a few representative sub- +graphs. +The results of missing link prediction with respect to Preci- +sion are shown in Table 8. For each network, the bold number +8 + +Table 4: Prediction measured by AUC (90% observed links). Bold numbers are the best results of all methods. +Data +USAir +NS +PB +Yeast +C.ele +Router +E.coli +WLK +96.63 ± 0.73 +98.57 ± 0.51 +93.83 ± 0.59 +95.86 ± 0.54 +89.72 ± 1.67 +87.42 ± 2.08 +96.94 ± 0.29 +WLNM +95.95 ± 1.10 +98.61 ± 0.49 +93.49 ± 0.47 +95.62 ± 0.52 +86.18 ± 1.72 +94.41 ± 0.88 +97.21 ± 0.27 +Node2Vec +91.44 ± 1.78 +91.52 ± 1.28 +85.79 ± 0.78 +93.67 ± 0.46 +84.11 ± 1.27 +65.46 ± 0.86 +90.82 ± 1.49 +LINE +81.47 ± 10.71 +80.63 ± 1.90 +76.95 ± 2.76 +87.45 ± 3.33 +69.21 ± 3.14 +67.15 ± 2.10 +82.38 ± 2.19 +SEAL +97.09 ± 0.7 +98.85 ± 0.41 +95.01 ± 0.34 +97.91 ± 0.52 +90.30 ± 1.35 +96.38 ± 1.45 +97.64 ± 0.22 +WP +98.68 ± 0.48 +98.95 ± 0.41 +95.60 ± 0.37 +98.37 ± 0.25 +95.79 ± 1.09 +97.27 ± 0.28 +98.58 ± 0.19 +GraphLP +99.26 ± 1.01 +99.64 ± 0.98 +99.73 ± 0.25 +99.41 ± 0.15 +99.90 ± 0.14 +99.02 ± 0.19 +99.23 ± 0.23 +Table 5: Prediction measured by AP (90% observed links). Bold numbers are the best results of all methods. +Data +USAir +NS +PB +Yeast +C.ele +Router +E.coli +WLK +96.82 ± 0.84 +98.79 ± 0.40 +93.34 ± 0.89 +96.82 ± 0.35 +88.96 ± 2.06 +86.59 ± 2.23 +97.25 ± 0.42 +WLNM +95.95 ± 1.13 +98.81 ± 0.49 +92.69 ± 0.64 +96.40 ± 0.38 +85.08 ± 2.05 +93.53 ± 1.09 +97.50 ± 0.23 +Node2Vec +89.71 ± 2.97 +94.28 ± 0.91 +84.79 ± 1.03 +94.90 ± 0.38 +83.12 ± 1.90 +68.66 ± 1.49 +90.87 ± 1.48 +LINE +97.70 ± 11.76 +85.17 ± 1.65 +78.82 ± 2.71 +90.55 ± 2.39 +67.51 ± 2.72 +71.92 ± 1.53 +86.45 ± 1.82 +SEAL +97.13 ± 0.80 +99.06 ± 0.37 +94.55 ± 0.43 +98.33 ± 0.37 +89.48 ± 1.85 +96.23 ± 1.71 +98.03 ± 0.20 +WP +98.66 ± 0.55 +99.09 ± 0.29 +95.28 ± 0.41 +98.64 ± 0.28 +91.53 ± 1.33 +97.20 ± 0.38 +98.79 ± 0.21 +GraphLP +99.91 ± 1.03 +98.94 ± 0.96 +98.32 ± 1.43 +98.74 ± 0.16 +99.41 ± 0.42 +79.30 ± 0.19 +98.96 ± 0.19 +Table 6: Prediction measured by AUC ( 50% observed links). Bold numbers are the best results of all methods. +Data +USAir +NS +PB +Yeast +C.ele +Router +E.coli +WLK +91.93 ± 0.71 +87.27 ± 1.71 +92.54 ± 0.33 +91.15 ± 0.35 +83.29 ± 0.89 +71.25 ± 4.37 +92.38 ± 0.46 +WLNM +91.42 ± 0.95 +87.61 ± 1.63 +90.93 ± 0.23 +92.22 ± 0.32 +75.72 ± 1.33 +86.10 ± 0.52 +92.81 ± 0.30 +Node2Vec +84.63 ± 1.58 +80.29 ± 1.20 +79.29 ± 0.67 +90.18 ± 0.17 +75.53 ± 1.23 +62.45 ± 0.81 +84.73 ± 0.81 +LINE +72.51 ± 12.19 +65.96 ± 1.60 +75.53 ± 1.78 +79.44 ± 7.90 +59.46 ± 7.08 +62.43 ± 3.10 +74.50 ± 11.10 +SEAL +93.36 ± 0.67 +90.88 ± 1.18 +93.79 ± 0.25 +93.90 ± 0.54 +82.33 ± 2.31 +86.64 ± 1.58 +94.18 ± 0.41 +WP +95.50 ± 0.74 +90.97 ± 0.96 +94.57 ± 0.16 +95.00 ± 0.21 +87.62 ± 1.39 +88.13 ± 0.61 +95.33 ± 0.30 +GraphLP +98.97 ± 0.15 +97.08 ± 0.14 +98.19 ± 0.10 +98.74 ± 0.25 +97.96 ± 0.14 +98.10 ± 0.15 +98.05 ± 0.14 +Table 7: Prediction measured by AP ( 50% observed links). Bold numbers are the best results of all methods. +Data +USAir +NS +PB +Yeast +C.ele +Router +E.coli +WLK +93.34 ± 0.51 +89.97 ± 1.02 +92.34 ± 0.34 +93.55 ± 0.46 +83.20 ± 0.90 +75.49 ± 3.43 +94.51 ± 0.32 +WLNM +92.54 ± 0.81 +90.10 ± 1.11 +91.01 ± 0.20 +93.93 ± 0.20 +76.12 ± 1.08 +86.12 ± 0.68 +94.47 ± 0.21 +Node2Vec +82.51 ± 2.08 +86.01 ± 0.87 +77.21 ± 0.97 +92.45 ± 0.23 +72.91 ± 1.74 +66.77 ± 0.57 +85.41 ± 0.94 +LINE +71.75 ± 11.85 +71.53 ± 0.97 +78.72 ± 1.24 +83.06 ± 9.70 +60.71 ± 6.26 +64.87 ± 6.76 +75.98 ± 14.45 +SEAL +94.15 ± 0.54 +92.21 ± 0.97 +93.42 ± 0.19 +95.32 ± 0.38 +81.99 ± 2.18 +87.79 ± 1.71 +95.67 ± 0.24 +WP +95.87 ± 0.74 +92.33 ± 0.76 +94.22 ± 0.27 +96.15 ± 0.13 +86.25 ± 1.42 +89.17 ± 0.55 +96.36 ± 0.34 +GraphLP +97.96 ± 0.09 +93.08 ± 0.08 +96.27 ± 0.10 +97.27 ± 0.09 +95.89 ± 0.11 +79.23 ± 0.14 +96.48 ± 0.13 +Table 8: The precision (90% observed links) of missing links prediction. Bold numbers are the best results of all methods. +Data +Macaque +Mangwet +Jazz +Metabolic +USAir +C.ele +E.coli +Yeast +RA +0.5099 +0.1292 +0.5547 +0.2451 +0.4443 +0.093 +0.4857 +0.2609 +CN +0.5695 +0.125 +0.5292 +0.1127 +0.3764 +0.091 +0.4399 +0.1334 +LP +0.5483 +0.1319 +0.5109 +0.1275 +0.3821 +0.089 +0.4837 +0.1454 +NMF +0.7316 +0.4398 +0.5309 +0.2315 +0.3981 +0.1270 +0.5013 +0.3812 +RPCA +0.7421 +0.5421 +0.6138 +0.1842 +0.3596 +0.098 +0.3418 +0.5359 +LFLP +0.7605 +0.5572 +0.5956 +0.3241 +0.4545 +0.2010 +0.5007 +0.60 +GraphLP +0.7881 +0.7986 +0.8212 +0.7030 +0.8208 +0.8224 +0.7169 +0.7374 +9 + +Table 9: The precision (90% observed links) of spurious links prediction. Bold numbers are the best results of all methods. +Data +Macaque +Mangwet +Jazz +Metabolic +USAir +C.ele +E.coli +Yeast +RA +0.5490 +0.1380 +0.5410 +0.140 +0.2650 +0.2790 +0.4171 +0.1110 +CN +0.5710 +0.2880 +0.5690 +0.1670 +0.2480 +0.2458 +0.3222 +0.073 +LP +0.5939 +0.3280 +0.7016 +0.6911 +0.6271 +0.4780 +0.6089 +0.4585 +NMF +0.8090 +0.5660 +0.6510 +0.2430 +0.4820 +0.4333 +0.4662 +0.2330 +RPCA +0.810 +0.5180 +0.5920 +0.074 +0.443 +0.2609 +0.3795 +0.4250 +LFLP +0.818 +0.583 +0.663 +0.2210 +0.5970 +0.4390 +0.4246 +0.5680 +GraphLP +0.9073 +0.8750 +0.9197 +0.8812 +0.9057 +0.9533 +0.8452 +0.6210 +Figure 4: The topology visualization of Club dataset. The experiment performs 10% link perturbation, i.e. 10% spurious links are +added and 10% missing links are deleted. +Figure 5: The topology visualization of Club dataset. The experiment performs 20% link perturbation, i.e. 20% spurious links are +added and 20% missing links are deleted. +in the corresponding column indicates the highest accuracy. The +results presented in Table 8 demonstrate that the proposed model +GraphLP model performs the best among the methods. Further- +more, the link prediction accuracy of the proposed model is far +higher than that of the other methods, which can be at least three +times higher than that of the best-performing method. For spuri- +ous links prediction, the results measured by Precision are listed +in Table 9. For all networks, GraphLP performs the best among +the methods and is remarkably better than the second best algo- +rithm. The results presented in Table 8 and 9 demonstrate that +GraphLP has stronger ability to learn structural features, and can +recover the structure of the original network more accurately. +Based on Table 2, it can be observed that the Precision of our +proposed model performs best, despite the large differences be- +tween the ACC and AD across all the datasets; thus indicates +that the proposed model performs well for heterogeneous graph +structures. +5.4. Recovered Graph Visualization +To verify the effectiveness of the proposed model of miss- +ing and spurious links inference, the topology of the recovered +graphs in the model training process on Club dataset is visually +compared, as depicted in Figure 4 and 5. The top half depicts the +topology of the graphs, wherein the red links denote the missing +10 + +missing links +spurious links +missing links +spurious links + spurious links +missing links +spurious links +missing links +spurious links +missing links +19): 0.4048 +(6 , 18): 0.4008 +(1 . +,19): 0.2513 +(6 , 18): 0.5258 +(1 +(1 , 19): 0.2069 +(6 , 18): 0.6525 +19): 0.2090 +(6 , 18): 0.6648 +19): 0.2329 +(6 , 18): 0.6827 +(14, 22): 0.4129 +(14, 22): 0.3667 +(19, 25): 0.4614 +(14, 22): 0.2958 +(14, 22): 0.2439 +(14, 22): 0.1988 +(19, 25): 0.4028 +(19, 25): 0.6381 +(19, 25): 0.7258 +(19, 25): 0.7441 +(6 , 26): 0.4099 +(6 , 17): 0.4039 +(6 , 26): 0.2985 +(6 , 17): 0.4416 +(6 , 26): 0.1294 +(6 , 17): 0.6539 +(6 , 26): 0.1033 +(6 , 17): 0.7358 +(6, +, 26): 0.0972 +(6 , 17): 0.7634 +(25, 27): 0.1967 +8): 0.4286 +(25, 27): 0.0915 +(25, 27): 0.4147 +(6 ，8): 0.4008 +(25, 27): 0.1181 +8): 0.6909 +(25, 27): 0.1132 +(6 ，8): 0.7282 +8): 0.7768 +(6 +(6 +(6 +，9): 0.2884 +(18, 32): 0.3396 +9): 0.1618 +(3 +9): 0.4123 +(18, 32): 0.4053 +(3 +(3 +9): 0.1895 +(18, 32): 0.6068 +(3, +9): 0.1828 +(18, 32): 0.7402 +(3 +(18, 32): 0.7749 +,19): 0.2591 +,19): 0.4038 +, 14): 0.4049 +14): 0.4613 +(6 , 19): 0.1851 +(2 , 14): 0.6174 +(6 , 19): 0.1719 +,14): 0.7085 +19): 0.1827 +(6 +(2. +(2 , 14): 0.7264 +(2 +7): 0.4149 +, 32): 0.4041 +, 17): 0.2157 +32): 0.3625 +(6 +17): 0.1525 +(6 , 32): 0.6537 +17): 0.1588 +(6 +32): 0.7956 +(3 +17): 0.1412 +(6 , 32): 0.8063 +(3 +(6 +3 +(3 , +(3 +(11, 18): 0.4092 +(1, 17): 0.4071 +(11, 18): 0.2697 +17): 0.3585 +(11, 18): 0.2444 +17): 0.4941 +(11, 18): 0.2197 +17): 0.6085 +(11, 18): 0.2203 +(1, +17): 0.5978 +25): 0.4102 +(13, 19): 0.3980 +, 25): 0.3579 +(13, 19): 0.2835 +(2 , 25): 0.2842 +3, 19): 0.3608 +(2 +(2 +, 25): 0.2526 +(13, 19): 0.4578 +(2 +25): 0.2606 +(13, 19): 0.5659 +, 32): 0.4138 +(18, 21): 0.4040 +(8 , 32): 0.1396 +(18, 21): 0.4487 +(8 +(8 , 32): 0.0680 +, 32): 0.0656 +(18, 21): 0.7626 +(8 +,32): 0.0697 +(18, 21): 0.7722 +(18, 21): 0.6938 +(8 +(11, 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+missing links performed 40 epochs +missing links performed 120 epochs +missing links performed 1 epochs +missing links performed 80 epochs +missing links performed 140 epochsnegative links + positive links + negative links +positive links + negative links +positive links +negative links +positive links +negative links +positive links +28): 0.5039 +(10, 25): 0.2858 +(5, +28): 0.5615 +(10, 25): 0.3655 +(10, 25): 0.1989 +(5. +28): 0.8291 +(10, 25): 0.1192 +(5. +28): 0.9795 +(10, 25): 0.0702 +(5. +28): 0.9960 +14): 0.5108 +32): 0.2786 +(0, +14): 0.6029 +32): 0.1386 +14): 0.7367 +32): 0.0366 +(0, +32): 0.4057 +(0, +(0, +(0, +(0, +(0, +(0, +14): 0.9393 +(0, +32): 0.0814 +(0, +14): 0.9787 +32): 0.3874 +32): 0.5822 +(5,32): 0.2812 +32): 0.6264 +(5, 32): 0.1954 +32): 0.7374 +32): 0.1713 +32): 0.8605 +32): 0.1426 +32): 0.9424 +(5. +(1, +(1, +(5, +(1, +(5, +(1, +19): 0.2930 +(18, 24): 0.4254 +19): 0.2342 +(18, 24): 0.4081 +19): 0.1049 +(18, 24): 0.5512 +19): 0.0263 +(18, 24): 0.8087 +19): 0.0058 +(18, 24): 0.8801 +(0. +(0, +(0, +(0, +(18, 33): 0.1842 +17): 0.6631 +(18, 33): 0.0816 +17): 0.8405 +(18, 33): 0.0169 +17): 0.9706 +(18, 33): 0.0029 +17): 0.9902 +(18, 33): 0.3128 +(2, +17): 0.5981 +(2. +(2, +(2, +(2, +21): 0.2887 +23): 0.5964 +21): 0.2166 +(0, +(0, +21): 0.0993 +(0, +23): 0.8907 +23): 0.9841 +(0, 21): 0.0085 +(0, +(0, +23): 0.6629 +(0, 21): 0.0200 +(0, +(0, +23): 0.9933 +(12, 18): 0.3138 +16): 0.4231 +(12, 18): 0.2342 +16): 0.4246 +(12, 18): 0.1588 +16): 0.4794 +(12, 18): 0.0733 +16): 0.6510 +(9, +(9. +(9, +(9, +(12, 18): 0.0378 +(9, +16): 0.8059 +(a)The similarity of negative and +(b)The similarity of negative and +(c)The similarity of negative and +(d) The similarity of negative and +(c)The similarity of negative and +positive links performed 20 epochs + positive links performed 40 epochs +positive links performed 60 epochs + positive links performed 100 epochs + positive links performed 80 epochsYeast +USAir +C.ele +E.coli +PB +NS +Router +Training networks +0.4 +0.6 +0.8 +1.0 +AUC +Layer=1 +Layer=2 +Layer=3 +Layer=4 +(a) +Yeast +USAir +C.ele +E.coli +PB +NS +Router +Training networks +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +AP +Layer=1 +Layer=2 +Layer=3 +Layer=4 +(b) +Figure 6: The performance of link prediction under various model depth. (a) AUC of link prediction method GraphLP with different +layer number. (b) AP of link prediction method GraphLP with different layer number. +Yeast +USAir +C.ele +E.coli +PB +NS +Router +Training networks +0.980 +0.985 +0.990 +0.995 +1.000 +AUC +=0.08 +=0.10 +=0.13 +=0.15 +=0.20 +(a) +Yeast +USAir +C.ele +E.coli +PB +NS +Router +Training networks +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +AP +=0.08 +=0.10 +=0.13 +=0.15 +=0.20 +(b) +Figure 7: The performance of link prediction under different values of λ. (a) AUC of link prediction method GraphLP under different +values of λ. (b) AP of link prediction method GraphLP under different values of λ. +0 +50 +100 +150 +epoch +0.40 +0.45 +0.50 +0.55 +train_loss +0 +50 +100 +150 +epoch +0.25 +0.50 +0.75 +1.00 +val_auc +0 +50 +100 +150 +epoch +0.0 +0.5 +1.0 +val_ap +0 +50 +100 +150 +epoch +0.2 +0.4 +val_loss +(a) +0 +50 +100 +150 +epoch +0.4 +0.5 +0.6 +train_loss +0 +50 +100 +150 +epoch +0.96 +0.98 +1.00 +val_auc +0 +50 +100 +150 +epoch +0.85 +0.90 +0.95 +val_ap +0 +50 +100 +150 +epoch +0.2 +0.4 +0.6 +val_loss +(b) +Figure 8: Visualization of model convergence. (a) Convergence of USAir dataset. (b) Convergence of C.ele dataset. +links, the blue links denote the spurious links, and the gray links +denote the original links. The bottom half depicts the likelihood +scores of missing and spurious links. Based on the results, it +can be concluded that with an increase in epoch times, the like- +lihood scores of missing links increases gradually, and the like- +lihood scores of spurious links decline gradually. The widths +of the lines indicate the following process: when the number of +epochs reaches 100, the likelihood scores of missing links ap- +proach 1.0, and the likelihood scores of spurious links approach +0.0. This proves that the proposed GraphLP model can distin- +guish between missing links and spurious links and infer them +effectively. Moreover, to further prove the effectiveness of the +proposed model, the topology of the recovered graphs in the +model training process is visualized when 20% of the links of +Club are perturbed, as depicted in Figure 5. Compared to Fig- +ure 4, we determined that the likelihood of missing and spuri- +ous links is weakened with an increase in the structure pertur- +bation ratio. However, with an increase in the training epochs, +the model is still able to distinguish and infer the missing and +spurious links according to the structural patterns. For instance, +when the training epoch reaches 140, the likelihood scores of +missing links are greater than 0.5, and those of spurious links +11 + +are less than 0.3, indicating that the proposed model can still +predict missing and spurious links with high accuracy. +5.5. Impact of Model Depth +Next, the performance of GraphLP is explored at various +model depths. As depicted in Figure 6, the performance of the +model in terms of the AUC and AP trained with one-layer neu- +ral network is poor; however, its performance improves signif- +icantly with an increase in the number of layers. In particular, +when the model depth is equal to two, a significant performance +improvement is noted. The primary reason for this is that the +model with two-layers multi-order global and local structural +features is integrated adaptively based on the MLP component, +which considerably improves the performance of the model. +Subsequently, as the layer number increases, a slight improve- +ment in model performance is still noted. When the number of +layers is four, the accuracy of GraphLP declines significantly on +NS and fluctuates on other datasets. A possible reason for this is +that the model with four layers becomes more complex, thereby +requiring more training iterations or an appropriate learning rate +[14]. In general, the performance of the proposed model is opti- +mal when the depth is three, and a deep architecture is necessary. +5.6. Impact of Trade-off Parameter +To examine the sensitivity of the proposed model to the trade- +off parameter, the AUC and AP values of link prediction meth- +ods with different λ are presented in Figure 7. Based on the re- +sults, it can be concluded that the performance of the proposed +model is not sensitive to λ for most datasets. In Figure 7(a), for +the USAir and NS datsets, the AUC value varies significantly +under different λ, but the performance is still better than other +of the other algorithms. In Figure 7(b), the AP value remains +stable for different λ values, indicating that the proposed model +is insensitive to different λ. Overall, our proposed algorithm ex- +hibited satisfactory performance on most datasets with various +λ. +5.7. The Convergence Analysis +Generally, GraphLP converges to optimal values after approx- +imitely 200 epochs on most datasets. In particular, Figure 8 plots +the learning curves of GraphLP on the USAir and C.ele datasets, +including the training loss, validation AUC, validation AP, and +validation loss. The results indicate that the AUC and AP values +increase rapidly with the decrease in training loss and validation +loss, and these values converge to the optimal value when the +validation loss approaches a minimum value. Additionally, we +discover that validation loss is lower than training loss, and the +difference between them remains relatively stable. A possible +reason for this is that the dropout manipulation is only applied +to the training process. +6. Conclusion +This paper aims to reconstruct graph structure to improve the +performance of link prediction. In particular, unlike existing +subgraph-classification-based discriminative methods, this work +achieves the aforementioned objective by developing a genera- +tive GNN, namely GraphLP, which considered both global and +local structure features and hierarchical structural patterns. Con- +currently, a novel collaborative inference operation and high- +order connectivity computation mechanism are developed. We +also present an analysis about the relation between GraphLP and +other classical link prediction methods. Extensive experimental +results demonstrate the superiority of the proposed method over +other state-of-the-art models and traditional baseline methods. +This could be a fruitful avenue for future research aimed at ad- +dressing graph learning tasks. +Acknowledgment +This work was partially supported by the National Natu- +ral Science Foundation of China under Grant Nos. 62106030, +61802039, 62272066; Chongqing Municipal Postdoctoral Sci- +ence Foundation under Grant No. +cstc2021jcyj-bsh0176; +Chongqing Municipal Natural Science Foundation under Grant +No. cstc2020jcyj-msxmX0804; the Chongqing Research Pro- +gram of Basic Research and Frontier Technology under Grant +No. cstc2021jcyj-msxmX0530. +References +[1] Robert Ackland et al. Mapping the us political blogosphere: Are conserva- +tive bloggers more prominent? In BlogTalk Downunder 2005 Conference, +Sydney. BlogTalk Downunder 2005 Conference, Sydney, 2005. +[2] Lada A Adamic and Eytan Adar. Friends and neighbors on the web. Social +networks, 25(3):211–230, 2003. +[3] Yong-Yeol Ahn, James P Bagrow, and Sune Lehmann. Link communi- +ties reveal multiscale complexity in networks. nature, 466(7307):761–764, +2010. +[4] James Atwood and Don Towsley. +Diffusion-convolutional neural net- +works. In Advances in neural information processing systems, pages 1993– +2001, 2016. +[5] Daniel Baird, J Luczkovich, and Robert R Christian. Assessment of spa- +tial and temporal variability in ecosystem attributes of the st marks national +wildlife refuge, apalachee bay, florida. Estuarine, Coastal and Shelf Sci- +ence, 47(3):329–349, 1998. +[6] Baruch Barzel and Albert-L´aszl´o Barab´asi. Network link prediction by +global silencing of indirect correlations. Nature biotechnology, 31(8):720– +725, 2013. +[7] Austin R Benson, David F Gleich, and Jure Leskovec. Higher-order orga- +nization of complex networks. Science, 353(6295):163–166, 2016. +[8] Lei Cai, Jundong Li, Jie Wang, and Shuiwang Ji. Line graph neural net- +works for link prediction. IEEE Transactions on Pattern Analysis and Ma- +chine Intelligence, 2021. +[9] Emmanuel J Cand`es, Xiaodong Li, Yi Ma, and John Wright. Robust prin- +cipal component analysis? Journal of the ACM (JACM), 58(3):1–37, 2011. +[10] Shaosheng Cao, Wei Lu, and Qiongkai Xu. +Deep neural networks for +learning graph representations. In Proceedings of the AAAI Conference on +Artificial Intelligence, volume 30, 2016. +[11] Zhiqian Chen, Fanglan Chen, Lei Zhang, Taoran Ji, Kaiqun Fu, Liang +Zhao, Feng Chen, Lingfei Wu, Charu Aggarwal, and Chang-Tien Lu. +Bridging the gap between spatial and spectral domains: A survey on graph +neural networks. arXiv preprint arXiv:2002.11867, 2020. +[12] Ara Cho, Junha Shin, Sohyun Hwang, Chanyoung Kim, Hongseok Shim, +Hyojin Kim, Hanhae Kim, and Insuk Lee. Wormnet v3: a network-assisted +hypothesis-generating server for caenorhabditis elegans. Nucleic acids re- +search, 42(W1):76–82, 2014. +[13] Aaron Clauset, Cristopher Moore, and Mark EJ Newman. Hierarchical +structure and the prediction of missing links in networks. +Nature, 453 +(7191):98–101, 2008. +[14] Weilin Cong, Morteza Ramezani, and Mehrdad Mahdavi. On provable +benefits of depth in training graph convolutional networks. Advances in +Neural Information Processing Systems, 34:9936–9949, 2021. +[15] Lucianoda F Costa, Marcus Kaiser, and Claus C Hilgetag. Predicting the +connectivity of primate cortical networks from topological and spatial node +properties. BMC systems biology, 1(1):1–17, 2007. +[16] Micha¨el Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolu- +tional neural networks on graphs with fast localized spectral filtering. In +Proceedings of the 30th International Conference on Neural Information +Processing Systems, pages 3844–3852, 2016. +[17] Jordi Duch and Alex Arenas. Community detection in complex networks +using extremal optimization. Physical review E, 72(2):027104, 2005. +[18] Zhihong Fang, Shaolin Tan, Yaonan Wang, and Jinhu Lu. Elementary sub- +graph features for link prediction with neural networks. IEEE Transactions +on Knowledge and Data Engineering, 2021. +12 + +[19] Pablo M Gleiser and Leon Danon. Community structure in jazz. Advances +in complex systems, 6(04):565–573, 2003. +[20] Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for +networks. In Proceedings of the 22nd ACM SIGKDD international confer- +ence on Knowledge discovery and data mining, pages 855–864, 2016. +[21] Roger Guimer`a and Marta Sales-Pardo. Missing and spurious interactions +and the reconstruction of complex networks. Proceedings of the National +Academy of Sciences, 106(52):22073–22078, 2009. +[22] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation +learning on large graphs. Advances in neural information processing sys- +tems, 30, 2017. +[23] Glen Jeh and Jennifer Widom. Simrank: a measure of structural-context +similarity. In Proceedings of the eighth ACM SIGKDD international con- +ference on Knowledge discovery and data mining, pages 538–543, 2002. +[24] Meng Jiang, Peng Cui, Alex Beutel, Christos Faloutsos, and Shiqiang +Yang. Detecting suspicious following behavior in multimillion-node so- +cial networks. In Proceedings of the 23rd International Conference on +World Wide Web, pages 305–306, 2014. +[25] Leo Katz. A new status index derived from sociometric analysis. Psy- +chometrika, 18(1):39–43, 1953. +[26] Daniel Lee and H Sebastian Seung. Algorithms for non-negative matrix +factorization. +Advances in neural information processing systems, 13, +2000. +[27] Boning Li, Yingce Xia, Shufang Xie, Lijun Wu, and Tao Qin. Distance- +enhanced graph neural network for link prediction. In ICML 2021 Work- +shop on Computational Biology, 2021. +[28] Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, and +Yi Ma. Robust recovery of subspace structures by low-rank representa- +tion. IEEE transactions on pattern analysis and machine intelligence, 35 +(1):171–184, 2012. +[29] Francois Lorrain and Harrison C White. Structural equivalence of indi- +viduals in social networks. The Journal of mathematical sociology, 1(1): +49–80, 1971. +[30] Paul Louis, Shweta Ann Jacob, and Amirali Salehi-Abari. Sampling en- +closing subgraphs for link prediction. arXiv preprint arXiv:2206.12004, +2022. +[31] Linyuan L¨u, Mat´uˇs Medo, Chi Ho Yeung, Yi-Cheng Zhang, Zi-Ke Zhang, +and Tao Zhou. +Recommender systems. +Physics reports, 519(1):1–49, +2012. +[32] Linyuan L¨u, Liming Pan, Tao Zhou, Yi-Cheng Zhang, and H Eugene Stan- +ley. Toward link predictability of complex networks. Proceedings of the +National Academy of Sciences, 112(8):2325–2330, 2015. +[33] Mark EJ Newman. Finding community structure in networks using the +eigenvectors of matrices. Physical review E, 74(3):036104, 2006. +[34] Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning +convolutional neural networks for graphs. In International conference on +machine learning, pages 2014–2023. PMLR, 2016. +[35] Gergely Palla, Imre Der´enyi, Ill´es Farkas, and Tam´as Vicsek. Uncovering +the overlapping community structure of complex networks in nature and +society. nature, 435(7043):814–818, 2005. +[36] Liming Pan, Cheng Shi, and Ivan Dokmani´c. Neural link prediction with +walk pooling. In International Conference on Learning Representations, +pages 5171–5181, 2021. +[37] Ratha Pech, Dong Hao, Liming Pan, Hong Cheng, and Tao Zhou. Link pre- +diction via matrix completion. EPL (Europhysics Letters), 117(3):38002, +2017. +[38] Ratha Pech, Dong Hao, Yan-Li Lee, Ye Yuan, and Tao Zhou. Link pre- +diction via linear optimization. Physica A: Statistical Mechanics and its +Applications, 528:121319, 2019. +[39] Erzs´ebet Ravasz and Albert-L´aszl´o Barab´asi. Hierarchical organization in +complex networks. Physical review E, 67(2):026112, 2003. +[40] Ryan Rossi and Nesreen Ahmed. The network data repository with inter- +active graph analytics and visualization. In Twenty-ninth AAAI conference +on artificial intelligence, 2015. +[41] Marta Sales-Pardo, Roger Guimera, Andr´e A Moreira, and Lu´ıs A Nunes +Amaral. +Extracting the hierarchical organization of complex systems. +Proceedings of the National Academy of Sciences, 104(39):15224–15229, +2007. +[42] Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt +Mehlhorn, and Karsten M Borgwardt. Weisfeiler-lehman graph kernels. +Journal of Machine Learning Research, 12(9), 2011. +[43] Neil Spring, Ratul Mahajan, and David Wetherall. Measuring isp topolo- +gies with rocketfuel. ACM SIGCOMM Computer Communication Review, +32(4):133–145, 2002. +[44] Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu +Mei. Line: Large-scale information network embedding. In Proceedings +of the 24th international conference on world wide web, pages 1067–1077, +2015. +[45] Ala Trusina, Sergei Maslov, Petter Minnhagen, and Kim Sneppen. Hi- +erarchy measures in complex networks. Physical review letters, 92(17): +178702, 2004. +[46] Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, +Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint +arXiv:1710.10903, 2017. +[47] Christian Von Mering, Roland Krause, Berend Snel, Michael Cornell, +Stephen G Oliver, Stanley Fields, and Peer Bork. Comparative assess- +ment of large-scale data sets of protein–protein interactions. Nature, 417 +(6887):399–403, 2002. +[48] Xu-Wen Wang, Yize Chen, and Yang-Yu Liu. Link prediction through +deep generative model. iScience, 23(10):101626, 2020. +[49] Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small- +world’networks. nature, 393(6684):440–442, 1998. +[50] Max Welling and Thomas N Kipf. +Semi-supervised classification with +graph convolutional networks. In J. International Conference on Learning +Representations (ICLR 2017), 2016. +[51] Tao Wu, Yuxiao Guo, Leiting Chen, and Yanbing Liu. Integrated struc- +ture investigation in complex networks by label propagation. Physica A: +Statistical Mechanics and its Applications, 448:68–80, 2016. +[52] Tao Wu, Hongyu Ma, Chao Wang, Shaojie Qiao, Liang Zhang, and Shui +Yu. Heterogeneous representation learning and matching for few-shot re- +lation prediction. Pattern Recognition, page 108830, 2022. +[53] Xingping Xian, Tao Wu, Shaojie Qiao, Xi-Zhao Wang, Wei Wang, and +Yanbing Liu. +Netsre: +Link predictability measuring and regulating. +Knowledge-Based Systems, 196:105800, 2020. +[54] Xingping Xian, Tao Wu, Shaojie Qiao, Wei Wang, Chao Wang, Yanbing +Liu, and Guangxia Xu. Deepec: Adversarial attacks against graph structure +prediction models. Neurocomputing, 437:168–185, 2021. +[55] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi +Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs +with jumping knowledge networks. In International conference on ma- +chine learning, pages 5453–5462. PMLR, 2018. +[56] Muhan Zhang and Yixin Chen. Weisfeiler-lehman neural machine for link +prediction. In Proceedings of the 23rd ACM SIGKDD international con- +ference on knowledge discovery and data mining, pages 575–583, 2017. +[57] Muhan Zhang and Yixin Chen. Link prediction based on graph neural +networks. In Proceedings of the 32nd International Conference on Neural +Information Processing Systems, pages 5171–5181, 2018. +[58] Muhan Zhang, Zhicheng Cui, Shali Jiang, and Yixin Chen. Beyond link +prediction: Predicting hyperlinks in adjacency space. In Proceedings of +the AAAI Conference on Artificial Intelligence, volume 32, 2018. +[59] Jiajun Zhou, Jie Shen, and Qi Xuan. Data augmentation for graph clas- +sification. In Proceedings of the 29th ACM International Conference on +Information & Knowledge Management, pages 2341–2344, 2020. +[60] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, +Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun. Graph +neural networks: A review of methods and applications. AI Open, 1:57– +81, 2020. +[61] Tao Zhou, Linyuan L¨u, and Yi-Cheng Zhang. Predicting missing links +via local information. The European Physical Journal B, 71(4):623–630, +2009. +13 + diff --git a/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/load_file.txt b/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8631b47221d87ed1a92571579d54d84cabb5efbe --- /dev/null +++ b/ZNAyT4oBgHgl3EQfWveE/content/tmp_files/load_file.txt @@ -0,0 +1,1575 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf,len=1574 +page_content='Generative Graph Neural Networks for Link Prediction Xingping Xiana, Tao Wua,∗, Xiaoke Mab, Shaojie Qiaoc, Yabin Shaod, Chao Wange, Lin Yuana, Yu Wua aSchool of Cybersecurity and Information Law, Chongqing University of Posts and Telecommunications, Chongqing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' bSchool of Computer Science and Technology, XiDian University, XiAn, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' cSchool of Software Engineering, Chengdu University of Information Technology, Chengdu, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' dSchool of Science, Chongqing University of Posts and Telecommunications, Chongqing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' eSchool of Computer and Information Science, Chongqing Normal University, Chongqing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Abstract Inferring missing links or detecting spurious ones based on observed graphs, known as link prediction, is a long-standing challenge in graph data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' With the recent advances in deep learning, graph neural networks have been used for link prediction and have achieved state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Nevertheless, existing methods developed for this purpose are typically discriminative, computing features of local subgraphs around two neighboring nodes and predicting potential links between them from the perspec- tive of subgraph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In this formalism, the selection of enclosing subgraphs and heuristic structural features for subgraph classification significantly affects the performance of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To overcome this limitation, this paper proposes a novel and rad- ically different link prediction algorithm based on the network reconstruction theory, called GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Instead of sampling positive and negative links and heuristically computing the features of their enclosing subgraphs, GraphLP utilizes the feature learning abil- ity of deep-learning models to automatically extract the structural patterns of graphs for link prediction under the assumption that real-world graphs are not locally isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, GraphLP explores high-order connectivity patterns to utilize the hierarchical organizational structures of graphs for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Our experimental results on all common benchmark datasets from different applications demonstrate that the proposed method consistently outperforms other state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Unlike the discriminative neural network models used for link prediction, GraphLP is generative, which provides a new paradigm for neural-network-based link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='com/star4455/GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Keywords: Graph Machine Learning, Graph Neural Networks, Link Prediction, Structural Patterns, Network Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Introduction Graphs provide an elegant representation for characterizing entities and their interrelations in complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Given that real-world graphs can usually only be partially observed and are often noisy, link prediction aimed at inferring missing and spurious links based on observed graphs is a paradigmatic and fundamental problem across many scientific domains, including knowledge graph completion [52], experimental design in bio- logical networks [6], fake account detection in online social net- works [24], and product recommendation on e-commerce web- sites [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To address the link prediction problem, numerous heuristic methods have been proposed, including local indices such as Common Neighbors (CN) [2], and Resource Allocation (RA) [61], global indices such as Katz [25], and SimRank [23], and quasi-local indices such as the Local Path Index (LP) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' How- ever, heuristic methods have a strong assumption on when two nodes are likely to be linked in real-world graphs and lack uni- versal applicability to diverse areas [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Subsequently, statistical learning-based algorithms have been proposed to obtain ground- breaking results, such as maximum likelihood-based hierarchi- cal structure model [13], stochastic block model [21], matrix factorization-based link prediction method [37], Linear Opti- mization (LO) link prediction method [38], and Low Frobenius norm-based Link Prediction (LFLP) method [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' With the pro- posal of network representation learning, various network em- ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Email addresses: xxp0213@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='com (Xingping Xian), wutaoadeny@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='com (Tao Wu) bedding algorithms have been put forth so that the likelihood of a non-observed links can be estimated based on the proximity of nodes in low-dimensional vector space, including LINE [44], Node2Vec [20], and DNGR [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Recently, driven by the dramatic advances in deep learning techniques, neural networks have gradually been used to solve the link prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [56] trained a fully-connected neu- ral network on the enclosing subgraphs of target links for link prediction, wherein a Weisfeiler-Lehman (WL) algorithm-based graph labeling mechanism was proposed to encode subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on the enclosing subgraphs extracted around links, [57] trained a Graph Neural Network (GNN) for link prediction to achieve a performance comparable to that of heuristic meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Along this line of research, [36] encoded subgraphs into random-walk transition probabilities and then computed fea- tures using these probabilities to classify positive and negative links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Although these subgraph classification-based methods have achieved state-of-the-art link prediction performance, the prediction results are found to be considerably affected by the extraction process of the k-hop enclosing subgraphs and the graph structure features for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For example, in representa- tion learning on graphs [55], the range of enclosing subgraphs strongly depends on the graph structure, and the effective range should be different for subgraphs with varying properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Typically, from the perspective of subgraph classification, link prediction methods treat subgraphs in real-world graphs inde- pendently and equivalently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' That is, the global structural in- formation of real-world graphs is totally neglected during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, extensive empirical analyses indicate that real-world graphs are not locally isolated but globally relevant Preprint submitted to Journal of LATEX Templates January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='00169v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='SI] 31 Dec 2022 Figure 1: An illustrative example depicting the global and high-order organizations of real-world graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (a) Gene network for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' elegans [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) Representative hierarchical star-like structure [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (c) Representative hierarchical modular organization [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) and (c) depict the representative structural patterns of real-world graphs such as (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [35, 51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' here, nodes and edges naturally portray different struc- tural roles and contribute differently to the global organization of real-world graphs [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, subgraph classification- based link prediction methods assume that real-world graphs ex- hibit low-order connectivity patterns and can be captured at the level of individual nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, empirical studies have discovered that real-world graphs exhibit high-order orga- nizations at the level of small subgraphs, which are recursively grouped into a hierarchical structure [7, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' An illustrative ex- ample of the global and high-order organization in real-world graphs is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hence, two challenges need to be addressed for link prediction: (i) how to learn good representa- tion preserving both local and global graph structural features?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' and (ii) how to characterize and utilize hierarchical structure pat- terns?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To address these challenges, instead of predicting poten- tial links through subgraph classification, this study designs a novel generative and multi-order GNN for link prediction, called GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Evidently, real-world graphs share some global prop- erties, such as low-rank and sparsity, that can be used to pro- vide guidance for graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hence, motivated by the network reconstruction theory [21], GraphLP defines a self- representation model-based collaborative inference operation to refine the observed graphs globally, which assumes that the orig- inal graph can be reconstructed utilizing the correlation between subgraph patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Assuming that the paths between a pair of nodes provide evidence for the existence of potential links, GraphLP extracts the local structural information via a high- order connectivity operation on the observed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, ev- ery neural network layer obtains the connectivity of node pairs within two-hop neighborhood, and a neural network with multi- ple connectivity layers captures the degree of connectivity be- tween node pairs with various path lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Meanwhile, the weighted adjacency matrices generated by the connectivity op- eration in every neural network layer reflect the multi-order con- nectivity pattern in the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Further, the hierarchical organi- zational structure of real-world graphs is explored by applying a collaborative inference operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The contributions of this study can be summarized as follows: Generative framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Rather than subgraph classi- fication based discriminative schemes, a novel network reconstruction-based generative GNN is proposed for link prediction, which provides a new paradigm for the applica- tion of neural networks in link prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' End-to-end learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Instead of designing heuristic graph structural features for subgraph representation, local and global structural patterns are extracted and fused in an end- to-end fashion for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A novel collaborative inference operation and high-order connectivity computation mechanism are devel- oped to characterize the structural patterns in real-world graphs at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Extensive experiments on real-world datasets from different areas reveal that the proposed method, GraphLP, achieves promising performance and consistently outperforms other state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Paper Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The rest of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Section 2 discusses related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Section 3 presents the problem definitions and describes the preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Section 4 describes the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Section 5 presents the experi- mental results, and finally, Section 6 presents the conclusion and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Related Work GNNs and link prediction task have been extensively investi- gated in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A brief review of related studies is pro- vided in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Graph Neural Networks Owing to their potential in modeling the complex structures of non-Euclidean graphs, GNNs have achieved state-of-the-art performance on almost all graph-based tasks, such as node clas- sification, graph classification, link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on differ- ent theories and perspectives, a plethora of different GNNs have 2 (a) (b) (c) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='80been proposed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Generally, GNNs can be divided into two categories: spectral-based and spatial-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Of these, spectral-based GNNs are types of GNNs that design graph convolution operators in the spectral domain using Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The involved convolution operation is defined as fol- lows: f1 ∗ f2 = U[(UT f1) ⊙ (UT f2)], (1) where ⊙ denotes an element-wise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The spectral filter is defined as g = UT f1, and the node signal X can be processed as follows: Z = U[g(Λ) ⊙ (UTX)] = Ug(Λ)UTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) where U denotes a matrix of eigenvectors of the normalized Laplacian graph L = I − D− 1 2 AD− 1 2 = UΛUT [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Assum- ing that feature representation of node should be affected only by its k-hop neighborhood, [16] proposed a Chebyshev poly- nomial based k-localized convolution and developed a convolu- tional neural network, ChebNet, which eliminated the need to compute the eigenvectors of the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Subsequently, [50] simplified the Chebshev polynomial filter using its first-order approximation and proposed the popular spectral-based method called Graph Convolutional Networks (GCNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Notably, spatial- based GNNs define graph convolution operator based on graph topology wherein the feature vectors of node’s neighbors are ag- gregated via a permutation-invariant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Specifically, [22] proposed a GraphSAGE approach that sampled fixed size neigh- borhood nodes and used max pooling, mean pooling, and LSTM pooling scheme to aggregate neighbor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Considering the different weights of node’s neighbors, [46] proposed a Graph Attention Network (GAT) algorithm to calculate attention coeffi- cient and then aggregated the neighborhood information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Other related models include PATCHY-SAN [34], DCNN [4], and fur- ther details on GNNs can be found in the review [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Neural Networks based Link Prediction Following heuristic methods, matrix completion-based meth- ods and network embedding-based methods, neural networks have been gradually applied to link prediction problem and have achieved state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Specifically, [56] pro- posed a link prediction method called Weisfeiler-Lehman Neural Machine (WLNM), which labeled nodes using the Weisfeiler- Lehman algorithm and encoded subgraphs to construct a feed- forward neural network-based classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Next, from the perspective of subgraph classification, [57] proposed a novel GNN-based link prediction framework, SEAL, to learn subgraph structures and node features from local enclosing subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Along this line, to directly leverage the topology features of lo- cal subgraphs, [36] proposed a new random-walk-based pool- ing scheme, WalkPool, and built features for subgraph classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, [18] proposed a neural network-based link prediction method with only one-hop neighborhood informa- tion, which demonstrated almost equivalent performance to the WLNM and SEAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Instead of subgraph classification, [8] con- verted the original graph into a corresponding line graph and solved the node classification problem for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To perform link prediction for general directed or undirected com- plex networks, [48] represented the adjacency matrices of net- works as binary images and developed a generative adversarial networks (GANs)-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In addition, because existing GNN-based methods do not scale appropriately to large graphs, [30] extracted sparse enclosing subgraphs based on multiple ran- dom walks and presented a scalable link prediction solution, called ScaLed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To reduce the time required to determine the distances between two nodes, [27] defined an anchor-based dis- tance and proposed a new distance-enhanced GNN method for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Among all existing methods for link prediction, the work clos- est to the one condidered in this study is the GANs-based method [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, this method predicts potential links via image processing within the GANs framework, whereas the proposed method conducts link prediction via GNNs-based network re- construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Network Structure Analysis Real-world graphs, also known as complex networks, are ab- stract representation of complex systems and have been exten- sively studied in the field of network science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Consequently, numerous studies have revealed that complex networks exhibit rich and diverse connectivity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [32] augmented that the organization of real networks usually embodies both regularities and irregularities, where the former can be modeled and decides the extent to which the formation of a network can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Notably, link predictability reflects the structural regularities in real-world networks and denotes the inherent difficulty of link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [53] proposed a self-representation network model- based method, called NetSRE, for measuring and regulating link predictability of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [54] proposed a deep linear coding- based link prediction adversarial attack method by disturbing the underlying structural pattern of networks, which proved that links play global structural roles in network organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' More- over, [7] suggested that high-order connectivity patterns are es- sential for understanding the fundamental structures of networks and developed a framework that identified clusters of network motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [41] claimed that hierarchical structure plays an impor- tant role in complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To prove the existence of hierar- chical organization, an unsupervised method for extracting the hierarchical organization of complex networks was introduced and validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Although real-world graphs exhibit various structural pat- terns, most existing neural networks-based link prediction meth- ods simply assume that they are flattened and locally isolated, and these methods judge the existence of links explicitly based only on local enclosing subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' With the exception of local structural features, this study focuses on integrating global and hierarchical structural patterns into neural networks for link pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Problem Definition and Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Problem Definition Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Let G = (V, E) denote an undirected and un- weighted graph, where V = {v1, · · · , vN} denotes the set of nodes and E = {e1, · · · , eM} denotes the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The adjacency matrix of graph G is denoted as A ∈ {0, 1}N×N, where Ai j = 1 if nodes i and j are connected and Ai j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Each edge e can be represented as a node pair (u, v, ), where u, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Let N (u) denote the neighbors of node u, N (u) = {v|(u, v) ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Given an observed graph Go = (V, Eo) that corresponds to the original graph G = (V, E), link prediction aims to infer the presence or absence of an edge between a pair of target nodes based on Go, thereby generating a recovered graph G∗ to approximate the original graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In particular, the prediction problem involves identifying a function that generates 3 a likelihood score for a pair of nodes (u, v) � E to infer the miss- ing link (u, v), or to produce a likelihood score for an existing edge (u, v) ∈ E to identify spurious links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the link predic- tion problem can be formulated as suv = f(u, v, A|θ), where θ denotes the parameter of link prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In this work, Em and Es denote the identified missing and spurious links, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Note that data augmentation is a set of techniques that in- creases the amount and diversity of data by creating reasonable virtual data points from existing data, such that better machine learning models can be constructed based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' According to [59], this study considers graph data augmentation and adopts a random mapping mechanism to produce augmented graph set D based on the observed graph Go = (V, Eo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Specifically, the set of all possible edges in the graph Go is denoted as Ω, the ex- isting edge set is denoted as Eo, and the non-existing edge set is denoted as Enon = Ω − Eo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the candidate sets for random mapping are defined as follows: Ec del = Eo, Ec add = Enon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) Thereafter, samples are randomly produced from the candidate sets to obtain the edge sets Edel and Eadd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Finally, a new aug- mented graph is generated by modifying the graph Go based on Edel and Eadd: G′ = (V, (E ∪ Eadd)\\Edel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) Each input graph can be viewed as an instance for link pre- diction, owing to the generative learning scheme of the models considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the dataset containing a series of augmented graphs can be denoted as D = {Gi|i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', t} and split to yield disjoint training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' These can be denoted as Dtrain and Dval respectively, wherein the missing and spurious links of the validation set are guaranteed not to appear in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The observed graph Go used to generate the augmented graphs is defined as test set Dtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Graph Convolutional Networks GCNs are a class of neural networks designed to general- ize traditional convolution operator for non-euclidean graph- structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In essence, GCNs aim to learn new feature rep- resentations of nodes in graphs by exploiting their structural in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Let adjacency matrix A ∈ {0, 1}N×N denote the struc- tural information of the graph G, and X ∈ RN×F denote the fea- ture matrix of all graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Mathematically, using the output of the l-th layer as the input for the next layer, each neural net- work layer can be formulated as a nonlinear function: H(l+1) = f(H(l), A) (5) where H(l) corresponds to the feature matrix of the l-th layer, and H(0) = X is the input feature matrix of the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Specific GCNs models differ only in the manner in which the nonlinear function f(·) is instantiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A simple example of f(·) is as fol- lows: f(H(l), A) = σ(AH(l)W(l)) (6) where σ(·) denotes a nonlinear activation function, such as a Rectified Linear Unit (ReLU), and W(l) denotes a trainable weight matrix for the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' With this propagation rule, the neighbour’s features are aggregated to represent each node at every layer, and the features become increasingly abstract by stacking layers on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, there exist two limitations: the propagation rule simply aggregates the features Table 1: Notations and meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Notations Descriptions G Original graph Go Observed graph A Adjacency matrix of graph Em Missing links Es Spurious links D Dataset that contains augmented graphs H(l) Feature matrix of l-th neural network layer W(l) Trainable weight matrix for the l-th layer || · ||0 ℓ0−norm || · ||2,1 ℓ2,1−norm of neighboring nodes but not the node itself, and the multiplica- tion with A expected to change the scale of the feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' That is, the nodes with a high degree will have a larger value, and the nodes with a low degree may have smaller values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To address the problems, a new propagation function, f(·), is pre- sented as follows: f(H(l), A) = σ( ˆD− 1 2 ˆA ˆD− 1 2 H(l)W(l)) (7) where ˆA is obtained by adding an identity matrix I to the adja- cency matrix ˆA = A + I, ˆD denotes the diagonal node degree matrix of ˆA, and ˆD− 1 2 ˆA ˆD− 1 2 denotes symmetric normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Low-rank and Sparse Modeling Traditionally, Principal component analysis (PCA) was pro- posed to determine a low-dimensional representation of data while retaining as much information as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, the PCA is particularly effective when dealing with Gaussian noise, which is independent and identically distributed with respect to the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hence, the Robust Principal Component Anal- ysis (RPCA) [9] has been proposed to eliminate the effect of erratic noise (outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' PCA and RPCA methods implicitly as- sume that the underlying data structure is a single low-rank sub- space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' however, real-world data may be drawn from a union of multiple subspaces, and therefore, modeling may be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To this end, Low-Rank Representation (LRR) [28] has been pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Considering the correlation between the connectivity patterns of nodes in real-world graphs, the adjacency matrix of the graphs should be low-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In other words, the rows or columns of the adjacency matrix must not be linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, as- suming that hidden non-zero entries representing missing links can be recovered according to the adjacency matrix, [37] pro- posed an RPCA-based link prediction method, which is formu- lated as the following optimization problem: min X∗,E rank(X∗) + γ||E||0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', A = X∗ + E (8) where rank(X∗) denotes the rank of matrix X∗, || · ||0 is the ℓ0−norm, and γ denots the balancing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The method searches for X∗ with a low rank as low as possible and E as sparse as possible from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, by representing a net- work structure with as few representative subgraphs as possible, [53] proposed an LRR-based link prediction method, wherein networks could be modeled via a low-rank and sparse represen- 4 Figure 2: Demonstration of our link prediction method, GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (a) Link prediction method, GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The original graph is per- turbed using a random mapping mechanism to obtain the observed graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' after this, the observed graph is further perturbed to generate augmented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' These augmented graphs are fed into GraphLP to learn the model using the observed graph as the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Subse- quently, the learned model is used to infer the original graph based on the observed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) Self-representation-based collaborative inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on the structural regularity of graphs, the original graph can be reconstructed by utilizing the correlation between subgraph patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (c) Example of high-order connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In addition to the 1-hop neighborhood, multi-hop connectivity influences the existence of links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The right graph represents the two-hop connectivity of the graph on the left, and the red dotted lines in the left graph provide an example of the two-hop connectivity path of node 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' tation, as follows: min Z,E rank(Z) + α||Z||0 + β||E||2,1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', A = AZ + E (9) where Z denotes the representation matrix reflecting the organi- zation principle of the network, and || · ||2,1 is the ℓ2,1−norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The notations used in this study are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The Proposed Method This section presents the proposed link prediction method, GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' As depicted in Figure 2, the framework of GraphLP consists of three main components: Collaborative inference operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' There exist certain sim- ilarities between the connection patterns of individuals in a complex system such that the perturbed structure of real- world graphs can be recovered globally based on the corre- lation between subgraph patterns (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' High-order connectivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The existence of a link between any two target nodes is intended to be primar- ily determined by the connectivity degree between nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', the number of paths and their length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the like- lihood of a link can be estimated locally by computing the connectivity (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Pattern fusion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In addition to the first-order adja- cency matrix, the connection patterns of nodes in the high- order adjacency matrix are also considered to be correlated, and the high-order connectivity can be reconstructed based on the collaborative inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the graph topology can be estimated by fusing the k-order (k ≥ 1) adjacency matrix (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Collaborative Inference Operation [32] suggested that link formation in real-world graphs is usu- ally driven by both regular and irregular factors, and the for- mer can be explained based on the mixture of multiple mech- anisms, such as homophily, triadic closure, preferential attach- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Meanwhile, assuming that high-dimensional data are a mixture of simple data and are drawn from a union of multiple low-dimensional linear subspaces, the LRR has been proposed to represent the data A = [a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', aN] as a linear combination of the basis in a ”dictionary” D = [d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', dM]: min Z rank(Z) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A = DZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='(a) Link Prediction Method GraphLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Target Graph for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='A* = AZ* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='(D 2AD 2H()W() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Original Graph Observed Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Model Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Collaborative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Collaborative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='High-order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Concat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Augmented Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Target Graph for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Process of Model Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Flatten of Adjacency Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='Model Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='→ Process of Link Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='(c) High-order Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='(b) Self-representation based Collaborative Inference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2-order Connectivity Path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='A Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='+Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' the optimal representation matrix Z∗ uncovers the under- lying subspaces in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' By using each subspace to model a homogeneous subset of the data, multiple subspaces in LRR can capture heterogeneous structures within the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' There- fore, considering the above ideas, the regular structure of real- world graphs can be described appropriately by the LRR model, wherein the generation mechanisms of graph organization essen- tially corresponds to subspaces and the low rankness constraint captures the global correlation in graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Meanwhile, based on the generation mechanisms of graph organization, individual nodes may have similar connection patterns, and substructures that follow the same generation mechanism can be represented by each other, as depicted in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Therefore, by using the adjacency matrix A as the dictionary, the real-world graph can be represented by itself, as follows: min Z rank(Z) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', A = AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (11) In addition to their regular structure, real-world graphs also contain irregular components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, we let matrix E denote such irregular connections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' then, the proposed self-representation model can be modified as A = AZ + E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' According to the LRR, data are considered to be ”sample specific”, and the ℓ21−norm is adopted to constrain the matrix E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', ||E||2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, al- though the proposed method can be used to model real-world graphs, the low-rank model and ℓ21−norm constraints are usu- ally solved using Alternating direction method (ADM), which requires a large number of iterations and has high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Therefore, a reasonable strategy is to relax the constraints with Frobenius norm: min Z ||Z||2 F + λ||A − AZ||2 F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' , A = AZ + E (12) Let L = ||Z||2 F + λ||A − AZ||2 F denote the partial derivative of L with respect to Z is ∂L/∂Z = 2Z + λ(−2ATA + 2ATAZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' By setting ∂L/∂Z = 0, the optimal representation Z∗ can be obtained as follows: Z∗ = λ(λATA+I)−1ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (13) where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, in the case that the clean data is sufficient enough to represent the graph’s struc- tural patterns and the irregular connections are properly char- acterized, the structure perturbations can be inferred using AZ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hence, the collaborative inference operation is defined as fol- lows: CI(A) = λA(λATA+I)−1ATA (14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' High-order Connectivity Computation According to local similarity indices for link prediction, the more the number of paths two nodes possess, the greater the similarity between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Specifically, two nodes with a high mutual connectivity are more likely to generate a link between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, n-hop-based (n ≥ 2) paths must be explored to char- acterize the local structural features for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Using a deep learning framework, the n-hop computation can be decom- posed into two-hop operations on each neural layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hence, a high-order connectivity computation calculates the two hop con- nectivity of graph nodes in each layer, and the mutual connec- tivity of two nodes can be estimated by stacking the high-order connectivity computation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Assuming that the integer powers of the adjacency matrix characterizes the mutual connec- tivity of graph nodes, that is, [An]ij denotes the number of paths Figure 3: Illustration of high-order connectivity computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' with length n connecting nodes i and j, the high-order connectiv- ity computation in each neural layer can be defined based on the idea of the second power of adjacency matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' From the per- spective of graph convolution networks, high-order connectivity computation can be defined as HCCA(A) = ˆD− 1 2 ˆA ˆD− 1 2 CI(A), (15) where the weighted adjacency matrix generated by the proposed collaborative inference operation is viewed as the features of graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Figure 3 illustrates a high-order connectivity com- putation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' As presented in Equation (15), the global and local structural features can be captured for link prediction at the level of individual nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, the nonlinear propagation function can be defined as follows: H(l+1) = ˆD− 1 2 ˆA ˆD− 1 2 CI(H(l))W(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (16) Thus, the hierarchical structure of real-world graphs can be char- acterized by executing the nonlinear propagation function itera- tively, in which HCCA(H(l)) = ˆD− 1 2 ˆA ˆD− 1 2 CI(H(l)) represents the high-order connectivity of graph nodes, as depicted in Figure 2(c), and CI(H(l+1)) denotes the collaborative inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Pattern Fusion Operation To estimate the likelihood of potential links, the output of the (l − 1)-th layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', H(l), is fed as the input of the l-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on CI(H(l)) and HCCA(H(l)), the shallow layers extract the low-order global and local structure features, while the deep layers extract the high-order global and local structure features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Meanwhile, the effective range that the local structure features drawn from increases as the model depth increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Therefore, the structure features in different range at various order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', HCCA(H(l)) and CI(H(l)), 0 ≤ l ≤ L, all contribute to the inference of potential links, although the exact extent of their contribution depends on the graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To overcome the issues mentioned above, in addition to being used as the inputs of the next layer, the outputs of neural net- work layers are mapped to skip a block of several layers based on residual connections, as illustrated in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Next, all outputs are concatenated and used as the input of a two-layer Multi-layer Perceptron (MLP), which is defined as: O= MLP(concat(CI(H(l)), HCCA(H(l)))),0 ≤ l ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (17) where O is a vector containing the probabilities of links between all possible node pairs, and missing and spurious links can be inferred based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 6 Central node 1-hop neighbor nodes 2-hop neighbor nodes Node features representing link weights to 2-hop neighbors Computing connectivity of central node to 2-hop neighbors Adjacency relations of central node Weighted links produced by C(A)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Model Training To train the proposed model, augmented graphs generated based on the observed graph are used as training data, and the adjacency matrix of the observed graph is flatten as its labels Y, where Yi∗N+j denotes the existence of the link between nodes i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Correspondingly, O represents the prediction results ob- tained by the proposed model for all possible links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Here, the Binary Cross-Entropy (BCE) is used as the loss function: L = − 1 N2 N2 � i=1 Yi log(Oi) + (1 − Yi) log(1 − Oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (18) The learned model is then deployed on the observed graph to reconstruct the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The training process of GraphLP is outlined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Algorithm 1 Training Process of GraphLP Input: Training set Dtrain, validation set Dval, and test set Dtest, number of neural network layers L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Output: The well-trained model GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 1: while not convergence do 2: for 0 ≤ l ≤ L do 3: Conduct collaborative inference operation using (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 4: Compute high-order connectivity using (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5: end for 6: Fuse the outputs based on MLP using (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 7: Update the model by minimizing the loss function (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 8: end while 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Model Analysis (1) Generalized local similarity indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The high-order con- nectivity computation HCCA(H(l)) in every neural network layer is essentially the second power of the adjacency matrix, and it obtains the connectivity of node pairs within two-hop neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' As the model depth increases, the connectivity of node pairs in a wider range is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Thus, GraphLP can degenerate to S i j = A2 + αA3 + βA4 + · · · when collaborative inference and deep learning mechanism are abolished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) Connection to WalkPool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' WalkPool [36] first generates node representations based on GNN and encodes them into edge weights of the extracted enclosing subgraphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' following this, it uses the edge weights to compute the transition probabilities of random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Next, the method calculates a list of features based on the transition probabilities to classify the subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, for an enclosing subgraph G = (V, E), its variants G+ = (V, E ∪ {i, j}) and G− = (V, E\\{i, j}) are used as positive and negative samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In essence, this method dis- criminates only those subgraphs that differ by a single edge and is not suitable for practical link prediction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In contrast, GraphLP can predict any potential links based on graph structure features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) Connection to LFLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The LFLP [53] constructs an ad- jacency matrix based on a self-representation model and then combines it with the observed network to identify missing and spurious links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The collaborative inference operation CI(H(l)) of our work is similar to that in the LFLP with respect to model- ing the global structure of graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' however, the difference is that only low-order global structural features are considered in LFLP, whereas multi-order global and local structural features are char- acterized based on the deep-learning framework in GraphLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Experiments Further, extensive experiments are conducted on real-world graphs to evaluate the performance of the proposed method GraphLP: (1) Compare GraphLP with state-of-the-art meth- ods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) Compare GraphLP with traditional baseline methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) Model architecture analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) Model sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Here, Area Under Curve (AUC) and Average Precision (AP) are adopted to evaluate the performance of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Further- more, Precision is used to verify the superiority of GraphLP over traditional link prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on the link prediction results O, the scores are sorted in descending and ascending or- ders, and following this, their top-L links are taken as the pre- dicted missing and spurious links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Note that Precision is defined by calculating the ratio of accurately discovered links to the total number of links in the probe set: Precision = T/R (19) where T is the number of accurately identified links, and R is the total number of links in the probe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Experimental Settings 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Experimental Datasets Herein, seven widely used graph datasets are used for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (1) USAir [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This is the transportation network of the United States, including 332 airports as nodes and 2,126 airlines as edges, connecting the United States worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This is a neu- ral network of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' elegans, with 297 neurons representing nodes and 2,148 synaptic connections representing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) PB [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This dataset is a network of hyperlinks between weblogs on US political blogs, with 1,222 blogs on US politics as nodes and 16,714 hyperlinks between blogs as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) NS [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This is an undirected co-authorship network with 1,589 nodes and 2,742 edges, where the nodes denote the scientists engaged in network science research, and the edges denote two scientists have co-authored a publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (5) Yeast [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This represents a protein-protein interaction net- work formed in yeast with 2,375 proteins as nodes and 11,693 protein-protein interactions as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (6) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This is a pairwise reaction network of metabolites with 1,805 nodes and 14,660 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (7) Router [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' It is a snapshot of the In- ternet structure at the level of autonomous systems, with 5,022 nodes and 6,258 edges, in which the nodes represent routers and the edges represent the data transmission between routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The average node degree is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The properties of the datasets are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' To extensively validate the performance of the proposed method, 90% and 50% of the links of the original graph are se- lected randomly to first construct the observed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' There- after, based on the observed graph Go, 10% nonexisting links are add randomly as spurious links, and 10% existing links are removed randomly as missing links, denoted as Edel and Eadd respectively, to generate the augmented graph set D = {Gi|i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=', t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Following this, 90% and 10% graphs are randomly se- lect from D as the training and validation set, respectively, and the observed graph Go is used as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 7 Table 2: Summary of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' ACC is the average clustering coefficient, and AD is the average node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Dataset USAir NS PB Yeast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele Router E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli Node 332 1589 1222 2375 297 5022 1805 Edges 2126 2742 16714 11693 2148 6258 14660 ACC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='516 AD 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Comparison Methods The proposed method was compared with six state-of-the-art deep learning-based link prediction methods, including: (1) Weisfeiler-Lehman graph kernel (WLK) [42] is a fast fea- ture extraction scheme based on the WL test for graph isomor- phism, which maps the original graph to a graph sequence and adds the pair-wise similarities between the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) Weifeiler-Lehmam Neural Machine (WLNM) [56] is a subgraph classification-based link prediction method that lever- age deep learning to automatically learn topological features from enclosing subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) Node2Vec [20] is a network embedding method that en- codes proximity information into low-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The node features and low-dimensional vectors are then fed into the MLP for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) LINE [44] learns network embeddings that preserve the first-order and second-order proximity, and the resulting low- dimensional vectors are used for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (5) SEAL [57] extracts the enclosing subgraphs of positive and negative links and marks different roles of their nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The method then trains a GNN based on the node information matrix to classify subgraphs for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (6) WalkPool (WP) [36] is a subgraph classification-based link prediction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' It encodes node feature and graph topol- ogy into the transition probabilities of random walk, and follow- ing this, a list of features is computed to classify subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Parameter Settings GraphLP is implemented on a Pytorch platform with a NVIDIA GeForce RTX GPU and optimized using Adam op- timizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' All models are implemented using Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0012 for the NS dataset and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0005 for the other graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For all the datasets, the weight decay is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The number of epochs on the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli and Yeast dataset is 300, whereas it was 200 on the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Dropout is ap- plied to the MLP, and the dropout rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5 on Router and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2 on the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The trade-off parameter λ is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13, and the number of neural network layers in the GraphLP is set to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The detailed hyperparameter settings for the model are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Table 3: Hyperparameter setting for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Name Value optimizer Adam loss function binary cross entropy learning rate NS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0012, others=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0005 weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0 epochs Ecoli, Yeast=300, others=200 dropout Router=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5, others=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 number of network layers 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Experimental Result For 90% of the observed links, the results about the AUC and AP with standard deviations are presented in Table 4 and 5, which indicate that GraphLP significantly outperforms other state-of-the-art algorithms in terms of both AUC and AP, with exception of the NS and Router datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results demon- strate that the learning of local and global graph structure en- tirely characterizes the underlying structural patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' thus, the missing links and spurious links can be better identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Ta- ble 4 indicates that GraphLP significantly improves the AUC on the PB, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele, and Router datasets, with approximately 4%, 7%, and 3% performance improvement, respectively, compared to the WP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In addition, the proposed method still per- forms better than other state-of-the-art methods on the USAir, Yeast and NS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, the results for the AP pre- sented in Table 5 also indicate that GraphLP outperforms state- of-the-art methods on most of datasets, and GraphLP achieves a maximum performance enhancement of approximately 9% com- pared to the best performing graph neural network method WP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For 50% of the observed links, the results also demonstrate that the proposed model achieves remarkable performance com- pared to the methods, as described in Table 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results illustrate that, as the amount of structure perturbation increases, GraphLP can still appropriately learn the real graph structure, thus recovering the original graph effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Therefore, the values of the AUC and AP decreased to a lower extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Fur- thermore, by comparing Table 4 with Table 6 and Table 5 with Table 7, we can infer that the AUC and AP values drop faster for the other state-of-the-art methods than those for GraphLP, which demonstrates that GraphLP can better capture the under- lying structural patterns to demonstrate better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Compared with Traditional Link Prediction Methods To further verify the proposed method, the precision of GraphLP and traditional link prediction methods are calculated based on the following datasets: (1) Macaque [15], cortical net- works of the macaque monkey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) Mangwet [5], the food web in Mangrove Estuary during the wet season;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) Jazz [19], a collaboration network of jazz musicians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) Metabolic [17], a metabolic network of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='elegans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (5) USAir, (6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele, (7) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli and (8) Yeast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Here, six representative traditional link prediction methods are selected for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (1) The CN [29] metric is among the most widely used meth- ods for link prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' It assumes that two nodes will be more easily connected if they share more common neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (2) The RA [61] metric is inspired by the physical processes involved in resource allocation, which suppresses the contribu- tion of high-degree common neighbors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (3) The LP [61] index measures the structural similarity of node pairs within three-hops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (4) The Non-negative Matrix Factorization (NMF) [26] model is used for structure prediction by learning the latent features of real-world graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (5) The Robust Principal Component Analysis (RPCA) [37] represents a real-world graph based on the sparsity and low rank property of its adjacency matrix and infers potential links based on matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (6) The LFLP [53] uses a self-representation model to re- construct the original graph based on a few representative sub- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results of missing link prediction with respect to Preci- sion are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For each network, the bold number 8 Table 4: Prediction measured by AUC (90% observed links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Bold numbers are the best results of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Data USAir NS PB Yeast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele Router E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli WLK 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='73 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='59 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='54 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='72 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='42 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='29 WLNM 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='47 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='52 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='18 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='72 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='88 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='27 Node2Vec 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='44 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='78 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='28 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='78 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='11 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='27 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='86 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='82 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 LINE 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='47 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='76 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='14 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 SEAL 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='34 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='52 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='30 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='35 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='22 WP 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='48 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='37 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='09 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='28 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 GraphLP 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='01 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='14 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 Table 5: Prediction measured by AP (90% observed links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Bold numbers are the best results of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Data USAir NS PB Yeast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele Router E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli WLK 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='84 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='40 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='89 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='35 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='06 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='59 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='42 WLNM 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='64 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='05 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='09 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 Node2Vec 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='97 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='91 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='03 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='12 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='87 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='48 LINE 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='70 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='76 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='17 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='65 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='82 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='39 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='72 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='92 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='53 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='82 SEAL 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='80 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='37 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='43 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='37 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='48 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 WP 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='29 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='28 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 GraphLP 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='03 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='32 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='43 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='16 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='42 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 Table 6: Prediction measured by AUC ( 50% observed links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Bold numbers are the best results of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Data USAir NS PB Yeast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele Router E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli WLK 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='27 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='71 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='35 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='89 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='37 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 WLNM 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='61 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='63 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='32 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='72 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='52 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='30 Node2Vec 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='58 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='29 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='67 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='17 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 LINE 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='60 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='53 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='78 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='44 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='43 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='50 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 SEAL 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='67 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='88 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='18 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='54 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='31 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='64 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='58 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 WP 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='74 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='16 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='62 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='39 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='61 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='30 GraphLP 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='14 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='14 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='14 Table 7: Prediction measured by AP ( 50% observed links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Bold numbers are the best results of all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Data USAir NS PB Yeast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele Router E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli WLK 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='02 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='34 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='46 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='49 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='43 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='32 WLNM 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='81 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='11 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='12 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='68 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 Node2Vec 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='51 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='87 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='97 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='23 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='74 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='57 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='94 LINE 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='75 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The experiment performs 20% link perturbation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 20% spurious links are added and 20% missing links are deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' in the corresponding column indicates the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results presented in Table 8 demonstrate that the proposed model GraphLP model performs the best among the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Further- more, the link prediction accuracy of the proposed model is far higher than that of the other methods, which can be at least three times higher than that of the best-performing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For spuri- ous links prediction, the results measured by Precision are listed in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For all networks, GraphLP performs the best among the methods and is remarkably better than the second best algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results presented in Table 8 and 9 demonstrate that GraphLP has stronger ability to learn structural features, and can recover the structure of the original network more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on Table 2, it can be observed that the Precision of our proposed model performs best, despite the large differences be- tween the ACC and AD across all the datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' thus indicates that the proposed model performs well for heterogeneous graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='4.' metadata={'source': 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+page_content='0733 16): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='6510 (9, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (9, (9, (12, 18): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0378 (9, 16): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='8059 (a)The similarity of negative and (b)The similarity of negative and (c)The similarity of negative and (d) The similarity of negative and (c)The similarity of negative and positive links performed 20 epochs positive links performed 40 epochs positive links performed 60 epochs positive links performed 100 epochs positive links performed 80 epochsYeast USAir C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0 AP Layer=1 Layer=2 Layer=3 Layer=4 (b) Figure 6: The performance of link prediction under various model depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (a) AUC of link prediction method GraphLP with different layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) AP of link prediction method GraphLP with different layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Yeast USAir C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli PB NS Router Training networks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='000 AUC =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 (a) Yeast USAir C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='coli PB NS Router Training networks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='00 AP =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='08 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='13 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='15 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='20 (b) Figure 7: The performance of link prediction under different values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (a) AUC of link prediction method GraphLP under different values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) AP of link prediction method GraphLP under different values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 0 50 100 150 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='55 train_loss 0 50 100 150 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='25 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='00 val_auc 0 50 100 150 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='95 val_ap 0 50 100 150 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='6 val_loss (b) Figure 8: Visualization of model convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (a) Convergence of USAir dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' (b) Convergence of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' links, the blue links denote the spurious links, and the gray links denote the original links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The bottom half depicts the likelihood scores of missing and spurious links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on the results, it can be concluded that with an increase in epoch times, the like- lihood scores of missing links increases gradually, and the like- lihood scores of spurious links decline gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The widths of the lines indicate the following process: when the number of epochs reaches 100, the likelihood scores of missing links ap- proach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0, and the likelihood scores of spurious links approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This proves that the proposed GraphLP model can distin- guish between missing links and spurious links and infer them effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Moreover, to further prove the effectiveness of the proposed model, the topology of the recovered graphs in the model training process is visualized when 20% of the links of Club are perturbed, as depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Compared to Fig- ure 4, we determined that the likelihood of missing and spuri- ous links is weakened with an increase in the structure pertur- bation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' However, with an increase in the training epochs, the model is still able to distinguish and infer the missing and spurious links according to the structural patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' For instance, when the training epoch reaches 140, the likelihood scores of missing links are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5, and those of spurious links 11 are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='3, indicating that the proposed model can still predict missing and spurious links with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Impact of Model Depth Next, the performance of GraphLP is explored at various model depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' As depicted in Figure 6, the performance of the model in terms of the AUC and AP trained with one-layer neu- ral network is poor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' however, its performance improves signif- icantly with an increase in the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In particular, when the model depth is equal to two, a significant performance improvement is noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The primary reason for this is that the model with two-layers multi-order global and local structural features is integrated adaptively based on the MLP component, which considerably improves the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Subsequently, as the layer number increases, a slight improve- ment in model performance is still noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' When the number of layers is four, the accuracy of GraphLP declines significantly on NS and fluctuates on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A possible reason for this is that the model with four layers becomes more complex, thereby requiring more training iterations or an appropriate learning rate [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In general, the performance of the proposed model is opti- mal when the depth is three, and a deep architecture is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Impact of Trade-off Parameter To examine the sensitivity of the proposed model to the trade- off parameter, the AUC and AP values of link prediction meth- ods with different λ are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Based on the re- sults, it can be concluded that the performance of the proposed model is not sensitive to λ for most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Figure 7(a), for the USAir and NS datsets, the AUC value varies significantly under different λ, but the performance is still better than other of the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Figure 7(b), the AP value remains stable for different λ values, indicating that the proposed model is insensitive to different λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Overall, our proposed algorithm ex- hibited satisfactory performance on most datasets with various λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The Convergence Analysis Generally, GraphLP converges to optimal values after approx- imitely 200 epochs on most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In particular, Figure 8 plots the learning curves of GraphLP on the USAir and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='ele datasets, including the training loss, validation AUC, validation AP, and validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The results indicate that the AUC and AP values increase rapidly with the decrease in training loss and validation loss, and these values converge to the optimal value when the validation loss approaches a minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Additionally, we discover that validation loss is lower than training loss, and the difference between them remains relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A possible reason for this is that the dropout manipulation is only applied to the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Conclusion This paper aims to reconstruct graph structure to improve the performance of link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In particular, unlike existing subgraph-classification-based discriminative methods, this work achieves the aforementioned objective by developing a genera- tive GNN, namely GraphLP, which considered both global and local structure features and hierarchical structural patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Con- currently, a novel collaborative inference operation and high- order connectivity computation mechanism are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' We also present an analysis about the relation between GraphLP and other classical link prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Extensive experimental results demonstrate the superiority of the proposed method over other state-of-the-art models and traditional baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' This could be a fruitful avenue for future research aimed at ad- dressing graph learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Acknowledgment This work was partially supported by the National Natu- ral Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 62106030, 61802039, 62272066;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Chongqing Municipal Postdoctoral Sci- ence Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' cstc2021jcyj-bsh0176;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Chongqing Municipal Natural Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' cstc2020jcyj-msxmX0804;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' the Chongqing Research Pro- gram of Basic Research and Frontier Technology under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' cstc2021jcyj-msxmX0530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' References [1] Robert Ackland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Mapping the us political blogosphere: Are conserva- tive bloggers more prominent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In BlogTalk Downunder 2005 Conference, Sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' BlogTalk Downunder 2005 Conference, Sydney, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [2] Lada A Adamic and Eytan Adar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Friends and neighbors on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Social networks, 25(3):211–230, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [3] Yong-Yeol Ahn, James P Bagrow, and Sune Lehmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link communi- ties reveal multiscale complexity in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' nature, 466(7307):761–764, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [4] James Atwood and Don Towsley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Diffusion-convolutional neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Advances in neural information processing systems, pages 1993– 2001, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [5] Daniel Baird, J Luczkovich, and Robert R Christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Assessment of spa- tial and temporal variability in ecosystem attributes of the st marks national wildlife refuge, apalachee bay, florida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Estuarine, Coastal and Shelf Sci- ence, 47(3):329–349, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [6] Baruch Barzel and Albert-L´aszl´o Barab´asi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Network link prediction by global silencing of indirect correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Nature biotechnology, 31(8):720– 725, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [7] Austin R Benson, David F Gleich, and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Higher-order orga- nization of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Science, 353(6295):163–166, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [8] Lei Cai, Jundong Li, Jie Wang, and Shuiwang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Line graph neural net- works for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Ma- chine Intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [9] Emmanuel J Cand`es, Xiaodong Li, Yi Ma, and John Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Robust prin- cipal component analysis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Journal of the ACM (JACM), 58(3):1–37, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [10] Shaosheng Cao, Wei Lu, and Qiongkai Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Deep neural networks for learning graph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [11] Zhiqian Chen, Fanglan Chen, Lei Zhang, Taoran Ji, Kaiqun Fu, Liang Zhao, Feng Chen, Lingfei Wu, Charu Aggarwal, and Chang-Tien Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Bridging the gap between spatial and spectral domains: A survey on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='11867, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [12] Ara Cho, Junha Shin, Sohyun Hwang, Chanyoung Kim, Hongseok Shim, Hyojin Kim, Hanhae Kim, and Insuk Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Wormnet v3: a network-assisted hypothesis-generating server for caenorhabditis elegans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Nucleic acids re- search, 42(W1):76–82, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [13] Aaron Clauset, Cristopher Moore, and Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hierarchical structure and the prediction of missing links in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Nature, 453 (7191):98–101, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [14] Weilin Cong, Morteza Ramezani, and Mehrdad Mahdavi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' On provable benefits of depth in training graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:9936–9949, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [15] Lucianoda F Costa, Marcus Kaiser, and Claus C Hilgetag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Predicting the connectivity of primate cortical networks from topological and spatial node properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' BMC systems biology, 1(1):1–17, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [16] Micha¨el Defferrard, Xavier Bresson, and Pierre Vandergheynst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Convolu- tional neural networks on graphs with fast localized spectral filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 30th International Conference on Neural Information Processing Systems, pages 3844–3852, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [17] Jordi Duch and Alex Arenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Community detection in complex networks using extremal optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physical review E, 72(2):027104, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [18] Zhihong Fang, Shaolin Tan, Yaonan Wang, and Jinhu Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Elementary sub- graph features for link prediction with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' IEEE Transactions on Knowledge and Data Engineering, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 12 [19] Pablo M Gleiser and Leon Danon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Community structure in jazz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Advances in complex systems, 6(04):565–573, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [20] Aditya Grover and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' node2vec: Scalable feature learning for networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 22nd ACM SIGKDD international confer- ence on Knowledge discovery and data mining, pages 855–864, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [21] Roger Guimer`a and Marta Sales-Pardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Missing and spurious interactions and the reconstruction of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 106(52):22073–22078, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [22] Will Hamilton, Zhitao Ying, and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Inductive representation learning on large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Advances in neural information processing sys- tems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [23] Glen Jeh and Jennifer Widom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Simrank: a measure of structural-context similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the eighth ACM SIGKDD international con- ference on Knowledge discovery and data mining, pages 538–543, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [24] Meng Jiang, Peng Cui, Alex Beutel, Christos Faloutsos, and Shiqiang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Detecting suspicious following behavior in multimillion-node so- cial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 23rd International Conference on World Wide Web, pages 305–306, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [25] Leo Katz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' A new status index derived from sociometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Psy- chometrika, 18(1):39–43, 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [26] Daniel Lee and H Sebastian Seung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Algorithms for non-negative matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Advances in neural information processing systems, 13, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [27] Boning Li, Yingce Xia, Shufang Xie, Lijun Wu, and Tao Qin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Distance- enhanced graph neural network for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In ICML 2021 Work- shop on Computational Biology, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [28] Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, and Yi Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Robust recovery of subspace structures by low-rank representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 35 (1):171–184, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [29] Francois Lorrain and Harrison C White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Structural equivalence of indi- viduals in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The Journal of mathematical sociology, 1(1): 49–80, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [30] Paul Louis, Shweta Ann Jacob, and Amirali Salehi-Abari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Sampling en- closing subgraphs for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='12004, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [31] Linyuan L¨u, Mat´uˇs Medo, Chi Ho Yeung, Yi-Cheng Zhang, Zi-Ke Zhang, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physics reports, 519(1):1–49, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [32] Linyuan L¨u, Liming Pan, Tao Zhou, Yi-Cheng Zhang, and H Eugene Stan- ley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Toward link predictability of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 112(8):2325–2330, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [33] Mark EJ Newman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Finding community structure in networks using the eigenvectors of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physical review E, 74(3):036104, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [34] Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Learning convolutional neural networks for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In International conference on machine learning, pages 2014–2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' PMLR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [35] Gergely Palla, Imre Der´enyi, Ill´es Farkas, and Tam´as Vicsek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Uncovering the overlapping community structure of complex networks in nature and society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' nature, 435(7043):814–818, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [36] Liming Pan, Cheng Shi, and Ivan Dokmani´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Neural link prediction with walk pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In International Conference on Learning Representations, pages 5171–5181, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [37] Ratha Pech, Dong Hao, Liming Pan, Hong Cheng, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link pre- diction via matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' EPL (Europhysics Letters), 117(3):38002, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [38] Ratha Pech, Dong Hao, Yan-Li Lee, Ye Yuan, and Tao Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link pre- diction via linear optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physica A: Statistical Mechanics and its Applications, 528:121319, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [39] Erzs´ebet Ravasz and Albert-L´aszl´o Barab´asi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hierarchical organization in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physical review E, 67(2):026112, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [40] Ryan Rossi and Nesreen Ahmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The network data repository with inter- active graph analytics and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Twenty-ninth AAAI conference on artificial intelligence, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [41] Marta Sales-Pardo, Roger Guimera, Andr´e A Moreira, and Lu´ıs A Nunes Amaral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Extracting the hierarchical organization of complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 104(39):15224–15229, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [42] Nino Shervashidze, Pascal Schweitzer, Erik Jan Van Leeuwen, Kurt Mehlhorn, and Karsten M Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Weisfeiler-lehman graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Journal of Machine Learning Research, 12(9), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [43] Neil Spring, Ratul Mahajan, and David Wetherall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Measuring isp topolo- gies with rocketfuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' ACM SIGCOMM Computer Communication Review, 32(4):133–145, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [44] Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Line: Large-scale information network embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 24th international conference on world wide web, pages 1067–1077, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [45] Ala Trusina, Sergei Maslov, Petter Minnhagen, and Kim Sneppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Hi- erarchy measures in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physical review letters, 92(17): 178702, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [46] Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Graph attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content='10903, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [47] Christian Von Mering, Roland Krause, Berend Snel, Michael Cornell, Stephen G Oliver, Stanley Fields, and Peer Bork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Comparative assess- ment of large-scale data sets of protein–protein interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Nature, 417 (6887):399–403, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [48] Xu-Wen Wang, Yize Chen, and Yang-Yu Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link prediction through deep generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' iScience, 23(10):101626, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [49] Duncan J Watts and Steven H Strogatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Collective dynamics of ‘small- world’networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' nature, 393(6684):440–442, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [50] Max Welling and Thomas N Kipf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Semi-supervised classification with graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' International Conference on Learning Representations (ICLR 2017), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [51] Tao Wu, Yuxiao Guo, Leiting Chen, and Yanbing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Integrated struc- ture investigation in complex networks by label propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Physica A: Statistical Mechanics and its Applications, 448:68–80, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [52] Tao Wu, Hongyu Ma, Chao Wang, Shaojie Qiao, Liang Zhang, and Shui Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Heterogeneous representation learning and matching for few-shot re- lation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Pattern Recognition, page 108830, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [53] Xingping Xian, Tao Wu, Shaojie Qiao, Xi-Zhao Wang, Wei Wang, and Yanbing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Netsre: Link predictability measuring and regulating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Knowledge-Based Systems, 196:105800, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [54] Xingping Xian, Tao Wu, Shaojie Qiao, Wei Wang, Chao Wang, Yanbing Liu, and Guangxia Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Deepec: Adversarial attacks against graph structure prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Neurocomputing, 437:168–185, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [55] Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Representation learning on graphs with jumping knowledge networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In International conference on ma- chine learning, pages 5453–5462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [56] Muhan Zhang and Yixin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Weisfeiler-lehman neural machine for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 23rd ACM SIGKDD international con- ference on knowledge discovery and data mining, pages 575–583, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [57] Muhan Zhang and Yixin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Link prediction based on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 5171–5181, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [58] Muhan Zhang, Zhicheng Cui, Shali Jiang, and Yixin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Beyond link prediction: Predicting hyperlinks in adjacency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [59] Jiajun Zhou, Jie Shen, and Qi Xuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Data augmentation for graph clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' In Proceedings of the 29th ACM International Conference on Information & Knowledge Management, pages 2341–2344, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [60] Jie Zhou, Ganqu Cui, Shengding Hu, Zhengyan Zhang, Cheng Yang, Zhiyuan Liu, Lifeng Wang, Changcheng Li, and Maosong Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Graph neural networks: A review of methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' AI Open, 1:57– 81, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' [61] Tao Zhou, Linyuan L¨u, and Yi-Cheng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' Predicting missing links via local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' The European Physical Journal B, 71(4):623–630, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQfWveE/content/2301.00169v1.pdf'} diff --git a/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/2301.11300v1.pdf.txt b/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/2301.11300v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..81e28e4f15db744746331d290e4fa4cfd0aee6f3 --- /dev/null +++ b/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/2301.11300v1.pdf.txt @@ -0,0 +1,3332 @@ +Published as a conference paper at ICLR 2023 +ZICO: ZERO-SHOT NAS VIA INVERSE COEFFICIENT +OF VARIATION ON GRADIENTS +Guihong Li1, Yuedong Yang1, Kartikeya Bhardwaj2∗, Radu Marculescu1 +1The University of Texas at Austin, 2Qualcomm +{lgh,albertyoung,radum}@utexas.edu, kbhardwa@qti.qualcomm.com +ABSTRACT +Neural Architecture Search (NAS) is widely used to automatically design the neu- +ral network with the best performance among a large number of candidate archi- +tectures. To reduce the search time, zero-shot NAS aims at designing training-free +proxies that can predict the test performance of a given architecture. However, as +shown recently, none of the zero-shot proxies proposed to date can actually work +consistently better than a naive proxy, namely, the number of network parameters +(#Params). To improve this state of affairs, as the main theoretical contribution, we +first reveal how some specific gradient properties across different samples impact +the convergence rate and generalization capacity of neural networks. Based on +this theoretical analysis, we propose a new zero-shot proxy, ZiCo, the first proxy +that works consistently better than #Params. We demonstrate that ZiCo works bet- +ter than State-Of-The-Art (SOTA) proxies on several popular NAS-Benchmarks +(NASBench101, NATSBench-SSS/TSS, TransNASBench-101) for multiple ap- +plications (e.g., image classification/reconstruction and pixel-level prediction). Fi- +nally, we demonstrate that the optimal architectures found via ZiCo are as compet- +itive as the ones found by one-shot and multi-shot NAS methods, but with much +less search time. For example, ZiCo-based NAS can find optimal architectures +with 78.1%, 79.4%, and 80.4% test accuracy under inference budgets of 450M, +600M, and 1000M FLOPs on ImageNet within 0.4 GPU days. +1 +INTRODUCTION +During the last decade, deep learning has achieved great success in many areas, such as computer +vision and natural language modeling Krizhevsky et al. (2012); Liu & Deng (2015); Huang et al. +(2017); He et al. (2016); Dosovitskiy et al. (2021); Brown et al. (2020); Vaswani et al. (2017). In +recent years, neural architecture search (NAS) has been proposed to search for optimal architectures, +while reducing the trial-and-error (manual) network design efforts Baker et al. (2017); Zoph & Le +(2017); Elsken et al. (2019). Moreover, the neural architectures found via NAS show better perfor- +mance than the manually-designed networks in many mainstream applications Real et al. (2017); +Gong et al. (2019); Xie et al. (2019); Wu et al. (2019); Wan et al. (2020); Li & Talwalkar (2020); +Kandasamy et al. (2018); Yu et al. (2020b); Liu et al. (2018b); Cai et al. (2018); Zhang et al. (2019a); +Zhou et al. (2019); Howard et al. (2019). +Despite these advantages, most of the existing NAS approaches involve a time-consuming and +resource-intensive search process. For example, multi-shot NAS uses a controller or an accuracy +predictor to conduct the search process and it requires training of multiple networks; thus, multi-shot +NAS is extremely time-consuming (typically thousands of GPU hours) Real et al. (2019); Chiang +et al. (2019). Alternatively, one-shot NAS merges all possible networks from the search space into +a supernet and thus only needs to train the supernet once Dong & Yang (2019); Zela et al. (2020); +Chen et al. (2019); Cai et al. (2019); Stamoulis et al. (2019); Chu et al. (2021); Guo et al. (2020); +this enables one-shot NAS to find a good architecture with much less search time. Though the one- +shot NAS has significantly improved the time efficiency of NAS, training is still required during the +search process. +∗Work done while Kartikeya Bhardwaj was at Arm, Inc. +1 +arXiv:2301.11300v1 [cs.LG] 26 Jan 2023 + +Published as a conference paper at ICLR 2023 +In the last few years, the zero-shot approaches have been proposed to liberate NAS from training +entirely Wu et al. (2021); Zhou et al. (2022; 2020); Ingolfsson et al. (2022); Tran & Bae (2021); Do & +Luong (2021); Tran et al. (2021); Shu et al. (2022b). Essentially, zero-shot NAS utilizes some proxy +that can predict the test performance of a given network without training. Moreover, the design of +the proxy in zero-shot NAS is usually based on some theoretical analysis of deep networks. Hence, +zero-shot approaches can not only significantly improve the time efficiency of NAS, but also deepen +the theoretical understanding on why certain networks work well. Nonetheless, as revealed in Ning +et al. (2021); White et al. (2022), the zero-shot proxies proposed to date cannot work consistently +better than a naive proxy, namely, the number of parameters (#Params); in fact, #Params often +achieves the best performance on most of popular NAS benchmarks. These results may undermine +the effectiveness of zero-shot NAS approaches. +To address the limitations of existing zero-shot proxies, we target the following key questions: +1. How do some specific gradient properties, i.e., mean value and standard deviation across +different samples, impact the training convergence of neural networks? +2. Can we use these two gradient properties to design a new theoretically-grounded proxy that +works better than #Params consistently? +To this end, we first theoretically analyze how the mean value and standard deviation of gradients +across different training batches impact the training convergence of neural networks. Based on +our analysis, we propose ZiCo, a new proxy for zero-shot NAS. We demonstrate that, compared +to all existing proxies (including #Params), ZiCo has either a higher or at least on-par correlation +with the test accuracy on popular NAS-Benchmarks (NASBench101, NATS-Bench-SSS/TSS) for +multiple datasets (CIFAR10/100, ImageNet16-120). Finally, we demonstrate that ZiCo enables +a zero-shot NAS framework that can efficiently find the network architectures with highest test +accuracy compared to other zero-shot baselines. In fact, our ZiCo-based zero-shot NAS framework +achieves competitive FLOPs-accuracy tradeoffs compared to multiple one-shot and multi-shot NAS, +but with much lower time costs. To summarize, we make the following major contributions: +• We theoretically reveal how the mean value and variance of gradients across multiple sam- +ples impact the training convergence and generalization capacity of neural networks. +• We propose a new zero-shot proxy, ZiCo, that works better than existing proxies on popu- +lar NAS-Benchmarks (NASBench101, NATS-Bench-SSS/TSS, TransNASBench-101) for +multiple applications (e.g. image classification/reconstruction and pixel-level prediction). +• We demonstrate that our proposed zero-shot NAS achieves competitive test accuracy with +representative one-shot and multi-shot NAS with much less search time. +The rest of the paper is organized as follows. We discuss related work in Section 2. In Section 3, we +introduce our theoretical analysis. We introduce our proposed zero-shot proxy (ZiCo) and the NAS +framework in Section 4. Section 5 validates our analysis and presents our results with the proposed +zero-shot NAS. We conclude the paper in Section 6 with remarks on our main contribution. +2 +RELATED WORK +2.1 +ZERO-SHOT NAS +The goal of zero-shot NAS is to rank the accuracy of various candidate network architectures without +training, such that we can replace the expensive training process in NAS with some computation- +efficient proxies Xiang et al. (2021a); Javaheripi et al. (2022); Bhardwaj et al. (2021); Li et al. (2021); +Bhardwaj et al. (2022a). Hence, the quality of the proxy determines the effectiveness of zero-shot +NAS. Several works use the number of linear regions to approximately measure the expressivity of a +deep neural network Mellor et al. (2021); Chen et al. (2021b); Bhardwaj et al. (2022b). Alternatively, +most of the existing proxies are derived from the gradient of deep networks. For example, Synflow, +SNIP, and GraSP rely on the gradient w.r.t the parameters of neural networks; they are proved to be +the different approximations of Taylor expansion of deep neural networks Abdelfattah et al. (2021); +Lee et al. (2019b); Tanaka et al. (2020); Wang et al. (2020). Moreover, the Zen-score approximates +the gradient w.r.t featuremaps and measures the complexity of neural networks Lin et al. (2021); +Sun et al. (2021). Furthermore, Jacob cov leverages the Jacobian matrix between the loss and mul- +tiple input samples to quantify the capacity of modeling the complex functions Lopes et al. (2021). +Though zero-shot NAS can significantly accelerate the NAS process, it has been revealed that the +naive proxy #Params generally works better than all the proxies proposed to date Ning et al. (2021); +White et al. (2022). These limitations of existing proxies motivate us to look for a new proxy that +2 + +Published as a conference paper at ICLR 2023 +can consistently work better than #Params and address the limitations of existing zero-shot NAS +approaches. +2.2 +KERNEL METHODS AND CONVERGENCE ANALYSIS +Kernel methods are widely explored to analyze the convergence property of networks trained with +gradient descent Neal (1996); Williams (1996); Du et al. (2019a); Lu et al. (2020); Allen-Zhu et al. +(2019); Hanin & Nica (2020); Golikov et al. (2022). For example, the training of wide neural +networks is proved to be equivalent to the optimization of a specific kernel function Arora et al. +(2019a); Lee et al. (2019a); Chizat et al. (2019); Arora et al. (2019b); Cho & Saul (2009). Moreover, +given the networks with specific width constraints, researchers proved that the training convergence +of networks can be described by some corresponding kernels and the convergence rates of training +are highly coupled with the eigenvalues of the kernel-based covariance matrix Mei et al. (2019); +Zhang et al. (2019b); Garriga-Alonso et al. (2019); Du et al. (2019b). In our work, we extend such +kernel-based analysis to reveal the relationships between the gradient properties and the training +convergence for neural networks. +3 +CONVERGENCE AND GENERALIZATION VIA GRADIENT ANALYSIS +We consider the mean value and standard deviation of gradients across different samples and first +explore how these two metrics impact the training convergence of linear regression tasks. +3.1 +LINEAR REGRESSION +Inspired by Du et al. (2019b), we use the training set S with M samples as follows: +S = {(xi, yi), i = 1, ..., M, xi ∈ Rd, yi ∈ R, ||xi|| = 1, |yi| ≤ R, M > 1} +(1) +where R is a positive constant and || · || denotes the L2-norm of a given vector; xi ∈ Rd is the ith +input sample and normalized by its L2-norm (i.e., ||xi|| = 1), and yi is the corresponding label. We +define the following linear model f = aT x optimized with an MSE-based loss function L: +mina +� +i +L(yi, f(xi; a)) = mina +� +i +1 +2(aT xi − yi)2 +(2) +where a ∈ Rd is the initial weight vector of f. We denote the gradient of L w.r.t to a as g(xi) when +taking (xi, yi) as the training sample: +g(xi) = ∂L(yi, f(xi; a)) +∂a +(3) +We denote the jth element of g(xi) as gj(xi). We compute the mean value (µj) and standard +deviation (σj) of gj(xi) across all training samples as follows: +µj = 1 +M +M +� +i +gj(xi) +σj = +� +� +� +� 1 +M +M +� +i +(gj(xi) − µj)2 +(4) +Theorem 3.1. We denote the updated weight vector as ˆa and denote � +ij[gj(xi)]2 = G. Assume +we use the accumulated gradient of all training samples and learning rate η to update the initial +weight vector a, i.e., ˆa = a − η � +i g(xi). If the learning rate 0 < η < 2, then the total training +loss is bounded as follows: +� +i +L(yi, f(xi; ˆa)) ≤ G +2 − η +2M 2(2 − η) +� +j +µ2 +j +(5) +In particular, if the learning rate η = +1 +M , then L(ˆa) is bounded by: +� +i +L(yi, f(xi; ˆa)) ≤ M +2 +� +j +σ2 +j +(6) +We provide the proof in Appendix A and the experimental results to validate this theorem in Sec 5.2. +Remark 3.1 Intuitively, Theorem. 3.1 tells us that the higher the gradient absolute mean across +different training samples, the lower the training loss the model converges to; i.e., the network +converges at a faster rate. Similarly, if ηM < 1, the smaller the gradient standard deviation across +different training samples/batches, the lower the training loss the model can achieve. +3 + +Published as a conference paper at ICLR 2023 +3.2 +MLPS WITH RELU +In this section, we generalize the linear model to a network with ReLU activation functions. We +primarily consider the standard deviation of gradients in the Gaussian kernel space. We still focus +on the regression task on the training set S defined in Eq. 1. We consider a neural network in the +same form as Du et al. (2019b): +h(x; s, W ) = +1 +√m +m +� +i +srReLU(wT +r x) +(7) +where m is the number of output neurons of the first layer; sr is the rth element in the output weight +vector s; W ∈ Rm×d is the input weight matrix, and wr ∈ Rd is the rth row weight vector in W . +For training on the dataset S with M samples defined in Eq. 1, we minimize the following loss +function: +L(s, W ) = +M +� +i=1 +1 +2(h(xi; s, W ) − yi)2 +(8) +Following the common practice Du et al. (2019b), we fix the second layer (s) and use gradient +descent to optimize the first layer (W ) with a learning rate η: +wr(t) = wr(t − 1) − η +t +� +i=0 +∂L(s, W (t − 1)) +∂wr(t − 1) +(9) +where W (t − 1) denote the input weight matrix after t − 1 training steps; wr(t) denote the rth row +weight vector after t training steps. +Definition 1. (Gram Matrix) A Gram Matrix H(t) ∈ RM×M on the training set {(xi, yi), i = +1, ..., M} after t training steps is defined as follows: +Hij(t) = 1 +mxT +i xj +m +� +r=1 +I{xT +i wr(t) ≥ 0, xT +j wr(t) ≥ 0} +(10) +where I is the indicator function and I{A} = 1 if and only if event A happens. We denote the +λmin(H) as the minimal eigenvalue of a given matrix H. We denote the λ0 = λmin(H(∞)). +Theorem 3.2. Given a neural network with ReLU activation function optimized by minimizing Eq. 8, +we assume that each initial weight vector {wr(0), r = 1, ..., n} is i.i.d. generated from N(0, I) and +the gradient for each weight follows i.i.d. N(0, σ), where the σ is measured across different training +steps. For some positive constants δ and ϵ, if the learning rate η satisfies η < +λ0 +√πδ +2M 2√ +2Φ(1−ϵ)tσ, +then with with probability at least (1 − δ)(1 − ϵ), the following holds true: for any r ∈ [m], +||wr(0) − wr(t)|| ≤ C = ηtσ +� +Φ(1 − ϵ), and at training step t the Gram matrix H(t) satisfies: +λmin(H(t)) ≥ λmin(H(0)) − 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +> 0 +(11) +Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d). +We provide the proof in Appendix B. We now introduce the following conclusion from Du et al. +(2019b) to further help our analysis. +Lemma 1. Du et al. (2019b) Assume we set the number of output neurons of the first layer m = +Ω( M 6 +λ4 +0δ3 ) and we i.i.d. initialize wr ∼ N(0, I) and sr ∼ uniform[{−1, 1}], for r ∈ [m]. When +minimizing the loss function in Eq. 8 on the training set S in Eq. 1, with probability at least 1 − δ +over the initialization, the training loss after t training steps is bounded by: +L(s, (W (t)) ≤ e−λmin(H(t))L(s, (W (t − 1)) +(12) +Theorem 3.3. Under the assumptions of Theorem 3.2 and Lemma 1, with probability at least (1 − +δ)(1 − ϵ), the following inequality holds true: +L(s, (W (t)) ≤ e−λmin(H(0))e +2 +√ +2M2ηtσ√ +Φ(1−ϵ) +√πδ +L(s, (W (t − 1)) +(13) +4 + +Published as a conference paper at ICLR 2023 +The proof consists of replacing λmin(H(t)) in Eq. 12 with its lower bound given by Theorem 3.2. +Remark 3.4 Theorem. 3.3 shows that after some training steps t, the network with a smaller standard +deviation (σ) of gradients will have a smaller training loss; i.e., the network has a faster convergence +rate at each training step. We further validate this theorem in Sec. 5.2. +Several prior works show that the generalization capacity of a neural network is highly correlated +with its sharpness of the loss function Keskar et al. (2017b;a); Li et al. (2018); Liang et al. (2019). +Usually, a flatter loss landscape leads to a better generalization capacity. Moreover, it has also been +shown that the largest eigenvalue of the Gram matrix of loss can be used to describe the sharpness +of the loss landscape Sagun et al. (2018); more precisely: +Proposition 3.4. The lower the largest eigenvalue of the Gram matrix, the higher the generalization +capacity of the network. [Lewkowycz et al. (2020); Sagun et al. (2016)] +Next, we analyze how the gradient of a neural network impacts the largest eigenvalues of the Gram +matrix and the generalization capacity of the given network. +Theorem 3.5. Under the assumptions of Theorem 3.2, for some positive constants δ and ϵ, if the +learning rate η satisfies η < +λ0 +√πδ +2M 2√ +2Φ(1−ϵ)tσ, then with with probability at least (1 − δ)(1 − ϵ), for +any r ∈ [m], ||wr(0) − wr(t)|| ≤ C = ηtσ +� +Φ(1 − ϵ), and at training step t, the Gram matrix +H(t) satisfies: +λmax(H(t)) ≤ λmax(H(0)) + 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +(14) +Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d). +We provide the proof in Appendix C. +Remark 3.5 Theorem. 3.5 shows that after some training steps t, the network with a smaller standard +deviation (σ) of gradients will have a lower largest eigenvalues of the Gram matrix; i.e., the network +has a flatter loss landscape rate at each training step. Therefore, based on Proposition 3.4, the model +will generalize better. We further validate this theorem in the following section. +3.3 +SUMMARY OF OUR THEORETICAL ANALYSIS +Theorem 3.1, Theorem 3.3 and Theorem 3.5 tell us that the network with high training convergence +speed and generalization capacity should have high absolute mean values and low standard deviation +values for the gradient, w.r.t the parameters across different training samples/batches. Inspired by +these theoretical insights, we next propose a proxy that jointly considers both absolute mean and +standard deviation values. +4 +NEW ZERO-SHOT PROXY AND NAS FRAMEWORK +In this section, we first define our proxy (ZiCo) and then introduce our zero-shot NAS framework. +Following the standard practice, we consider convolutional neural networks (CNNs) as candidate +networks. +4.1 +PROPOSED ZERO-SHOT PROXY: ZICO +Definition 2. Given a neural network with D layers and loss function L, the Zero-shot inverse +Coefficient of Variation (ZiCo) is defined as follows: +ZiCo = +D +� +l=1 +log( +� +ω∈θl +|E[∇ωL(Xi, yi; Θ)]| +� +V ar(∇ωL(Xi, yi; Θ)) +), +i ∈ {1, ..., N} +(15) +where Θ denote the initial parameters of the given network; θl denote the parameters of the lth +layer of the network, and ω represents each element in θl; Xi and yi are the ith input batch and +corresponding labels from the training set; N is number of training batches used to compute ZiCo. +We incorporate log to stabilize the computation by regularizing the extremely large or small values. +5 + +Published as a conference paper at ICLR 2023 +0.9900 +0.9905 +Square Sum of Mean Value ( +j +2 +j ) +0.350 +0.375 +0.400 +0.425 +0.450 +Total Training Loss +Loss vs. Mean value +(a) Loss vs. Mean (linear) +0.5 +1.0 +1.5 +Square Sum of Variance ( +j +2 +j ) +5 +10 +15 +Total Training Loss +Loss vs. Variance +(b) Loss vs. variance (linear) +Figure 1: Training loss vs. square sum of mean gradients and the sum of gradients variances for +linear networks on MNIST after one epoch. Clearly, larger mean gradient values lead to lower loss +values; also, networks with smaller � +j σ2 +j have lower loss values. +0.000 0.005 0.010 0.015 0.020 0.025 +Standard Deviation ( ) +0.2 +0.4 +0.6 +0.8 +Training Loss +Loss vs. Standard Deviation +(a) Training Loss vs. std. dev (w/ ReLU) +0.0000.0050.0100.0150.0200.0250.030 +Standard Deviation ( ) +0 +5 +10 +15 +20 +25 +30 +35 +Test Loss +Loss vs. Standard Deviation +(b) Test Loss vs. std. dev (w/ ReLU) +Figure 2: Training loss and Test loss vs. standard deviation of gradients for two-layer MLPs with +ReLU on MNIST after one training epoch. Networks with smaller σ tend to have lower training loss +and test loss values. We provide more results in Sec C.1. +Of note, our metric is applicable to general CNNs; i.e., there’s no restriction w.r.t. the neural ar- +chitecture when calculating ZiCo. As discussed in Section 3.3, the networks with higher ZiCo tend +to have better convergence rates and higher generalization capacity. Hence, the architectures with +higher ZiCo are better architectures. +We remark that the loss values in Eq. 15 are all computed with the initial parameters Θ; that is, +we never update the value of the parameters when computing ZiCo for a given network (hence it +follows the basic principle of zero-shot NAS, i.e., never train, and only use the initial parameters). In +practice, two batches are enough to make ZiCo achieve the SOTA performance among all previously +proposed accuracy proxies (see Sec. 5.5). Hence, we use only two input batches (N = 2) to compute +ZiCo; this makes ZiCo highly time efficient for a given network. +5 +EXPERIMENTAL RESULTS +5.1 +EXPERIMENTAL SETUP +We conduct the following types of experiments: (i) Empirical validation of Theorem 3.1, The- +orem 3.3 and Theorem 3.5; (ii) Evaluation of the proposed ZiCo on multiple NAS benchmarks; +(iii) Illustration of ZiCo-based zero-shot NAS on ImageNet. +For the experiments (i), to validate Theorem 3.1, we optimize a linear model as in Eq. 2 on the +MNIST dataset, the mean gradient values and the standard deviation vs. the total training loss. +Moreover, we also optimize the model defined by Eq. 7 on MNIST and report the training loss vs. +the standard deviation in order to validate Theorem 3.2 and Theorem 3.5. +For experiments (ii), we compare our proposed ZiCo against existing proxies on three mainstream +NAS benchmarks: NATSBench is a popular cell-based search space with two different search +spaces: (1) NATSBench-TSS consisting of 15625 total architectures with different cell structures +6 + +Published as a conference paper at ICLR 2023 +Grad_norm +SNIP +GraSP +Fisher +Synflow +Zen-score +FLOPs +#Params +ZiCo +0.0 +0.5 +Correlation +-0.17 +-0.12 +0.2 +-0.2 +0.23 +0.46 +0.31 +0.31 +0.46 +-0.25 +-0.17 +0.29 +-0.28 +0.35 +0.63 +0.44 +0.43 +0.63 +Correlation Coefficients between Proxies and Test Accuracy +Spearman's +Kendall's +Figure 3: The correlation coefficients between various zero-cost proxies vs. test accuracy on NAS- +Bench101 search space for CIFAR10 dataset. As shown, our proposed ZiCo correlates best with the +real test accuracy and is significantly better than all other proxies except for Zen-score. +trained on CIFAR10, CIFAR100, and ImageNet16-120 (Img16-120) datasets, which is just renamed +from NASBench-201 Dong & Yang (2020); (2) NATSBench-SSS contains includes 32768 architec- +tures (which differ only in the width values of each layer) and is also trained on the same three above +datasets Dong et al. (2021). NASBench101 provides users with 423k neural architectures with their +test accuracy on CIFAR10 dataset; the architectures are built by stacking the same cell multiple +times Ying et al. (2019). TransNASBench- 101-Mirco contains 4096 networks with different cell +structures on various downstream applications (see Appendix E.2) Duan et al. (2021). +For experiments (iii), we use ZiCo to conduct the zero-shot NAS (see Algorithm 1) on ImageNet. +We first use Algorithm 1 to find the networks with the highest ZiCo under various FLOPs budgets. +We conduct the search for 100k steps; this takes 10 hours on a single NVIDIA 3090 GPU (i.e., 0.4 +GPU days). Then, we train the obtained network with the exact same training setup as Lin et al. +(2021). Specifically, we train the neural network for 480 epochs with the batch size 512 and input +resolution 224. We also use the distillation-based training loss functions by taking Efficient-B3 as +the teacher. Finally, we set the initial learning rate as 0.1 with a cosine annealing scheduling scheme. +5.2 +VALIDATION OF THEOREM 3.1&3.3&3.5 +To empirically validate Theorem 3.1, we first create the training set S by normalizing randomly +sampled 1000 training samples in MNIST and normalizing them with their L2-norm. We compute +the gradient w.r.t. the network parameters for each individual training sample. Next, as discussed in +Theorem 3.1, we use the accumulated gradient over these samples to update the network parameters +with learning rate η = 1. Then, we calculate the square sum of mean gradients and the total +training loss. We repeat the above process 1000 times on the same S. As shown in Fig. 1(a), +we plot the total training loss vs. square sum of mean gradients as defined in Eq. 5. Clearly, the +networks with the higher square sum of mean gradients values tend to have lower training loss. In +comparison, Fig. 1(b) shows that networks with a lower square sum of variance value tend to have +lower training loss values, which coincides with the conclusion drawn from Eq. 6. These results +empirically validate our Theorem 3.1. +Moreover, to optimize a two-layer MLP with ReLU activation functions as defined in Eq. 7, we use +the entire training set of MNIST and apply the gradient descent (Eq. 9) to update the weights. We +set the batch size as 256 and measure the standard deviation of gradients (σ) w.r.t. parameters across +different training batches. We set a very small learning rate η = 10−8 to satisfy the assumption in +Theorem 3.2 and Theorem 3.5. We plot the training loss and test loss after one training epoch vs. +standard deviation of gradients (σ) in Fig. 2(a). Clearly, the results show that if a network has a +lower gradient standard deviation, then it tends to have lower training loss values, and thus, a faster +convergence rate. These results empirically prove our claims in Theorem 3.3. Similarly, Fig. 2(a) +show that if a network has a lower gradient standard deviation, then it tends to have lower test loss +values, which empirically validates Theorem 3.5. +5.3 +ZICO VS. OTHER PROXIES ON NAS BENCHMARKS +We first calculate the correlation coefficients between various proxies and the test accuracy on +CIFAR10, CIFAR100, and ImageNet16-120 datasets for NATSBench. +As shown in Table. 1, +ZiCo achieves the highest correlation with the real rest accuracy, except for CIFAR10/100 on +NATSBench-SSS. Moreover, we observe that most of the previously proposed proxies work well +for some specific scenarios, but do not generalize well to other scenarios. For example, though +Synflow is slightly better than ZiCo for CIFAR10/100 on NATSBench-SSS, it has poor correlation +scores in Img16-120. Similarly, Zen-score performs well for Img16-120 on NATSBench-SSS, but +it doesn’t work well on NATSBench-TSS. In contrast, ZiCo has either the highest or second highest +correlation coefficients under all these scenarios in Table 1. We provide more results in Appendix E. +7 + +Published as a conference paper at ICLR 2023 +Table 1: The correlation coefficients between various zero-cost proxies and two naive proxies +(#Params and FLOPs) vs. test accuracy on NATSBench-SSS and NATSBench-TSS (KT and SPR +represent Kendall’s τ and Spearman’s ρ, respectively). The results in italics represent the values of +#Params’ correlation coefficients. The best results are shown with bold fonts. Clearly, our proposed +ZiCo is the only proxy that works consistently better than #Params and is generally the best among +all these proxies. We provide more results in Appendix E.1 and Table 4. +NATSBench-TSS (NASBench201) +Dataset +CIFAR10 +CIFAR100 +Img16-120 +Proxy +Correlation +KT +SPR +KT +SPR +KT +SPR +Grad norm Abdelfattah et al. (2021) +0.46 +0.63 +0.47 +0.63 +0.43 +0.58 +SNIP Lee et al. (2019b) +0.46 +0.63 +0.46 +0.63 +0.43 +0.58 +GraSP Wang et al. (2020) +0.37 +0.54 +0.36 +0.51 +0.40 +0.56 +Fisher Liu et al. (2021) +0.40 +0.55 +0.41 +0.55 +0.37 +0.50 +Synflow Tanaka et al. (2020) +0.54 +0.73 +0.57 +0.76 +0.56 +0.75 +Zen-score Lin et al. (2021) +0.29 +0.38 +0.28 +0.36 +0.29 +0.40 +FLOPs +0.54 +0.73 +0.51 +0.71 +0.49 +0.67 +#Params +0.57 +0.75 +0.55 +0.73 +0.52 +0.69 +ZiCo +0.61 +0.80 +0.61 +0.81 +0.60 +0.79 +NATSBench-SSS +Dataset +CIFAR10 +CIFAR100 +Img16-120 +Proxy +Correlation +KT +SPR +KT +SPR +KT +SPR +Grad norm Abdelfattah et al. (2021) +0.35 +0.51 +0.34 +0.49 +0.49 +0.67 +SNIP Lee et al. (2019b) +0.42 +0.59 +0.46 +0.62 +0.57 +0.76 +GraSP Wang et al. (2020) +-0.09 +-0.13 +0.01 +0.01 +0.29 +0.42 +Fisher Liu et al. (2021) +0.30 +0.44 +0.41 +0.55 +0.33 +0.47 +Synflow Tanaka et al. (2020) +0.61 +0.81 +0.60 +0.80 +0.39 +0.57 +Zen-score Lin et al. (2021) +0.50 +0.69 +0.52 +0.71 +0.69 +0.87 +FLOPs +0.19 +0.28 +0.21 +0.30 +0.38 +0.53 +#Params +0.53 +0.72 +0.54 +0.73 +0.65 +0.84 +ZiCo +0.54 +0.73 +0.55 +0.75 +0.70 +0.88 +Table 2: The test accuracy of optimal architectures obtained by various zero-shot proxies (average +on 5 runs) on NATSBench-TSS search space. The best results are shown with bold fonts. +CIFAR100 +Groud Truth +Grad norm +SNIP +GraSP +Fisher +Jacob cov +Synflow +Zen-score +#Params +FLOPs +ZiCo +73.5 +60.0 +60.0 +60.0 +60.0 +68.9 +71.1 +68.1 +71.1 +71.1 +71.1±0.3 +Img16-120 +Groud Truth +Grad norm +SNIP +GraSP +Fisher +Jacob cov +Synflow +Zen-score +#Params +FLOPs +ZiCo +47.3 +29.3 +29.3 +5.5 +29.3 +25.1 +41.2 +40.8 +41.4 +41.4 +41.8±0.3 +CIFAR10 +Groud Truth +Grad norm +SNIP +GraSP +Fisher +Jacob cov +Synflow +Zen-score +#Params +FLOPs +ZiCo +94.5 +89.5 +89.5 +89.5 +89.5 +88.4 +90.4 +90.6 +93.7 +93.7 +94.0±0.4 +For NASBench101, as shown in Fig. 3, ZiCo has a significantly higher correlation score with the real +test accuracy than all the other proxies, except Zen-score. For example, ZiCo has a 0.46 Kendall’s τ +score, while #Params is only 0.31. In general, ZiCo has the highest correlation coefficients among +all existing proxies for various search spaces and datasets of NATSBench and NASBench101. +Beside the correlation coefficients, we also report the optimal architectures found with various prox- +ies. As shown in Table 2, the architectures found via ZiCo have the highest test accuracy on all +these three datasets. To our best knowledge, ZiCo is the first proxy that shows a consistently higher +correlation coefficient compared to #Params. +The above results validate the effectiveness of our proposed ZiCo; thus, ZiCo can be directly used +to search for optimal networks for various budgets. Next, we describe the search results in detail. +5.4 +ZICO ON IMAGENET +Search Space We use the commonly used MobileNetv2-based search space where the candidate +networks are built by stacking multiple Inverted Bottleneck Blocks (IBNs) with SE modules Sandler +et al. (2018); Pham et al. (2018); Lin et al. (2021). As for each IBN, the kernel size of the depth- +wise convolutional layer is sampled from {3,5,7} and the expansion ratio is randomly selected from +{1,2,4,6}. We primarily consider ReLU as the activation function. We use standard Kaiming Init +to initialize all linear and convolution layers for every candidate networks He et al. (2015). More +details of the search space are given in Appendix D. +We use Algorithm 1 to search networks under various FLOPs budgets (450M, 600M, and 1000M) +within the above search space. As shown in Table 3, ZiCo outperforms most previous NAS ap- +8 + +Published as a conference paper at ICLR 2023 +Table 3: Comparison of Top-1 accuracy of our ZiCo-based NAS against SOTA NAS methods on +ImageNet under various FLOP budgets (averages over three runs). For the ‘Method’ column, ‘MS’ +means multi-shot NAS; ‘OS’ is short for one-shot NAS; Scaling represents network scaling methods; +‘ZS’ is short for zero-shot NAS. OFA‡ is trained from scratch and reported in Moons et al. (2021). +Budget (maximal #FLOPs) +Approach +FLOPs +Top-1 [%] +Method +Costs [GPU Days] +450M +EfficientNet-B0 Tan & Le (2019) +390M +77.1 +Scaling +3800 +MnasNet-A3 Tan et al. (2019) +403M +76.7 +MS +- +OFA‡ Cai et al. (2020) +406M +77.7 +OS +50 +BN-NAS Chen et al. (2021a) +470M +75.7 +MS +0.8 +RLNAS Zhang et al. (2021) +473M +75.6 +OS +- +NASNet-B Zoph et al. (2018) +488M +72.8 +MS +1800 +CARS-D Yang et al. (2020) +496M +73.3 +MS +0.4 +DONNA Moons et al. (2021) +501M +78.0 +OS +405 +#Params +451M +63.5 +ZS +0.02 +ZiCo (Ours) +448M +78.1±0.3 +ZS +0.4 +600M +DARTS Liu et al. (2019) +574M +73.3 +OS +4 +NAO Luo et al. (2018) +584M +75.5 +MS +58.3 +PC-DARTS Xu et al. (2019) +586M +75.8 +OS +3.8 +BigNAS-L Yu et al. (2020a) +586M +79.5 +OS +2304 (TPU days) +PNAS Liu et al. (2018a) +588M +74.2 +MS +224 +CARS-I Yang et al. (2020) +591M +75.2 +MS +0.4 +EnTranNAS Yang et al. (2021) +594M +76.2 +OS +2.1 +ProxylessNAS Cai et al. (2019) +595M +76.0 +OS +8.3 +RLNAS Zhang et al. (2021) +597M +75.9 +OS +- +MAGIC-AT Xu et al. (2022) +598M +76.8 +OS +2 +SemiNAS Luo et al. (2020) +599M +76.5 +MS +4 +DONNA Moons et al. (2021) +599M +78.4 +OS +405 +Zen-score Lin et al. (2021) +611M +79.1 +ZS +0.5 +OFA‡ Cai et al. (2020) +662M +78.7 +OS +50 +EfficientNet-B1 Tan & Le (2019) +700M +79.1 +Scaling +3800 +ZiCo (Ours) +603M +79.4±0.3 +ZS +0.4 +1000M +sharpDARTS Hundt et al. (2019) +950M +76.0 +OS +- +Zen-score Lin et al. (2021) +934M +80.8 +ZS +0.5 +EfficientNet-B2 Tan & Le (2019) +1000M +80.1 +Scaling +3800 +ZiCo (Ours) +1005M +80.5±0.2 +ZS +0.4 +proaches by a large margin. For example, when the FLOPs budget is around 450M, ZiCo achieves +78.1% Top-1 accuracy, which is competitive with one of the SOTA NAS methods (DONNA), but +with fewer FLOPs and 648× faster search speed Moons et al. (2021). Moreover, if the FLOPs is +600M, ZiCo achieves 2.6% higher Top-1 Accuracy than the latest one-shot NAS method (MAGIC- +AT) with a 3× reduction in terms of search time Xu et al. (2022). +To make further comparison with #Params, we also use #Params as the proxy and Algorithm 1 +to conduct the search under a 450M FLOPs budget. As shown in Table 3, the obtained network +by #Params has a 14.6% lower accuracy than ours (63.5% vs. 78.1%). Hence, even though the +correlations for ZiCo and #Params in Table 1 and the optimal networks in Table 2 are similar for +small-scale datasets, ZiCo significantly outperforms naive baselines like #Params for large datasets +like ImageMet. To conclude, ZiCo achieves SOTA results for Zero-Shot NAS and outperforms naive +methods, existing zero-shot proxies, as well as several one-shot and multi-shot methods. +We remark that these results demonstrate two benefits of our proposed ZiCo: (i) Lightweight com- +putation costs. As discussed in Sec 3, during the search process, to evaluate a given architecture, we +only need to conduct the backward propagation twice (only takes 0.3s on an NVIDIA 3090 GPU). +The computation efficiency and exemption of training enable ZiCo to significantly reduce the search +time of NAS. (ii) High correlation with the real test accuracy. As demonstrated in Sec 5.3, ZiCo +has a very high correlation score with real accuracy for architectures from various search spaces and +datasets. Hence, ZiCo can accurately predict the test accuracy of diverse neural architectures, thus +helping find the optimal architectures with the best test performance. +5.5 +ABLATION STUDY +Number of batches We randomly select 2000 networks from NATSBench-TSS on CIFAR100 +dataset and compute ZiCo under varying number of training batches (N in Eq. 15) from {2,...,10}. +We then calculate the correlation score between ZiCo computed under different N values and the +real test accuracy. Fig. 4(a) shows that using two batches to compute ZiCo generates the highest +score. Hence, in our work, we always use two batches (N = 2) to compute ZiCo since it is both +accurate and time-efficient. +9 + +Published as a conference paper at ICLR 2023 +2 +4 +6 +8 +10 +The Number of Training Batches +0.6 +0.7 +0.8 +Correlation +Correlation Coefficients vs. #Batch +Kendall's +Spearman's +(a) Correlation vs. #Batch +0 +20 +40 +60 +80 +100 +120 +Batch Size +0.6 +0.7 +0.8 +Correlation +Correlation Coefficients vs. Batch Size +Kendall's +Spearman's +(b) Correlation vs. Batch Size +Figure 4: Ablation study. (a) The correlation coefficients between ZiCo computed under varying +number of batches and real test accuracy. (b) The correlation coefficients between ZiCo computed +with varying batch size and real test accuracy. +Batch size We pick 2000 networks from NATSBench-TSS on CIFAR100 and compute ZiCo with +two batches under varying batch size {1,2,4,8,16,32,64,128}. We then calculate the correlation score +between ZiCo computed under various batch sizes and the real test accuracy. As shown in Fig. 4(b), +batch size 64 is enough to stabilize the coefficient. Therefore, we set the batch size as 128 and use +two batches to compute ZiCo. We provide more ablation studies in Appendix F. +6 +CONCLUSION +In this work, we have proposed ZiCo, a new SOTA proxy for zero-shot NAS. As the main theoretical +contribution, we first reveal how the mean value and standard deviation of gradients impact the train- +ing convergence of a given architecture. Based on this theoretical analysis, we have shown that ZiCo +works better than all zero-shot NAS proxies proposed so far on multiple popular NAS-Benchmarks +(NASBench101, NATSBench-SSS/TSS) for multiple datasets (CIFAR10/100, ImageNet16-120). In +particular, we have demonstrated that ZiCo is consistently better than (#Params) and existing zero- +shot proxies. Moreover, ZiCo enables us to find architectures with competitive test performance to +representative one-shot and multi-shot NAS methods, but with much lower search costs. For exam- +ple, ZiCo-based NAS can find the architectures with 78.1%, 79.4%, and 80.4% test accuracies under +450M, 600M, and 1000M FLOPs budgets on ImageNet within 0.4 GPU days. +REFERENCES +Mohamed S. Abdelfattah, Abhinav Mehrotra, Lukasz Dudziak, and Nicholas Donald Lane. Zero- +cost proxies for lightweight NAS. In 9th International Conference on Learning Representations, +ICLR 2021, Virtual Event, Austria, May 3-7, 2021, 2021. +Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over- +parameterization. In Proceedings of the 36th International Conference on Machine Learning, +ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine +Learning Research, pp. 242–252. PMLR, 2019. +Sanjeev Arora, Simon S. Du, Wei Hu, Zhiyuan Li, Ruslan Salakhutdinov, and Ruosong Wang. +On exact computation with an infinitely wide neural net. In Advances in Neural Information +Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, +NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 8139–8148, 2019a. +Sanjeev Arora, Simon S. Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-grained analysis of +optimization and generalization for overparameterized two-layer neural networks. In Proceedings +of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long +Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp. 322–332. +PMLR, 2019b. +Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architec- +tures using reinforcement learning. In 5th International Conference on Learning Representations, +ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017. +Friedrich L Bauer and Charles T Fike. Norms and exclusion theorems. Numerische Mathematik, 2 +(1):137–141, 1960. +10 + +Published as a conference paper at ICLR 2023 +Kartikeya Bhardwaj, Guihong Li, and Radu Marculescu. How does topology influence gradient +propagation and model performance of deep networks with densenet-type skip connections? In +IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, +2021, pp. 13498–13507. Computer Vision Foundation / IEEE, 2021. +Kartikeya Bhardwaj, James Ward, Caleb Tung, Dibakar Gope, Lingchuan Meng, Igor Fedorov, Alex +Chalfin, Paul Whatmough, and Danny Loh. Restructurable activation networks. arXiv preprint +arXiv:2208.08562, 2022a. +Kartikeya Bhardwaj, James Ward, Caleb Tung, Dibakar Gope, Lingchuan Meng, Igor Fedorov, +Alex Chalfin, Paul N. Whatmough, and Danny Loh. Restructurable activation networks. CoRR, +abs/2208.08562, 2022b. +Tom B. Brown et al. Language models are few-shot learners. In Advances in Neural Information +Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, +NeurIPS 2020, December 6-12, 2020, virtual, 2020. +Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Efficient architecture search by +network transformation. In Proceedings of the AAAI Conference on Artificial Intelligence, 2018. +Han Cai, Ligeng Zhu, and Song Han. ProxylessNAS: Direct neural architecture search on target +task and hardware. In International Conference on Learning Representations, 2019. +Han Cai, Chuang Gan, Tianzhe Wang, Zhekai Zhang, and Song Han. +Once-for-all: Train one +network and specialize it for efficient deployment. +In International Conference on Learning +Representations, 2020. +Boyu Chen, Peixia Li, Baopu Li, Chen Lin, Chuming Li, Ming Sun, Junjie Yan, and Wanli Ouyang. +BN-NAS: neural architecture search with batch normalization. In 2021 IEEE/CVF International +Conference on Computer Vision, ICCV 2021, Montreal, QC, Canada, October 10-17, 2021, pp. +307–316. IEEE, 2021a. +Wuyang Chen, Xinyu Gong, and Zhangyang Wang. Neural architecture search on imagenet in four +GPU hours: A theoretically inspired perspective. In 9th International Conference on Learning +Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021, 2021b. +Xiangning Chen and Cho-Jui Hsieh. Stabilizing differentiable architecture search via perturbation- +based regularization. In Proceedings of the 37th International Conference on Machine Learning, +ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning +Research, pp. 1554–1565. PMLR, 2020. +Xin Chen, Lingxi Xie, Jun Wu, and Qi Tian. Progressive differentiable architecture search: Bridging +the depth gap between search and evaluation. In Proceedings of the IEEE/CVF international +conference on computer vision, pp. 1294–1303, 2019. +Wei-Lin Chiang et al. Cluster-gcn: An Efficient Algorithm for Training Deep and Large Graph +Convolutional Networks. In Proceedings of the 25th ACM SIGKDD International Conference on +Knowledge Discovery & Data Mining, pp. 257–266, 2019. +L´ena¨ıc Chizat, Edouard Oyallon, and Francis R. Bach. On lazy training in differentiable program- +ming. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural +Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, +Canada, pp. 2933–2943, 2019. +Youngmin Cho and Lawrence K. Saul. Kernel methods for deep learning. In Advances in Neural +Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing +Systems 2009. Proceedings of a meeting held 7-10 December 2009, Vancouver, British Columbia, +Canada, pp. 342–350. Curran Associates, Inc., 2009. +Xiangxiang Chu, Bo Zhang, and Ruijun Xu. Fairnas: Rethinking evaluation fairness of weight +sharing neural architecture search. In Proceedings of the IEEE/CVF International Conference on +Computer Vision, pp. 12239–12248, 2021. +11 + +Published as a conference paper at ICLR 2023 +Tu Do and Ngoc Hoang Luong. Training-free multi-objective evolutionary neural architecture search +via neural tangent kernel and number of linear regions. In International Conference on Neural +Information Processing, pp. 335–347. Springer, 2021. +Xuanyi Dong and Yi Yang. Searching for a robust neural architecture in four gpu hours. In Proceed- +ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1761–1770, +2019. +Xuanyi Dong and Yi Yang. Nas-bench-201: Extending the scope of reproducible neural architecture +search. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, +Ethiopia, April 26-30, 2020, 2020. +Xuanyi Dong, Lu Liu, Katarzyna Musial, and Bogdan Gabrys. Nats-bench: Benchmarking nas +algorithms for architecture topology and size. IEEE transactions on pattern analysis and machine +intelligence, 2021. +Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas +Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszko- +reit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at +scale. In International Conference on Learning Representations, 2021. +Simon S. Du, Jason D. Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds +global minima of deep neural networks. In Proceedings of the 36th International Conference on +Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of +Proceedings of Machine Learning Research, pp. 1675–1685. PMLR, 2019a. +Simon S. Du, Xiyu Zhai, Barnab´as P´oczos, and Aarti Singh. Gradient descent provably optimizes +over-parameterized neural networks. In 7th International Conference on Learning Representa- +tions, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019b. +Yawen Duan, Xin Chen, Hang Xu, Zewei Chen, Xiaodan Liang, Tong Zhang, and Zhenguo Li. +Transnas-bench-101: Improving transferability and generalizability of cross-task neural archi- +tecture search. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, +virtual, June 19-25, 2021, pp. 5251–5260. Computer Vision Foundation / IEEE, 2021. +Stanley C Eisenstat and Ilse CF Ipsen. Three absolute perturbation bounds for matrix eigenvalues +imply relative bounds. SIAM Journal on Matrix Analysis and Applications, 20(1):149–158, 1998. +Thomas Elsken, Jan Hendrik Metzen, and Frank Hutter. Neural architecture search: A survey. The +Journal of Machine Learning Research, 20(1):1997–2017, 2019. +Adri`a Garriga-Alonso, Carl Edward Rasmussen, and Laurence Aitchison. Deep convolutional net- +works as shallow gaussian processes. In 7th International Conference on Learning Representa- +tions, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019. +Eugene Golikov, Eduard Pokonechnyy, and Vladimir Korviakov. Neural tangent kernel: A survey. +CoRR, abs/2208.13614, 2022. +Xinyu Gong, Shiyu Chang, Yifan Jiang, and Zhangyang Wang. Autogan: Neural architecture search +for generative adversarial networks. In Proceedings of the IEEE/CVF International Conference +on Computer Vision, pp. 3224–3234, 2019. +Zichao Guo, Xiangyu Zhang, Haoyuan Mu, Wen Heng, Zechun Liu, Yichen Wei, and Jian Sun. +Single path one-shot neural architecture search with uniform sampling. In European conference +on computer vision, pp. 544–560. Springer, 2020. +Boris Hanin and Mihai Nica. Finite depth and width corrections to the neural tangent kernel. In 8th +International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April +26-30, 2020, 2020. +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing +human-level performance on imagenet classification. In 2015 IEEE International Conference +on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015, pp. 1026–1034. IEEE +Computer Society, 2015. +12 + +Published as a conference paper at ICLR 2023 +Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image +Recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, +pp. 770–778, 2016. +Andrew Howard, Mark Sandler, Grace Chu, Liang-Chieh Chen, Bo Chen, Mingxing Tan, Weijun +Wang, Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al. Searching for mobilenetv3. In Pro- +ceedings of the IEEE/CVF international conference on computer vision, pp. 1314–1324, 2019. +Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely Connected +Convolutional Networks. In Proceedings of the IEEE conference on computer vision and pattern +recognition, pp. 4700–4708, 2017. +Andrew Hundt, Varun Jain, and Gregory D. Hager. sharpdarts: Faster and more accurate differen- +tiable architecture search. CoRR, abs/1903.09900, 2019. +Thorir Mar Ingolfsson, Mark Vero, Xiaying Wang, Lorenzo Lamberti, Luca Benini, and Matteo +Spallanzani. Reducing neural architecture search spaces with training-free statistics and compu- +tational graph clustering. In CF ’22: 19th ACM International Conference on Computing Frontiers, +Turin, Italy, May 17 - 22, 2022, pp. 213–214. ACM, 2022. +Arthur Jacot, Cl´ement Hongler, and Franck Gabriel. Neural tangent kernel: Convergence and gener- +alization in neural networks. In Advances in Neural Information Processing Systems 31: Annual +Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, +Montr´eal, Canada, pp. 8580–8589, 2018. +Mojan Javaheripi, Shital Shah, Subhabrata Mukherjee, Tomasz L. Religa, Caio C. T. Mendes, Gus- +tavo H. de Rosa, S´ebastien Bubeck, Farinaz Koushanfar, and Debadeepta Dey. Litetransform- +ersearch: Training-free on-device search for efficient autoregressive language models. CoRR, +abs/2203.02094, 2022. +Kirthevasan Kandasamy, Willie Neiswanger, Jeff Schneider, Barnab´as P´oczos, and Eric P. Xing. +Neural architecture search with bayesian optimisation and optimal transport. In Advances in Neu- +ral Information Processing Systems 31: Annual Conference on Neural Information Processing +Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada, pp. 2020–2029, 2018. +Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Pe- +ter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In 5th +International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, +2017, Conference Track Proceedings, 2017a. +Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Pe- +ter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. In 5th +International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, +2017, Conference Track Proceedings, 2017b. +Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo- +lutional neural networks. In Advances in Neural Information Processing Systems, 2012. +Jaehoon Lee, Lechao Xiao, Samuel S. Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl- +Dickstein, and Jeffrey Pennington. Wide neural networks of any depth evolve as linear models +under gradient descent. In Advances in Neural Information Processing Systems 32: Annual Con- +ference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, +Vancouver, BC, Canada, pp. 8570–8581, 2019a. +Namhoon Lee, Thalaiyasingam Ajanthan, and Philip Torr. +SNIP: SINGLE-SHOT NETWORK +PRUNING BASED ON CONNECTION SENSITIVITY. In International Conference on Learn- +ing Representations, 2019b. +Aitor Lewkowycz, Yasaman Bahri, Ethan Dyer, Jascha Sohl-Dickstein, and Guy Gur-Ari. The large +learning rate phase of deep learning: the catapult mechanism. CoRR, abs/2003.02218, 2020. +Guihong Li, Sumit K. Mandal, ¨Umit Y. Ogras, and Radu Marculescu. FLASH: fast neural architec- +ture search with hardware optimization. ACM Trans. Embed. Comput. Syst., 20(5s):63:1–63:26, +2021. +13 + +Published as a conference paper at ICLR 2023 +Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the loss land- +scape of neural nets. In Advances in Neural Information Processing Systems 31: Annual Con- +ference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, +Montr´eal, Canada, pp. 6391–6401, 2018. +Liam Li and Ameet Talwalkar. Random search and reproducibility for neural architecture search. In +Uncertainty in artificial intelligence, pp. 367–377. PMLR, 2020. +Tengyuan Liang, Tomaso A. Poggio, Alexander Rakhlin, and James Stokes. +Fisher-rao metric, +geometry, and complexity of neural networks. In The 22nd International Conference on Artificial +Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 +of Proceedings of Machine Learning Research, pp. 888–896. PMLR, 2019. +Ming Lin, Pichao Wang, Zhenhong Sun, Hesen Chen, Xiuyu Sun, Qi Qian, Hao Li, and Rong +Jin. Zen-nas: A zero-shot nas for high-performance image recognition. In Proceedings of the +IEEE/CVF International Conference on Computer Vision, pp. 347–356, 2021. +Chenxi Liu et al. Progressive Neural Architecture Search. In Proceedings of the European Confer- +ence on Computer Vision (ECCV), pp. 19–34, 2018a. +Hanxiao Liu, Karen Simonyan, Oriol Vinyals, Chrisantha Fernando, and Koray Kavukcuoglu. Hi- +erarchical representations for efficient architecture search. In 6th International Conference on +Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Confer- +ence Track Proceedings, 2018b. +Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: differentiable architecture search. In +7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, +May 6-9, 2019, 2019. +Liyang Liu, Shilong Zhang, Zhanghui Kuang, Aojun Zhou, Jing-Hao Xue, Xinjiang Wang, Yimin +Chen, Wenming Yang, Qingmin Liao, and Wayne Zhang. Group fisher pruning for practical +network compression. In Proceedings of the 38th International Conference on Machine Learning, +ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning +Research, pp. 7021–7032. PMLR, 2021. +Shuying Liu and Weihong Deng. Very deep convolutional neural network based image classification +using small training sample size. In 2015 3rd IAPR Asian Conference on Pattern Recognition +(ACPR), pp. 730–734, 2015. +Vasco Lopes, Saeid Alirezazadeh, and Lu´ıs A Alexandre. Epe-nas: Efficient performance estimation +without training for neural architecture search. In International Conference on Artificial Neural +Networks, pp. 552–563. Springer, 2021. +Yiping Lu, Chao Ma, Yulong Lu, Jianfeng Lu, and Lexing Ying. A mean-field analysis of deep +resnet and beyond: Towards provable optimization via overparameterization from depth. CoRR, +abs/2003.05508, 2020. +Renqian Luo, Fei Tian, Tao Qin, Enhong Chen, and Tie-Yan Liu. Neural architecture optimization. +Advances in neural information processing systems, 31, 2018. +Renqian Luo, Xu Tan, Rui Wang, Tao Qin, Enhong Chen, and Tie-Yan Liu. Semi-supervised neural +architecture search. In Advances in Neural Information Processing Systems 33: Annual Con- +ference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, +virtual, 2020. +Song Mei, Theodor Misiakiewicz, and Andrea Montanari. Mean-field theory of two-layers neural +networks: dimension-free bounds and kernel limit. In Conference on Learning Theory, COLT +2019, 25-28 June 2019, Phoenix, AZ, USA, volume 99 of Proceedings of Machine Learning Re- +search, pp. 2388–2464. PMLR, 2019. +Joe Mellor, Jack Turner, Amos Storkey, and Elliot J Crowley. Neural architecture search without +training. In International Conference on Machine Learning, pp. 7588–7598. PMLR, 2021. +14 + +Published as a conference paper at ICLR 2023 +Bert Moons, Parham Noorzad, Andrii Skliar, Giovanni Mariani, Dushyant Mehta, Chris Lott, and +Tijmen Blankevoort. Distilling optimal neural networks: Rapid search in diverse spaces. In 2021 +IEEE/CVF International Conference on Computer Vision, ICCV 2021, Montreal, QC, Canada, +October 10-17, 2021, pp. 12209–12218. IEEE, 2021. +Radford M Neal. Priors for infinite networks. In Bayesian Learning for Neural Networks, pp. 29–53. +Springer, 1996. +Xuefei Ning, Changcheng Tang, Wenshuo Li, Zixuan Zhou, Shuang Liang, Huazhong Yang, and +Yu Wang. Evaluating efficient performance estimators of neural architectures. In Advances in +Neural Information Processing Systems 34: Annual Conference on Neural Information Process- +ing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pp. 12265–12277, 2021. +Hieu Pham, Melody Guan, Barret Zoph, Quoc Le, and Jeff Dean. Efficient neural architecture search +via parameters sharing. In International conference on machine learning, pp. 4095–4104. PMLR, +2018. +Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le. Regularized evolution for image +classifier architecture search. In Proceedings of the aaai conference on artificial intelligence, pp. +4780–4789, 2019. +Esteban Real et al. Large-scale Evolution of Image Classifiers. In International Conference on +Machine Learning, pp. 2902–2911. PMLR, 2017. +Levent Sagun, Leon Bottou, and Yann LeCun. Eigenvalues of the hessian in deep learning: Singu- +larity and beyond. arXiv preprint arXiv:1611.07476, 2016. +Levent Sagun, Utku Evci, V. Ugur G¨uney, Yann N. Dauphin, and L´eon Bottou. Empirical analysis +of the hessian of over-parametrized neural networks. In 6th International Conference on Learning +Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Workshop Track +Proceedings, 2018. +Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mo- +bilenetv2: Inverted Residuals and Linear Bottlenecks. In Proceedings of the IEEE conference on +computer vision and pattern recognition, pp. 4510–4520, 2018. +Yao Shu, Shaofeng Cai, Zhongxiang Dai, Beng Chin Ooi, and Bryan Kian Hsiang Low. NASI: +label- and data-agnostic neural architecture search at initialization. In The Tenth International +Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022, 2022a. +Yao Shu, Zhongxiang Dai, Zhaoxuan Wu, and Bryan Kian Hsiang Low. Unifying and boosting +gradient-based training-free neural architecture search. CoRR, abs/2201.09785, 2022b. +Dimitrios Stamoulis, Ruizhou Ding, Di Wang, Dimitrios Lymberopoulos, Bodhi Priyantha, Jie Liu, +and Diana Marculescu. Single-path NAS: designing hardware-efficient convnets in less than 4 +hours. In Machine Learning and Knowledge Discovery in Databases - European Conference, +ECML PKDD 2019, W¨urzburg, Germany, September 16-20, 2019, Proceedings, Part II, volume +11907 of Lecture Notes in Computer Science, pp. 481–497. Springer, 2019. +Zhenhong Sun, Ming Lin, Xiuyu Sun, Zhiyu Tan, and Rong Jin. Revisiting efficient object detection +backbones from zero-shot neural architecture search. arXiv preprint arXiv:2111.13336, 2021. +Mingxing Tan and Quoc V. Le. Efficientnet: Rethinking model scaling for convolutional neural +networks. In Proceedings of the 36th International Conference on Machine Learning, ICML +2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine +Learning Research, pp. 6105–6114. PMLR, 2019. +Mingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, Mark Sandler, Andrew Howard, and +Quoc V Le. Mnasnet: Platform-aware neural architecture search for mobile. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2820–2828, 2019. +Hidenori Tanaka, Daniel Kunin, Daniel L Yamins, and Surya Ganguli. Pruning neural networks +without any data by iteratively conserving synaptic flow. In Advances in Neural Information +Processing Systems, volume 33, pp. 6377–6389. Curran Associates, Inc., 2020. +15 + +Published as a conference paper at ICLR 2023 +Linh Tam Tran and Sung-Ho Bae. Training-free hardware-aware neural architecture search with +reinforcement learning. Journal of Broadcast Engineering, 26(7):855–861, 2021. +Linh-Tam Tran, Muhammad Salman Ali, and Sung-Ho Bae. A feature fusion based indicator for +training-free neural architecture search. IEEE Access, 9:133914–133923, 2021. +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, +Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural informa- +tion processing systems, 30, 2017. +Alvin Wan, Xiaoliang Dai, Peizhao Zhang, Zijian He, Yuandong Tian, Saining Xie, Bichen Wu, +Matthew Yu, Tao Xu, Kan Chen, et al. Fbnetv2: Differentiable neural architecture search for +spatial and channel dimensions. In Proceedings of the IEEE/CVF Conference on Computer Vision +and Pattern Recognition, pp. 12965–12974, 2020. +Chaoqi Wang, Guodong Zhang, and Roger B. Grosse. Picking winning tickets before training by +preserving gradient flow. In 8th International Conference on Learning Representations, ICLR +2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020. +Colin White, Mikhail Khodak, Renbo Tu, Shital Shah, S´ebastien Bubeck, and Debadeepta Dey. A +deeper look at zero-cost proxies for lightweight nas. In ICLR Blog Track, 2022. URL https: +//iclr-blog-track.github.io/2022/03/25/zero-cost-proxies/. +Christopher K. I. Williams. Computing with infinite networks. In Advances in Neural Information +Processing Systems 9, NIPS, Denver, CO, USA, December 2-5, 1996, pp. 295–301. MIT Press, +1996. +Bichen Wu, Xiaoliang Dai, Peizhao Zhang, Yanghan Wang, Fei Sun, Yiming Wu, Yuandong Tian, +Peter Vajda, Yangqing Jia, and Kurt Keutzer. Fbnet: Hardware-aware efficient convnet design +via differentiable neural architecture search. In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pp. 10734–10742, 2019. +Meng-Ting Wu, Hung-I Lin, and Chun-Wei Tsai. A training-free genetic neural architecture search. +In ACM ICEA ’21: 2021 ACM International Conference on Intelligent Computing and its Emerg- +ing Applications, Jinan, China, December 28 - 29, 2022, pp. 65–70. ACM, 2021. +Lichuan Xiang, Lukasz Dudziak, Mohamed S. Abdelfattah, Thomas Chau, Nicholas D. Lane, and +Hongkai Wen. Zero-cost proxies meet differentiable architecture search. CoRR, abs/2106.06799, +2021a. +Lichuan Xiang, Lukasz Dudziak, Mohamed S. Abdelfattah, Thomas Chau, Nicholas D. Lane, and +Hongkai Wen. Zero-cost proxies meet differentiable architecture search. CoRR, abs/2106.06799, +2021b. +Sirui Xie, Hehui Zheng, Chunxiao Liu, and Liang Lin. SNAS: stochastic neural architecture search. +In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, +USA, May 6-9, 2019, 2019. +Jin Xu, Xu Tan, Kaitao Song, Renqian Luo, Yichong Leng, Tao Qin, Tie-Yan Liu, and Jian Li. +Analyzing and mitigating interference in neural architecture search. In International Conference +on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA, volume 162 of +Proceedings of Machine Learning Research, pp. 24646–24662. PMLR, 2022. +Jingjing Xu, Liang Zhao, Junyang Lin, Rundong Gao, Xu Sun, and Hongxia Yang. KNAS: green +neural architecture search. +In Proceedings of the 38th International Conference on Machine +Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine +Learning Research, pp. 11613–11625. PMLR, 2021. +Yuhui Xu, Lingxi Xie, Xiaopeng Zhang, Xin Chen, Guo-Jun Qi, Qi Tian, and Hongkai Xiong. +Pc-darts: Partial channel connections for memory-efficient architecture search. arXiv preprint +arXiv:1907.05737, 2019. +16 + +Published as a conference paper at ICLR 2023 +Yibo Yang, Shan You, Hongyang Li, Fei Wang, Chen Qian, and Zhouchen Lin. Towards improving +the consistency, efficiency, and flexibility of differentiable neural architecture search. In IEEE +Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, +pp. 6667–6676. Computer Vision Foundation / IEEE, 2021. +Zhaohui Yang, Yunhe Wang, Xinghao Chen, Boxin Shi, Chao Xu, Chunjing Xu, Qi Tian, and Chang +Xu. CARS: continuous evolution for efficient neural architecture search. In 2020 IEEE/CVF +Conference on Computer Vision and Pattern Recognition, CVPR 2020, Seattle, WA, USA, June +13-19, 2020, pp. 1826–1835. Computer Vision Foundation / IEEE, 2020. +Chris Ying, Aaron Klein, Eric Christiansen, Esteban Real, Kevin Murphy, and Frank Hutter. Nas- +bench-101: Towards reproducible neural architecture search. +In International Conference on +Machine Learning, pp. 7105–7114. PMLR, 2019. +Jiahui Yu, Pengchong Jin, Hanxiao Liu, Gabriel Bender, Pieter-Jan Kindermans, Mingxing Tan, +Thomas Huang, Xiaodan Song, Ruoming Pang, and Quoc Le. Bignas: Scaling up neural archi- +tecture search with big single-stage models. In European Conference on Computer Vision, pp. +702–717. Springer, 2020a. +Kaicheng Yu, Christian Sciuto, Martin Jaggi, Claudiu Musat, and Mathieu Salzmann. Evaluating +the search phase of neural architecture search. +In 8th International Conference on Learning +Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020b. +Arber Zela, Thomas Elsken, Tonmoy Saikia, Yassine Marrakchi, Thomas Brox, and Frank Hutter. +Understanding and robustifying differentiable architecture search. In 8th International Confer- +ence on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020. +Chris Zhang, Mengye Ren, and Raquel Urtasun. +Graph hypernetworks for neural architecture +search. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, +LA, USA, May 6-9, 2019, 2019a. +Xiao Zhang, Yaodong Yu, Lingxiao Wang, and Quanquan Gu. +Learning one-hidden-layer relu +networks via gradient descent. In The 22nd International Conference on Artificial Intelligence and +Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 of Proceedings +of Machine Learning Research, pp. 1524–1534. PMLR, 2019b. +Xuanyang Zhang, Pengfei Hou, Xiangyu Zhang, and Jian Sun. Neural architecture search with +random labels. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, +virtual, June 19-25, 2021, pp. 10907–10916. Computer Vision Foundation / IEEE, 2021. +Zhihao Zhang and Zhihao Jia. Gradsign: Model performance inference with theoretical insights. +In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, +April 25-29, 2022, 2022. +Dongzhan Zhou, Xinchi Zhou, Wenwei Zhang, Chen Change Loy, Shuai Yi, Xuesen Zhang, and +Wanli Ouyang. Econas: Finding proxies for economical neural architecture search. In Proceed- +ings of the IEEE/CVF Conference on computer vision and pattern recognition, pp. 11396–11404, +2020. +Hongpeng Zhou, Minghao Yang, Jun Wang, and Wei Pan. Bayesnas: A bayesian approach for +neural architecture search. In International conference on machine learning, pp. 7603–7613. +PMLR, 2019. +Qinqin Zhou, Kekai Sheng, Xiawu Zheng, Ke Li, Xing Sun, Yonghong Tian, Jie Chen, and Ron- +grong Ji. Training-free transformer architecture search. CoRR, abs/2203.12217, 2022. +Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In 5th In- +ternational Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, +2017, Conference Track Proceedings, 2017. +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning Transferable Archi- +tectures for Scalable Image Recognition. In Proceedings of the IEEE Conference on Computer +Vision and Pattern Recognition (CVPR), pp. 8697–8710, 2018. +17 + +Published as a conference paper at ICLR 2023 +A +PROOF OF THEOREM 3.1 +Theorem 3.1 We denote the updated weight vector as ˆa and � +ij[gj(xi)]2 = G. Assume we use the +accumulated gradient of all training samples and learning rate η to update the initial weight vector +a, i.e., ˆa = a − η � +i g(xi). If the learning rate 0 < η < 2, then the total training loss is bounded +as follows: +� +i +L(yi, f(xi; ˆa)) ≤ G +2 − η +2M 2(2 − η) +� +j +µ2 +j +(16) +In particular, if the learning rate η = +1 +M , then L(ˆa) is bounded by: +� +i +L(yi, f(xi; ˆa)) ≤ M +2 +� +j +σ2 +j +(17) +Proof. Given each training sample (xi, yi) the gradient of L w.r.t to a when taking (xi, yi) as the +input is as follows: +g(xi) = ∂L(yi, f(xi; a)) +∂a += xixT +i a − yixi +(18) +We note that: +(a − g(xi))T xi − yi = aT xi − aT xixT +i xi + yixT +i xi − yi += aT xi − (aT xi)(xT +i xi) += aT xi − aT xi += 0 =⇒ yi = (a − g(xi))T xi +(19) +Then the total training loss among all training samples is given by: +M +� +i=1 +1 +2(ˆaT xi − yi)2 +(20) +By using Eq. 19, we can rewrite Eq. 20 as follows: +M +� +i=1 +1 +2(ˆaT xi − yi)2 = +M +� +i=1 +1 +2(ˆaT xi − (a − g(xi))T xi))2 += +M +� +i=1 +1 +2((ˆa − a + g(xi))T xi))2 +(21) +Recall the assumption that ˆa = a − η � +i g(xi); we rewrite Eq. 21 as follows: +M +� +i=1 +1 +2(ˆaT xi − yi)2 = +M +� +i=1 +1 +2(g(xi) − η +� +i +g(xi))T xi)2 +(22) +18 + +Published as a conference paper at ICLR 2023 +According to the Cauchy–Schwarz inequality and ||xi|| = 1, the total training loss is bounded by: +M +� +i=1 +1 +2(ˆaT xi − yi)2 ≤ 1 +2 +M +� +i=1 +||(g(xi) − η +� +i +g(xi)||2 ∗ ||xi||2 += 1 +2 +M +� +i=1 +||(g(xi) − η +� +i +g(xi)||2 += 1 +2 +M +� +i=1 +d +� +j=1 +((gj(xi) − ηMµj)2 += 1 +2 +M +� +i=1 +d +� +j=1 +([gj(xi)]2 − 2ηMµjgj(xi) + η2M 2µ2 +j) += 1 +2 +� +ij +[gj(xi)]2 + +� +j +η2M 2µ2 +j − 2 +� +j +(ηMµj +� +i +gj(xi)) += 1 +2 +� +ij +[gj(xi)]2 + +� +j +η2M 2µ2 +j − 2 +� +j +(ηMµjMµj) += 1 +2G + +� +j +(η2M 2µ2 +j − 2ηM 2µ2 +j) += 1 +2G − ηM 2(2 − η) +� +j +µ2 +j +(23) +Since �M +i=1 +1 +2(ˆaT xi −yi)2 is always non-negative, the above upper bound of training loss satisfies: +1 +2G − ηM 2(2 − η) +� +j +µ2 +j ≥ +M +� +i=1 +1 +2(ˆaT xi − yi)2 ≥ 0 +(24) +Note that, if 0 < η < 2, then η(2 − η) > 0. Therefore, the larger � +j µ2 +j term would make the +upper bound of training loss in Eq. 23 closer to 0. In other words, the higher the gradient absolute +mean values across different training samples/batches, the lower the training loss values the model +converges to; i.e., the network converges at a faster rate. +In particular, if η = +1 +M , the Eq. 23 can be rewritten as: +M +� +i=1 +1 +2(ˆaT xi − yi)2 ≤ 1 +2 +M +� +i=1 +d +� +j=1 +((gj(xi) − µj)2 += 1 +2 +� +j +Mσ2 +j += M +2 +� +j +σ2 +j +(25) +This completes our proof. +B +PROOF OF THEOREM 3.2 +Theorem 3.2 Given a neural network with ReLU activation function optimized by minimizing Eq. 8, +we assume that each initial weight vector {wr(0), r = 1, ..., n} is i.i.d. generated from N(0, I) and +the gradient for each weight follows an i.i.d. N(0, σ). For some positive constants δ and ϵ, if the +learning rate η satisfies η < +λ0 +√πδ +2M 2√ +2Φ(1−ϵ)tσ, then with with probability at least (1 − δ)(1 − ϵ), the +following holds true: for any r ∈ [m], ||wr(0) − wr(t)|| ≤ C = ηtσ +� +Φ(1 − ϵ), and at training +step t the Gram matrix H(t) satisfies: +19 + +Published as a conference paper at ICLR 2023 +λmin(H(t)) ≥ λmin(H(0)) − 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +> 0 +(26) +Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d). +Proof. We first compute the probability of ||wr(0) − wr(t)|| ≤ C. Based on the assumption +wi(0), i = 1, ..., n} follows i.i.d. N(0, I) and the gradient for each weight follows i.i.d. N(0, σ), +considering the weight updating rule defined in Eq. 9, each element in wr(0)−wr(t) follows a i.i.d. +N(0, ηtσ). Therefore, ||wr(0)−wr||2 +η2t2σ2 +follows the chi-squared distribution with d degrees of freedom +χ2(d). +P(||wr(0) − wr|| ≤ C) = P(||wr(0) − wr(t)||2 ≤ C2) += P(||wr(0) − wr(t)||2 +η2t2σ2 +≤ +C2 +η2t2σ2 ) += P(||wr(0) − wr(t)||2 +η2t2σ2 +≤ Φ(1 − ϵ)) += 1 − ϵ +(27) +Given an input sample xi and a weight vector wr(t) from W (t), we define the following event: +Air = {||wr(t) − wr(0)|| ≤ C} ∩ {I{xT +i wr(0) ≥ 0} ̸= I{xT +i wr(t) ≥ 0}} +(28) +If ||wr(t) − wr(0)|| ≤ C holds true, +xT +i wr(t) = xT +i (wr(t) − wr(0)) + xT +i wr(0) += sign(xT +i (wr(t) − wr(0)))||wr(t) − wr(0)|| + sign(xT +i wr(0))||wr(0)|| +(29) +Eq. 29 tells us that if ||wr(0)|| is larger than ||wr(t) − wr(0)||, then xT +i wr(0) determines the sign +value of xT +i wr(t); in other words, xT +i wr(t) always has the same sign values with xT +i wr(0); i.e., +I{xT +i wr(0) ≥ 0} = I{xT +i wr(t) ≥ 0}. That is, if ||wr(t) − wr(0)|| ≤ C and I{xT +i wr(0) ≥ 0} ̸= +I{xT +i wr(t) ≥ 0} hold true, then ||wr(0)|| ≤ C. Therefore, the probability of event Air: +P(Air) ≤ P({||wr(0)|| ≤ C}) +(30) +By anti-concentration inequality of Gaussian distribution Du et al. (2019b), we have: +P(Air) ≤ P({||wr(0)|| ≤ C}) ≤ +√ +2C +√π +(31) +Therefore, if any weight vector w1, ..., wm satisfies ||wr(0) − wr(t)|| ≤ C, we can bound the +entry-wise deviation on the Gram matrix H(t) at training step t: for any (i, j) ∈ [n] × [n]: +E[|Hij(0) − Hij(t)|] +=E[ 1 +m|xT +i xj +m +� +r=1 +(I{xT +i wr(0) ≥ 0, xT +j wr(0) ≥ 0} − I{xT +i wr(t) ≥ 0, xT +j wr(t) ≥ 0})|] +=E[ 1 +m|xT +i xj +m +� +r=1 +(I{xT +i wr(0) ≥ 0}I{xT +j wr(0) ≥ 0} − I{xT +i wr(t) ≥ 0}I{xT +j wr(t) ≥ 0})|] +≤E[ 1 +m +m +� +r=1 +(I{Air ∪ Ajr}] ≤ P(Air) + P(Ajr) +≤2 +√ +2C +√π +(32) +where the expectation is summing over the initial weight w(0). Hence, considering all the elements +in H, we have: +E[ +M,M +� +i=1,j=1 +|Hij(0) − Hij(t)|] ≤ 2M 2√ +2C +√π +(33) +20 + +Published as a conference paper at ICLR 2023 +Therefore, by Markov’s inequality, given the probability 1 − δ, we get: +M,M +� +i=1,j=1 +|Hij(0) − Hij(t)| ≤ 2M 2√ +2C +√πδ +(34) +In Du et al. (2019b), the authors prove that, given a small perturbation K: +if [ +� +ij +|Hij(0) − Hij|] ≤ K, then λmin(H) ≥ λmin(H(0)) − K +(35) +In our case, K in Eq. 35 is given by 2M 2√ +2C +√πδ +. Therefore, +λmin(H(t)) ≥ λmin(H(0)) − 2M 2√ +2C +√πδ += λmin(H(0)) − 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +(36) +We replace the term η in Eq.36 with η’s upper bound given in the assumption of Theorem 3.2, i.e., +η < +λ0 +√πδ +2M 2√ +2Φ(1−ϵ)tσ, we can get that λmin(H(t)) is always larger than 0; that is: +λmin(H(t)) ≥ λmin(H(0)) − 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +> 0 +(37) +This completes our proof. +C +PROOF OF THEOREM 3.5 +Theorem 3.5 Given a neural network with ReLU activation function optimized by minimizing Eq. 8, +we assume that each initial weight vector {wr(0), r = 1, ..., n} is i.i.d. generated from N(0, I) +and the gradient for each weight follows an i.i.d. distribution N(0, σ). For some positive constants +δ and ϵ, if the learning rate η satisfies η < +λ0 +√πδ +2M 2√ +2Φ(1−ϵ)tσ, then with with probability at least +(1−δ)(1−ϵ), the following holds true: for any r ∈ [m], ||wr(0)−wr(t)|| ≤ C = ηtσ +� +Φ(1 − ϵ), +and at training step t, the Gram matrix H(t) satisfies: +λmax(H(t)) ≤ λmax(H(0)) + 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +(38) +Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d). +The proof is similar to the proof of Theorem 3.2 (see Appendix B). We provide the entire proof +below. +Proof. We first compute the probability of ||wr(0) − wr(t)|| ≤ C. Based on the assumption that +{wi(0), i = 1, ..., n} follow i.i.d. N(0, I) and the gradient of each weight follows i.i.d. N(0, σ), +considering the weight updating rule defined in Eq. 9 with learning rate η, each element in wr(0) − +wr(t) follows an i.i.d. N(0, ηtσ). Therefore, ||wr(0)−wr||2 +η2t2σ2 +follows a chi-distribution with d degrees +of freedom χ2(d): +P(||wr(0) − wr|| ≤ C) = P(||wr(0) − wr(t)||2 ≤ C2) += P(||wr(0) − wr(t)||2 +η2t2σ2 +≤ +C2 +η2t2σ2 ) += P(||wr(0) − wr(t)||2 +η2t2σ2 +≤ Φ(1 − ϵ)) += 1 − ϵ +(39) +Given an input sample xi and a weight vector wr(t) from W (t), we define the following event: +Air = {||wr(t) − wr(0)|| ≤ C} ∩ {I{xT +i wr(0) ≥ 0} ̸= I{xT +i wr(t) ≥ 0}} +(40) +21 + +Published as a conference paper at ICLR 2023 +If ||wr(t) − wr(0)|| ≤ C holds true, then: +xT +i wr(t) = xT +i (wr(t) − wr(0)) + xT +i wr(0) += sign(xT +i (wr(t) − wr(0)))||wr(t) − wr(0)|| + sign(xT +i wr(0))||wr(0)|| +(41) +Eq. 41 implies that if ||wr(0)|| is larger than ||wr(t) − wr(0)||, then xT +i wr(0) determines the sign +value of xT +i wr(t). In other words, xT +i wr(t) always has the same sign values as xT +i wr(0); that is, +I{xT +i wr(0) ≥ 0} = I{xT +i wr(t) ≥ 0}. Hence, if ||wr(t) − wr(0)|| ≤ C and I{xT +i wr(0) ≥ 0} ̸= +I{xT +i wr(t) ≥ 0} hold true, then ||wr(0)|| ≤ C. Therefore, the probability of event Air: +P(Air) ≤ P({||wr(0)|| ≤ C}) +(42) +By the anti-concentration inequality of a Gaussian distribution Du et al. (2019b), we have: +P(Air) ≤ P({||wr(0)|| ≤ C}) ≤ +√ +2C +√π +(43) +Therefore, if any weight vector w1, ..., wm satisfies ||wr(0) − wr(t)|| ≤ C, we can bound the +entry-wise deviation on the Gram matrix H(t) at the training step t: for any (i, j) ∈ [n] × [n]: +E[|Hij(0) − Hij(t)|] +=E[ 1 +m|xT +i xj +m +� +r=1 +(I{xT +i wr(0) ≥ 0, xT +j wr(0) ≥ 0} − I{xT +i wr(t) ≥ 0, xT +j wr(t) ≥ 0})|] +=E[ 1 +m|xT +i xj +m +� +r=1 +(I{xT +i wr(0) ≥ 0}I{xT +j wr(0) ≥ 0} − I{xT +i wr(t) ≥ 0}I{xT +j wr(t) ≥ 0})|] +(44) +We note that all the samples in the training set S (Eq. 1) are normalized with their L2-norm. Hence, +we have both ||xi|| = 1 and ||xj|| = 1. Therefore, using the Cauchy–Schwarz inequality, the above +equation is bounded as follows: +E[|Hij(0) − Hij(t)|] ≤E[ 1 +m +m +� +r=1 +(I{Air ∪ Ajr}] ≤ P(Air) + P(Ajr)] ≤ +2 +√ +2C +√π +(45) +where the expectation is over the initial weight wr(0), r = {1, ..., m}. Hence, considering all the +elements in H, we have: +E[ +M,M +� +i=1,j=1 +|Hij(0) − Hij(t)|] ≤ 2M 2√ +2C +√π +(46) +Therefore, by the Markov’s inequality, given the probability 1 − δ, we get: +M,M +� +i=1,j=1 +|Hij(0) − Hij(t)| ≤ 2M 2√ +2C +√πδ +(47) +Based on the matrix perturbation theory Bauer & Fike (1960); Eisenstat & Ipsen (1998), given a +small perturbation K: +if [ +� +ij +|Hij(0) − Hij(t)|] ≤ K, then λmax(H(t)) ≤ λmax(H(0)) + K +(48) +In our case, K in Eq. 48 is given by 2M 2√ +2C +√πδ +; that is: +λmax(H(t)) ≤ λmax(H(0)) + 2 +√ +2M 2ηtσ +� +Φ(1 − ϵ) +√πδ +(49) +This completes our proof. +22 + +Published as a conference paper at ICLR 2023 +0.0000.0050.0100.0150.0200.0250.030 +Standard Deviation ( ) +0 +5 +10 +15 +20 +25 +30 +35 +Test Loss +Loss vs. Standard Deviation +(a) Batch size=64 +0.0000.0050.0100.0150.0200.0250.0300.035 +Standard Deviation ( ) +0 +5 +10 +15 +20 +25 +30 +35 +Test Loss +Loss vs. Standard Deviation +(b) Batch size=128 +0.0000.0050.0100.0150.0200.0250.0300.035 +Standard Deviation ( ) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Test Loss +Loss vs. Standard Deviation +(c) Batch size=256 +Figure 5: Test loss vs. standard deviation of gradients (σ in Eq. 13) for randomly sampled 500 two- +layer MLPs with ReLU on MNIST after one training epoch. We train these networks by minimizing +the MSE loss between the output of networks and the real labels. As shown, the Networks with +smaller σ tend to have lower test loss values and thus have a better generalization capacity. +C.1 +SUPPLEMENTARY RESULTS: VALIDATION OF THEOREM 3.5 +To empirically validate Theorem 3.5, we first create the training set S by normalizing the training +samples in MNIST with their L2-norm. Next, we optimize a two-layer MLP with ReLU activation +functions as defined in Eq. 7. We use the entire training set of MNIST and apply the gradient +descent (Eq. 9) to update the weights. We vary the batch size as {64,128,256} and measure the +standard deviation of gradients (σ) w.r.t. parameters across different training batches. A very small +learning rate of η = 10−8 is set to satisfy the assumption in Theorem 3.5. Fig. 5 demonstrates the +training loss after one epoch vs. standard deviation of gradients (σ). Clearly, the results show that if +a network has a lower gradient standard deviation, then it tends to have lower test loss values, and +thus, a better generalization capacity. These results empirically prove our claims in Theorem 3.5. +D +EXPERIMENTAL SETUP OF ZICO ON IMAGENET +D.1 +SEARCH SPACE +We use the commonly used MobileNetv2-based search space where the candidate networks are built +by stacking multiple Inverted Bottleneck Blocks (IBNs) with SE modules Sandler et al. (2018); +Pham et al. (2018); Lin et al. (2021); all the SE modules share the same se ratio as 0.25. For each +IBN, we vary the kernel size of the depth-wise convolutional layer from {3,5,7} and sample the +expansion ratio from {1,2,4,6}. We primarily consider ReLU as the activation function. For each +point-wise convolutional layer, the range of the number of channels is from 8 to 1024 with a step size +of 8. We use standard Kaiming Init to initialize all linear and convolution layers for every candidate +networks He et al. (2015). +23 + +Published as a conference paper at ICLR 2023 +Algorithm 1 ZiCo-based zero-shot NAS framework +INPUT: Number of search steps T +Inference budget B, Search space S +Set of input batch Z = {(Xi, yi), i = 1, 2} +Population size E, Initial network F0 ∈ S +OUTPUT: Optimal network FP +SEARCH: +Initialize F = {F0} +for i = 1 to T do +Randomly sample network Ft from F +Fi = randomly mutated architecture based on Ft from S +if Fi meets the inference budget B then +Compute ZiCo for Fi on Z by Eq. 15 +Add Fi to F +if |F| > E then +Remove network with the smallest ZiCo from F +end if +end if +end for +FP = the network of the highest ZiCo in F. +D.2 +SEARCH ALGORITHM +We use an Evolutionary Algorithm (EA) to conduct the zero-shot NAS because it is concise and easy +to implement1 As shown in Algorithm 1, we search for the neural architectures with the highest ZiCo +within the search space, given a specific budget B (e.g., FLOPs). We repeat the search T times; at +each search step, we randomly select a structure from the candidate set F and mutate its architectures +(e.g., kernel size, block type, number of blocks, and layer width) to generate a new network Fi ∈ S. +If the generated network Fi meets the inference budget B, we calculate its ZiCo on Z and add Fi +to the candidate set F. We remove the network with the smallest ZiCo from F, if the number of +architectures in F exceeds the threshold E. After T steps, we select the network with the largest +ZiCo as the final (optimal) architecture FP . +Specifically, We repeat the search 105 times (i.e., T = 105) with the population size E = 512. For +each of the candidate architectures, we compute ZiCo with two batches randomly sampled from the +training set of ImageNet with batch size 128. In total, it takes 10 hours on a single NVIDIA 3090 +GPU for 105 search steps. +D.3 +TRAINING DETAILS +: We use the same data augmentations configurations as in Pham et al. (2018): mix-up, label- +smoothing, random erasing, random crop/resize/flip/lighting, and AutoAugment. We use the SGD +optimizer with momentum 0.9 and weight decay 4e-5. We take EfficientNet-B3 as a teacher network +and use the knowledge distillation method to train the network. We set the initial learning rate as +0.1 and used the cosine annealing scheme to adjust the learning rate during training. We train the +obtained network 480 epochs, which takes 83 hours on a 40-core Intel Xeon CPU and 8 NVIDIA +3090 GPU-powered server. +E +SUPPLEMENTARY RESULTS ON NAS BENCHMARKS +E.1 +COMPARISON WITH MORE PROXIES +In this section, we further provide the comparison between our proposed ZiCo and more proxies +proposed recently: KNAS (Xu et al. (2021)), NASWOT (Lopes et al. (2021)), GradSign ( Zhang +& Jia (2022)), and NTK (TE-NAS Chen et al. (2021b), NASI Shu et al. (2022a)). To compute the +correlations, we use the official code released by the authors of the above papers to obtain the values +1One can also use other methods to perform the search; check Appendix F.2. +24 + +Published as a conference paper at ICLR 2023 +Table 4: The correlation coefficients between various zero-cost proxies and two naive proxies +(#Params and FLOPs) vs. test accuracy on NATSBench-SSS and NATSBench-TSS (KT and SPR +represent Kendall’s τ and Spearman’s ρ, respectively). The results in italics represent the values +of #Params’ correlation coefficients. The results better than #Params are shown with bold fonts. +Clearly, our proposed ZiCo is the only proxy that works consistently better than #Params and is gen- +erally the best among all these proxies. ‡Both TE-NAS (Chen et al. (2021b)) and NASI (Chen et al. +(2021b)) use NTK (Jacot et al. (2018)) as the accuracy proxy to build their own search algorithms. +NATSBench-TSS (NASBench201) +Dataset +CIFAR10 +CIFAR100 +Img16-120 +Proxy +Correlation +KT +SPR +KT +SPR +KT +SPR +Grad norm Abdelfattah et al. (2021) +0.46 +0.63 +0.47 +0.63 +0.43 +0.58 +SNIP Lee et al. (2019b) +0.46 +0.63 +0.46 +0.63 +0.43 +0.58 +GraSP Wang et al. (2020) +0.37 +0.54 +0.36 +0.51 +0.40 +0.56 +Fisher Liu et al. (2021) +0.40 +0.55 +0.41 +0.55 +0.37 +0.50 +Synflow Tanaka et al. (2020) +0.54 +0.73 +0.57 +0.76 +0.56 +0.75 +KNAS Xu et al. (2021) +0.14 +0.20 +0.24 +0.35 +0.30 +0.42 +NASWOT Mellor et al. (2021) +0.58 +0.77 +0.62 +0.80 +0.60 +0.78 +NTK [TE-NAS Chen et al. (2021b), NASI Shu et al. (2022a)]‡ +0.33 +0.44 +0.33 +0.43 +0.46 +0.63 +GradSign Zhang & Jia (2022) +0.58 +0.77 +0.59 +0.79 +0.59 +0.78 +Zen-score Lin et al. (2021) +0.29 +0.38 +0.28 +0.36 +0.29 +0.40 +FLOPs +0.54 +0.73 +0.51 +0.71 +0.49 +0.67 +#Params +0.57 +0.75 +0.55 +0.73 +0.52 +0.69 +ZiCo +0.61 +0.80 +0.61 +0.81 +0.60 +0.79 +NATSBench-SSS +Dataset +CIFAR10 +CIFAR100 +Img16-120 +Proxy +Correlation +KT +SPR +KT +SPR +KT +SPR +Grad norm Abdelfattah et al. (2021) +0.35 +0.51 +0.34 +0.49 +0.49 +0.67 +SNIP Lee et al. (2019b) +0.42 +0.59 +0.46 +0.62 +0.57 +0.76 +GraSP Wang et al. (2020) +-0.09 +-0.13 +0.01 +0.01 +0.29 +0.42 +Fisher Liu et al. (2021) +0.30 +0.44 +0.41 +0.55 +0.33 +0.47 +Synflow Tanaka et al. (2020) +0.61 +0.81 +0.60 +0.80 +0.39 +0.57 +KNAS Xu et al. (2021) +0.25 +0.37 +0.12 +0.18 +0.32 +0.46 +NASWOT Mellor et al. (2021) +0.45 +0.63 +0.43 +0.59 +0.42 +0.59 +NTK [TE-NAS Chen et al. (2021b), NASI Shu et al. (2022a)]‡ +0.17 +0.26 +0.04 +0.06 +0.20 +0.30 +GradSign Zhang & Jia (2022) +0.21 +0.30 +0.16 +0.27 +0.04 +0.05 +Zen-score Lin et al. (2021) +0.50 +0.69 +0.52 +0.71 +0.69 +0.87 +FLOPs +0.19 +0.28 +0.21 +0.30 +0.38 +0.53 +#Params +0.53 +0.72 +0.54 +0.73 +0.65 +0.84 +ZiCo +0.54 +0.73 +0.55 +0.75 +0.70 +0.88 +of these proxies2. As shown in Table 4, our proposed ZiCo performs better than all these proxies. For +example, NASWOT and GradSign achieve a similar correlation score as ZiCo on NATSBench-TSS; +however, ZiCo has a significantly higher correlation score than these two proxies on NATSBench- +SSS. +2NASI uses NTK to build their own search algorithms. Here we directly compute the correlation between +NTK and the real test accuracy. +25 + +Published as a conference paper at ICLR 2023 +Table 5: The correlation coefficients under different proxies vs. test performance on TransNAS- +Bench-101-Mirco. Clearly, our proposed ZiCo is consistently very close to the best score (only +0.01 or 0.02 lower score) except for Autoencoding (still, ZiCo is the second best on Autoencoding). +Though Fisher works better than ZiCo on Autoencoding, ZiCo has a significantly higher score on the +rest of tasks. We note that existing proxies do not achieve a high correlation on all tasks consistently. +Autoencoding +Scene Classification +Proxy +Kendall’s τ +Spearman’s ρ +Kendall’s τ +Spearman’s ρ +Grad norm Abdelfattah et al. (2021) +0.24 +0.32 +0.47 +0.65 +SNIP Lee et al. (2019b) +0.20 +0.27 +0.52 +0.71 +Grasp Wang et al. (2020) +0.09 +0.14 +0.19 +0.28 +Fisher Liu et al. (2021) +0.42 +0.59 +0.49 +0.67 +Synflow Tanaka et al. (2020) +0.00 +0.00 +0.53 +0.72 +NASWOT Lopes et al. (2021) +0.01 +0.02 +0.43 +0.60 +Zen-score Lin et al. (2021) +0.09 +0.14 +0.52 +0.72 +GradSign Zhang & Jia (2022) +0.01 +0.02 +0.32 +0.46 +Params +0.01 +0.01 +0.46 +0.64 +FLOPs +0.02 +0.02 +0.47 +0.65 +ZiCo (Ours) +0.24 +0.35 +0.51 +0.71 +Jigsaw +Surface Normal +Proxy +Kendall’s τ +Spearman’s ρ +Kendall’s τ +Spearman’s ρ +Grad norm Abdelfattah et al. (2021) +0.23 +0.35 +0.24 +0.36 +SNIP Lee et al. (2019b) +0.27 +0.41 +0.32 +0.49 +Grasp Wang et al. (2020) +0.07 +0.11 +0.01 +0.01 +Fisher Liu et al. (2021) +0.19 +0.30 +0.10 +0.14 +Synflow Wang et al. (2020) +0.32 +0.47 +0.00 +0.00 +NASWOT Lopes et al. (2021) +0.29 +0.42 +0.41 +0.57 +Zen-score Lin et al. (2021) +0.35 +0.50 +0.52 +0.71 +GradSign Zhang & Jia (2022) +0.38 +0.53 +0.29 +0.40 +Params +0.29 +0.44 +0.45 +0.63 +FLOPs +0.30 +0.45 +0.46 +0.64 +ZiCo (Ours) +0.36 +0.52 +0.50 +0.68 +E.2 +COMPARISON ON TRANSNAS-BENCH-101-MICRO +In this section, we compare our proposed ZiCo against existing proxies on more diverse tasks. We +compare our proposed ZiCo against existing proxies on one mainstream NAS benchmark TransNAS- +Bench-101 Duan et al. (2021). We pick the largest search space TransNAS-Bench-101-Micro which +contains 4096 total architectures with different cell structures. We compare ZiCo with various prox- +ies under the following four tasks: +• Scene Classification. Scene classification is a 47-class classification task that predicts the +room type in the image. +• Jigsaw. In the Jigsaw task, the input image is divided into nine patches and shuffled based +on one of 1,000 predefined permutations. The target here is to classify which permutation +is used. +• Autoencoding. Autoencoding is a pixel-level prediction task that encodes an input im- +age into a low-dimension embedding vector and then reconstructs the raw image from the +vector. +• Surface Normal. Similar to autoencoding, surface normal is a pixel-level prediction task +that predicts surface normal statistics. +As shown in Table 5, ZiCo consistently works well on Scene Classification, Jigsaw, and Surface +Normal; ZiCo has only 0.01 or 0.02 lower correlation scores than the highest scores. Though +Fisher works better than ZiCo on Autoencoding, ZiCo has significantly higher correlation scores +than Fisher on the remaining three tasks. One possibility why Fisher works best on Autoencoding +is that Autoencoding is an image-to-image task; Fisher is the only proxy that is built on the gradient +26 + +Published as a conference paper at ICLR 2023 +Table 6: The test performance of optimal architectures obtained by various zero-shot proxies (aver- +age on 5 runs) on TransNAS-Bench-101-Micro search space. The best results are shown with bold +fonts. +Autoencoding +Scene Classification +Jigsaw +Surface Normal +Metric +SSIM +Accuracy +Accuracy +SSIM +Ground Truth +0.58 +54.9 +95.4 +0.59 +Grad norm +0.36± 0.03 +48.7±0.7 +80.3±0.3 +0.53±0.00 +SNIP +0.33±0.04 +48.7±1.1 +80.3±0.1 +0.53±0.01 +Grasp +0.33±0.06 +50.2±1.6 +91.1±0.3 +0.38±0.06 +Fisher +0.49±0.01 +48.7±0.6 +83.5±1.2 +0.31±0.03 +Synflow +0.46±0.07 +53.7±1.2 +90.9±0.4 +0.57±0.06 +NASWOT +0.43±0.02 +53.2±0.6 +92.3±0.3 +0.53±0.02 +Zen-score +0.46±0.01 +53.7±0.2 +87.5±0.4 +0.55±0.00 +GradSign +0.35±0.03 +53.6±0.4 +93.1±0.4 +0.57±0.02 +Params +0.46 +53.70 +85.90 +0.55 +FLOPs +0.46 +53.70 +85.90 +0.55 +ZiCo (Ours) +0.48±0.02 +53.7±0.4 +93.2±0.4 +0.57±0.01 +Table 7: The correlation coefficients under three different proxies vs. test accuracy on NATSBench- +SSS (KT and SPR represent Kendall’s τ and Spearman’s ρ, respectively). Clearly, our proposed +ZiCo works consistently better than using mean only and STD only on all these datasets. +Dataset +CIFAR10 +CIFAR100 +Img16-120 +Method +KT +SPR +KT +SPR +KT +SPR +Mean Only +0.25 +0.37 +0.39 +0.55 +0.61 +0.81 +STD only +0.39 +0.55 +0.42 +0.6 +0.45 +0.62 +ZiCo (Mean + STD) +0.54 +0.73 +0.55 +0.75 +0.70 +0.88 +w.r.t. feature maps and thus can better extract the information between the input and output images. +Although Fisher works better than ZiCo on Autoencoding (we are still second best), ZiCo has a +significantly higher score on the remaining tasks. As shown in the main paper, we again note that +existing proxies do not achieve a high correlation on all tasks consistently. +Table 6 demonstrates the test accuracy of the best architectures found using various proxies on each +of the above tasks in TransNAS-Bench-101-Micro. Once again, we see that ZiCo significantly out- +performs existing proxies on all tasks except Autoencoding, where we trail Fisher by only 0.01 +SSIM. Nonetheless, ZiCo is second best on the Autoencoding task. Note that, similar to the correla- +tion results in Table 5, other proxies do not consistently achieve high accuracy. For instance, while +methods like Synflow or Zenscore achieve results close to ours on Scene Classification and Surface +Normal, they produce poor results on other tasks like Jigsaw. Therefore, ZiCo consistently performs +well on highly different tasks. +E.3 +ILLUSTRATION OF VARIOUS PROXIES VS. REAL TEST ACCURACY +We provide some illustration figures of real test accuracy vs. various proxies on NATSBench-SSS +search space for CIFAR10 (Fig. 6) and ImageNet16-120 datasets(Fig. 7). We also show the same +illustrative results (real test accuracy vs. various proxies) on NASBench101 search space in Fig. 8. +F +ABLATION STUDY +F.1 +IMPACT OF MEAN AND STD +We randomly select 2000 networks from NATSBench-SSS on CIFAR10, CIFAR100, and Img16- +120 datasets and compute the following proxies: (i) Mean value of gradients only; (ii) Standard +deviation (STD) value of gradients only; (iii) Combination of mean and std value, i.e., our proposed +ZiCo. We then calculate the correlation coefficients between these proxies and the real test accuracy. +As shown in Table. 7, our proposed ZiCo performs better on these three datasets than either using +27 + +Published as a conference paper at ICLR 2023 +Table 8: The test accuracy of optimal architectures obtained by various zero-shot proxies (average +on 5 runs) on NATSBench-TSS search space. The best results are shown with bold fonts. +Proxy +CIFAR10 +CIFAR100 +Img16-120 +Costs(GPU hours) +Zero-PT+SNIP Lee et al. (2019b) +93.52±0.18 +70.75±0.19 +44.45±0.14 +0.10 +Zero-PT+NASWOT Lopes et al. (2021) +93.42±0.07 +70.77±0.51 +45.11±0.26 +0.11 +Zero-PT+Synflow Tanaka et al. (2020) +87.68±0.16 +58.92±0.17 +32.20±0.00 +0.13 +Zero-PT+KNAS Xu et al. (2021) +93.95±0.03 +72.44±0.26 +46.01±0.12 +0.10 +Zero-PT+Grad norm Abdelfattah et al. (2021) +93.52±0.18 +70.75±0.30 +44.48±0.11 +0.07 +Zero-PT+Zen-score Lin et al. (2021) +93.84±0.05 +71.63±0.06 +46.67±0.16 +0.02 +Zero-PT+GradSign Zhang & Jia (2022) +93.76±0.12 +71.11±0.23 +42.95±1.29 +0.06 +Zero-PT+ZiCo (Ours) +94.15±0.22 +72.77±0.66 +46.39±0.23 +0.12 +mean only or STD only. Therefore, our proposed ZiCo is a better-designed proxy than using mean +or STD individually. +F.2 +SEARCH ALGORITHMS: ZERO-COST PT +In this section, we demonstrate that our proposed ZiCo can be combined with other search algo- +rithms. We take the Zero-Cost-PT (Zero-PT) as an example Xiang et al. (2021b) because it is +specifically designed for zero-shot proxies and is very time-efficient. Essentially, Zero-PT first +integrates all candidate networks into a supernet and assigns learnable weights to each candidate +operation (same as one-shot NAS). Then Zero-PT uses the zero-cost proxy instead of the training +accuracy to update the weights for each candidate operation. The final architecture is generated by +selecting the operations with the highest weight values. +We combine different accuracy proxies with Zero-PT under the NASBench-201 and report the op- +timal architectures found with various proxies3. As shown in Table 8, the architectures found via +ZiCo have the highest test accuracy except for Img16-120 datasets (ZiCo is the second best on +Img16-120)). +F.3 +TRAINING RECIPE: WITHOUT DISTILLATION +In this section, we train the obtained network under various FLOPs budgets with the exact same +training setup as Xu et al. (2022); Cai et al. (2019). Specifically, we train the neural network for +150 epochs with batch size 512 and input resolution 224. We train the network without knowledge +distillation and do not use advanced data augmentation methods (e.g., mixup, RandAugment, etc). +Finally, we set the initial learning rate as 0.4 with a cosine annealing scheduling scheme. Moreover, +we train EfficientNets and the previous SOTA zero-shot NAS approach (Zen-score) under the same +setup. +As shown in Table 9, ZiCo outperforms all of the previous zero-shot NAS approaches. For exam- +ple, when the FLOPs budget is around 600M, ZiCo achieves 77.1% Top-1 accuracy, which is 1.0% +and 1.6% higher than previous SOTA zero-shot NAS methods, i.e., Zen-score, and TE-NAS, re- +spectively. Moreover, ZiCo finds a model with similar accuracy as EfficientNet-B1, but with 100M +fewer FLOPs and much less search cost. Overall, compared to the regular one-shot or multi-shot +NAS methods, ZiCo achieves comparable or higher test accuracy with 5-9500× less search time. +F.4 +SEARCH SPACE: DARTS +In this section, we use ZiCo to conduct the zero-shot NAS on the DARTS search space. We first use +Algorithm 1 to find the networks with the highest ZiCo without FLOPs budgets on the CIFAR10 +dataset. We conduct the search for 100k steps; this takes 0.7 hours on a single NVIDIA 3090 GPU +(i.e., 0.03 GPU days). Then, we train the obtained network with the exact same training setup as the +original DARTS paper Liu et al. (2019)4; specifically, we train the neural network for 600 epochs +3We implement the code ourselves since the authors have not released the code yet. The difference between +Table 2 and Table 8 comes from the search algorithm: Table 2 uses traversal search among all candidate +networks; Table 8 uses perturbation-based zero-cost PT Xiang et al. (2021b). +4This is the same setup as most of the baseline works in Table 10. +28 + +Published as a conference paper at ICLR 2023 +Table 9: Comparison of Top-1 accuracy of our ZiCo-based NAS against NAS methods with stan- +dalone training on ImageNet under various FLOP budgets. For the ‘Method’ column, ‘MS’ repre- +sents multi-shot NAS; ‘OS’ is short for one-shot NAS; Scaling represents network scaling methods; +‘ZS’ is short for zero-shot NAS. ‘no KD’ means we train the network without Knowledge Distilla- +tion (KD); ‘150E’ means we train the network with 150 epochs, similar for 350E. The results are +averaged over three suns. We note that some NAS methods use knowledge distillation to improve +the test accuracy; hence, we remove those methods from this table. The results are averaged over +three runs. +Budget (maximal #FLOPs) +Approach +FLOPs +Top-1 +Method +Costs[GPU Days] +450M +EfficientNet-B0 Tan & Le (2019) [350E] +390M +77.1 +Scaling +3800 +EfficientNet-B0 Tan et al. (2019)[150E] +390M +76.0 +Scaling +3800 +MnasNet-A3 Tan et al. (2019) +403M +76.7 +MS +- +BN-NAS Chen et al. (2021a) +470M +75.7 +MS +0.8 +RLNAS Zhang et al. (2021) +473M +75.6 +OS +- +NASNet-B Zoph et al. (2018) +488M +72.8 +MS +1800 +CARS-D Yang et al. (2020) +496M +73.3 +MS +0.4 +Zen-score Lin et al. (2021) [no KD; 150E] +410M +75.6 +ZS +0.5 +#Params +451M +63.5 +ZS +0.02 +ZiCo (Ours) [no KD; 150E] +448M +76.5±0.2 +ZS +0.4 +600M +DARTS Liu et al. (2019) +574M +73.3 +OS +4 +NAO Luo et al. (2018) +584M +75.5 +MS +58.3 +PC-DARTS Xu et al. (2019) +586M +75.8 +OS +3.8 +PNAS Liu et al. (2018a) +588M +74.2 +MS +224 +CARS-I Yang et al. (2020) +591M +75.2 +MS +0.4 +EnTranNAS Yang et al. (2021) +594M +76.2 +OS +2.1 +ProxylessNAS Cai et al. (2019) +595M +76.0 +OS +8.3 +RLNAS Zhang et al. (2021) +597M +75.9 +OS +- +MAGIC-AT Xu et al. (2022) +598M +76.8 +OS +2 +SemiNAS Luo et al. (2020) +599M +76.5 +MS +4 +EfficientNet-B1 Tan et al. (2019)[350E] +700M +79.1 +Scaling +3800 +EfficientNet-B1 Tan et al. (2019)[150E] +700M +77.4 +Scaling +3800 +TE-NAS Chen et al. (2021b) +599M +75.5 +ZS +0.17 +Zen-score Lin et al. (2021) [no KD; 150E] +611M +76.1 +ZS +0.5 +ZiCo (Ours) [no KD; 150E] +603M +77.1±0.3 +ZS +0.4 +Table 10: Comparison of Top-1 accuracy of our ZiCo-based NAS against NAS methods with stan- +dalone training on CIFAR10 in DARTS search space. For the ‘Method’ column,‘MS’ represents +multi-shot NAS; ‘OS’ is short for one-shot NAS; ‘ZS’ is short for zero-shot NAS. ‘600E’ means we +train the network with 600 epochs, similar to 800E. The results are averaged over three suns. The +results are averaged over three runs. +Approach +Test Error (%) +Method +Cost(GPU days) +AmoebaNet-A Real et al. (2019) +3.34±0.06 +MS +3150 +PNAS Liu et al. (2018a) +3.41±0.09 +MS +225 +ENAS Tan & Le (2019) +2.89 +MS +0.5 +NASNet-A Zoph et al. (2018) +2.65 +MS +2000 +DARTS-v1 Liu et al. (2019) +3.00±0.14 F +OS +0.4 +DARTS-v2 Liu et al. (2019) +2.76±0.09 +OS +1 +SNAS Xie et al. (2019) +2.85±0.02 +OS +1.5 +GDAS Dong & Yang (2019) +2.82 +OS +0.17 +BayesNAS Zhou et al. (2019) +2.81±0.04 +OS +0.2 +ProxylessNAS Cai et al. (2019) +2.08 +OS +4 +P-DARTS Chen et al. (2019) +2.5 +OS +0.3 +PC-DARTS Xu et al. (2019) +2.57±0.07 +OS +0.1 +SDARTS-ADV Chen & Hsieh (2020) +2.61±0.02 +OS +1.3 +Zen-score Lin et al. (2021) +2.55±0.04 +ZS +0.01 +TE-NAS Chen et al. (2021b) +2.63±0.064 +ZS +0.05 +ZiCo(ours) +2.45±0.11 +ZS +0.03 +29 + +Published as a conference paper at ICLR 2023 +with a batch size of 128. We only use the standard data augmentation (normalization, cropping, and +random flipping) together with the cutout tricks. We don’t use knowledge distillation or any other +advanced data augmentation tricks. Finally, we set the initial learning rate as 0.025 with a cosine +annealing scheduling scheme. We repeat the same experiments for Zen-score. +As shown in Table 10, ZiCo outperforms previous zero-shot NAS approaches, e.g, Zen-score and +TE-NAS. Moreover, compared to the regular one-shot or multi-shot NAS methods, ZiCo achieves +comparable or higher test accuracy with at least 10× less search time. +30 + +Published as a conference paper at ICLR 2023 +3 +4 +5 +6 +Grad_norm +40 +50 +60 +70 +Test accuracy +Test acc vs. Grad_norm ( = 0.36 += 0.51) +(a) Grad norm +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +SNIP +40 +50 +60 +70 +Test accuracy +Test acc vs. SNIP ( = 0.42 += 0.59) +(b) SNIP +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +GraSP +40 +50 +60 +70 +Test accuracy +Test acc vs. GraSP ( = +0.09 += +0.13) +(c) GraSP +0.000250.000500.000750.001000.001250.001500.00175 +Fisher +40 +50 +60 +70 +Test accuracy +Test acc vs. Fisher ( = 0.30 += 0.44) +(d) Fisher +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Synflow +40 +50 +60 +70 +Test accuracy +Test acc vs. Synflow ( = 0.61 += 0.81) +(e) Synflow +25 +30 +35 +40 +45 +50 +55 +Zen-score +40 +50 +60 +70 +Test accuracy +Test acc vs. Zen-score ( = 0.50 += 0.69) +(f) Zen-score +0 +200000 +400000 +600000 +#Params +40 +50 +60 +70 +Test accuracy +Test acc vs. #Params ( = 0.53 += 0.72) +(g) #Params +200 +220 +240 +260 +280 +300 +ZiCo +40 +50 +60 +70 +Test accuracy +Test acc vs. ZiCo ( = 0.54 += 0.73) +(h) ZiCo +Figure 6: Real test accuracy vs. various proxies on NATSBench-SSS search space for CIFAR10 +dataset. τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively. +31 + +Published as a conference paper at ICLR 2023 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +Grad_norm +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. Grad_norm ( = 0.49 += 0.67) +(a) Grad norm +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +SNIP +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. SNIP ( = 0.57 += 0.76) +(b) SNIP +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +GraSP +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. GraSP ( = 0.29 += 0.42) +(c) GraSP +0.0002 +0.0004 +0.0006 +0.0008 +Fisher +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. Fisher ( = 0.41 += 0.57) +(d) Fisher +100 +150 +200 +Synflow +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. Synflow ( = 0.39 += 0.57) +(e) Synflow +25 +30 +35 +40 +45 +50 +55 +Zen-score +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. Zen-score ( = 0.69 += 0.87) +(f) Zen-score +0 +200000 +400000 +600000 +#Params +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. #Params ( = 0.65 += 0.84) +(g) #Params +200 +220 +240 +260 +280 +300 +320 +ZiCo +20 +25 +30 +35 +40 +45 +Test accuracy +Test acc vs. ZiCo ( = 0.70 += 0.88) +(h) ZiCo +Figure 7: Real test accuracy vs. various proxies on NATSBench-SSS search space for ImageNet16- +120 dataset. τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively. +32 + +Published as a conference paper at ICLR 2023 +0 +200 +400 +600 +800 +1000 +1200 +Grad_norm +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. Grad_norm ( = +0.17 += +0.25) +(a) Grad norm +0 +2000 +4000 +6000 +SNIP +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. SNIP ( = +0.12 += +0.17) +(b) SNIP +0 +5000 10000 15000 20000 25000 +GraSP +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. GraSP ( = 0.20 += 0.29) +(c) GraSP +0 +250 +500 +750 +1000 +1250 +Fisher +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. Fisher ( = +0.20 += +0.28) +(d) Fisher +0 +50000 +100000 +150000 +Synflow +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. Synflow ( = 0.23 += 0.35) +(e) Synflow +50 +75 +100 +125 +150 +175 +Zen-score +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. Zen-score ( = 0.46 += 0.63) +(f) Zen-score +0 +1 +2 +3 +4 +#Params +1e7 +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. #Params ( = 0.31 += 0.43) +(g) #Params +200 +400 +600 +800 +1000 +1200 +ZiCo +0.2 +0.4 +0.6 +0.8 +Test accuracy +Test acc vs. ZiCo ( = 0.46 += 0.63) +(h) ZiCo +Figure 8: Real test accuracy vs. various proxies on NASBench101 search space for CIFAR10 +dataset. τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively. +33 + diff --git a/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/load_file.txt b/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..50577c0db2c0faf1c4350044a43a5a68f6446b50 --- /dev/null +++ b/ZNFIT4oBgHgl3EQfkSuq/content/tmp_files/load_file.txt @@ -0,0 +1,2550 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf,len=2549 +page_content='Published as a conference paper at ICLR 2023 ZICO: ZERO-SHOT NAS VIA INVERSE COEFFICIENT OF VARIATION ON GRADIENTS Guihong Li1, Yuedong Yang1, Kartikeya Bhardwaj2∗, Radu Marculescu1 1The University of Texas at Austin, 2Qualcomm {lgh,albertyoung,radum}@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='edu, kbhardwa@qti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='qualcomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='com ABSTRACT Neural Architecture Search (NAS) is widely used to automatically design the neu- ral network with the best performance among a large number of candidate archi- tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To reduce the search time, zero-shot NAS aims at designing training-free proxies that can predict the test performance of a given architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' However, as shown recently, none of the zero-shot proxies proposed to date can actually work consistently better than a naive proxy, namely, the number of network parameters (#Params).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To improve this state of affairs, as the main theoretical contribution, we first reveal how some specific gradient properties across different samples impact the convergence rate and generalization capacity of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Based on this theoretical analysis, we propose a new zero-shot proxy, ZiCo, the first proxy that works consistently better than #Params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We demonstrate that ZiCo works bet- ter than State-Of-The-Art (SOTA) proxies on several popular NAS-Benchmarks (NASBench101, NATSBench-SSS/TSS, TransNASBench-101) for multiple ap- plications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', image classification/reconstruction and pixel-level prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fi- nally, we demonstrate that the optimal architectures found via ZiCo are as compet- itive as the ones found by one-shot and multi-shot NAS methods, but with much less search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, ZiCo-based NAS can find optimal architectures with 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1%, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4%, and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4% test accuracy under inference budgets of 450M, 600M, and 1000M FLOPs on ImageNet within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 GPU days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1 INTRODUCTION During the last decade, deep learning has achieved great success in many areas, such as computer vision and natural language modeling Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Liu & Deng (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In recent years, neural architecture search (NAS) has been proposed to search for optimal architectures, while reducing the trial-and-error (manual) network design efforts Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zoph & Le (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Elsken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, the neural architectures found via NAS show better perfor- mance than the manually-designed networks in many mainstream applications Real et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Li & Talwalkar (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kandasamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Despite these advantages, most of the existing NAS approaches involve a time-consuming and resource-intensive search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, multi-shot NAS uses a controller or an accuracy predictor to conduct the search process and it requires training of multiple networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' thus, multi-shot NAS is extremely time-consuming (typically thousands of GPU hours) Real et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Alternatively, one-shot NAS merges all possible networks from the search space into a supernet and thus only needs to train the supernet once Dong & Yang (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Stamoulis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' this enables one-shot NAS to find a good architecture with much less search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Though the one- shot NAS has significantly improved the time efficiency of NAS, training is still required during the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ∗Work done while Kartikeya Bhardwaj was at Arm, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11300v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='LG] 26 Jan 2023 Published as a conference paper at ICLR 2023 In the last few years, the zero-shot approaches have been proposed to liberate NAS from training entirely Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Ingolfsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Tran & Bae (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Do & Luong (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Essentially, zero-shot NAS utilizes some proxy that can predict the test performance of a given network without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, the design of the proxy in zero-shot NAS is usually based on some theoretical analysis of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, zero-shot approaches can not only significantly improve the time efficiency of NAS, but also deepen the theoretical understanding on why certain networks work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nonetheless, as revealed in Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022), the zero-shot proxies proposed to date cannot work consistently better than a naive proxy, namely, the number of parameters (#Params);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' in fact, #Params often achieves the best performance on most of popular NAS benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' These results may undermine the effectiveness of zero-shot NAS approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To address the limitations of existing zero-shot proxies, we target the following key questions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' How do some specific gradient properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', mean value and standard deviation across different samples, impact the training convergence of neural networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Can we use these two gradient properties to design a new theoretically-grounded proxy that works better than #Params consistently?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To this end, we first theoretically analyze how the mean value and standard deviation of gradients across different training batches impact the training convergence of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Based on our analysis, we propose ZiCo, a new proxy for zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We demonstrate that, compared to all existing proxies (including #Params), ZiCo has either a higher or at least on-par correlation with the test accuracy on popular NAS-Benchmarks (NASBench101, NATS-Bench-SSS/TSS) for multiple datasets (CIFAR10/100, ImageNet16-120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Finally, we demonstrate that ZiCo enables a zero-shot NAS framework that can efficiently find the network architectures with highest test accuracy compared to other zero-shot baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In fact, our ZiCo-based zero-shot NAS framework achieves competitive FLOPs-accuracy tradeoffs compared to multiple one-shot and multi-shot NAS, but with much lower time costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To summarize, we make the following major contributions: We theoretically reveal how the mean value and variance of gradients across multiple sam- ples impact the training convergence and generalization capacity of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We propose a new zero-shot proxy, ZiCo, that works better than existing proxies on popu- lar NAS-Benchmarks (NASBench101, NATS-Bench-SSS/TSS, TransNASBench-101) for multiple applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' image classification/reconstruction and pixel-level prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We demonstrate that our proposed zero-shot NAS achieves competitive test accuracy with representative one-shot and multi-shot NAS with much less search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We discuss related work in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Section 3, we introduce our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We introduce our proposed zero-shot proxy (ZiCo) and the NAS framework in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Section 5 validates our analysis and presents our results with the proposed zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We conclude the paper in Section 6 with remarks on our main contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ZERO-SHOT NAS The goal of zero-shot NAS is to rank the accuracy of various candidate network architectures without training, such that we can replace the expensive training process in NAS with some computation- efficient proxies Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Javaheripi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, the quality of the proxy determines the effectiveness of zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Several works use the number of linear regions to approximately measure the expressivity of a deep neural network Mellor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bhardwaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Alternatively, most of the existing proxies are derived from the gradient of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, Synflow, SNIP, and GraSP rely on the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t the parameters of neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' they are proved to be the different approximations of Taylor expansion of deep neural networks Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, the Zen-score approximates the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t featuremaps and measures the complexity of neural networks Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Furthermore, Jacob cov leverages the Jacobian matrix between the loss and mul- tiple input samples to quantify the capacity of modeling the complex functions Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Though zero-shot NAS can significantly accelerate the NAS process, it has been revealed that the naive proxy #Params generally works better than all the proxies proposed to date Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' These limitations of existing proxies motivate us to look for a new proxy that 2 Published as a conference paper at ICLR 2023 can consistently work better than #Params and address the limitations of existing zero-shot NAS approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 KERNEL METHODS AND CONVERGENCE ANALYSIS Kernel methods are widely explored to analyze the convergence property of networks trained with gradient descent Neal (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Williams (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Allen-Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hanin & Nica (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Golikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, the training of wide neural networks is proved to be equivalent to the optimization of a specific kernel function Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chizat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Cho & Saul (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, given the networks with specific width constraints, researchers proved that the training convergence of networks can be described by some corresponding kernels and the convergence rates of training are highly coupled with the eigenvalues of the kernel-based covariance matrix Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Garriga-Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In our work, we extend such kernel-based analysis to reveal the relationships between the gradient properties and the training convergence for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3 CONVERGENCE AND GENERALIZATION VIA GRADIENT ANALYSIS We consider the mean value and standard deviation of gradients across different samples and first explore how these two metrics impact the training convergence of linear regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 LINEAR REGRESSION Inspired by Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b), we use the training set S with M samples as follows: S = {(xi, yi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', M, xi ∈ Rd, yi ∈ R, ||xi|| = 1, |yi| ≤ R, M > 1} (1) where R is a positive constant and || · || denotes the L2-norm of a given vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' xi ∈ Rd is the ith input sample and normalized by its L2-norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', ||xi|| = 1), and yi is the corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We define the following linear model f = aT x optimized with an MSE-based loss function L: mina � i L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' a)) = mina � i 1 2(aT xi − yi)2 (2) where a ∈ Rd is the initial weight vector of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We denote the gradient of L w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t to a as g(xi) when taking (xi, yi) as the training sample: g(xi) = ∂L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' a)) ∂a (3) We denote the jth element of g(xi) as gj(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We compute the mean value (µj) and standard deviation (σj) of gj(xi) across all training samples as follows: µj = 1 M M � i gj(xi) σj = � � � � 1 M M � i (gj(xi) − µj)2 (4) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We denote the updated weight vector as ˆa and denote � ij[gj(xi)]2 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Assume we use the accumulated gradient of all training samples and learning rate η to update the initial weight vector a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', ˆa = a − η � i g(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' If the learning rate 0 < η < 2, then the total training loss is bounded as follows: � i L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ˆa)) ≤ G 2 − η 2M 2(2 − η) � j µ2 j (5) In particular, if the learning rate η = 1 M , then L(ˆa) is bounded by: � i L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ˆa)) ≤ M 2 � j σ2 j (6) We provide the proof in Appendix A and the experimental results to validate this theorem in Sec 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Intuitively, Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 tells us that the higher the gradient absolute mean across different training samples, the lower the training loss the model converges to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', the network converges at a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Similarly, if ηM < 1, the smaller the gradient standard deviation across different training samples/batches, the lower the training loss the model can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3 Published as a conference paper at ICLR 2023 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 MLPS WITH RELU In this section, we generalize the linear model to a network with ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We primarily consider the standard deviation of gradients in the Gaussian kernel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We still focus on the regression task on the training set S defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We consider a neural network in the same form as Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b): h(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' s, W ) = 1 √m m � i srReLU(wT r x) (7) where m is the number of output neurons of the first layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' sr is the rth element in the output weight vector s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' W ∈ Rm×d is the input weight matrix, and wr ∈ Rd is the rth row weight vector in W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For training on the dataset S with M samples defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1, we minimize the following loss function: L(s, W ) = M � i=1 1 2(h(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' s, W ) − yi)2 (8) Following the common practice Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b), we fix the second layer (s) and use gradient descent to optimize the first layer (W ) with a learning rate η: wr(t) = wr(t − 1) − η t � i=0 ∂L(s, W (t − 1)) ∂wr(t − 1) (9) where W (t − 1) denote the input weight matrix after t − 1 training steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' wr(t) denote the rth row weight vector after t training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (Gram Matrix) A Gram Matrix H(t) ∈ RM×M on the training set {(xi, yi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', M} after t training steps is defined as follows: Hij(t) = 1 mxT i xj m � r=1 I{xT i wr(t) ≥ 0, xT j wr(t) ≥ 0} (10) where I is the indicator function and I{A} = 1 if and only if event A happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We denote the λmin(H) as the minimal eigenvalue of a given matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We denote the λ0 = λmin(H(∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Given a neural network with ReLU activation function optimized by minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8, we assume that each initial weight vector {wr(0), r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', n} is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' generated from N(0, I) and the gradient for each weight follows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, σ), where the σ is measured across different training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For some positive constants δ and ϵ, if the learning rate η satisfies η < λ0 √πδ 2M 2√ 2Φ(1−ϵ)tσ, then with with probability at least (1 − δ)(1 − ϵ), the following holds true: for any r ∈ [m], ||wr(0) − wr(t)|| ≤ C = ηtσ � Φ(1 − ϵ), and at training step t the Gram matrix H(t) satisfies: λmin(H(t)) ≥ λmin(H(0)) − 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ > 0 (11) Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide the proof in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We now introduce the following conclusion from Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) to further help our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) Assume we set the number of output neurons of the first layer m = Ω( M 6 λ4 0δ3 ) and we i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' initialize wr ∼ N(0, I) and sr ∼ uniform[{−1, 1}], for r ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' When minimizing the loss function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8 on the training set S in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1, with probability at least 1 − δ over the initialization, the training loss after t training steps is bounded by: L(s, (W (t)) ≤ e−λmin(H(t))L(s, (W (t − 1)) (12) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 and Lemma 1, with probability at least (1 − δ)(1 − ϵ), the following inequality holds true: L(s, (W (t)) ≤ e−λmin(H(0))e 2 √ 2M2ηtσ√ Φ(1−ϵ) √πδ L(s, (W (t − 1)) (13) 4 Published as a conference paper at ICLR 2023 The proof consists of replacing λmin(H(t)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12 with its lower bound given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 shows that after some training steps t, the network with a smaller standard deviation (σ) of gradients will have a smaller training loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', the network has a faster convergence rate at each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We further validate this theorem in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Several prior works show that the generalization capacity of a neural network is highly correlated with its sharpness of the loss function Keskar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Usually, a flatter loss landscape leads to a better generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, it has also been shown that the largest eigenvalue of the Gram matrix of loss can be used to describe the sharpness of the loss landscape Sagun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' more precisely: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The lower the largest eigenvalue of the Gram matrix, the higher the generalization capacity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' [Lewkowycz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Sagun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2016)] Next, we analyze how the gradient of a neural network impacts the largest eigenvalues of the Gram matrix and the generalization capacity of the given network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Under the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2, for some positive constants δ and ϵ, if the learning rate η satisfies η < λ0 √πδ 2M 2√ 2Φ(1−ϵ)tσ, then with with probability at least (1 − δ)(1 − ϵ), for any r ∈ [m], ||wr(0) − wr(t)|| ≤ C = ηtσ � Φ(1 − ϵ), and at training step t, the Gram matrix H(t) satisfies: λmax(H(t)) ≤ λmax(H(0)) + 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ (14) Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide the proof in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 shows that after some training steps t, the network with a smaller standard deviation (σ) of gradients will have a lower largest eigenvalues of the Gram matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', the network has a flatter loss landscape rate at each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, based on Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4, the model will generalize better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We further validate this theorem in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 SUMMARY OF OUR THEORETICAL ANALYSIS Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 tell us that the network with high training convergence speed and generalization capacity should have high absolute mean values and low standard deviation values for the gradient, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t the parameters across different training samples/batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Inspired by these theoretical insights, we next propose a proxy that jointly considers both absolute mean and standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4 NEW ZERO-SHOT PROXY AND NAS FRAMEWORK In this section, we first define our proxy (ZiCo) and then introduce our zero-shot NAS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Following the standard practice, we consider convolutional neural networks (CNNs) as candidate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 PROPOSED ZERO-SHOT PROXY: ZICO Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Given a neural network with D layers and loss function L, the Zero-shot inverse Coefficient of Variation (ZiCo) is defined as follows: ZiCo = D � l=1 log( � ω∈θl |E[∇ωL(Xi, yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Θ)]| � V ar(∇ωL(Xi, yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Θ)) ), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', N} (15) where Θ denote the initial parameters of the given network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' θl denote the parameters of the lth layer of the network, and ω represents each element in θl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xi and yi are the ith input batch and corresponding labels from the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N is number of training batches used to compute ZiCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We incorporate log to stabilize the computation by regularizing the extremely large or small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5 Published as a conference paper at ICLR 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9905 Square Sum of Mean Value ( j 2 j ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='450 Total Training Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mean value (a) Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mean (linear) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 Square Sum of Variance ( j 2 j ) 5 10 15 Total Training Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Variance (b) Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' variance (linear) Figure 1: Training loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' square sum of mean gradients and the sum of gradients variances for linear networks on MNIST after one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, larger mean gradient values lead to lower loss values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' also, networks with smaller � j σ2 j have lower loss values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='025 Standard Deviation ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 Training Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Standard Deviation (a) Training Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' dev (w/ ReLU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='030 Standard Deviation ( ) 0 5 10 15 20 25 30 35 Test Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Standard Deviation (b) Test Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' dev (w/ ReLU) Figure 2: Training loss and Test loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' standard deviation of gradients for two-layer MLPs with ReLU on MNIST after one training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Networks with smaller σ tend to have lower training loss and test loss values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide more results in Sec C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Of note, our metric is applicable to general CNNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', there’s no restriction w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the neural ar- chitecture when calculating ZiCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3, the networks with higher ZiCo tend to have better convergence rates and higher generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, the architectures with higher ZiCo are better architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We remark that the loss values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 15 are all computed with the initial parameters Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' that is, we never update the value of the parameters when computing ZiCo for a given network (hence it follows the basic principle of zero-shot NAS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', never train, and only use the initial parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In practice, two batches are enough to make ZiCo achieve the SOTA performance among all previously proposed accuracy proxies (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, we use only two input batches (N = 2) to compute ZiCo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' this makes ZiCo highly time efficient for a given network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5 EXPERIMENTAL RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 EXPERIMENTAL SETUP We conduct the following types of experiments: (i) Empirical validation of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1, The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (ii) Evaluation of the proposed ZiCo on multiple NAS benchmarks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (iii) Illustration of ZiCo-based zero-shot NAS on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For the experiments (i), to validate Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1, we optimize a linear model as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2 on the MNIST dataset, the mean gradient values and the standard deviation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the total training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, we also optimize the model defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7 on MNIST and report the training loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the standard deviation in order to validate Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For experiments (ii), we compare our proposed ZiCo against existing proxies on three mainstream NAS benchmarks: NATSBench is a popular cell-based search space with two different search spaces: (1) NATSBench-TSS consisting of 15625 total architectures with different cell structures 6 Published as a conference paper at ICLR 2023 Grad_norm SNIP GraSP Fisher Synflow Zen-score FLOPs #Params ZiCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content="63 Correlation Coefficients between Proxies and Test Accuracy Spearman's Kendall's Figure 3: The correlation coefficients between various zero-cost proxies vs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' test accuracy on NAS- Bench101 search space for CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown, our proposed ZiCo correlates best with the real test accuracy and is significantly better than all other proxies except for Zen-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' trained on CIFAR10, CIFAR100, and ImageNet16-120 (Img16-120) datasets, which is just renamed from NASBench-201 Dong & Yang (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2) NATSBench-SSS contains includes 32768 architec- tures (which differ only in the width values of each layer) and is also trained on the same three above datasets Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' NASBench101 provides users with 423k neural architectures with their test accuracy on CIFAR10 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the architectures are built by stacking the same cell multiple times Ying et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' TransNASBench- 101-Mirco contains 4096 networks with different cell structures on various downstream applications (see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2) Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For experiments (iii), we use ZiCo to conduct the zero-shot NAS (see Algorithm 1) on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We first use Algorithm 1 to find the networks with the highest ZiCo under various FLOPs budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We conduct the search for 100k steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' this takes 10 hours on a single NVIDIA 3090 GPU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 GPU days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Then, we train the obtained network with the exact same training setup as Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Specifically, we train the neural network for 480 epochs with the batch size 512 and input resolution 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We also use the distillation-based training loss functions by taking Efficient-B3 as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Finally, we set the initial learning rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 with a cosine annealing scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 VALIDATION OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1&3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3&3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 To empirically validate Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1, we first create the training set S by normalizing randomly sampled 1000 training samples in MNIST and normalizing them with their L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We compute the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the network parameters for each individual training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Next, as discussed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1, we use the accumulated gradient over these samples to update the network parameters with learning rate η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Then, we calculate the square sum of mean gradients and the total training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We repeat the above process 1000 times on the same S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1(a), we plot the total training loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' square sum of mean gradients as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, the networks with the higher square sum of mean gradients values tend to have lower training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1(b) shows that networks with a lower square sum of variance value tend to have lower training loss values, which coincides with the conclusion drawn from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' These results empirically validate our Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, to optimize a two-layer MLP with ReLU activation functions as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7, we use the entire training set of MNIST and apply the gradient descent (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 9) to update the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We set the batch size as 256 and measure the standard deviation of gradients (σ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' parameters across different training batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We set a very small learning rate η = 10−8 to satisfy the assumption in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We plot the training loss and test loss after one training epoch vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' standard deviation of gradients (σ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, the results show that if a network has a lower gradient standard deviation, then it tends to have lower training loss values, and thus, a faster convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' These results empirically prove our claims in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2(a) show that if a network has a lower gradient standard deviation, then it tends to have lower test loss values, which empirically validates Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 ZICO VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' OTHER PROXIES ON NAS BENCHMARKS We first calculate the correlation coefficients between various proxies and the test accuracy on CIFAR10, CIFAR100, and ImageNet16-120 datasets for NATSBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1, ZiCo achieves the highest correlation with the real rest accuracy, except for CIFAR10/100 on NATSBench-SSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, we observe that most of the previously proposed proxies work well for some specific scenarios, but do not generalize well to other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, though Synflow is slightly better than ZiCo for CIFAR10/100 on NATSBench-SSS, it has poor correlation scores in Img16-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Similarly, Zen-score performs well for Img16-120 on NATSBench-SSS, but it doesn’t work well on NATSBench-TSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In contrast, ZiCo has either the highest or second highest correlation coefficients under all these scenarios in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide more results in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7 Published as a conference paper at ICLR 2023 Table 1: The correlation coefficients between various zero-cost proxies and two naive proxies (#Params and FLOPs) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' test accuracy on NATSBench-SSS and NATSBench-TSS (KT and SPR represent Kendall’s τ and Spearman’s ρ, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results in italics represent the values of #Params’ correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The best results are shown with bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, our proposed ZiCo is the only proxy that works consistently better than #Params and is generally the best among all these proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide more results in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' NATSBench-TSS (NASBench201) Dataset CIFAR10 CIFAR100 Img16-120 Proxy Correlation KT SPR KT SPR KT SPR Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='58 SNIP Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='56 Fisher Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 Synflow Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='67 #Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 ZiCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='79 NATSBench-SSS Dataset CIFAR10 CIFAR100 Img16-120 Proxy Correlation KT SPR KT SPR KT SPR Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='34 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42 Fisher Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 Synflow Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='87 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 #Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='84 ZiCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='88 Table 2: The test accuracy of optimal architectures obtained by various zero-shot proxies (average on 5 runs) on NATSBench-TSS search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The best results are shown with bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CIFAR100 Groud Truth Grad norm SNIP GraSP Fisher Jacob cov Synflow Zen-score #Params FLOPs ZiCo 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 Img16-120 Groud Truth Grad norm SNIP GraSP Fisher Jacob cov Synflow Zen-score #Params FLOPs ZiCo 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 CIFAR10 Groud Truth Grad norm SNIP GraSP Fisher Jacob cov Synflow Zen-score #Params FLOPs ZiCo 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 For NASBench101, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3, ZiCo has a significantly higher correlation score with the real test accuracy than all the other proxies, except Zen-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, ZiCo has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 Kendall’s τ score, while #Params is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In general, ZiCo has the highest correlation coefficients among all existing proxies for various search spaces and datasets of NATSBench and NASBench101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Beside the correlation coefficients, we also report the optimal architectures found with various prox- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 2, the architectures found via ZiCo have the highest test accuracy on all these three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To our best knowledge, ZiCo is the first proxy that shows a consistently higher correlation coefficient compared to #Params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The above results validate the effectiveness of our proposed ZiCo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' thus, ZiCo can be directly used to search for optimal networks for various budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Next, we describe the search results in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 ZICO ON IMAGENET Search Space We use the commonly used MobileNetv2-based search space where the candidate networks are built by stacking multiple Inverted Bottleneck Blocks (IBNs) with SE modules Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As for each IBN, the kernel size of the depth- wise convolutional layer is sampled from {3,5,7} and the expansion ratio is randomly selected from {1,2,4,6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We primarily consider ReLU as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We use standard Kaiming Init to initialize all linear and convolution layers for every candidate networks He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' More details of the search space are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We use Algorithm 1 to search networks under various FLOPs budgets (450M, 600M, and 1000M) within the above search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 3, ZiCo outperforms most previous NAS ap- 8 Published as a conference paper at ICLR 2023 Table 3: Comparison of Top-1 accuracy of our ZiCo-based NAS against SOTA NAS methods on ImageNet under various FLOP budgets (averages over three runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For the ‘Method’ column, ‘MS’ means multi-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘OS’ is short for one-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Scaling represents network scaling methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘ZS’ is short for zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' OFA‡ is trained from scratch and reported in Moons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Budget (maximal #FLOPs) Approach FLOPs Top-1 [%] Method Costs [GPU Days] 450M EfficientNet-B0 Tan & Le (2019) 390M 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Scaling 3800 MnasNet-A3 Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 403M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 MS OFA‡ Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 406M 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 OS 50 BN-NAS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021a) 470M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 RLNAS Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 473M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 OS NASNet-B Zoph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018) 488M 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 MS 1800 CARS-D Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 496M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 DONNA Moons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 501M 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 OS 405 #Params 451M 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 ZiCo (Ours) 448M 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 600M DARTS Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 574M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 OS 4 NAO Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018) 584M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 MS 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 PC-DARTS Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 586M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 OS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 BigNAS-L Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020a) 586M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 OS 2304 (TPU days) PNAS Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018a) 588M 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 MS 224 CARS-I Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 591M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 EnTranNAS Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 594M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 OS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ProxylessNAS Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 595M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 OS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 RLNAS Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 597M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9 OS MAGIC-AT Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022) 598M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 OS 2 SemiNAS Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 599M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 MS 4 DONNA Moons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 599M 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 OS 405 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 611M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 OFA‡ Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 662M 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 OS 50 EfficientNet-B1 Tan & Le (2019) 700M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Scaling 3800 ZiCo (Ours) 603M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 1000M sharpDARTS Hundt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 950M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 OS Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 934M 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 EfficientNet-B2 Tan & Le (2019) 1000M 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Scaling 3800 ZiCo (Ours) 1005M 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 proaches by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, when the FLOPs budget is around 450M, ZiCo achieves 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1% Top-1 accuracy, which is competitive with one of the SOTA NAS methods (DONNA), but with fewer FLOPs and 648× faster search speed Moons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, if the FLOPs is 600M, ZiCo achieves 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6% higher Top-1 Accuracy than the latest one-shot NAS method (MAGIC- AT) with a 3× reduction in terms of search time Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To make further comparison with #Params, we also use #Params as the proxy and Algorithm 1 to conduct the search under a 450M FLOPs budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 3, the obtained network by #Params has a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6% lower accuracy than ours (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, even though the correlations for ZiCo and #Params in Table 1 and the optimal networks in Table 2 are similar for small-scale datasets, ZiCo significantly outperforms naive baselines like #Params for large datasets like ImageMet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To conclude, ZiCo achieves SOTA results for Zero-Shot NAS and outperforms naive methods, existing zero-shot proxies, as well as several one-shot and multi-shot methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We remark that these results demonstrate two benefits of our proposed ZiCo: (i) Lightweight com- putation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As discussed in Sec 3, during the search process, to evaluate a given architecture, we only need to conduct the backward propagation twice (only takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3s on an NVIDIA 3090 GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The computation efficiency and exemption of training enable ZiCo to significantly reduce the search time of NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (ii) High correlation with the real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As demonstrated in Sec 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3, ZiCo has a very high correlation score with real accuracy for architectures from various search spaces and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, ZiCo can accurately predict the test accuracy of diverse neural architectures, thus helping find the optimal architectures with the best test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 ABLATION STUDY Number of batches We randomly select 2000 networks from NATSBench-TSS on CIFAR100 dataset and compute ZiCo under varying number of training batches (N in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 15) from {2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=',10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We then calculate the correlation score between ZiCo computed under different N values and the real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4(a) shows that using two batches to compute ZiCo generates the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, in our work, we always use two batches (N = 2) to compute ZiCo since it is both accurate and time-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 9 Published as a conference paper at ICLR 2023 2 4 6 8 10 The Number of Training Batches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 Correlation Correlation Coefficients vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=" #Batch Kendall's Spearman's (a) Correlation vs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' #Batch 0 20 40 60 80 100 120 Batch Size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 Correlation Correlation Coefficients vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=" Batch Size Kendall's Spearman's (b) Correlation vs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Batch Size Figure 4: Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (a) The correlation coefficients between ZiCo computed under varying number of batches and real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (b) The correlation coefficients between ZiCo computed with varying batch size and real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Batch size We pick 2000 networks from NATSBench-TSS on CIFAR100 and compute ZiCo with two batches under varying batch size {1,2,4,8,16,32,64,128}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We then calculate the correlation score between ZiCo computed under various batch sizes and the real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4(b), batch size 64 is enough to stabilize the coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, we set the batch size as 128 and use two batches to compute ZiCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide more ablation studies in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6 CONCLUSION In this work, we have proposed ZiCo, a new SOTA proxy for zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As the main theoretical contribution, we first reveal how the mean value and standard deviation of gradients impact the train- ing convergence of a given architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Based on this theoretical analysis, we have shown that ZiCo works better than all zero-shot NAS proxies proposed so far on multiple popular NAS-Benchmarks (NASBench101, NATSBench-SSS/TSS) for multiple datasets (CIFAR10/100, ImageNet16-120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In particular, we have demonstrated that ZiCo is consistently better than (#Params) and existing zero- shot proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, ZiCo enables us to find architectures with competitive test performance to representative one-shot and multi-shot NAS methods, but with much lower search costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For exam- ple, ZiCo-based NAS can find the architectures with 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1%, 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4%, and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4% test accuracies under 450M, 600M, and 1000M FLOPs budgets on ImageNet within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 GPU days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' REFERENCES Mohamed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Abdelfattah, Abhinav Mehrotra, Lukasz Dudziak, and Nicholas Donald Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zero- cost proxies for lightweight NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A convergence theory for deep learning via over- parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 242–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Sanjeev Arora, Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du, Wei Hu, Zhiyuan Li, Ruslan Salakhutdinov, and Ruosong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' On exact computation with an infinitely wide neural net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8139–8148, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Sanjeev Arora, Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du, Wei Hu, Zhiyuan Li, and Ruosong Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 322–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Designing neural network architec- tures using reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Friedrich L Bauer and Charles T Fike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Norms and exclusion theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Numerische Mathematik, 2 (1):137–141, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 10 Published as a conference paper at ICLR 2023 Kartikeya Bhardwaj, Guihong Li, and Radu Marculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' How does topology influence gradient propagation and model performance of deep networks with densenet-type skip connections?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 13498–13507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computer Vision Foundation / IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kartikeya Bhardwaj, James Ward, Caleb Tung, Dibakar Gope, Lingchuan Meng, Igor Fedorov, Alex Chalfin, Paul Whatmough, and Danny Loh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Restructurable activation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='08562, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kartikeya Bhardwaj, James Ward, Caleb Tung, Dibakar Gope, Lingchuan Meng, Igor Fedorov, Alex Chalfin, Paul N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Whatmough, and Danny Loh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Restructurable activation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='08562, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Efficient architecture search by network transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Han Cai, Ligeng Zhu, and Song Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ProxylessNAS: Direct neural architecture search on target task and hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Han Cai, Chuang Gan, Tianzhe Wang, Zhekai Zhang, and Song Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Once-for-all: Train one network and specialize it for efficient deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Boyu Chen, Peixia Li, Baopu Li, Chen Lin, Chuming Li, Ming Sun, Junjie Yan, and Wanli Ouyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' BN-NAS: neural architecture search with batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision, ICCV 2021, Montreal, QC, Canada, October 10-17, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 307–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' IEEE, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wuyang Chen, Xinyu Gong, and Zhangyang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search on imagenet in four GPU hours: A theoretically inspired perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xiangning Chen and Cho-Jui Hsieh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Stabilizing differentiable architecture search via perturbation- based regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1554–1565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xin Chen, Lingxi Xie, Jun Wu, and Qi Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Progressive differentiable architecture search: Bridging the depth gap between search and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF international conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1294–1303, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wei-Lin Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Cluster-gcn: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 257–266, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' L´ena¨ıc Chizat, Edouard Oyallon, and Francis R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' On lazy training in differentiable program- ming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2933–2943, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Youngmin Cho and Lawrence K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Saul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kernel methods for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Proceedings of a meeting held 7-10 December 2009, Vancouver, British Columbia, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 342–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xiangxiang Chu, Bo Zhang, and Ruijun Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fairnas: Rethinking evaluation fairness of weight sharing neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12239–12248, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 11 Published as a conference paper at ICLR 2023 Tu Do and Ngoc Hoang Luong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Training-free multi-objective evolutionary neural architecture search via neural tangent kernel and number of linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Neural Information Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 335–347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xuanyi Dong and Yi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Searching for a robust neural architecture in four gpu hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1761–1770, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xuanyi Dong and Yi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nas-bench-201: Extending the scope of reproducible neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xuanyi Dong, Lu Liu, Katarzyna Musial, and Bogdan Gabrys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nats-bench: Benchmarking nas algorithms for architecture topology and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszko- reit, and Neil Houlsby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' An image is worth 16x16 words: Transformers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du, Jason D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Gradient descent finds global minima of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1675–1685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Du, Xiyu Zhai, Barnab´as P´oczos, and Aarti Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Gradient descent provably optimizes over-parameterized neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 7th International Conference on Learning Representa- tions, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yawen Duan, Xin Chen, Hang Xu, Zewei Chen, Xiaodan Liang, Tong Zhang, and Zhenguo Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Transnas-bench-101: Improving transferability and generalizability of cross-task neural archi- tecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5251–5260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computer Vision Foundation / IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Stanley C Eisenstat and Ilse CF Ipsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Three absolute perturbation bounds for matrix eigenvalues imply relative bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' SIAM Journal on Matrix Analysis and Applications, 20(1):149–158, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Thomas Elsken, Jan Hendrik Metzen, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The Journal of Machine Learning Research, 20(1):1997–2017, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Adri`a Garriga-Alonso, Carl Edward Rasmussen, and Laurence Aitchison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Deep convolutional net- works as shallow gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 7th International Conference on Learning Representa- tions, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Eugene Golikov, Eduard Pokonechnyy, and Vladimir Korviakov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural tangent kernel: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='13614, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xinyu Gong, Shiyu Chang, Yifan Jiang, and Zhangyang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Autogan: Neural architecture search for generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 3224–3234, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zichao Guo, Xiangyu Zhang, Haoyuan Mu, Wen Heng, Zechun Liu, Yichen Wei, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Single path one-shot neural architecture search with uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In European conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 544–560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Boris Hanin and Mihai Nica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Finite depth and width corrections to the neural tangent kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Delving deep into rectifiers: Surpassing human-level performance on imagenet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 2015 IEEE International Conference on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1026–1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' IEEE Computer Society, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12 Published as a conference paper at ICLR 2023 Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Deep Residual Learning for Image Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Andrew Howard, Mark Sandler, Grace Chu, Liang-Chieh Chen, Bo Chen, Mingxing Tan, Weijun Wang, Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Searching for mobilenetv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Pro- ceedings of the IEEE/CVF international conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1314–1324, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Densely Connected Convolutional Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4700–4708, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Andrew Hundt, Varun Jain, and Gregory D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' sharpdarts: Faster and more accurate differen- tiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09900, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Thorir Mar Ingolfsson, Mark Vero, Xiaying Wang, Lorenzo Lamberti, Luca Benini, and Matteo Spallanzani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Reducing neural architecture search spaces with training-free statistics and compu- tational graph clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In CF ’22: 19th ACM International Conference on Computing Frontiers, Turin, Italy, May 17 - 22, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 213–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ACM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Arthur Jacot, Cl´ement Hongler, and Franck Gabriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural tangent kernel: Convergence and gener- alization in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8580–8589, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mojan Javaheripi, Shital Shah, Subhabrata Mukherjee, Tomasz L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Religa, Caio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mendes, Gus- tavo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' de Rosa, S´ebastien Bubeck, Farinaz Koushanfar, and Debadeepta Dey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Litetransform- ersearch: Training-free on-device search for efficient autoregressive language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02094, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kirthevasan Kandasamy, Willie Neiswanger, Jeff Schneider, Barnab´as P´oczos, and Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search with bayesian optimisation and optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neu- ral Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2020–2029, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Pe- ter Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' On large-batch training for deep learning: Generalization gap and sharp minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Pe- ter Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' On large-batch training for deep learning: Generalization gap and sharp minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Imagenet classification with deep convo- lutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Jaehoon Lee, Lechao Xiao, Samuel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Schoenholz, Yasaman Bahri, Roman Novak, Jascha Sohl- Dickstein, and Jeffrey Pennington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Wide neural networks of any depth evolve as linear models under gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 32: Annual Con- ference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8570–8581, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Namhoon Lee, Thalaiyasingam Ajanthan, and Philip Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' SNIP: SINGLE-SHOT NETWORK PRUNING BASED ON CONNECTION SENSITIVITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Learn- ing Representations, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Aitor Lewkowycz, Yasaman Bahri, Ethan Dyer, Jascha Sohl-Dickstein, and Guy Gur-Ari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The large learning rate phase of deep learning: the catapult mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02218, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Guihong Li, Sumit K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mandal, ¨Umit Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Ogras, and Radu Marculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' FLASH: fast neural architec- ture search with hardware optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', 20(5s):63:1–63:26, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Visualizing the loss land- scape of neural nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 31: Annual Con- ference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6391–6401, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Liam Li and Ameet Talwalkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Random search and reproducibility for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Uncertainty in artificial intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 367–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Tengyuan Liang, Tomaso A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Poggio, Alexander Rakhlin, and James Stokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fisher-rao metric, geometry, and complexity of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 888–896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Ming Lin, Pichao Wang, Zhenhong Sun, Hesen Chen, Xiuyu Sun, Qi Qian, Hao Li, and Rong Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zen-nas: A zero-shot nas for high-performance image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 347–356, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chenxi Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Progressive Neural Architecture Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the European Confer- ence on Computer Vision (ECCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 19–34, 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hanxiao Liu, Karen Simonyan, Oriol Vinyals, Chrisantha Fernando, and Koray Kavukcuoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hi- erarchical representations for efficient architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Confer- ence Track Proceedings, 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hanxiao Liu, Karen Simonyan, and Yiming Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' DARTS: differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Liyang Liu, Shilong Zhang, Zhanghui Kuang, Aojun Zhou, Jing-Hao Xue, Xinjiang Wang, Yimin Chen, Wenming Yang, Qingmin Liao, and Wayne Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Group fisher pruning for practical network compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7021–7032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Shuying Liu and Weihong Deng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Very deep convolutional neural network based image classification using small training sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 2015 3rd IAPR Asian Conference on Pattern Recognition (ACPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 730–734, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Vasco Lopes, Saeid Alirezazadeh, and Lu´ıs A Alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Epe-nas: Efficient performance estimation without training for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Artificial Neural Networks, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 552–563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yiping Lu, Chao Ma, Yulong Lu, Jianfeng Lu, and Lexing Ying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A mean-field analysis of deep resnet and beyond: Towards provable optimization via overparameterization from depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='05508, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Renqian Luo, Fei Tian, Tao Qin, Enhong Chen, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Renqian Luo, Xu Tan, Rui Wang, Tao Qin, Enhong Chen, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Semi-supervised neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 33: Annual Con- ference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Song Mei, Theodor Misiakiewicz, and Andrea Montanari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Conference on Learning Theory, COLT 2019, 25-28 June 2019, Phoenix, AZ, USA, volume 99 of Proceedings of Machine Learning Re- search, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2388–2464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Joe Mellor, Jack Turner, Amos Storkey, and Elliot J Crowley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7588–7598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 Bert Moons, Parham Noorzad, Andrii Skliar, Giovanni Mariani, Dushyant Mehta, Chris Lott, and Tijmen Blankevoort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Distilling optimal neural networks: Rapid search in diverse spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 2021 IEEE/CVF International Conference on Computer Vision, ICCV 2021, Montreal, QC, Canada, October 10-17, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12209–12218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Radford M Neal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Priors for infinite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Bayesian Learning for Neural Networks, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 29–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xuefei Ning, Changcheng Tang, Wenshuo Li, Zixuan Zhou, Shuang Liang, Huazhong Yang, and Yu Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Evaluating efficient performance estimators of neural architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Process- ing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12265–12277, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hieu Pham, Melody Guan, Barret Zoph, Quoc Le, and Jeff Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Efficient neural architecture search via parameters sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4095–4104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Regularized evolution for image classifier architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the aaai conference on artificial intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4780–4789, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Esteban Real et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Large-scale Evolution of Image Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2902–2911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Levent Sagun, Leon Bottou, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Eigenvalues of the hessian in deep learning: Singu- larity and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='07476, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Levent Sagun, Utku Evci, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Ugur G¨uney, Yann N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Dauphin, and L´eon Bottou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Empirical analysis of the hessian of over-parametrized neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Workshop Track Proceedings, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mo- bilenetv2: Inverted Residuals and Linear Bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4510–4520, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yao Shu, Shaofeng Cai, Zhongxiang Dai, Beng Chin Ooi, and Bryan Kian Hsiang Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' NASI: label- and data-agnostic neural architecture search at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yao Shu, Zhongxiang Dai, Zhaoxuan Wu, and Bryan Kian Hsiang Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Unifying and boosting gradient-based training-free neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09785, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Dimitrios Stamoulis, Ruizhou Ding, Di Wang, Dimitrios Lymberopoulos, Bodhi Priyantha, Jie Liu, and Diana Marculescu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Single-path NAS: designing hardware-efficient convnets in less than 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2019, W¨urzburg, Germany, September 16-20, 2019, Proceedings, Part II, volume 11907 of Lecture Notes in Computer Science, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 481–497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhenhong Sun, Ming Lin, Xiuyu Sun, Zhiyu Tan, and Rong Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Revisiting efficient object detection backbones from zero-shot neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='13336, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mingxing Tan and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Efficientnet: Rethinking model scaling for convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6105–6114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mingxing Tan, Bo Chen, Ruoming Pang, Vijay Vasudevan, Mark Sandler, Andrew Howard, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Mnasnet: Platform-aware neural architecture search for mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2820–2828, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hidenori Tanaka, Daniel Kunin, Daniel L Yamins, and Surya Ganguli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Pruning neural networks without any data by iteratively conserving synaptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, volume 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6377–6389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 15 Published as a conference paper at ICLR 2023 Linh Tam Tran and Sung-Ho Bae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Training-free hardware-aware neural architecture search with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Journal of Broadcast Engineering, 26(7):855–861, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Linh-Tam Tran, Muhammad Salman Ali, and Sung-Ho Bae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A feature fusion based indicator for training-free neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' IEEE Access, 9:133914–133923, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Advances in neural informa- tion processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Alvin Wan, Xiaoliang Dai, Peizhao Zhang, Zijian He, Yuandong Tian, Saining Xie, Bichen Wu, Matthew Yu, Tao Xu, Kan Chen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fbnetv2: Differentiable neural architecture search for spatial and channel dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 12965–12974, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chaoqi Wang, Guodong Zhang, and Roger B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Grosse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Picking winning tickets before training by preserving gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Colin White, Mikhail Khodak, Renbo Tu, Shital Shah, S´ebastien Bubeck, and Debadeepta Dey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A deeper look at zero-cost proxies for lightweight nas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In ICLR Blog Track, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' URL https: //iclr-blog-track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='io/2022/03/25/zero-cost-proxies/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Christopher K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computing with infinite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 9, NIPS, Denver, CO, USA, December 2-5, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 295–301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' MIT Press, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bichen Wu, Xiaoliang Dai, Peizhao Zhang, Yanghan Wang, Fei Sun, Yiming Wu, Yuandong Tian, Peter Vajda, Yangqing Jia, and Kurt Keutzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fbnet: Hardware-aware efficient convnet design via differentiable neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 10734–10742, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Meng-Ting Wu, Hung-I Lin, and Chun-Wei Tsai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A training-free genetic neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In ACM ICEA ’21: 2021 ACM International Conference on Intelligent Computing and its Emerg- ing Applications, Jinan, China, December 28 - 29, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 65–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ACM, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lichuan Xiang, Lukasz Dudziak, Mohamed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Abdelfattah, Thomas Chau, Nicholas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lane, and Hongkai Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zero-cost proxies meet differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='06799, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lichuan Xiang, Lukasz Dudziak, Mohamed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Abdelfattah, Thomas Chau, Nicholas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lane, and Hongkai Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zero-cost proxies meet differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='06799, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Sirui Xie, Hehui Zheng, Chunxiao Liu, and Liang Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' SNAS: stochastic neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Jin Xu, Xu Tan, Kaitao Song, Renqian Luo, Yichong Leng, Tao Qin, Tie-Yan Liu, and Jian Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Analyzing and mitigating interference in neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA, volume 162 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 24646–24662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Jingjing Xu, Liang Zhao, Junyang Lin, Rundong Gao, Xu Sun, and Hongxia Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' KNAS: green neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 11613–11625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Yuhui Xu, Lingxi Xie, Xiaopeng Zhang, Xin Chen, Guo-Jun Qi, Qi Tian, and Hongkai Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Pc-darts: Partial channel connections for memory-efficient architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='05737, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 16 Published as a conference paper at ICLR 2023 Yibo Yang, Shan You, Hongyang Li, Fei Wang, Chen Qian, and Zhouchen Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Towards improving the consistency, efficiency, and flexibility of differentiable neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6667–6676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computer Vision Foundation / IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhaohui Yang, Yunhe Wang, Xinghao Chen, Boxin Shi, Chao Xu, Chunjing Xu, Qi Tian, and Chang Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CARS: continuous evolution for efficient neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2020, Seattle, WA, USA, June 13-19, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1826–1835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computer Vision Foundation / IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chris Ying, Aaron Klein, Eric Christiansen, Esteban Real, Kevin Murphy, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nas- bench-101: Towards reproducible neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7105–7114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Jiahui Yu, Pengchong Jin, Hanxiao Liu, Gabriel Bender, Pieter-Jan Kindermans, Mingxing Tan, Thomas Huang, Xiaodan Song, Ruoming Pang, and Quoc Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bignas: Scaling up neural archi- tecture search with big single-stage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 702–717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Springer, 2020a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Kaicheng Yu, Christian Sciuto, Martin Jaggi, Claudiu Musat, and Mathieu Salzmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Evaluating the search phase of neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Arber Zela, Thomas Elsken, Tonmoy Saikia, Yassine Marrakchi, Thomas Brox, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Understanding and robustifying differentiable architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 8th International Confer- ence on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Chris Zhang, Mengye Ren, and Raquel Urtasun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Graph hypernetworks for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xiao Zhang, Yaodong Yu, Lingxiao Wang, and Quanquan Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Learning one-hidden-layer relu networks via gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1524–1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Xuanyang Zhang, Pengfei Hou, Xiangyu Zhang, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search with random labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2021, virtual, June 19-25, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 10907–10916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Computer Vision Foundation / IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zhihao Zhang and Zhihao Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Gradsign: Model performance inference with theoretical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Dongzhan Zhou, Xinchi Zhou, Wenwei Zhang, Chen Change Loy, Shuai Yi, Xuesen Zhang, and Wanli Ouyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Econas: Finding proxies for economical neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceed- ings of the IEEE/CVF Conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 11396–11404, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hongpeng Zhou, Minghao Yang, Jun Wang, and Wei Pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Bayesnas: A bayesian approach for neural architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7603–7613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Qinqin Zhou, Kekai Sheng, Xiawu Zheng, Ke Li, Xing Sun, Yonghong Tian, Jie Chen, and Ron- grong Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Training-free transformer architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' CoRR, abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='12217, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Barret Zoph and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Neural architecture search with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In 5th In- ternational Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Learning Transferable Archi- tectures for Scalable Image Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8697–8710, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 17 Published as a conference paper at ICLR 2023 A PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 We denote the updated weight vector as ˆa and � ij[gj(xi)]2 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Assume we use the accumulated gradient of all training samples and learning rate η to update the initial weight vector a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', ˆa = a − η � i g(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' If the learning rate 0 < η < 2, then the total training loss is bounded as follows: � i L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ˆa)) ≤ G 2 − η 2M 2(2 − η) � j µ2 j (16) In particular, if the learning rate η = 1 M , then L(ˆa) is bounded by: � i L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ˆa)) ≤ M 2 � j σ2 j (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Given each training sample (xi, yi) the gradient of L w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t to a when taking (xi, yi) as the input is as follows: g(xi) = ∂L(yi, f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' a)) ∂a = xixT i a − yixi (18) We note that: (a − g(xi))T xi − yi = aT xi − aT xixT i xi + yixT i xi − yi = aT xi − (aT xi)(xT i xi) = aT xi − aT xi = 0 =⇒ yi = (a − g(xi))T xi (19) Then the total training loss among all training samples is given by: M � i=1 1 2(ˆaT xi − yi)2 (20) By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 19, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 20 as follows: M � i=1 1 2(ˆaT xi − yi)2 = M � i=1 1 2(ˆaT xi − (a − g(xi))T xi))2 = M � i=1 1 2((ˆa − a + g(xi))T xi))2 (21) Recall the assumption that ˆa = a − η � i g(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 21 as follows: M � i=1 1 2(ˆaT xi − yi)2 = M � i=1 1 2(g(xi) − η � i g(xi))T xi)2 (22) 18 Published as a conference paper at ICLR 2023 According to the Cauchy–Schwarz inequality and ||xi|| = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the total training loss is bounded by: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2(ˆaT xi − yi)2 ≤ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='||(g(xi) − η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g(xi)||2 ∗ ||xi||2 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g(xi)||2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='((gj(xi) − ηMµj)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='η2M 2µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='(ηMµj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='η2M 2µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='(ηMµjMµj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2G + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='(η2M 2µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j − 2ηM 2µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2G − ηM 2(2 − η) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='(23) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='Since �M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2(ˆaT xi −yi)2 is always non-negative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the above upper bound of training loss satisfies: 1 2G − ηM 2(2 − η) � j µ2 j ≥ M � i=1 1 2(ˆaT xi − yi)2 ≥ 0 (24) Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' if 0 < η < 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' then η(2 − η) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, the larger � j µ2 j term would make the upper bound of training loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 23 closer to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In other words, the higher the gradient absolute mean values across different training samples/batches, the lower the training loss values the model converges to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', the network converges at a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In particular, if η = 1 M , the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 23 can be rewritten as: M � i=1 1 2(ˆaT xi − yi)2 ≤ 1 2 M � i=1 d � j=1 ((gj(xi) − µj)2 = 1 2 � j Mσ2 j = M 2 � j σ2 j (25) This completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' B PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 Given a neural network with ReLU activation function optimized by minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8, we assume that each initial weight vector {wr(0), r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', n} is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' generated from N(0, I) and the gradient for each weight follows an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For some positive constants δ and ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' if the learning rate η satisfies η < λ0 √πδ 2M 2√ 2Φ(1−ϵ)tσ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' then with with probability at least (1 − δ)(1 − ϵ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' the following holds true: for any r ∈ [m],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ||wr(0) − wr(t)|| ≤ C = ηtσ � Φ(1 − ϵ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' and at training step t the Gram matrix H(t) satisfies: 19 Published as a conference paper at ICLR 2023 λmin(H(t)) ≥ λmin(H(0)) − 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ > 0 (26) Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We first compute the probability of ||wr(0) − wr(t)|| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Based on the assumption wi(0), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', n} follows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, I) and the gradient for each weight follows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, σ), considering the weight updating rule defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 9, each element in wr(0)−wr(t) follows a i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, ηtσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, ||wr(0)−wr||2 η2t2σ2 follows the chi-squared distribution with d degrees of freedom χ2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' P(||wr(0) − wr|| ≤ C) = P(||wr(0) − wr(t)||2 ≤ C2) = P(||wr(0) − wr(t)||2 η2t2σ2 ≤ C2 η2t2σ2 ) = P(||wr(0) − wr(t)||2 η2t2σ2 ≤ Φ(1 − ϵ)) = 1 − ϵ (27) Given an input sample xi and a weight vector wr(t) from W (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' we define the following event: Air = {||wr(t) − wr(0)|| ≤ C} ∩ {I{xT i wr(0) ≥ 0} ̸= I{xT i wr(t) ≥ 0}} (28) If ||wr(t) − wr(0)|| ≤ C holds true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' xT i wr(t) = xT i (wr(t) − wr(0)) + xT i wr(0) = sign(xT i (wr(t) − wr(0)))||wr(t) − wr(0)|| + sign(xT i wr(0))||wr(0)|| (29) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 29 tells us that if ||wr(0)|| is larger than ||wr(t) − wr(0)||, then xT i wr(0) determines the sign value of xT i wr(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' in other words, xT i wr(t) always has the same sign values with xT i wr(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', I{xT i wr(0) ≥ 0} = I{xT i wr(t) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' That is, if ||wr(t) − wr(0)|| ≤ C and I{xT i wr(0) ≥ 0} ̸= I{xT i wr(t) ≥ 0} hold true, then ||wr(0)|| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, the probability of event Air: P(Air) ≤ P({||wr(0)|| ≤ C}) (30) By anti-concentration inequality of Gaussian distribution Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b), we have: P(Air) ≤ P({||wr(0)|| ≤ C}) ≤ √ 2C √π (31) Therefore, if any weight vector w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' wm satisfies ||wr(0) − wr(t)|| ≤ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' we can bound the entry-wise deviation on the Gram matrix H(t) at training step t: for any (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' j) ∈ [n] × [n]: E[|Hij(0) − Hij(t)|] =E[ 1 m|xT i xj m � r=1 (I{xT i wr(0) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' xT j wr(0) ≥ 0} − I{xT i wr(t) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' xT j wr(t) ≥ 0})|] =E[ 1 m|xT i xj m � r=1 (I{xT i wr(0) ≥ 0}I{xT j wr(0) ≥ 0} − I{xT i wr(t) ≥ 0}I{xT j wr(t) ≥ 0})|] ≤E[ 1 m m � r=1 (I{Air ∪ Ajr}] ≤ P(Air) + P(Ajr) ≤2 √ 2C √π (32) where the expectation is summing over the initial weight w(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, considering all the elements in H, we have: E[ M,M � i=1,j=1 |Hij(0) − Hij(t)|] ≤ 2M 2√ 2C √π (33) 20 Published as a conference paper at ICLR 2023 Therefore, by Markov’s inequality, given the probability 1 − δ, we get: M,M � i=1,j=1 |Hij(0) − Hij(t)| ≤ 2M 2√ 2C √πδ (34) In Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b), the authors prove that, given a small perturbation K: if [ � ij |Hij(0) − Hij|] ≤ K, then λmin(H) ≥ λmin(H(0)) − K (35) In our case, K in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 35 is given by 2M 2√ 2C √πδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, λmin(H(t)) ≥ λmin(H(0)) − 2M 2√ 2C √πδ = λmin(H(0)) − 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ (36) We replace the term η in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 with η’s upper bound given in the assumption of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', η < λ0 √πδ 2M 2√ 2Φ(1−ϵ)tσ, we can get that λmin(H(t)) is always larger than 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' that is: λmin(H(t)) ≥ λmin(H(0)) − 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ > 0 (37) This completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' C PROOF OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 Given a neural network with ReLU activation function optimized by minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8, we assume that each initial weight vector {wr(0), r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', n} is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' generated from N(0, I) and the gradient for each weight follows an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' distribution N(0, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For some positive constants δ and ϵ, if the learning rate η satisfies η < λ0 √πδ 2M 2√ 2Φ(1−ϵ)tσ, then with with probability at least (1−δ)(1−ϵ), the following holds true: for any r ∈ [m], ||wr(0)−wr(t)|| ≤ C = ηtσ � Φ(1 − ϵ), and at training step t, the Gram matrix H(t) satisfies: λmax(H(t)) ≤ λmax(H(0)) + 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ (38) Φ(·) is the inverse cumulative distribution function for a d-degree chi-squared distribution χ2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The proof is similar to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We provide the entire proof below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We first compute the probability of ||wr(0) − wr(t)|| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Based on the assumption that {wi(0), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', n} follow i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, I) and the gradient of each weight follows i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, σ), considering the weight updating rule defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 9 with learning rate η, each element in wr(0) − wr(t) follows an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' N(0, ηtσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ||wr(0)−wr||2 η2t2σ2 follows a chi-distribution with d degrees of freedom χ2(d): P(||wr(0) − wr|| ≤ C) = P(||wr(0) − wr(t)||2 ≤ C2) = P(||wr(0) − wr(t)||2 η2t2σ2 ≤ C2 η2t2σ2 ) = P(||wr(0) − wr(t)||2 η2t2σ2 ≤ Φ(1 − ϵ)) = 1 − ϵ (39) Given an input sample xi and a weight vector wr(t) from W (t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' we define the following event: Air = {||wr(t) − wr(0)|| ≤ C} ∩ {I{xT i wr(0) ≥ 0} ̸= I{xT i wr(t) ≥ 0}} (40) 21 Published as a conference paper at ICLR 2023 If ||wr(t) − wr(0)|| ≤ C holds true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' then: xT i wr(t) = xT i (wr(t) − wr(0)) + xT i wr(0) = sign(xT i (wr(t) − wr(0)))||wr(t) − wr(0)|| + sign(xT i wr(0))||wr(0)|| (41) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 41 implies that if ||wr(0)|| is larger than ||wr(t) − wr(0)||, then xT i wr(0) determines the sign value of xT i wr(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In other words, xT i wr(t) always has the same sign values as xT i wr(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' that is, I{xT i wr(0) ≥ 0} = I{xT i wr(t) ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, if ||wr(t) − wr(0)|| ≤ C and I{xT i wr(0) ≥ 0} ̸= I{xT i wr(t) ≥ 0} hold true, then ||wr(0)|| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, the probability of event Air: P(Air) ≤ P({||wr(0)|| ≤ C}) (42) By the anti-concentration inequality of a Gaussian distribution Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b), we have: P(Air) ≤ P({||wr(0)|| ≤ C}) ≤ √ 2C √π (43) Therefore, if any weight vector w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', wm satisfies ||wr(0) − wr(t)|| ≤ C, we can bound the entry-wise deviation on the Gram matrix H(t) at the training step t: for any (i, j) ∈ [n] × [n]: E[|Hij(0) − Hij(t)|] =E[ 1 m|xT i xj m � r=1 (I{xT i wr(0) ≥ 0, xT j wr(0) ≥ 0} − I{xT i wr(t) ≥ 0, xT j wr(t) ≥ 0})|] =E[ 1 m|xT i xj m � r=1 (I{xT i wr(0) ≥ 0}I{xT j wr(0) ≥ 0} − I{xT i wr(t) ≥ 0}I{xT j wr(t) ≥ 0})|] (44) We note that all the samples in the training set S (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 1) are normalized with their L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, we have both ||xi|| = 1 and ||xj|| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, using the Cauchy–Schwarz inequality, the above equation is bounded as follows: E[|Hij(0) − Hij(t)|] ≤E[ 1 m m � r=1 (I{Air ∪ Ajr}] ≤ P(Air) + P(Ajr)] ≤ 2 √ 2C √π (45) where the expectation is over the initial weight wr(0), r = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Hence, considering all the elements in H, we have: E[ M,M � i=1,j=1 |Hij(0) − Hij(t)|] ≤ 2M 2√ 2C √π (46) Therefore, by the Markov’s inequality, given the probability 1 − δ, we get: M,M � i=1,j=1 |Hij(0) − Hij(t)| ≤ 2M 2√ 2C √πδ (47) Based on the matrix perturbation theory Bauer & Fike (1960);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Eisenstat & Ipsen (1998), given a small perturbation K: if [ � ij |Hij(0) − Hij(t)|] ≤ K, then λmax(H(t)) ≤ λmax(H(0)) + K (48) In our case, K in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 48 is given by 2M 2√ 2C √πδ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' that is: λmax(H(t)) ≤ λmax(H(0)) + 2 √ 2M 2ηtσ � Φ(1 − ϵ) √πδ (49) This completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 22 Published as a conference paper at ICLR 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='030 Standard Deviation ( ) 0 5 10 15 20 25 30 35 Test Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Standard Deviation (a) Batch size=64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='035 Standard Deviation ( ) 0 5 10 15 20 25 30 35 Test Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Standard Deviation (b) Batch size=128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='035 Standard Deviation ( ) 0 5 10 15 20 25 30 35 40 Test Loss Loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Standard Deviation (c) Batch size=256 Figure 5: Test loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' standard deviation of gradients (σ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 13) for randomly sampled 500 two- layer MLPs with ReLU on MNIST after one training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We train these networks by minimizing the MSE loss between the output of networks and the real labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown, the Networks with smaller σ tend to have lower test loss values and thus have a better generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 SUPPLEMENTARY RESULTS: VALIDATION OF THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 To empirically validate Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5, we first create the training set S by normalizing the training samples in MNIST with their L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Next, we optimize a two-layer MLP with ReLU activation functions as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We use the entire training set of MNIST and apply the gradient descent (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 9) to update the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We vary the batch size as {64,128,256} and measure the standard deviation of gradients (σ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' parameters across different training batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' A very small learning rate of η = 10−8 is set to satisfy the assumption in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 5 demonstrates the training loss after one epoch vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' standard deviation of gradients (σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, the results show that if a network has a lower gradient standard deviation, then it tends to have lower test loss values, and thus, a better generalization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' These results empirically prove our claims in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' D EXPERIMENTAL SETUP OF ZICO ON IMAGENET D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 SEARCH SPACE We use the commonly used MobileNetv2-based search space where the candidate networks are built by stacking multiple Inverted Bottleneck Blocks (IBNs) with SE modules Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' all the SE modules share the same se ratio as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For each IBN, we vary the kernel size of the depth-wise convolutional layer from {3,5,7} and sample the expansion ratio from {1,2,4,6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We primarily consider ReLU as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For each point-wise convolutional layer, the range of the number of channels is from 8 to 1024 with a step size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We use standard Kaiming Init to initialize all linear and convolution layers for every candidate networks He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 23 Published as a conference paper at ICLR 2023 Algorithm 1 ZiCo-based zero-shot NAS framework INPUT: Number of search steps T Inference budget B, Search space S Set of input batch Z = {(Xi, yi), i = 1, 2} Population size E, Initial network F0 ∈ S OUTPUT: Optimal network FP SEARCH: Initialize F = {F0} for i = 1 to T do Randomly sample network Ft from F Fi = randomly mutated architecture based on Ft from S if Fi meets the inference budget B then Compute ZiCo for Fi on Z by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 15 Add Fi to F if |F| > E then Remove network with the smallest ZiCo from F end if end if end for FP = the network of the highest ZiCo in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 SEARCH ALGORITHM We use an Evolutionary Algorithm (EA) to conduct the zero-shot NAS because it is concise and easy to implement1 As shown in Algorithm 1, we search for the neural architectures with the highest ZiCo within the search space, given a specific budget B (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', FLOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We repeat the search T times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' at each search step, we randomly select a structure from the candidate set F and mutate its architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', kernel size, block type, number of blocks, and layer width) to generate a new network Fi ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' If the generated network Fi meets the inference budget B, we calculate its ZiCo on Z and add Fi to the candidate set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We remove the network with the smallest ZiCo from F, if the number of architectures in F exceeds the threshold E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' After T steps, we select the network with the largest ZiCo as the final (optimal) architecture FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Specifically, We repeat the search 105 times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', T = 105) with the population size E = 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For each of the candidate architectures, we compute ZiCo with two batches randomly sampled from the training set of ImageNet with batch size 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In total, it takes 10 hours on a single NVIDIA 3090 GPU for 105 search steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 TRAINING DETAILS : We use the same data augmentations configurations as in Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018): mix-up, label- smoothing, random erasing, random crop/resize/flip/lighting, and AutoAugment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We use the SGD optimizer with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9 and weight decay 4e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We take EfficientNet-B3 as a teacher network and use the knowledge distillation method to train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We set the initial learning rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 and used the cosine annealing scheme to adjust the learning rate during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We train the obtained network 480 epochs, which takes 83 hours on a 40-core Intel Xeon CPU and 8 NVIDIA 3090 GPU-powered server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' E SUPPLEMENTARY RESULTS ON NAS BENCHMARKS E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 COMPARISON WITH MORE PROXIES In this section, we further provide the comparison between our proposed ZiCo and more proxies proposed recently: KNAS (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021)), NASWOT (Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021)), GradSign ( Zhang & Jia (2022)), and NTK (TE-NAS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b), NASI Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' To compute the correlations, we use the official code released by the authors of the above papers to obtain the values 1One can also use other methods to perform the search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' check Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 24 Published as a conference paper at ICLR 2023 Table 4: The correlation coefficients between various zero-cost proxies and two naive proxies (#Params and FLOPs) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' test accuracy on NATSBench-SSS and NATSBench-TSS (KT and SPR represent Kendall’s τ and Spearman’s ρ, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results in italics represent the values of #Params’ correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results better than #Params are shown with bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, our proposed ZiCo is the only proxy that works consistently better than #Params and is gen- erally the best among all these proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‡Both TE-NAS (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b)) and NASI (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b)) use NTK (Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018)) as the accuracy proxy to build their own search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' NATSBench-TSS (NASBench201) Dataset CIFAR10 CIFAR100 Img16-120 Proxy Correlation KT SPR KT SPR KT SPR Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='58 SNIP Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='58 GraSP Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='56 Fisher Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 Synflow Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 KNAS Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 GradSign Zhang & Jia (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='05 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='87 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 #Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='84 ZiCo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='88 of these proxies2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 4, our proposed ZiCo performs better than all these proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For example, NASWOT and GradSign achieve a similar correlation score as ZiCo on NATSBench-TSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' however, ZiCo has a significantly higher correlation score than these two proxies on NATSBench- SSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 2NASI uses NTK to build their own search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Here we directly compute the correlation between NTK and the real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 25 Published as a conference paper at ICLR 2023 Table 5: The correlation coefficients under different proxies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' test performance on TransNAS- Bench-101-Mirco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, our proposed ZiCo is consistently very close to the best score (only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 lower score) except for Autoencoding (still, ZiCo is the second best on Autoencoding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Though Fisher works better than ZiCo on Autoencoding, ZiCo has a significantly higher score on the rest of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We note that existing proxies do not achieve a high correlation on all tasks consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Autoencoding Scene Classification Proxy Kendall’s τ Spearman’s ρ Kendall’s τ Spearman’s ρ Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 SNIP Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 Grasp Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='28 Fisher Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='67 Synflow Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='72 NASWOT Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='60 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='72 GradSign Zhang & Jia (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='64 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 ZiCo (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 Jigsaw Surface Normal Proxy Kendall’s τ Spearman’s ρ Kendall’s τ Spearman’s ρ Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 SNIP Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='49 Grasp Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 Fisher Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 Synflow Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00 NASWOT Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='71 GradSign Zhang & Jia (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='40 Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='64 ZiCo (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='68 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 COMPARISON ON TRANSNAS-BENCH-101-MICRO In this section, we compare our proposed ZiCo against existing proxies on more diverse tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We compare our proposed ZiCo against existing proxies on one mainstream NAS benchmark TransNAS- Bench-101 Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We pick the largest search space TransNAS-Bench-101-Micro which contains 4096 total architectures with different cell structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We compare ZiCo with various prox- ies under the following four tasks: Scene Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Scene classification is a 47-class classification task that predicts the room type in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Jigsaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' In the Jigsaw task, the input image is divided into nine patches and shuffled based on one of 1,000 predefined permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The target here is to classify which permutation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Autoencoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Autoencoding is a pixel-level prediction task that encodes an input im- age into a low-dimension embedding vector and then reconstructs the raw image from the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Surface Normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Similar to autoencoding, surface normal is a pixel-level prediction task that predicts surface normal statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 5, ZiCo consistently works well on Scene Classification, Jigsaw, and Surface Normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ZiCo has only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 lower correlation scores than the highest scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Though Fisher works better than ZiCo on Autoencoding, ZiCo has significantly higher correlation scores than Fisher on the remaining three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' One possibility why Fisher works best on Autoencoding is that Autoencoding is an image-to-image task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fisher is the only proxy that is built on the gradient 26 Published as a conference paper at ICLR 2023 Table 6: The test performance of optimal architectures obtained by various zero-shot proxies (aver- age on 5 runs) on TransNAS-Bench-101-Micro search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The best results are shown with bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Autoencoding Scene Classification Jigsaw Surface Normal Metric SSIM Accuracy Accuracy SSIM Ground Truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='58 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='59 Grad norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='03 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 Params 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 FLOPs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 ZiCo (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 Table 7: The correlation coefficients under three different proxies vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' test accuracy on NATSBench- SSS (KT and SPR represent Kendall’s τ and Spearman’s ρ, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Clearly, our proposed ZiCo works consistently better than using mean only and STD only on all these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Dataset CIFAR10 CIFAR100 Img16-120 Method KT SPR KT SPR KT SPR Mean Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='81 STD only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='62 ZiCo (Mean + STD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='88 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' feature maps and thus can better extract the information between the input and output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Although Fisher works better than ZiCo on Autoencoding (we are still second best), ZiCo has a significantly higher score on the remaining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in the main paper, we again note that existing proxies do not achieve a high correlation on all tasks consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Table 6 demonstrates the test accuracy of the best architectures found using various proxies on each of the above tasks in TransNAS-Bench-101-Micro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Once again, we see that ZiCo significantly out- performs existing proxies on all tasks except Autoencoding, where we trail Fisher by only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Nonetheless, ZiCo is second best on the Autoencoding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Note that, similar to the correla- tion results in Table 5, other proxies do not consistently achieve high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For instance, while methods like Synflow or Zenscore achieve results close to ours on Scene Classification and Surface Normal, they produce poor results on other tasks like Jigsaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, ZiCo consistently performs well on highly different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 ILLUSTRATION OF VARIOUS PROXIES VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' REAL TEST ACCURACY We provide some illustration figures of real test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' various proxies on NATSBench-SSS search space for CIFAR10 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 6) and ImageNet16-120 datasets(Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We also show the same illustrative results (real test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' various proxies) on NASBench101 search space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' F ABLATION STUDY F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 IMPACT OF MEAN AND STD We randomly select 2000 networks from NATSBench-SSS on CIFAR10, CIFAR100, and Img16- 120 datasets and compute the following proxies: (i) Mean value of gradients only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (ii) Standard deviation (STD) value of gradients only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (iii) Combination of mean and std value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', our proposed ZiCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We then calculate the correlation coefficients between these proxies and the real test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 7, our proposed ZiCo performs better on these three datasets than either using 27 Published as a conference paper at ICLR 2023 Table 8: The test accuracy of optimal architectures obtained by various zero-shot proxies (average on 5 runs) on NATSBench-TSS search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The best results are shown with bold fonts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Proxy CIFAR10 CIFAR100 Img16-120 Costs(GPU hours) Zero-PT+SNIP Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019b) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='18 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='19 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='10 Zero-PT+NASWOT Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='07 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11 Zero-PT+Synflow Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='16 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='13 Zero-PT+KNAS Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='03 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='26 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='10 Zero-PT+Grad norm Abdelfattah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='18 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='07 Zero-PT+Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='05 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='06 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 Zero-PT+GradSign Zhang & Jia (2022) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='23 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='95±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='06 Zero-PT+ZiCo (Ours) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='22 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='66 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='12 mean only or STD only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Therefore, our proposed ZiCo is a better-designed proxy than using mean or STD individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 SEARCH ALGORITHMS: ZERO-COST PT In this section, we demonstrate that our proposed ZiCo can be combined with other search algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We take the Zero-Cost-PT (Zero-PT) as an example Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b) because it is specifically designed for zero-shot proxies and is very time-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Essentially, Zero-PT first integrates all candidate networks into a supernet and assigns learnable weights to each candidate operation (same as one-shot NAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Then Zero-PT uses the zero-cost proxy instead of the training accuracy to update the weights for each candidate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The final architecture is generated by selecting the operations with the highest weight values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We combine different accuracy proxies with Zero-PT under the NASBench-201 and report the op- timal architectures found with various proxies3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 8, the architectures found via ZiCo have the highest test accuracy except for Img16-120 datasets (ZiCo is the second best on Img16-120)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 TRAINING RECIPE: WITHOUT DISTILLATION In this section, we train the obtained network under various FLOPs budgets with the exact same training setup as Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Specifically, we train the neural network for 150 epochs with batch size 512 and input resolution 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We train the network without knowledge distillation and do not use advanced data augmentation methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', mixup, RandAugment, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Finally, we set the initial learning rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 with a cosine annealing scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, we train EfficientNets and the previous SOTA zero-shot NAS approach (Zen-score) under the same setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 9, ZiCo outperforms all of the previous zero-shot NAS approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For exam- ple, when the FLOPs budget is around 600M, ZiCo achieves 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1% Top-1 accuracy, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6% higher than previous SOTA zero-shot NAS methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', Zen-score, and TE-NAS, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, ZiCo finds a model with similar accuracy as EfficientNet-B1, but with 100M fewer FLOPs and much less search cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Overall, compared to the regular one-shot or multi-shot NAS methods, ZiCo achieves comparable or higher test accuracy with 5-9500× less search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 SEARCH SPACE: DARTS In this section, we use ZiCo to conduct the zero-shot NAS on the DARTS search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We first use Algorithm 1 to find the networks with the highest ZiCo without FLOPs budgets on the CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We conduct the search for 100k steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' this takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 hours on a single NVIDIA 3090 GPU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='03 GPU days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Then, we train the obtained network with the exact same training setup as the original DARTS paper Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019)4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' specifically, we train the neural network for 600 epochs 3We implement the code ourselves since the authors have not released the code yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The difference between Table 2 and Table 8 comes from the search algorithm: Table 2 uses traversal search among all candidate networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Table 8 uses perturbation-based zero-cost PT Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 4This is the same setup as most of the baseline works in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 28 Published as a conference paper at ICLR 2023 Table 9: Comparison of Top-1 accuracy of our ZiCo-based NAS against NAS methods with stan- dalone training on ImageNet under various FLOP budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For the ‘Method’ column, ‘MS’ repre- sents multi-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘OS’ is short for one-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Scaling represents network scaling methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘ZS’ is short for zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘no KD’ means we train the network without Knowledge Distilla- tion (KD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘150E’ means we train the network with 150 epochs, similar for 350E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results are averaged over three suns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We note that some NAS methods use knowledge distillation to improve the test accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' hence, we remove those methods from this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results are averaged over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Budget (maximal #FLOPs) Approach FLOPs Top-1 Method Costs[GPU Days] 450M EfficientNet-B0 Tan & Le (2019) [350E] 390M 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Scaling 3800 EfficientNet-B0 Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019)[150E] 390M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 Scaling 3800 MnasNet-A3 Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 403M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 MS BN-NAS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021a) 470M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='7 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 RLNAS Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 473M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 OS NASNet-B Zoph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018) 488M 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 MS 1800 CARS-D Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 496M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) [no KD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 150E] 410M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 #Params 451M 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 ZiCo (Ours) [no KD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 150E] 448M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 600M DARTS Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 574M 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 OS 4 NAO Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018) 584M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 MS 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 PC-DARTS Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 586M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 OS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 PNAS Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018a) 588M 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 MS 224 CARS-I Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 591M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 EnTranNAS Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 594M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 OS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ProxylessNAS Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 595M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 OS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 RLNAS Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 597M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='9 OS MAGIC-AT Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2022) 598M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 OS 2 SemiNAS Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2020) 599M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 MS 4 EfficientNet-B1 Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019)[350E] 700M 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 Scaling 3800 EfficientNet-B1 Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019)[150E] 700M 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 Scaling 3800 TE-NAS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b) 599M 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) [no KD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 150E] 611M 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 ZiCo (Ours) [no KD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 150E] 603M 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 Table 10: Comparison of Top-1 accuracy of our ZiCo-based NAS against NAS methods with stan- dalone training on CIFAR10 in DARTS search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' For the ‘Method’ column,‘MS’ represents multi-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘OS’ is short for one-shot NAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘ZS’ is short for zero-shot NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ‘600E’ means we train the network with 600 epochs, similar to 800E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results are averaged over three suns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' The results are averaged over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Approach Test Error (%) Method Cost(GPU days) AmoebaNet-A Real et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='06 MS 3150 PNAS Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018a) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09 MS 225 ENAS Tan & Le (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='89 MS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 NASNet-A Zoph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2018) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 MS 2000 DARTS-v1 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='14 F OS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 DARTS-v2 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09 OS 1 SNAS Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 OS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 GDAS Dong & Yang (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='82 OS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 BayesNAS Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='04 OS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 ProxylessNAS Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='08 OS 4 P-DARTS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 OS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 PC-DARTS Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='07 OS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 SDARTS-ADV Chen & Hsieh (2020) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='02 OS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 Zen-score Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='04 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='01 TE-NAS Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' (2021b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='064 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='05 ZiCo(ours) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='11 ZS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='03 29 Published as a conference paper at ICLR 2023 with a batch size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We only use the standard data augmentation (normalization, cropping, and random flipping) together with the cutout tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We don’t use knowledge distillation or any other advanced data augmentation tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Finally, we set the initial learning rate as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='025 with a cosine annealing scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' We repeat the same experiments for Zen-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' As shown in Table 10, ZiCo outperforms previous zero-shot NAS approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='g, Zen-score and TE-NAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Moreover, compared to the regular one-shot or multi-shot NAS methods, ZiCo achieves comparable or higher test accuracy with at least 10× less search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 30 Published as a conference paper at ICLR 2023 3 4 5 6 Grad_norm 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Grad_norm ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='36 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='51) (a) Grad norm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 SNIP 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' SNIP ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='59) (b) SNIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 GraSP 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' GraSP ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='09 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='13) (c) GraSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='000250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='000500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='000750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='001000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='001250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='001500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='00175 Fisher 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fisher ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='30 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='44) (d) Fisher 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 Synflow 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Synflow ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='61 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='81) (e) Synflow 25 30 35 40 45 50 55 Zen-score 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zen-score ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='50 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69) (f) Zen-score 0 200000 400000 600000 #Params 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' #Params ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='53 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='72) (g) #Params 200 220 240 260 280 300 ZiCo 40 50 60 70 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ZiCo ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='54 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='73) (h) ZiCo Figure 6: Real test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' various proxies on NATSBench-SSS search space for CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 31 Published as a conference paper at ICLR 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 Grad_norm 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Grad_norm ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='49 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='67) (a) Grad norm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 SNIP 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' SNIP ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='76) (b) SNIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='5 GraSP 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' GraSP ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='29 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='42) (c) GraSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='0008 Fisher 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Fisher ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='41 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57) (d) Fisher 100 150 200 Synflow 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Synflow ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='39 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='57) (e) Synflow 25 30 35 40 45 50 55 Zen-score 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Zen-score ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='69 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='87) (f) Zen-score 0 200000 400000 600000 #Params 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' #Params ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='65 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='84) (g) #Params 200 220 240 260 280 300 320 ZiCo 20 25 30 35 40 45 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ZiCo ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='70 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='88) (h) ZiCo Figure 7: Real test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' various proxies on NATSBench-SSS search space for ImageNet16- 120 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' 32 Published as a conference paper at ICLR 2023 0 200 400 600 800 1000 1200 Grad_norm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' Grad_norm ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='17 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='25) (a) Grad norm 0 2000 4000 6000 SNIP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': 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+page_content=' Zen-score ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63) (f) Zen-score 0 1 2 3 4 #Params 1e7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='8 Test accuracy Test acc vs.' metadata={'source': 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Test accuracy Test acc vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' ZiCo ( = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='46 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content='63) (h) ZiCo Figure 8: Real test accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' various proxies on NASBench101 search space for CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNFIT4oBgHgl3EQfkSuq/content/2301.11300v1.pdf'} +page_content=' τ and ρ are short for Kendall’s τ and Spearman’s ρ, respectively.' metadata={'source': 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Apolloni2,3 + +1 Institut Sénégalais de Recherches Agricoles/Laboratoire National de l’Elevage et de Recherches +Vétérinaires BP 2057 Dakar-Hann, Sénégal +2 CIRAD, UMR ASTRE, Montpellier, France +3 CIRAD, UMR ASTRE, Univ Montpellier, INRAE, Montpellier, France +4 Department of Biosciences, Swansea University, Swansea, SA2 8PP, UK +5 Direction des Services Vétérinaires, Dakar, Sénégal +§ M. C. and A. G. contributed equally + +Corresponding author: Alessandra Giacomini; a.giacomini.2156511@swansea.ac.uk + + +Abstract +Livestock mobility, particularly that of small and large ruminants, is one of the main pillars of +production and trade in West Africa: livestock is moved around in search of better grazing or sold in +markets for domestic consumption and for festival-related activities. These movements cover several +thousand kilometers and have the capability of connecting the whole West African region thus +facilitating the diffusion of many animal and zoonotic diseases. Several factors shape mobility +patterns even in normal years and surveillance systems need to account for such changes. In this +paper, we present a procedure based on temporal network theory to identify possible sentinel locations +using two indicators: vulnerability (i.e. the probability of being reached by the disease) and time of +infection (i.e. the time of first arrival of the disease). Using these indicators in our structural analysis + +of the changing network enabled us to identify a set of nodes that could be used in an early warning +system. +As a case study we simulated the introduction of F.A.S.T. (Foot and Mouth Similar Transboundary) +diseases in Senegal and used data taken from 2020 Sanitary certificates (LPS – laissez-passer +sanitaire) issued by the Senegalese Veterinary Services to reconstruct the national mobility network. +Our analysis showed that a static approach can significantly overestimate the speed and the extent of +disease propagation, whereas temporal analysis revealed that the reachability and vulnerability of the +different administrative departments (used as nodes of the mobility network) change over the course +of the year. For this reason, several sets of sentinel nodes were identified in different periods of the +year, underlining the role of temporality in shaping patterns of disease diffusion. + +Keywords: network analysis, livestock mobility, epidemiology, livestock production + +1. Introduction + +The West African region includes the southern part of the bulge in the African continent and is crossed +by the Sahel, a transitional strip between the Sahara Desert in the north and the Sudanic zone in the +south (Bossard, 2009). The region is composed of 18 countries and is bounded in the north by +Mauritania, Mali and Niger, in the east by Chad and Cameroon, in the south and west by the Atlantic +Ocean. The region is characterized by different climates, and hence, different agro-ecological zones +and different livestock farming systems (Missohou et al., 2016). Livestock farming (particularly +cattle and small ruminants) is one of the most important economic activities in this area. + +In West Africa, livestock mobility is an intrinsic component of livestock production and trade. The +harsh environmental conditions, as well as the absence of the facilities required to slaughter animals +and store meat, means livestock has to be mobile. To optimize the use of natural resources such as +pasture and surface water, whose availability varies throughout the year, livestock farmers are forced +to move their herds around: these movements occur all the year round (nomadism) or in specific +periods (transhumance). Because of the lack of storage facilities and infrastructure, the majority of +animals are sold alive at markets all year round. Most animals are concentrated in the northern part +of West Africa, notably in Mali, Chad, Niger, and Mauritania, where the vast uninhabited areas are +unsuitable for cropping but allow extensive livestock raising, and the animals are moved towards the +greener southern coastal areas. These movements are seasonal, and depend both on the availability +of resources and on other socio-cultural factors, and the mobility patterns and the distribution of the + +volume of animals involved change over the course of the year (Apolloni et al., 2019; Bouslikhane, +2015). These movements integrate the region, connect contrasted agro-ecological areas, and, in +addition, generate income for many supply chain actors, including producers, traders, transporters, +and vendors, and contribute to the food and nutrition security of the region (Valerio, 2020). +As it is, mobility in West Africa is a complex phenomenon involving different temporal scales (from +a few days to several months) and spatial scales (from a few kilometers to reach local markets to +international transhumance and/or international trade), and whose determinants range from +environmental factors, e.g. the availability of natural resources, commercial factors, e.g. market +demand and prices, to social factors, such as religious festivals (Apolloni et al., 2019). + +In Senegal, livestock production is one of the main economic activities, it involves 28% of the +population (ANSD, 2013) and provides almost 4% (ANSD, 2020) of gross national domestic product. +Due to the different agro-ecological zones, several production systems co-exist. Senegal is located on +the Atlantic Coast axes of the transhumance routes (from Mauritania and Mali to Guinea and Guinea- +Bissau) and is involved in international trade movements. Within Senegal, trade follows a strict +market hierarchy: from village markets to consumption markets in coastal areas. National +transhumance involves movements from the central and southern predominantly agricultural area +towards the area of Ferlo in the north. +Like in other West African countries, movements within and towards Senegal vary over the course +of the year. This is particularly true of the Tabaski religious festival, an important Muslim festival +characterized by the sacrifice of rams, and the Grand Magal of Touba, during which the consumption +of beef increases significantly. The two festivals mean imports of livestock increase enormously in a +short period of time (Apolloni et al., 2019; Cesaro et al., 2010). +Animal movements also mean pathogens can be introduced and spread at national and international +scales. Such pathogens spread very rapidly across national borders and have serious socio-economic +and public health consequences. Some of these, such as Contagious Bovine Pleuropneumonia +(CBPP), Foot-and-Mouth Disease (FMD), Peste des Petits Ruminants (PPR), and Rift Valley Fever +(RVF) are currently a major problem in West Africa (Apolloni et al., 2019; Bouslikhane, 2015; +Chaters et al., 2019; Di Nardo et al., 2011). +The porosity of the border, the absence of an animal identification system, together with the lack of +coordinated control and surveillance systems hinders the development of a regional surveillance +system and increases the risk of epidemics (Apolloni et al., 2019). Understanding mobility patterns, +as well as their variations, is of the uttermost importance to optimize surveillance and control systems. +Senegal is one of the few countries in West Africa already equipped with a system for mapping and + +controlling animal movements within its borders. Movements are regulated through the use of +sanitary certificates (LPS – laissez-passer sanitaire), issued by the veterinary services to livestock +transporters every time they move animals. The certificates are also routinely collected and +centralized by the veterinary services. The information that can be retrieved from these data provides +a snapshot of the livestock mobility network at each period of the year and could be used to develop +tools to improve the surveillance system, adapted to the period concerned. +Network-based approaches are widely used in veterinary epidemiology to study the role of animal +mobility in the spread of diseases, with the aim of developing effective strategies for disease +surveillance and control (Dubé et al., 2009; Motta et al., 2017). Network-based approaches make it +possible to depict livestock movements as a spatial network in which the nodes represent villages, +administrative units, markets or herds, and a link is created each time at least one animal is moved +from one node to another. However, while network methods have been extensively applied to +engineer surveillance system in European countries thanks to the existence of vast live animal +movement traceability datasets (i.e. Lentz et al. (Lentz et al., 2016) and Schirdewahn et al. +(Schirdewahn et al., 2021)), little has been done in West Africa due to the scarcity of such information +(Muwonge et al., 2021). Only a few articles that report network analysis in West Africa have been +published recently, including Apolloni et al. (Apolloni et al., 2018), and Nicolas et al. (Nicolas et al., +2018) for Mauritania, Jahel et al. (Jahel et al., 2020) for Senegal and Mauritania, and Valerio et al. +(Valerio et al., 2020) for the whole West Africa region. +Static network approaches may not be the best way to create effective surveillance and control tools +against the spread of infectious diseases, as a static approach can overestimate or underestimate the +rate and extent of outbreaks (Masuda & Holme, 2013). The influence of temporality on the structure +of the network can significantly affect the spread of a disease, which consequently can only be +accurately predicted if the chronology of links is accurately represented (Masuda & Holme, 2013; +Williams & Musolesi, 2016). + +In this work, we used a temporal network approach to assess the influence of change on the diffusion +of animal diseases over time. We used data collected in 2020 by the Senegalese Veterinary Services +to build a representation of the network, and adapted tools from complex networks to assess the risk +of being infected and the role of different Senegalese areas in spreading infections over the course of +the year. To this end, we relied on measures of the “vulnerability” and “reachability” of nodes. +Vulnerability gives an indication of the likelihood a node will be infected, while reachability gives +an indication of the time to infection. This approach takes changes in the network over time into +account as well as the network structure, and differs markedly from the static, individual-centric + +approaches used in previous risk assessments. The objective of the present work is to provide a +theoretical basis for improving the Senegalese surveillance system by identifying different potential +geographical spots that contribute to the spread of pathogens at different times of the year and that +could be used as sentinel nodes. + +2. Materials and methods + + +2.1 Study area +Bordering Mauritania to the north, Mali to the east, Guinea and Guinea-Bissau to the south, and the +Gambia and the Atlantic Ocean to the west, Senegal occupies an area of 196,722 km2 and in 2020, +had an estimated population of more than 16.7 million (World Bank, 2022). + +The administration of the Senegalese territory is organized in 14 regions, 45 departments, and 123 +arrondissements (Ministère de l’Intérieur du Sénégal, 2017). There is a clear contrast between the +empty area in the east (hosting around 10% of the human population of Senegal) and the populated +and urbanized central and areas in the west, where 90% of human population is concentrated, of +which 25% in the Dakar area (ANSD, 2020; World Bank, 2022). + +Senegal’s climate is very varied and distinct climatic zones are characterized by different levels of +rainfall and different types of vegetation. This diverse climate strongly influences the livestock +farming sector, whose different farming systems depending on agro-climatic gradients, among other +factors (Cesaro et al., 2010). + +As mentioned above, the livestock trade is organized in a strict hierarchical system starting at village +weekly markets (Lumo), the animals are collected by traders to be sold at collection markets before +being sent on to consumer markets, where they are sold to be slaughtered. Because there are +practically no meat storage facilities, most trade involves live animals. Livestock trade routes +converge on the Dakar region, the main consumer market, with stops in smaller markets such as Saint- +Louis, Touba, Thiès, and Kaolack. Before reaching the urban markets, the vast majority of the animals +originating from northern Senegal, Mauritania, and Mali, are grouped in Dahra, called the “livestock +capital” of Senegal. Another collection market in the southeastern part of the country also plays a +major role in the livestock trade: Tambacounda, the point of convergence for animals from eastern + +and southern Senegal, as well as from southern Mauritania and Mali. In addition to the movement of +animals for sale, transhumance is widespread in Senegal, both at national scale from the central area +to the north (in particular the Ferlo region), and, due to its location, international, from Mali and +Mauritania to the Senegalese coast (Apolloni et al., 2019; Cesaro et al., 2010) (SI Figure 1). + +2.2 Data +In Senegal, a certification system based on a sanitary pass named “Laissez-Passer Sanitaire” (LPS) +is used to track animal mobility and to map the most important axes of movement in the region. +Veterinary posts belonging to the Senegalese Ministry of Livestock and Animal Production provide +an LPS each time a herd is moved, the document states the origin of the movement (village, +department, region, country), the destination (village, department, region, country), the date, the +species and number of animals involved, and the means of transport. Copies of the LPS are centralized +and stored in electronic form. + +We focused our analyses on movements of cattle and small ruminants (goats and sheep), either +separately or together. For analytical purposes, the two were aggregated on the spatial scale of an +administrative department (all 45 Senegalese departments are involved in this trade) and on a time +scale of one month or one week, depending on the type of analysis: we chose a month as the temporal +unit for the general description of the data and for cluster analysis, and a week to simulate the disease +spread, as a week is a more realistic unit to study disease propagation. + +3. Methods +3.1 Descriptive and network analyses +We analyzed mobility data using a complex network approach. LPS data were used to build three +oriented and weighted networks, one for each species plus one species-independent network: the +nodes corresponded to the departments of origin and destination; a direct link existed between two +nodes if at least one animal was moved from the department of origin to the destination department; +the link was weighted according to the number of animals moved along it. + +A cluster analysis was performed to explore the behavior of the different nodes over the study period. +The nodes were ranked based on their activity, defined as the effective number of animals traded each +month; the number being positive if the inflow of animals was greater than the outflow (importing +behavior), otherwise negative (exporting behavior). + + +Clustering was performed using HCPC (Hierarchical Clustering on Principal Components), which +successively applies three standard methods used in multivariate analyses: (i) Principal Component +Analysis (PCA), which identifies the principal components, (ii) hierarchical clustering, which defines +the optimal number of clusters of nodes according to their score on the principal components, and +(iii) non-hierarchical clustering (in particular the k-means algorithm), which associates a cluster with +each node (Celebi, 2015). +To study the structure of the livestock network, we conducted a spatio-temporal analysis of link’s +frequency, defined as the number of months in the year in which movements occur on the link. We +considered a link to be active when at least one trade movement was recorded in a given month. We +then categorized the links according to the number of months in which they were active. In particular, +we identified four frequency categories, which were, starting from the least frequent: occasional +(activity only occurred in one month per year), intermediate (activity occurred in two or three months +per year), frequent (activity in four to nine months per year), and backbone (activity in 10 to 12 +months). + +To compare the risk of diffusion over the course of the year, we used the epidemic threshold q +(Volkova et al., 2010). This measure provides information on the minimum probability for a virus to +spread throughout the network: the lower the value of the epidemic threshold, the higher the risk of +propagation. +For a weighted network, this parameter can be estimated as follows: +𝑞𝑤 = +〈𝑤𝑜𝑢𝑡〉 +〈𝑤𝑖𝑛 × 𝑤𝑜𝑢𝑡〉 +where 〈 〉 indicates the average value, win and wout indicate the nodes’ in-weight and out-weight, +respectively (Nicolas et al., 2018). + +Following the procedures of Lancelot et al. (Lancelot et al., 2017) and Nicolas et al. (Nicolas et al. +2018), for each of the three mobility networks considered (All species, Cattle, Small ruminants +separately), the epidemic threshold was estimated for each monthly snapshot of the network, to assess +the risk of an epidemic occurring over the course of the year. + +3.2 Simulation of disease spread +Temporality, i.e. the variation in time of the mobility network, affects disease spread. Figure 1 – A +shows an example of a temporal network and its static counterpart. The network is composed of seven + +nodes and eight possible links, whose direction is indicated by the arrows. In this case, the temporal +network is characterized by three temporal snapshots that contain the same nodes but different links. +A link that is present and active in a snapshot is not necessarily the same in the previous or the +following snapshots. If we disregard the information on timing, we obtain an aggregated/static +network composed of the same nodes and links as the temporal network, all present and active at the +same time. If we simulate an outbreak in the two networks (temporal and static) (Figure 1 – B), we +can see that the potential diffusion of the pathogen differs in the two situations. In this case, there is +significantly more propagation in the static network than in the temporal one. This happens because, +in the temporal network, the disease can only propagate through temporal paths. In other words, if a +link connecting an infected node to a susceptible one is active in a specific temporal snapshot, the +disease can spread to the latter; conversely, if the link is not active in the temporal window concerned, +disease propagation stops. + +To study the influence of temporality on disease propagation, we simulated the spread of an animal +disease transmitted by direct contact through the livestock mobility networks. We used a SI +(Susceptible-Infected) model: the disease was transmitted from an infected node to a susceptible one +with a probability of 1, and the infected nodes remained infected for the entire period of analysis, and +were consequently able to continue to spread the disease even weeks after being infected. The aim of +this procedure was to estimate the number of potentially infected nodes when the underlying structure +varied. Because of our focus on the control of transboundary animal diseases, the departments of Mali +and Mauritania, which export livestock directly to Senegal, were chosen as sources of the disease, as +the majority of Senegalese imports of small ruminants and cattle are from these two countries +(Apolloni et al., 2019; Cesaro et al., 2010). +To explore the effect of temporality on the structure of the network, and hence on diffusion of the +disease, we compared results obtained with a static representation (in which the structure of the +network remains unchanged throughout the year) with results obtained with a temporal +representation. In the first case, all the links recorded in the dataset were present at the same time, the +time of activation was not taken into account, while in the second case, we included changes in the +structure in every week of the study period. To this end, we used temporal path formalism, according +to which a temporal path is a sequence of links connecting two nodes with each link in the path +coming temporally after the one before it (Masuda & Holme, 2013). This approach enabled us to +estimate the infection time: that is the minimum number of timesteps (i.e. weeks) needed to create a +temporal path between an infected node and the node under observation. + + + + +Figure 1: (A) An example of a directed temporal network and its static counterpart. The dark links are those in the temporal snapshot, +while the pale links are those that are possible but are not present in the temporal snapshot. (B) Simple simulation of disease spread +in the temporal network (on the left) and the static network (on the right). +Among all the possible temporal paths between the sources and the other nodes, we decided to +consider the “earliest arriving” paths, which represent the first time a node is infected by the disease +(Bender-deMoll et al., 2021; Berlingerio et al., 2013). The speed/rate at which a node became infected +was estimated by the infection time, i.e. the number of weeks that elapsed between the onset of the +disease and the time at which the department concerned was reached for the first time. For static +networks, the speed/rate of infection was estimated from the length of the shortest paths, converting +the links into temporal units, specifically, weeks. If a node was reached by more than one source, the +shortest infection time (for temporal networks) or the shortest path (for static networks) was chosen. + +All descriptive analyses and static/temporal network analyses were carried out using R software with +the following packages: ggplot2 for graphs (Wickham, 2016), ggplot2 and tmap for maps (Tennekes, +2018); FactoMineR (Lê et al., 2008) and factoextra (Kassambara & Mundt, 2020) for cluster analysis, +sna (Butts, 2020), and tsna (Bender-deMoll et al., 2021) for static and temporal network analysis, +respectively. + +B +A + +Temporal network +Static network +t2 +t1,3 +ti +t2 +t3 +t1,3 +Susceptiblenode +Infectednode +Sourceofthe disease4. Results +4.1 Summary statistics +The database contained information on 8,861 livestock trade movements from January to December +2020. The network is composed of a total of 88 nodes, corresponding to an Administrative Unit of +level 2, of which 45 are Senegalese (Departments), and 590 unique links, i.e. origin-destination +combinations. The movements concerned Senegal as the origin and/or destination of 87% of the +movements, and eight other countries: Mali (9%), Gambia (2%), Mauritania (1%), Guinea, Guinea- +Bissau, Burkina Faso, Niger and Nigeria (<1% each) as either the origin or as the destination of +movements. Focusing on Senegal, a total of 6,511 national movements and 2,350 international +movements, over respectively 458 and 132 unique links, involving 87,017 cattle and 553,718 small +ruminants were recorded in the dataset. Despite the large number of national trades, the majority of +animals were moved for the purpose of international trade. More than 95% of these movements were +in trucks, which is the most widely used means of transport for animals in the region concerned. More +than 600,000 animals were transported by truck, the remainder mainly on foot (Table 1). + +The livestock network was analyzed as static but also took temporality into account, which influences +the presence/absence of links. +Table 1: summary of the characteristics of the data analyzed in the study. The number of movements, the number of animals and the +number of unique links are given for each species, type of trade, and means of transport. + + +Trade movements Headcount +Number of +unique links +Species + + + + + +Cattle +3,186 +87,017 +328 + +Small Ruminants 5,675 +553,718 +502 +Type + + + + + +International +2,350 +365,903 +132 + +National +6,511 +274,832 +458 +Means of +transport + + + + + +Train +4 +170 +2 + +Truck +8,239 +608,816 +552 + +On foot +587 +30,354 +85 + +Boat +31 +1,395 +9 + +As shown in Figure 2, all Senegalese administrative departments are involved in animal trade either +as the origin, the destination, or both. Movements are both national and international, and, while +Senegal is the final destination of almost all the trade, many animals are moved not only from other +Senegalese departments, but also from Mali and Mauritania, the main exporters, with some +departments, particularly in Mali, exporting a significant number of animals (Figure 2– A). + +The departments in north-eastern Senegal (Podor, Matam, Kanel, and Ranérou Ferlo), are notable for +their high level of animal “exports”. Other Senegalese departments (Tambacounda in the south, +Koungheul and Gossas in the center, Louga and Kébémer in the north) also export considerable +numbers of animals. + +Concerning imports, the departments that import the most animals are located in the Dakar region, in +particular Pikine, Rufisque, Thiès, and M’bour, where the majority of consumer markets are located. +Other Senegalese departments that import large numbers of animals are Saint-Louis in the north, +Mbacké and Guinguinéo in the center, Ziguinchor in the south-west, Tambacounda and Sayara in the +south-east, on the border with Mali (Figure 2– B). + +Figure 2: Distribution of the volume of animals in the administrative departments of Senegal, according to whether the department is +the origin (A) or the destination (B) of livestock movement. The miniature pie charts show the percentage of cattle (yellow) and small +ruminants (green) in the total number of animals. Quartiles were chosen for the colors representing the volume of animals traded. +Only countries that account for at least 1% of exports (A) or imports (B) are shown. +Figure 3 shows the number of movements and the volume of animals traded in each species (cattle or +small ruminants) per month. Overall, movements of animals for the purpose of trade were less +frequent in the first six months of the year, but increased in July, particularly trade in small ruminants. +Similarly, July was the month with the most trade in small ruminants in the study period, involving +N +0 +50 +100 +150 +200 km +Volume of animals +0 to 9 +10 to 109 +110 to 724 +725 to 6337 +6338 to 209485 +Species +Cattle +Small Ruminants +0 +50 100 150 200km +A +N +0 +50 +100 +150 +200 km +Volume of animals +0 to 20 +21 to 560 +561 to 2438 +2439 to 8464 +8465 to 199744 +Species +Cattle +Small Ruminants +B + +more than 300,000 animals. In August and September, the volume of small ruminants decreased, +while both the movement and volume of cattle traded increased, overtaking those of small ruminants. +In October, November and December, the number of cattle trades decreased, but remained higher +than in the rest of the year, while the number and volume of trade in small ruminants increased, +although less sharply. In 2020, two important Muslin festivals took place at the end of July (Tabaski) +and at the beginning of October (Grand Magal of Touba) and are represented on the chart by a dashed +line and a dotted line, respectively (Figure 3). + + +Figure 3: Number of trade movements (line plot) and volume of livestock traded (bar plot) recorded in 2020, per species and per +month. Data concerning cattle are in yellow, data concerning small ruminants are in green. The dashed line represents the Tabaski +festival (July 31), the dotted line represents the Grand Magal of Touba festival (October 6). + +In the whole year, trades of small ruminants were concentrated on 503 links and trades in cattle on +329 links, including 242 links trades of both species. Like for small ruminants, the highest number of +unique trade links occurred in July, followed by, in decreasing order, December, November and +0 +500 +1,000 +1,500 +0 +100,000 +200,000 +300,000 +Jan. +Feb. +March +April +May +June +July +Aug. +Sept. +Oct. +Nov. +Dec. +Month +Number of trade movements +Volume of animals +Number of trades +Cattle +Small ruminants +Species +Cattle +Small ruminants + +October, also the months with the highest number of trade links for cattle. The links used by the two +species also increased in the last three months of the year (SI Table 1). +Concerning the means of transport, trucks were used for almost all movements of animals for sale +throughout the year. The number of movements peaked in July, and, then after a significant drop, +started to increase again in September (SI Table 2). + +4.2 Cluster analysis +Figure 5 shows the nodes of the livestock network in three (3) clusters: +• Cluster 1, composed of 7 nodes and characterized by a “weak”1 exporting behavior; +• Cluster 2, composed of 61 nodes and characterized by a “strong”1 exporting behavior; +• Cluster 3, composed by 20 nodes and characterized by a “strong”1 importing behavior. +Cluster 1 (in red) aggregates seven nodes, of which four are located on the north-eastern border of +Senegal (Matam, Podor, Kanel, and Ranérou Ferlo) with high volumes of animals traded, while the +other three (Foundiougne, Kaffrine, Gossas) are located on the southern border of Dakar region +(Figure 4 – A). However, in September, Cluster 1 imports are “weak”, with a slightly less than 3,000 +animals imported (Figure 4 – B). + +Cluster 2 (in green) aggregates all the foreign nodes, except Banjul (Gambia), and several nodes +across Senegal, accounting for a total of 61 nodes out of 88. Cluster 2 is “strong” in terms of volume +of animals exported over the year, despite the fact some nodes import more than export. Some nodes +that export large numbers of livestock include Bamako (Mali, 208,462) and Nouakchott (Mauritania, +43,472), while Tambacounda (Senegal, 70,845) is a good example of an importing node (Figure 4 – +A). The highest number of exports by this cluster occurred in July, when the number of animals +exported was slightly under 200,000. (Figure 4 – B). +Cluster 3 (in blue) aggregates 20 nodes of which the majority is concentrated in the Dakar region but +includes some nodes in southern Senegal and one foreign node, Banjul (Gambia). Of the nodes +located in southern Senegal, two are on the border with Mali (Saraya and Kédougou), while the other +four are located farther west. All the nodes in this cluster were characterized by strong import trade, +with most imported animals via Pikine (Senegal, 199,703), but also via Thiès (Senegal, 51,939) and +Kaolack (Senegal, 46,432) (Figure 4 – A). Reflecting the movements of livestock for export, this +cluster shows a peak of imported animals in July, with a volume of around 250,000 animals, and + +1 We introduce the terms importing and exporting behaviour to indicate those nodes whose net flow of animals (difference +between inflow and outflow) through them is positive and negative respectively. Weak and strong refer to magnitude of +the net flow (small or large). + +another, +less +significant +increase +from +September +to +October +(Figure +4 +– +B). + +Figure 4: Clustering of livestock network. (A) Spatial representation of nodes colored according to the cluster to which they belong. +The size of each dot indicates the volume of animals traded over the course of the year and the division is made in quantiles. (B) +Temporal representation of trade by the three clusters over the course of the year, in terms of the volume of animals traded. Imported +animals are represented as positive numbers, exported animals as negative numbers. The dashed line represents the Tabaski festival +on July 31, the dotted line represents the Grand Magal of Touba festival on October 6. +4.3 Frequency of links +Figure 5 shows the trade links divided by the frequency of their activity over the course of the year. +In general, far more links were characterized by low and very low activity than by very high activity. + +The backbone links are three national, short-range connections between north-eastern and north- +western nodes: Kanel – Pikine, Ranérou Ferlo – Linguère, Ranérou Ferlo – Mbacké. The majority of +frequent links is concentrated in the north of Senegal, where several connections link eastern and +western nodes, but some connections link northern and southern nodes. Moreover, some international +links are frequent, in particular four originating from Mali and one from Guinea-Bissau. The number +of intermediate links is significantly larger than that of the two previous categories, with several +connections between Senegal and Mali, and between Senegal and Mauritania. Occasional links are +extremely numerous and dense, with several connections in Senegal but also links to all its +neighboring countries (Figure 5). The majority of intermediate and occasional links are active in July +and October, due to the Tabaski and the Grand Magal of Touba religious festivals. However, +considering the whole study period, frequent links are the most common (SI Figure 4). +10°N +12°N +14°N +16°N +18°N +20°N +15°W +10°W + 5°W + 0° + 5°E +Longitude +Latitude +Cluster +1 +2 +3 +Volume +202 +1892 +8912 +208462 +A +−200,000 +−100,000 +0 +100,000 +200,000 +Jan. +Feb. March April May June July +Aug. Sept. Oct. Nov. Dec. +Month +Volume of animals +Cluster +1 +2 +3 +B + + +Figure 5: Geographical representation of the livestock network links, divided by the frequency of their activity over the year. Backbone +links were active in more than nine months, frequent links were active between four and nine months, intermediate links were active +for two or three months, and occasional links were active for only one month of the study period. We decided to only show the +administrative departments of Senegal on these maps. Therefore, concerning national trade, the origin and the destination are both +Senegalese departments, while for international trades, they may be a Senegalese department or a central point in a foreign country. + +Backbone +18°N +Mauritania +17N +16°N +15°N +Mall +No +13°N +12°N +11°N +Guinea +Frequent +18°N +Mauritania +17°N +16°N +15"N +Mali +14°N +Gambia +13°N +12°N +Guinea +Frequency category +Backbone +Intermediate +Frequent +18°N +Mauritania +intermediate +Occasional +17°N +16°N +15°N +14°N +Gambia +13-N +12°N +11°N +Guinea +Occasional +18°N +Mauritania +17°N +16°N +Niger +15°N +Mali +14°N +Gambia +13°N +Burkina Faso +12°N +Nigeria +11°N +Guinea +18W +12°W +10W +M.8 +Mot +Longitude4.4 Epidemic threshold +Overall, the values of the epidemic threshold of all three networks are extremely low, particularly for +the combined livestock and the small ruminants network, whose results are almost identical. April is +the only month with a significantly higher value than in the rest of the year. On the other hand, the +epidemic threshold values of the cattle network (yellow curve in Figure 6), are higher overall, with a +value of 1 in April. The lowest values of this network were measured in January, June, and from +October until the end of the year (Figure 6). + + +Figure 6: Logarithmic representation of changes in the epidemic threshold over the course of the year in the three livestock mobility +networks. The zero on the left extremity of the horizontal axis identifies the value calculated for the whole year. The small ruminants +network is in yellow, the cattle network in green, and the combined livestock network in blue. + +4.5 Simulation of disease spread + +Maps focused on Senegalese departments were drawn to compare the results of the simulations run +on the three networks in an efficient and easily understandable way (Figure 7). To assess the role of +changes in the structure of the networks over time, and hence changes in disease spread, we compared +the results of a static representation (in the column on the left) with those of a temporal representation +1e−04 +1e−02 +1e+00 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Epidemic Threshold +Network +Cattle +Cattle and small ruminants +Small ruminants + +(in the other seven columns). For the static representation, considering the time the outbreak began is +meaningless, whereas for the temporal one, it is important, as the network structure can change over +time. Therefore, each element in the seven columns representing the temporal networks corresponds +to the results of an outbreak that began in a specific week of the year (the number of the week is given +in the header of each map). + +The departments are colored according to their infection time, i.e. the length of the period before they +were reached by the virus. For each scenario, the infection time was estimated as the time (number of +weeks) elapsed since the outbreak of the epidemic. We created four categories of infection time, each +category is shown in a different color: red for departments infected less than one month from the +beginning of the disease spread (less than 5 weeks), orange for departments infected after 1-2 months +(between 5 and 9 weeks), yellow for departments infected after more than two months (more than 9 +weeks), and green for those never reached by the disease. If a node was reached by infections from +several sources, the shortest infection time was chosen. + +For the static networks, we considered the number of links in the shortest path between the source +and the node as weeks: red for the shortest paths with less than 5 links, orange for paths with between +5 and 9 links, yellow for paths with more than 9 links, and green for nodes that were never reached +by the disease. + +The results presented are those of the simulation of a disease spreading from Mali, the origin of most +animals imported into Senegal in 2020. For the temporal networks, we present only a few weeks +characterized by activity, in order to be able to simultaneously show changes over time and +differences between the three networks. The complete results of the spread of a disease from Mali +plus for a disease spreading from Mauritania can be found in Supplementary Information (SI Figure +5 – 10). + +In general, in all three networks, maps representing aggregate networks strongly overestimated both +the quantity of potentially infected nodes and the earliness of infection, compared to those of temporal +networks. In addition, there is a difference in the potential sanitary risk between the cattle temporal +network and small ruminants temporal network, the latter showing on average wider and potentially +greater disease propagation. However, the combined network of cattle and small ruminants (the +livestock network) is under the greatest sanitary risk. + + +Figure 7 also shows that, particularly for the livestock network and the small ruminants network, the +periods around religious festivals (weeks 30 and 31 for the Tabaski, and weeks 40 and 41 for the +Grand Magal of Touba) are characterized by a large number of infected departments, some of which +are infected early. + + +Figure 7: Geographical representation of infection time in the case of disease propagation from Mali. For each mobility network, the +first column on the left represents the spread in the static network, the other seven columns represent the seven worst scenarios of +transmission if that specific week represents the beginning of the disease spread. The colors indicate the infection time of the disease: +red for less than one month, orange for less than two months, yellow for more than two months, green for nodes that have never been +touched in one year time. For the static networks in the first column on the left, the colors are based on the number of links in the path: +up to 5 in red, green for nodes that have never been touched in one year time. The squares outlined in blue identify the week of the +Tabaski festival (week 31) and the week of the Grand Magal of Touba festival (week 41). +5. Discussion and conclusions +In 2020, during the COVID-19 pandemic, several restrictive measures were introduced in Senegal +that affected both human and livestock mobility. Borders were closed for both humans and animals +in March 2020 and, at the same time, movements between regions were regulated. To supply markets +and families in preparation for the Tabaski festival on July 31, borders were reopened 45 days before +Tabaski and measures were eased for national and international movement (lettre circulaire n° 01806 +PR/MESG/CT-PSS du 17 juin 2020). Similar decisions were taken on the occasion of the Grand +Magal of Touba, a religious pilgrimage during which a large number of cattle in particular are sold +and consumed. The application of these restrictions had a huge effect on the structure of the networks +and on the risk of introduction and diffusion of pathogens, as did re-opening the border. To assess the +impact of these measures on the spread of livestock diseases, and for possible future use, we used +tools from temporal network theory to identify the area with the highest risk of introduction. It is + +Static +All +merimportant to note that normally, there are other religious festivals, like Gamou of Tivaouane, in +addition to the Grand Magal of Touba, but these were cancelled due to COVID-19 pandemic. + +In our study, we considered the diffusion of a generic direct animal disease transmission and +estimated the vulnerability and the reachability of nodes when the underlined network changes over +time. In this way, we were able to identify Senegalese departments that could be infected at the earliest +stage of an epidemic. With a few modifications, our approach could be extended to include vector- +borne diseases. +The structure of the Senegalese livestock network varies widely over the course of the year due to the +seasonality of transhumance and the effect of religious festivals (Apolloni et al., 2019; Jahel et al., +2020) but, in 2020, these effects were exacerbated by the restrictive measures introduced as a result +of Covid-19. In fact, around June and July, we noted a pickup in the movement and exchange of +animals (mainly small ruminants) mainly due to the easing of the restrictive measures in preparation +for the Tabaski festival and (mainly cattle) for the Grand Magal of Touba festival. We also noted that +the dynamics of the small ruminants trade strongly drive the dynamics of the network as a whole. + +Dakar is the main consumption area of Senegal because almost a quarter of the population of the +country live in the city. Consequently, the main markets of Dakar and Pikine (at the entrance of +Dakar) are the main destination of livestock movements. In particular, regular movements occur +between the areas of Kanel, Ranerou Ferlo, Dahra and the Senegalese capital. In these areas, there is +a high concentration of collection markets (local name luma), where traders frequently buy animals +to be sold directly to Dakar, or to the other collection markets in Dahra or Thiès before being sent on +to the capital city. Overall, the majority of northern links end in the Dakar region, or in some smaller +but nevertheless important markets such as Saint-Louis, Thiès, Mbour, but also Ziguinchor in the +south. Some links in the southeast start from Tambacounda, an important point of convergence for +animals from eastern Senegal, as well as from Mauritania and Mali. Our analysis revealed that the +role played by the different departments changes over the course of the year. Locations that are idle +for a large part of the year become active during the Tabaski period and continue to be active until +the end of the year, in particular, departments that produce small ruminants. Occasional and +intermediate links that are active a few times a year, are usually located near festival centers, to +support the increased supply of livestock, thereby increasing the sanitary risk. + +Analysis of the threshold parameters showed that the network is prone to disease spread, but that the +risk fluctuates over the course of the year. The risk increases significantly on the occasion of festivals + +due to the introduction of large numbers of animals and the creation of new commercial routes, and +diseases can then spread easily across the network. However, the potential infected areas, and the +reachable time does not remain stable over the course of the year and this information cannot be +captured using a static representation of the network. In fact, a static representation of the mobility +patterns may largely overestimate the speed and the extent of disease diffusion: when the simple static +approach is used, diseases appear to spread throughout the country in less than a month, whereas +temporal analysis shows that reachability and vulnerability of departments varies over the course of +the year. In most cases, and depending on the species involved, few departments are reached in a +month, although during the months around Tabaski and Grand Magal, the number of departments that +can be reached increases drastically, and for some departments (like Dakar, Thiès, Tambacounda and +Dahra) where the main markets are located, and at the border, this risk is even higher. These results +could be of great interest not only for risk-based surveillance but also for optimizing the distribution +of resources and personnel needed for control at specific times of the year by focusing on the areas +that are most likely to be reached. The fact that departments located at the border are most prone to +early infection, means that sanitary control at the border should be strengthened and surveillance and +control measures should be harmonized at regional level. + +Previous works already underlined the importance of mobility and of data collection as a tool to +improve surveillance and control in Africa (Chaters et al., 2019; Motta et al., 2017; Nicolas et al., +2018). Our work fits into this strand, emphasizing the importance of collecting data on animal +mobility on a regular basis in order to retrieve information on structural changes. The objective of the +present study is to provide theoretical tools to assess the importance of network dynamics when +planning control and surveillance policies. A more detailed analysis focused on specific diseases and +that accounts for volume distribution may reduce the list of departments to monitor. To this end, +further simulations are needed and their results will depend to a large extent on the characteristics of +the disease concerned, e.g. it transmissibility and incubation period, that could shape the spatio- +temporal pattern of the epidemics and hence the involvement of the different departments. Future +works should thus also consider stochastic models like Kim et al. (2018) (Kim et al., 2018) for specific +diseases. In the model we used here, we aggregated data at the spatial scale of an administrative +department, based on the assumption that the diffusion within a department is homogeneous. In +practice, the presence of markets or transhumance corridors could attract movements in specific parts +of the department, thereby increasing the risk in certain locations over the risk in other parts of the +same department. Data on movements within departments were rare in our dataset (because of the +way data were collected) and further field studies are recommended to collect data at a finer scale. + + + +6. Acknowledgments +The authors are grateful to Dr Mbargou Lô, head of the veterinary Services Directorate, and Dr +Mathioro Fall, head of Animal Health Protection Division. The authors acknowledge the support of +Yves Amevoin, Alioune Ka, Fallou Niakh, and Khady Ndiaye, and all those who assisted in the LPS +collection. We thank all the donors who supported the CGIAR Livestock research program through +their contributions to the CGIAR trust fund. +7. Funding +This study was partially funded by the Project Eco-PPR (European Commission through the +International Fund for Agricultural Development (grant number 2000002577) and the CGIAR +Livestock research program, RVF OIE twinning program (CIRAD-ISRA) granted through the EBO- +SURSY project (European Union FOOD/2016/379-660). The funders had no role in study design, +data collection and analysis, decision to publish, or preparation of the manuscript. + +8. Conflict of interest statement +All authors declare that they have no conflicts of interest. + +9. References +ANSD. (2013). Recensement Général de la Population et de l’Habitat, de l’Agriculture et de l’Elevage +(RGPHAE). +ANSD. (2020). Situation économique et sociale du Sénégal Ed. 2017/2018 (p. 413). +Apolloni, A., Corniaux, C., Coste, C., Lancelot, R., & Toure, I. (2019). Livestock Mobility in West +Africa and Sahel and Transboundary Animal Diseases. In Transboundary Animal Diseases in +Sahelian Africa and Connected Regions (p. 31:52). Springer International Publishing. +https://doi.org/10.1007/978-3-030-25385-1 +Apolloni, A., Nicolas, G., Coste, C., EL Mamy, A. B., Yahya, B., EL Arbi, A. S., Gueya, M. B., +Baba, D., Gilbert, M., & Lancelot, R. (2018). Towards the description of livestock mobility + +in Sahelian Africa: Some results from a survey in Mauritania. PLoS ONE, 13(1). +https://doi.org/10.1371/journal.pone.0191565 +Bender-deMoll, S., Morris, M., & Moody, J. (2021, aprile 23). Tools for Temporal Social Network +Analysis [R package tsna version 0.3.3]. Comprehensive R Archive Network (CRAN). +https://CRAN.R-project.org/package=tsna +Berlingerio, M., Coscia, M., Giannotti, F., Monreale, A., & Pedreschi, D. (2013). Evolving networks: +Eras and turning points. Intell. Data Anal. https://doi.org/10.3233/IDA-120566 +Bossard, +L. +(2009). +Regional +Atlas +on +West +Africa. +Éditions +OCDE. +https://doi.org/10.1787/9789264056763-en +Bouslikhane, M. (2015). CROSS BORDER MOVEMENTS OF ANIMALS AND ANIMAL +PRODUCTS AND THEIR RELEVANCE TO THE EPIDEMIOLOGY OF ANIMAL +DISEASES IN AFRICA. OIE Regional Commission. +Butts, C. T. (2020, ottobre 6). Tools for Social Network Analysis [R package sna version 2.6]. +Comprehensive R Archive Network (CRAN). https://CRAN.R-project.org/package=sna +Celebi, M. E. (A c. Di). (2015). Partitional Clustering Algorithms. Springer International Publishing. +https://doi.org/10.1007/978-3-319-09259-1 +Cesaro, J. D., Magrin, G., & Ninot, O. (2010). Atlas de l’elevage au Senegal: Commerces et +territoires. PRODIG. http://publications.cirad.fr/une_notice.php?dk=558823 +Chaters, G., Johnson, P., Cleaveland, S., Crispell, J., De Glanville, W., Doherty, T., Matthews, L., +Mohr, S., Nyasebwa, O., Rossi, G., Salvador, L., Swai, E., & Kao, R. (2019). Analysing +livestock network data for infectious disease control: An argument for routine data collection +in emerging economies. Philosophical Transactions of the Royal Society B: Biological +Sciences, 374, 20180264. https://doi.org/10.1098/rstb.2018.0264 +Di Nardo, A., Knowles, N. j, & Paton, D. j. (2011). Combining livestock trade patterns with +phylogenetics to help understand the spread of foot and mouth disease in sub-Saharan Africa, +the Middle East and Southeast Asia. 30(1), 63. https://doi.org/10.20506/rst.30.1.2022 + +Dubé, C., Ribble, C., Kelton, D., & McNab, B. (2009). A Review of Network Analysis Terminology +and its Application to Foot-and-Mouth Disease Modelling and Policy Development. +Transboundary and Emerging Diseases, 56(3), 73–85. https://doi.org/10.1111/j.1865- +1682.2008.01064.x +Jahel, C., Lenormand, M., Seck, I., Apolloni, A., Toure, I., Faye, C., Sall, B., Lo, M., Diaw, C. S., +Lancelot, R., & Coste, C. (2020). Mapping livestock movements in Sahelian Africa. Scientific +Reports, 10. https://doi.org/10.1038/s41598-020-65132-8 +Kassambara, A., & Mundt, F. (2020, aprile 1). Extract and Visualize the Results of Multivariate Data +Analyses [R package factoextra version 1.0.7]. Comprehensive R Archive Network (CRAN). +https://CRAN.R-project.org/package=factoextra +Kim, Y., Dommergues, L., M’sa, A. B., Mérot, P., Cardinale, E., Edmunds, J., Pfeiffer, D., Fournié, +G., & Métras, R. (2018). Livestock trade network: Potential for disease transmission and +implications for risk-based surveillance on the island of Mayotte. Scientific Reports, 8(1), +11550. https://doi.org/10.1038/s41598-018-29999-y +Lancelot, R., Béral, M., Rakotoharinome, V. M., Andriamandimby, S.-F., Héraud, J.-M., Coste, C., +Apolloni, A., Squarzoni-Diaw, C., de La Rocque, S., Formenty, P. B. H., Bouyer, J., Wint, G. +R. W., & Cardinale, E. (2017). Drivers of Rift Valley fever epidemics in Madagascar. +Proceedings +of +the +National +Academy +of +Sciences, +114(5), +938–943. +https://doi.org/10.1073/pnas.1607948114 +Lê, S., Josse, J., & Husson, F. (2008). FactoMineR: A Package for Multivariate Analysis. Journal of +Statistical Software, 25(1), 1–18. https://doi.org/10.18637/jss.v025.i01 +Lentz, H. H. K., Koher, A., Hövel, P., Gethmann, J., Sauter-Louis, C., Selhorst, T., & Conraths, F. J. +(2016). Disease Spread through Animal Movements: A Static and Temporal Network +Analysis +of +Pig +Trade +in +Germany. +PloS +One, +11(5), +e0155196. +https://doi.org/10.1371/journal.pone.0155196 + +Masuda, N., & Holme, P. (2013). Predicting and controlling infectious disease epidemics using +temporal networks. F1000Prime Reports, 5, 6. https://doi.org/10.12703/P5-6 +Ministère de l’Intérieur du Sénégal. (2017). Politique de Gouvernance intérieure | Ministère de +l’Intérieur. https://interieur.sec.gouv.sn/administration-territoriale/politique-de-gouvernance- +interieure +Missohou, A., Nahimana, G., Ayssiwede, S. B., & Sembene, M. (2016). Elevage caprin en Afrique +de l’Ouest: Une synthèse. Revue d’élevage et de médecine vétérinaire des pays tropicaux, +69(1), 3. https://doi.org/10.19182/remvt.31167 +Motta, P., Porphyre, T., Handel, I., Hamman, S. M., Ngu Ngwa, V., Tanya, V., Morgan, K., Christley, +R., & Bronsvoort, B. M. deC. (2017). Implications of the cattle trade network in Cameroon +for +regional +disease +prevention +and +control. +Scientific +Reports, +7. +https://doi.org/10.1038/srep43932 +Muwonge, A., Bessell, P. R., Porphyre, T., Motta, P., Rydevik, G., Devailly, G., Egbe, N. F., Kelly, +R. F., Handel, I. G., Mazeri, S., & Bronsvoort, B. M. deC. (2021). Inferring livestock +movement networks from archived data to support infectious disease control in developing +countries. https://doi.org/10.1101/2021.03.18.435930 +Nicolas, G., Apolloni, A., Coste, C., Wint, G. R. W., Lancelot, R., & Gilbert, M. (2018). Predictive +gravity models of livestock mobility in Mauritania: The effects of supply, demand and cultural +factors. PLoS ONE, 13(7). https://doi.org/10.1371/journal.pone.0199547 +Schirdewahn, F., Lentz, H. H. K., Colizza, V., Koher, A., Hövel, P., & Vidondo, B. (2021). Early +warning of infectious disease outbreaks on cattle-transport networks. PLOS ONE, 16(1), +e0244999. https://doi.org/10.1371/journal.pone.0244999 +Tennekes, M. (2018). tmap: Thematic Maps in R. Journal of Statistical Software, 84(6). +https://doi.org/10.18637/jss.v084.i06 +Valerio, V. C. (2020). The structure of livestock trade in West Africa (West African Papers Fasc. 29; +West African Papers, Vol. 29). https://doi.org/10.1787/f8c71341-en + +Valerio, V. C., Walther, O. J., Eilittä, M., Cissé, B., Muneepeerakul, R., & Kiker, G. A. (2020). +Network analysis of regional livestock trade in West Africa. PLoS ONE, 15(5). +https://doi.org/10.1371/journal.pone.0232681 +Volkova, V. V., Howey, R., Savill, N. J., & Woolhouse, M. E. J. (2010). Sheep Movement Networks +and +the +Transmission +of +Infectious +Diseases. +PLoS +ONE, +5(6), +e11185. +https://doi.org/10.1371/journal.pone.0011185 +Wickham, H. (2016). ggplot2: Elegant Graphics for Data Analysis (2nd ed. 2016). Springer +International Publishing : Imprint: Springer. https://doi.org/10.1007/978-3-319-24277-4 +Williams, M. J., & Musolesi, M. (2016). Spatio-temporal networks: Reachability, centrality and +robustness. Royal Society Open Science, 3(6), 160196. https://doi.org/10.1098/rsos.160196 +World +Bank. +(2022). +Population +Senegal +[Text/HTML]. +World +Bank. +https://www.worldbank.org/en/country/senegal/overview + + + + +Supplementary information + + + +SI Figure 1: Map of Senegal colored based on the aridity index. The inset map shows all the countries involved in livestock mobility +network. + + + +V +Mauritania +Mali +Niger +SENEGAL +MAURITANIA +Gambia +Burkina Faso +Podor +suin +Nigeria +Senegal +Saint-Louis +RiverValley +Matam +Louga +Sylvopastoral +Dahra +zone +Kanel +Pikine +Tivaouane +Touba +Thies +Mbake +DAKAR +Rur +oue +Diourbel +M'bour +Groundnut +-Kaolack +basin +Tambacounda +Eastern +MALI +Senegal +GAMBIA +casamdne +OKolda +Ziguinchor +GUINEA-BISSAU +GUINEA +0 +100 +200km +Aridity Index +Maincities(population） +Agro-ecologicalzones +0.03 -0.2 +Morethan500,000 +Borders +0.2-0.35 +100,000-500,000 +0.35-0.5 +Main rivers +50000-100,000 +0.5-0.65 +o +Lessthan50,000 +0.65-0.8 + +Small Ruminants +Cattle +Links in common +Whole year + +503 +329 +242 +Months + + + + + +January +42 +27 +13 + +February +52 +27 +22 + +March +59 +30 +19 + +April +26 +18 +12 + +May +27 +22 +11 + +June +71 +28 +20 + +July +202 +47 +37 + +August +29 +21 +6 + +September +24 +41 +14 + +October +131 +123 +66 + +November +163 +113 +69 + +December +184 +128 +82 +SI Table 1: Number of unique trade links in the small ruminant network and in the cattle network. The values are represented divided +by month, while the “whole year” line corresponds to the number of unique links considering the network as static. The last column +shows the number of links that are used by the two species. + + + + + +Truck +Others +Links in common + + +Water +Walking +Train +Whole year + +552 +9 +85 +2 +55 +Months + + + + + + + +January +49 +0 +8 +0 +1 + +February +56 +0 +3 +0 +2 + +March +67 +0 +3 +0 +0 + +April +31 +0 +1 +0 +0 + +May +38 +0 +0 +0 +0 + +June +73 +0 +9 +0 +3 + +July +207 +0 +11 +0 +6 + +August +43 +0 +3 +0 +2 + +September +42 +3 +11 +0 +5 + +October +178 +1 +17 +1 +9 + +November +178 +3 +33 +0 +7 + +December +219 +3 +20 +1 +11 +SI Table 2: Number of unique trade links according to the means of transport. The values are given per month, while the “whole year” +line corresponds to the number of unique links considering the network as static. The last column shows the number of links that are +shared by the movements made by truck and those made by all other types of transport, including on foot. + + + + + + +SI Figure 2: volume of livestock traded in 2020 divided by week. Cattle are shown in yellow and small ruminants are in green. The +black dashed line represents the day of the Tabaski festival (July 31), the black dotted line represents the day of the Grand Magal of +Touba (October 6). The months are indicated by the gray dashed lines. + + + +SI Figure 3: Graphical visualization of methods chosen to assess the number of clusters: (A) Elbow method, (B) cluster dendrogram, +and (C) cluster division with the HCPC (Hierarchical Clustering on Principal Components) function of the FactoMineR package. + +120.000 +B0T000 +folume of animal +WCK +Spedes +Cefle +Smel rumnantOptimalnumberofclusters +Factormap +5e+05 +ofSqu +4e+05 +2e+05 +2 +3 +4 +5 +6 +10 +cluster +Numberofclustersk +ClusterDendrogram +4- +Height +-2. +Dim1 (50.9%) + +SI Figure 4: Activity of the links over the course of the year. For each month, the quantity of active links is shown, colored according +to their frequency. Backbone links are those active more than nine months a year, frequent links are those active from four to nine +months, intermediate links are those active two or three months, and occasional links are only active one month. The orange line +represents the Tabaski festival (July 31), the violet line represents the Grand Magal of Touba festival (October 6). + +200 +150 +100 +50 +Jan. +Fob. +April +May +Juy +Aug. +Sept. +Ort. +Nov. +Dec. +Month +Berckbone +I feguon +I reguency category +Intermeclafe +Occasional + +SI Figure 5: Geographical representation of infection time, in the case of a disease propagated from Mali through the livestock +network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the +path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been reached are in green. + +Infectiontime +Withinonemonth +Withintwomonths +Aftertwomonths +Never +42 + +SI Figure 6: Geographical representation of infection time, in the case of a disease propagated from Mali through the small ruminant +network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: +up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. + +Infectiontime +Withinonemonth +Withintwomonths +Aftertwomonths +Never + +SI Figure 7: Geographical representation of infection time in the case of a disease propagated from Mali through the cattle network. +Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: up to 5 +in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. + +Infectiontime +Withinonemonth +Withintwomonths +Aftertwomonths +Never + +SI Figure 8: Geographical representation of infection time in the case of a disease propagated from Mauritania through the livestock +network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the +path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. + +Infectiontime +Withinonemonth +Withintwomonths +Aftertwomonths +Never + + +SI Figure 9: Geographical representation of infection time in the case of a disease propagated from Mauritania through the small +ruminant network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links +in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. Nodes that have never been touched are colored green. + + + +Infectiontime +Withinonemonth +Withintwomonths +Aftertwomonths +Never +45 + +SI Figure 10: Geographical representation of infection time in the case of a disease propagated from Mauritania through the cattle +network. Temporal network on the left, static network on the right. For the static network, the colors are based on the links in the path: +up to 5 in red, between 5 and 9 in orange, more than 9 in yellow. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' UMR ASTRE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Univ Montpellier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' INRAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Montpellier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' France 4 Department of Biosciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Swansea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Swansea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' SA2 8PP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' UK 5 Direction des Services Vétérinaires,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dakar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Sénégal § M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' contributed equally Corresponding author: Alessandra Giacomini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='giacomini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2156511@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='uk Abstract Livestock mobility, particularly that of small and large ruminants, is one of the main pillars of production and trade in West Africa: livestock is moved around in search of better grazing or sold in markets for domestic consumption and for festival-related activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' These movements cover several thousand kilometers and have the capability of connecting the whole West African region thus facilitating the diffusion of many animal and zoonotic diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Several factors shape mobility patterns even in normal years and surveillance systems need to account for such changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In this paper, we present a procedure based on temporal network theory to identify possible sentinel locations using two indicators: vulnerability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the probability of being reached by the disease) and time of infection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the time of first arrival of the disease).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Using these indicators in our structural analysis of the changing network enabled us to identify a set of nodes that could be used in an early warning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' As a case study we simulated the introduction of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Foot and Mouth Similar Transboundary) diseases in Senegal and used data taken from 2020 Sanitary certificates (LPS – laissez-passer sanitaire) issued by the Senegalese Veterinary Services to reconstruct the national mobility network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Our analysis showed that a static approach can significantly overestimate the speed and the extent of disease propagation, whereas temporal analysis revealed that the reachability and vulnerability of the different administrative departments (used as nodes of the mobility network) change over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For this reason, several sets of sentinel nodes were identified in different periods of the year, underlining the role of temporality in shaping patterns of disease diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Keywords: network analysis, livestock mobility, epidemiology, livestock production 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Introduction The West African region includes the southern part of the bulge in the African continent and is crossed by the Sahel, a transitional strip between the Sahara Desert in the north and the Sudanic zone in the south (Bossard, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The region is composed of 18 countries and is bounded in the north by Mauritania, Mali and Niger, in the east by Chad and Cameroon, in the south and west by the Atlantic Ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The region is characterized by different climates, and hence, different agro-ecological zones and different livestock farming systems (Missohou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Livestock farming (particularly cattle and small ruminants) is one of the most important economic activities in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In West Africa, livestock mobility is an intrinsic component of livestock production and trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The harsh environmental conditions, as well as the absence of the facilities required to slaughter animals and store meat, means livestock has to be mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To optimize the use of natural resources such as pasture and surface water, whose availability varies throughout the year, livestock farmers are forced to move their herds around: these movements occur all the year round (nomadism) or in specific periods (transhumance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Because of the lack of storage facilities and infrastructure, the majority of animals are sold alive at markets all year round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Most animals are concentrated in the northern part of West Africa, notably in Mali, Chad, Niger, and Mauritania, where the vast uninhabited areas are unsuitable for cropping but allow extensive livestock raising, and the animals are moved towards the greener southern coastal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' These movements are seasonal, and depend both on the availability of resources and on other socio-cultural factors, and the mobility patterns and the distribution of the volume of animals involved change over the course of the year (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Bouslikhane, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' These movements integrate the region, connect contrasted agro-ecological areas, and, in addition, generate income for many supply chain actors, including producers, traders, transporters, and vendors, and contribute to the food and nutrition security of the region (Valerio, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' As it is, mobility in West Africa is a complex phenomenon involving different temporal scales (from a few days to several months) and spatial scales (from a few kilometers to reach local markets to international transhumance and/or international trade), and whose determinants range from environmental factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the availability of natural resources, commercial factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' market demand and prices, to social factors, such as religious festivals (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In Senegal, livestock production is one of the main economic activities, it involves 28% of the population (ANSD, 2013) and provides almost 4% (ANSD, 2020) of gross national domestic product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Due to the different agro-ecological zones, several production systems co-exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Senegal is located on the Atlantic Coast axes of the transhumance routes (from Mauritania and Mali to Guinea and Guinea- Bissau) and is involved in international trade movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Within Senegal, trade follows a strict market hierarchy: from village markets to consumption markets in coastal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' National transhumance involves movements from the central and southern predominantly agricultural area towards the area of Ferlo in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Like in other West African countries, movements within and towards Senegal vary over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This is particularly true of the Tabaski religious festival, an important Muslim festival characterized by the sacrifice of rams, and the Grand Magal of Touba, during which the consumption of beef increases significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The two festivals mean imports of livestock increase enormously in a short period of time (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cesaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Animal movements also mean pathogens can be introduced and spread at national and international scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Such pathogens spread very rapidly across national borders and have serious socio-economic and public health consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Some of these, such as Contagious Bovine Pleuropneumonia (CBPP), Foot-and-Mouth Disease (FMD), Peste des Petits Ruminants (PPR), and Rift Valley Fever (RVF) are currently a major problem in West Africa (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Bouslikhane, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Chaters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Di Nardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The porosity of the border, the absence of an animal identification system, together with the lack of coordinated control and surveillance systems hinders the development of a regional surveillance system and increases the risk of epidemics (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Understanding mobility patterns, as well as their variations, is of the uttermost importance to optimize surveillance and control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Senegal is one of the few countries in West Africa already equipped with a system for mapping and controlling animal movements within its borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Movements are regulated through the use of sanitary certificates (LPS – laissez-passer sanitaire), issued by the veterinary services to livestock transporters every time they move animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The certificates are also routinely collected and centralized by the veterinary services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The information that can be retrieved from these data provides a snapshot of the livestock mobility network at each period of the year and could be used to develop tools to improve the surveillance system, adapted to the period concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Network-based approaches are widely used in veterinary epidemiology to study the role of animal mobility in the spread of diseases, with the aim of developing effective strategies for disease surveillance and control (Dubé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Motta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Network-based approaches make it possible to depict livestock movements as a spatial network in which the nodes represent villages, administrative units, markets or herds, and a link is created each time at least one animal is moved from one node to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' However, while network methods have been extensively applied to engineer surveillance system in European countries thanks to the existence of vast live animal movement traceability datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2016) and Schirdewahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Schirdewahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2021)), little has been done in West Africa due to the scarcity of such information (Muwonge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Only a few articles that report network analysis in West Africa have been published recently, including Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2018), and Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2018) for Mauritania, Jahel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Jahel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2020) for Senegal and Mauritania, and Valerio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Valerio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2020) for the whole West Africa region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Static network approaches may not be the best way to create effective surveillance and control tools against the spread of infectious diseases, as a static approach can overestimate or underestimate the rate and extent of outbreaks (Masuda & Holme, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The influence of temporality on the structure of the network can significantly affect the spread of a disease, which consequently can only be accurately predicted if the chronology of links is accurately represented (Masuda & Holme, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Williams & Musolesi, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In this work, we used a temporal network approach to assess the influence of change on the diffusion of animal diseases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We used data collected in 2020 by the Senegalese Veterinary Services to build a representation of the network, and adapted tools from complex networks to assess the risk of being infected and the role of different Senegalese areas in spreading infections over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To this end, we relied on measures of the “vulnerability” and “reachability” of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Vulnerability gives an indication of the likelihood a node will be infected, while reachability gives an indication of the time to infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This approach takes changes in the network over time into account as well as the network structure, and differs markedly from the static, individual-centric approaches used in previous risk assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The objective of the present work is to provide a theoretical basis for improving the Senegalese surveillance system by identifying different potential geographical spots that contribute to the spread of pathogens at different times of the year and that could be used as sentinel nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Materials and methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1 Study area Bordering Mauritania to the north, Mali to the east, Guinea and Guinea-Bissau to the south, and the Gambia and the Atlantic Ocean to the west, Senegal occupies an area of 196,722 km2 and in 2020, had an estimated population of more than 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='7 million (World Bank, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The administration of the Senegalese territory is organized in 14 regions, 45 departments, and 123 arrondissements (Ministère de l’Intérieur du Sénégal, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' There is a clear contrast between the empty area in the east (hosting around 10% of the human population of Senegal) and the populated and urbanized central and areas in the west, where 90% of human population is concentrated, of which 25% in the Dakar area (ANSD, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' World Bank, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Senegal’s climate is very varied and distinct climatic zones are characterized by different levels of rainfall and different types of vegetation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This diverse climate strongly influences the livestock farming sector, whose different farming systems depending on agro-climatic gradients, among other factors (Cesaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' As mentioned above, the livestock trade is organized in a strict hierarchical system starting at village weekly markets (Lumo), the animals are collected by traders to be sold at collection markets before being sent on to consumer markets, where they are sold to be slaughtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Because there are practically no meat storage facilities, most trade involves live animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Livestock trade routes converge on the Dakar region, the main consumer market, with stops in smaller markets such as Saint- Louis, Touba, Thiès, and Kaolack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Before reaching the urban markets, the vast majority of the animals originating from northern Senegal, Mauritania, and Mali, are grouped in Dahra, called the “livestock capital” of Senegal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Another collection market in the southeastern part of the country also plays a major role in the livestock trade: Tambacounda, the point of convergence for animals from eastern and southern Senegal, as well as from southern Mauritania and Mali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In addition to the movement of animals for sale, transhumance is widespread in Senegal, both at national scale from the central area to the north (in particular the Ferlo region), and, due to its location, international, from Mali and Mauritania to the Senegalese coast (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cesaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2010) (SI Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2 Data In Senegal, a certification system based on a sanitary pass named “Laissez-Passer Sanitaire” (LPS) is used to track animal mobility and to map the most important axes of movement in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Veterinary posts belonging to the Senegalese Ministry of Livestock and Animal Production provide an LPS each time a herd is moved, the document states the origin of the movement (village, department, region, country), the destination (village, department, region, country), the date, the species and number of animals involved, and the means of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Copies of the LPS are centralized and stored in electronic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We focused our analyses on movements of cattle and small ruminants (goats and sheep), either separately or together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For analytical purposes, the two were aggregated on the spatial scale of an administrative department (all 45 Senegalese departments are involved in this trade) and on a time scale of one month or one week, depending on the type of analysis: we chose a month as the temporal unit for the general description of the data and for cluster analysis, and a week to simulate the disease spread, as a week is a more realistic unit to study disease propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1 Descriptive and network analyses We analyzed mobility data using a complex network approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' LPS data were used to build three oriented and weighted networks, one for each species plus one species-independent network: the nodes corresponded to the departments of origin and destination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' a direct link existed between two nodes if at least one animal was moved from the department of origin to the destination department;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the link was weighted according to the number of animals moved along it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' A cluster analysis was performed to explore the behavior of the different nodes over the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The nodes were ranked based on their activity, defined as the effective number of animals traded each month;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the number being positive if the inflow of animals was greater than the outflow (importing behavior), otherwise negative (exporting behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Clustering was performed using HCPC (Hierarchical Clustering on Principal Components),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' which successively applies three standard methods used in multivariate analyses: (i) Principal Component Analysis (PCA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' which identifies the principal components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (ii) hierarchical clustering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' which defines the optimal number of clusters of nodes according to their score on the principal components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' and (iii) non-hierarchical clustering (in particular the k-means algorithm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' which associates a cluster with each node (Celebi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To study the structure of the livestock network, we conducted a spatio-temporal analysis of link’s frequency, defined as the number of months in the year in which movements occur on the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We considered a link to be active when at least one trade movement was recorded in a given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We then categorized the links according to the number of months in which they were active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In particular, we identified four frequency categories, which were, starting from the least frequent: occasional (activity only occurred in one month per year), intermediate (activity occurred in two or three months per year), frequent (activity in four to nine months per year), and backbone (activity in 10 to 12 months).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To compare the risk of diffusion over the course of the year, we used the epidemic threshold q (Volkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This measure provides information on the minimum probability for a virus to spread throughout the network: the lower the value of the epidemic threshold, the higher the risk of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For a weighted network, this parameter can be estimated as follows: 𝑞𝑤 = 〈𝑤𝑜𝑢𝑡〉 〈𝑤𝑖𝑛 × 𝑤𝑜𝑢𝑡〉 where 〈 〉 indicates the average value, win and wout indicate the nodes’ in-weight and out-weight, respectively (Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Following the procedures of Lancelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Lancelot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2017) and Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2018), for each of the three mobility networks considered (All species, Cattle, Small ruminants separately), the epidemic threshold was estimated for each monthly snapshot of the network, to assess the risk of an epidemic occurring over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2 Simulation of disease spread Temporality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the variation in time of the mobility network, affects disease spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 1 – A shows an example of a temporal network and its static counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The network is composed of seven nodes and eight possible links, whose direction is indicated by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In this case, the temporal network is characterized by three temporal snapshots that contain the same nodes but different links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' A link that is present and active in a snapshot is not necessarily the same in the previous or the following snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' If we disregard the information on timing, we obtain an aggregated/static network composed of the same nodes and links as the temporal network, all present and active at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' If we simulate an outbreak in the two networks (temporal and static) (Figure 1 – B), we can see that the potential diffusion of the pathogen differs in the two situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In this case, there is significantly more propagation in the static network than in the temporal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This happens because, in the temporal network, the disease can only propagate through temporal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In other words, if a link connecting an infected node to a susceptible one is active in a specific temporal snapshot, the disease can spread to the latter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' conversely, if the link is not active in the temporal window concerned, disease propagation stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To study the influence of temporality on disease propagation, we simulated the spread of an animal disease transmitted by direct contact through the livestock mobility networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We used a SI (Susceptible-Infected) model: the disease was transmitted from an infected node to a susceptible one with a probability of 1, and the infected nodes remained infected for the entire period of analysis, and were consequently able to continue to spread the disease even weeks after being infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The aim of this procedure was to estimate the number of potentially infected nodes when the underlying structure varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Because of our focus on the control of transboundary animal diseases, the departments of Mali and Mauritania, which export livestock directly to Senegal, were chosen as sources of the disease, as the majority of Senegalese imports of small ruminants and cattle are from these two countries (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cesaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To explore the effect of temporality on the structure of the network, and hence on diffusion of the disease, we compared results obtained with a static representation (in which the structure of the network remains unchanged throughout the year) with results obtained with a temporal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In the first case, all the links recorded in the dataset were present at the same time, the time of activation was not taken into account, while in the second case, we included changes in the structure in every week of the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To this end, we used temporal path formalism, according to which a temporal path is a sequence of links connecting two nodes with each link in the path coming temporally after the one before it (Masuda & Holme, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' This approach enabled us to estimate the infection time: that is the minimum number of timesteps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' weeks) needed to create a temporal path between an infected node and the node under observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 1: (A) An example of a directed temporal network and its static counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The dark links are those in the temporal snapshot, while the pale links are those that are possible but are not present in the temporal snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (B) Simple simulation of disease spread in the temporal network (on the left) and the static network (on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Among all the possible temporal paths between the sources and the other nodes, we decided to consider the “earliest arriving” paths, which represent the first time a node is infected by the disease (Bender-deMoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Berlingerio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The speed/rate at which a node became infected was estimated by the infection time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the number of weeks that elapsed between the onset of the disease and the time at which the department concerned was reached for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For static networks, the speed/rate of infection was estimated from the length of the shortest paths, converting the links into temporal units, specifically, weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' If a node was reached by more than one source, the shortest infection time (for temporal networks) or the shortest path (for static networks) was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' All descriptive analyses and static/temporal network analyses were carried out using R software with the following packages: ggplot2 for graphs (Wickham, 2016), ggplot2 and tmap for maps (Tennekes, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' FactoMineR (Lê et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2008) and factoextra (Kassambara & Mundt, 2020) for cluster analysis, sna (Butts, 2020), and tsna (Bender-deMoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2021) for static and temporal network analysis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' B A Temporal network Static network t2 t1,3 ti t2 t3 t1,3 Susceptiblenode Infectednode Sourceofthe disease4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1 Summary statistics The database contained information on 8,861 livestock trade movements from January to December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The network is composed of a total of 88 nodes, corresponding to an Administrative Unit of level 2, of which 45 are Senegalese (Departments), and 590 unique links, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' origin-destination combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The movements concerned Senegal as the origin and/or destination of 87% of the movements, and eight other countries: Mali (9%), Gambia (2%), Mauritania (1%), Guinea, Guinea- Bissau, Burkina Faso, Niger and Nigeria (<1% each) as either the origin or as the destination of movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Focusing on Senegal, a total of 6,511 national movements and 2,350 international movements, over respectively 458 and 132 unique links, involving 87,017 cattle and 553,718 small ruminants were recorded in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Despite the large number of national trades, the majority of animals were moved for the purpose of international trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' More than 95% of these movements were in trucks, which is the most widely used means of transport for animals in the region concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' More than 600,000 animals were transported by truck, the remainder mainly on foot (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The livestock network was analyzed as static but also took temporality into account, which influences the presence/absence of links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Table 1: summary of the characteristics of the data analyzed in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The number of movements, the number of animals and the number of unique links are given for each species, type of trade, and means of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Trade movements Headcount Number of unique links Species Cattle 3,186 87,017 328 Small Ruminants 5,675 553,718 502 Type International 2,350 365,903 132 National 6,511 274,832 458 Means of transport Train 4 170 2 Truck 8,239 608,816 552 On foot 587 30,354 85 Boat 31 1,395 9 As shown in Figure 2, all Senegalese administrative departments are involved in animal trade either as the origin, the destination, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Movements are both national and international, and, while Senegal is the final destination of almost all the trade, many animals are moved not only from other Senegalese departments, but also from Mali and Mauritania, the main exporters, with some departments, particularly in Mali, exporting a significant number of animals (Figure 2– A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The departments in north-eastern Senegal (Podor, Matam, Kanel, and Ranérou Ferlo), are notable for their high level of animal “exports”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Other Senegalese departments (Tambacounda in the south, Koungheul and Gossas in the center, Louga and Kébémer in the north) also export considerable numbers of animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Concerning imports, the departments that import the most animals are located in the Dakar region, in particular Pikine, Rufisque, Thiès, and M’bour, where the majority of consumer markets are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Other Senegalese departments that import large numbers of animals are Saint-Louis in the north, Mbacké and Guinguinéo in the center, Ziguinchor in the south-west, Tambacounda and Sayara in the south-east, on the border with Mali (Figure 2– B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 2: Distribution of the volume of animals in the administrative departments of Senegal, according to whether the department is the origin (A) or the destination (B) of livestock movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The miniature pie charts show the percentage of cattle (yellow) and small ruminants (green) in the total number of animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Quartiles were chosen for the colors representing the volume of animals traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Only countries that account for at least 1% of exports (A) or imports (B) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 3 shows the number of movements and the volume of animals traded in each species (cattle or small ruminants) per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Overall, movements of animals for the purpose of trade were less frequent in the first six months of the year, but increased in July, particularly trade in small ruminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Similarly, July was the month with the most trade in small ruminants in the study period, involving N 0 50 100 150 200 km Volume of animals 0 to 9 10 to 109 110 to 724 725 to 6337 6338 to 209485 Species Cattle Small Ruminants 0 50 100 150 200km A N 0 50 100 150 200 km Volume of animals 0 to 20 21 to 560 561 to 2438 2439 to 8464 8465 to 199744 Species Cattle Small Ruminants B more than 300,000 animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In August and September, the volume of small ruminants decreased, while both the movement and volume of cattle traded increased, overtaking those of small ruminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In October, November and December, the number of cattle trades decreased, but remained higher than in the rest of the year, while the number and volume of trade in small ruminants increased, although less sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In 2020, two important Muslin festivals took place at the end of July (Tabaski) and at the beginning of October (Grand Magal of Touba) and are represented on the chart by a dashed line and a dotted line, respectively (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 3: Number of trade movements (line plot) and volume of livestock traded (bar plot) recorded in 2020, per species and per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Data concerning cattle are in yellow, data concerning small ruminants are in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The dashed line represents the Tabaski festival (July 31), the dotted line represents the Grand Magal of Touba festival (October 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In the whole year, trades of small ruminants were concentrated on 503 links and trades in cattle on 329 links, including 242 links trades of both species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Like for small ruminants, the highest number of unique trade links occurred in July, followed by, in decreasing order, December, November and 0 500 1,000 1,500 0 100,000 200,000 300,000 Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' March April May June July Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Month Number of trade movements Volume of animals Number of trades Cattle Small ruminants Species Cattle Small ruminants October, also the months with the highest number of trade links for cattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The links used by the two species also increased in the last three months of the year (SI Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Concerning the means of transport, trucks were used for almost all movements of animals for sale throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The number of movements peaked in July, and, then after a significant drop, started to increase again in September (SI Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2 Cluster analysis Figure 5 shows the nodes of the livestock network in three (3) clusters: • Cluster 1, composed of 7 nodes and characterized by a “weak”1 exporting behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' • Cluster 2, composed of 61 nodes and characterized by a “strong”1 exporting behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' • Cluster 3, composed by 20 nodes and characterized by a “strong”1 importing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cluster 1 (in red) aggregates seven nodes, of which four are located on the north-eastern border of Senegal (Matam, Podor, Kanel, and Ranérou Ferlo) with high volumes of animals traded, while the other three (Foundiougne, Kaffrine, Gossas) are located on the southern border of Dakar region (Figure 4 – A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' However, in September, Cluster 1 imports are “weak”, with a slightly less than 3,000 animals imported (Figure 4 – B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cluster 2 (in green) aggregates all the foreign nodes, except Banjul (Gambia), and several nodes across Senegal, accounting for a total of 61 nodes out of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cluster 2 is “strong” in terms of volume of animals exported over the year, despite the fact some nodes import more than export.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Some nodes that export large numbers of livestock include Bamako (Mali, 208,462) and Nouakchott (Mauritania, 43,472), while Tambacounda (Senegal, 70,845) is a good example of an importing node (Figure 4 – A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The highest number of exports by this cluster occurred in July, when the number of animals exported was slightly under 200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (Figure 4 – B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cluster 3 (in blue) aggregates 20 nodes of which the majority is concentrated in the Dakar region but includes some nodes in southern Senegal and one foreign node, Banjul (Gambia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Of the nodes located in southern Senegal, two are on the border with Mali (Saraya and Kédougou), while the other four are located farther west.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' All the nodes in this cluster were characterized by strong import trade, with most imported animals via Pikine (Senegal, 199,703), but also via Thiès (Senegal, 51,939) and Kaolack (Senegal, 46,432) (Figure 4 – A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Reflecting the movements of livestock for export, this cluster shows a peak of imported animals in July, with a volume of around 250,000 animals, and 1 We introduce the terms importing and exporting behaviour to indicate those nodes whose net flow of animals (difference between inflow and outflow) through them is positive and negative respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Weak and strong refer to magnitude of the net flow (small or large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' another, less significant increase from September to October (Figure 4 – B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 4: Clustering of livestock network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (A) Spatial representation of nodes colored according to the cluster to which they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The size of each dot indicates the volume of animals traded over the course of the year and the division is made in quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (B) Temporal representation of trade by the three clusters over the course of the year, in terms of the volume of animals traded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Imported animals are represented as positive numbers, exported animals as negative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The dashed line represents the Tabaski festival on July 31, the dotted line represents the Grand Magal of Touba festival on October 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='3 Frequency of links Figure 5 shows the trade links divided by the frequency of their activity over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In general, far more links were characterized by low and very low activity than by very high activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The backbone links are three national, short-range connections between north-eastern and north- western nodes: Kanel – Pikine, Ranérou Ferlo – Linguère, Ranérou Ferlo – Mbacké.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The majority of frequent links is concentrated in the north of Senegal, where several connections link eastern and western nodes, but some connections link northern and southern nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Moreover, some international links are frequent, in particular four originating from Mali and one from Guinea-Bissau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The number of intermediate links is significantly larger than that of the two previous categories, with several connections between Senegal and Mali, and between Senegal and Mauritania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Occasional links are extremely numerous and dense, with several connections in Senegal but also links to all its neighboring countries (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The majority of intermediate and occasional links are active in July and October, due to the Tabaski and the Grand Magal of Touba religious festivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' However, considering the whole study period, frequent links are the most common (SI Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 10°N 12°N 14°N 16°N 18°N 20°N 15°W 10°W 5°W 0° 5°E Longitude Latitude Cluster 1 2 3 Volume 202 1892 8912 208462 A −200,000 −100,000 0 100,000 200,000 Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' March April May June July Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Month Volume of animals Cluster 1 2 3 B Figure 5: Geographical representation of the livestock network links, divided by the frequency of their activity over the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Backbone links were active in more than nine months, frequent links were active between four and nine months, intermediate links were active for two or three months, and occasional links were active for only one month of the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We decided to only show the administrative departments of Senegal on these maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Therefore, concerning national trade, the origin and the destination are both Senegalese departments, while for international trades, they may be a Senegalese department or a central point in a foreign country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Backbone 18°N Mauritania 17N 16°N 15°N Mall No 13°N 12°N 11°N Guinea Frequent 18°N Mauritania 17°N 16°N 15"N Mali 14°N Gambia 13°N 12°N Guinea Frequency category Backbone Intermediate Frequent 18°N Mauritania intermediate Occasional 17°N 16°N 15°N 14°N Gambia 13-N 12°N 11°N Guinea Occasional 18°N Mauritania 17°N 16°N Niger 15°N Mali 14°N Gambia 13°N Burkina Faso 12°N Nigeria 11°N Guinea 18W 12°W 10W M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='8 Mot Longitude4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='4 Epidemic threshold Overall, the values of the epidemic threshold of all three networks are extremely low, particularly for the combined livestock and the small ruminants network, whose results are almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' April is the only month with a significantly higher value than in the rest of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' On the other hand, the epidemic threshold values of the cattle network (yellow curve in Figure 6), are higher overall, with a value of 1 in April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The lowest values of this network were measured in January, June, and from October until the end of the year (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 6: Logarithmic representation of changes in the epidemic threshold over the course of the year in the three livestock mobility networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The zero on the left extremity of the horizontal axis identifies the value calculated for the whole year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The small ruminants network is in yellow, the cattle network in green, and the combined livestock network in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='5 Simulation of disease spread Maps focused on Senegalese departments were drawn to compare the results of the simulations run on the three networks in an efficient and easily understandable way (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To assess the role of changes in the structure of the networks over time, and hence changes in disease spread, we compared the results of a static representation (in the column on the left) with those of a temporal representation 1e−04 1e−02 1e+00 0 1 2 3 4 5 6 7 8 9 10 11 12 Epidemic Threshold Network Cattle Cattle and small ruminants Small ruminants (in the other seven columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static representation, considering the time the outbreak began is meaningless, whereas for the temporal one, it is important, as the network structure can change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Therefore, each element in the seven columns representing the temporal networks corresponds to the results of an outbreak that began in a specific week of the year (the number of the week is given in the header of each map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The departments are colored according to their infection time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' the length of the period before they were reached by the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For each scenario, the infection time was estimated as the time (number of weeks) elapsed since the outbreak of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We created four categories of infection time, each category is shown in a different color: red for departments infected less than one month from the beginning of the disease spread (less than 5 weeks), orange for departments infected after 1-2 months (between 5 and 9 weeks), yellow for departments infected after more than two months (more than 9 weeks), and green for those never reached by the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' If a node was reached by infections from several sources, the shortest infection time was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static networks, we considered the number of links in the shortest path between the source and the node as weeks: red for the shortest paths with less than 5 links, orange for paths with between 5 and 9 links, yellow for paths with more than 9 links, and green for nodes that were never reached by the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The results presented are those of the simulation of a disease spreading from Mali, the origin of most animals imported into Senegal in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the temporal networks, we present only a few weeks characterized by activity, in order to be able to simultaneously show changes over time and differences between the three networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The complete results of the spread of a disease from Mali plus for a disease spreading from Mauritania can be found in Supplementary Information (SI Figure 5 – 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In general, in all three networks, maps representing aggregate networks strongly overestimated both the quantity of potentially infected nodes and the earliness of infection, compared to those of temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In addition, there is a difference in the potential sanitary risk between the cattle temporal network and small ruminants temporal network, the latter showing on average wider and potentially greater disease propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' However, the combined network of cattle and small ruminants (the livestock network) is under the greatest sanitary risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 7 also shows that, particularly for the livestock network and the small ruminants network, the periods around religious festivals (weeks 30 and 31 for the Tabaski, and weeks 40 and 41 for the Grand Magal of Touba) are characterized by a large number of infected departments, some of which are infected early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Figure 7: Geographical representation of infection time in the case of disease propagation from Mali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For each mobility network, the first column on the left represents the spread in the static network, the other seven columns represent the seven worst scenarios of transmission if that specific week represents the beginning of the disease spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The colors indicate the infection time of the disease: red for less than one month, orange for less than two months, yellow for more than two months, green for nodes that have never been touched in one year time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static networks in the first column on the left, the colors are based on the number of links in the path: up to 5 in red, green for nodes that have never been touched in one year time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The squares outlined in blue identify the week of the Tabaski festival (week 31) and the week of the Grand Magal of Touba festival (week 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Discussion and conclusions In 2020, during the COVID-19 pandemic, several restrictive measures were introduced in Senegal that affected both human and livestock mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Borders were closed for both humans and animals in March 2020 and, at the same time, movements between regions were regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To supply markets and families in preparation for the Tabaski festival on July 31, borders were reopened 45 days before Tabaski and measures were eased for national and international movement (lettre circulaire n° 01806 PR/MESG/CT-PSS du 17 juin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Similar decisions were taken on the occasion of the Grand Magal of Touba, a religious pilgrimage during which a large number of cattle in particular are sold and consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The application of these restrictions had a huge effect on the structure of the networks and on the risk of introduction and diffusion of pathogens, as did re-opening the border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To assess the impact of these measures on the spread of livestock diseases, and for possible future use, we used tools from temporal network theory to identify the area with the highest risk of introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' It is Static All merimportant to note that normally, there are other religious festivals, like Gamou of Tivaouane, in addition to the Grand Magal of Touba, but these were cancelled due to COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In our study, we considered the diffusion of a generic direct animal disease transmission and estimated the vulnerability and the reachability of nodes when the underlined network changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In this way, we were able to identify Senegalese departments that could be infected at the earliest stage of an epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' With a few modifications, our approach could be extended to include vector- borne diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The structure of the Senegalese livestock network varies widely over the course of the year due to the seasonality of transhumance and the effect of religious festivals (Apolloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Jahel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2020) but, in 2020, these effects were exacerbated by the restrictive measures introduced as a result of Covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In fact, around June and July, we noted a pickup in the movement and exchange of animals (mainly small ruminants) mainly due to the easing of the restrictive measures in preparation for the Tabaski festival and (mainly cattle) for the Grand Magal of Touba festival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We also noted that the dynamics of the small ruminants trade strongly drive the dynamics of the network as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dakar is the main consumption area of Senegal because almost a quarter of the population of the country live in the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Consequently, the main markets of Dakar and Pikine (at the entrance of Dakar) are the main destination of livestock movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In particular, regular movements occur between the areas of Kanel, Ranerou Ferlo, Dahra and the Senegalese capital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In these areas, there is a high concentration of collection markets (local name luma), where traders frequently buy animals to be sold directly to Dakar, or to the other collection markets in Dahra or Thiès before being sent on to the capital city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Overall, the majority of northern links end in the Dakar region, or in some smaller but nevertheless important markets such as Saint-Louis, Thiès, Mbour, but also Ziguinchor in the south.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Some links in the southeast start from Tambacounda, an important point of convergence for animals from eastern Senegal, as well as from Mauritania and Mali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Our analysis revealed that the role played by the different departments changes over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Locations that are idle for a large part of the year become active during the Tabaski period and continue to be active until the end of the year, in particular, departments that produce small ruminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Occasional and intermediate links that are active a few times a year, are usually located near festival centers, to support the increased supply of livestock, thereby increasing the sanitary risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Analysis of the threshold parameters showed that the network is prone to disease spread, but that the risk fluctuates over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The risk increases significantly on the occasion of festivals due to the introduction of large numbers of animals and the creation of new commercial routes, and diseases can then spread easily across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' However, the potential infected areas, and the reachable time does not remain stable over the course of the year and this information cannot be captured using a static representation of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In fact, a static representation of the mobility patterns may largely overestimate the speed and the extent of disease diffusion: when the simple static approach is used, diseases appear to spread throughout the country in less than a month, whereas temporal analysis shows that reachability and vulnerability of departments varies over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In most cases, and depending on the species involved, few departments are reached in a month, although during the months around Tabaski and Grand Magal, the number of departments that can be reached increases drastically, and for some departments (like Dakar, Thiès, Tambacounda and Dahra) where the main markets are located, and at the border, this risk is even higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' These results could be of great interest not only for risk-based surveillance but also for optimizing the distribution of resources and personnel needed for control at specific times of the year by focusing on the areas that are most likely to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The fact that departments located at the border are most prone to early infection, means that sanitary control at the border should be strengthened and surveillance and control measures should be harmonized at regional level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Previous works already underlined the importance of mobility and of data collection as a tool to improve surveillance and control in Africa (Chaters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Motta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nicolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Our work fits into this strand, emphasizing the importance of collecting data on animal mobility on a regular basis in order to retrieve information on structural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The objective of the present study is to provide theoretical tools to assess the importance of network dynamics when planning control and surveillance policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' A more detailed analysis focused on specific diseases and that accounts for volume distribution may reduce the list of departments to monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' To this end, further simulations are needed and their results will depend to a large extent on the characteristics of the disease concerned, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' it transmissibility and incubation period, that could shape the spatio- temporal pattern of the epidemics and hence the involvement of the different departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Future works should thus also consider stochastic models like Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2018) (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', 2018) for specific diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In the model we used here, we aggregated data at the spatial scale of an administrative department, based on the assumption that the diffusion within a department is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In practice, the presence of markets or transhumance corridors could attract movements in specific parts of the department, thereby increasing the risk in certain locations over the risk in other parts of the same department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Data on movements within departments were rare in our dataset (because of the way data were collected) and further field studies are recommended to collect data at a finer scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Acknowledgments The authors are grateful to Dr Mbargou Lô, head of the veterinary Services Directorate, and Dr Mathioro Fall, head of Animal Health Protection Division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The authors acknowledge the support of Yves Amevoin, Alioune Ka, Fallou Niakh, and Khady Ndiaye, and all those who assisted in the LPS collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' We thank all the donors who supported the CGIAR Livestock research program through their contributions to the CGIAR trust fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Funding This study was partially funded by the Project Eco-PPR (European Commission through the International Fund for Agricultural Development (grant number 2000002577) and the CGIAR Livestock research program, RVF OIE twinning program (CIRAD-ISRA) granted through the EBO- SURSY project (European Union FOOD/2016/379-660).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Conflict of interest statement All authors declare that they have no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' References ANSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Recensement Général de la Population et de l’Habitat, de l’Agriculture et de l’Elevage (RGPHAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' ANSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Situation économique et sociale du Sénégal Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2017/2018 (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 413).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Apolloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Corniaux, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Coste, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Lancelot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Toure, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Livestock Mobility in West Africa and Sahel and Transboundary Animal Diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' In Transboundary Animal Diseases in Sahelian Africa and Connected Regions (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 31:52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1007/978-3-030-25385-1 Apolloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Nicolas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Coste, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', EL Mamy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Yahya, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', EL Arbi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Gueya, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Baba, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Gilbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Lancelot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Towards the description of livestock mobility in Sahelian Africa: Some results from a survey in Mauritania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' PLoS ONE, 13(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='0191565 Bender-deMoll, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Moody, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2021, aprile 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Tools for Temporal Social Network Analysis [R package tsna version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Comprehensive R Archive Network (CRAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/package=tsna Berlingerio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Coscia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Giannotti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Monreale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Pedreschi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Evolving networks: Eras and turning points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Data Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='3233/IDA-120566 Bossard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Regional Atlas on West Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Éditions OCDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1787/9789264056763-en Bouslikhane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' CROSS BORDER MOVEMENTS OF ANIMALS AND ANIMAL PRODUCTS AND THEIR RELEVANCE TO THE EPIDEMIOLOGY OF ANIMAL DISEASES IN AFRICA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' OIE Regional Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Butts, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2020, ottobre 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Tools for Social Network Analysis [R package sna version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='6].' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Fournié, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Métras, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Livestock trade network: Potential for disease transmission and implications for risk-based surveillance on the island of Mayotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Scientific Reports, 8(1), 11550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1038/s41598-018-29999-y Lancelot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Béral, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Rakotoharinome, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Andriamandimby, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Héraud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Coste, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Apolloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Squarzoni-Diaw, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', de La Rocque, S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Cardinale, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Drivers of Rift Valley fever epidemics in Madagascar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 114(5), 938–943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1607948114 Lê, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Josse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Husson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' FactoMineR: A Package for Multivariate Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Journal of Statistical Software, 25(1), 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='18637/jss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='v025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='i01 Lentz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Koher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Hövel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Gethmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Sauter-Louis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Selhorst, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Conraths, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Disease Spread through Animal Movements: A Static and Temporal Network Analysis of Pig Trade in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' PloS One, 11(5), e0155196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': 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controlling infectious disease epidemics using temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' F1000Prime Reports, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='12703/P5-6 Ministère de l’Intérieur du Sénégal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Politique de Gouvernance intérieure | Ministère de l’Intérieur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://interieur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='gouv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='sn/administration-territoriale/politique-de-gouvernance- interieure Missohou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Nahimana, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Ayssiwede, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Sembene, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Elevage caprin en Afrique de l’Ouest: Une synthèse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Revue d’élevage et de médecine vétérinaire des pays tropicaux, 69(1), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='19182/remvt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='31167 Motta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Porphyre, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Handel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Hamman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Ngu Ngwa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Tanya, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Morgan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Christley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Bronsvoort, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' deC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Implications of the cattle trade network in Cameroon for regional disease prevention and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Scientific Reports, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1038/srep43932 Muwonge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Bessell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Porphyre, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Motta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Rydevik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Devailly, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Egbe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Kelly, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Handel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Mazeri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} 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Wint, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Lancelot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Gilbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Predictive gravity models of livestock mobility in Mauritania: The effects of supply, demand and cultural factors.' metadata={'source': 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Lentz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Colizza, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Koher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Hövel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Vidondo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Early warning of infectious disease outbreaks on cattle-transport networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' PLOS ONE, 16(1), e0244999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='0244999 Tennekes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' tmap: Thematic Maps in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Journal of Statistical Software, 84(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='18637/jss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='v084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='i06 Valerio, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The structure of livestock trade in West Africa (West African Papers Fasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 29;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' West African Papers, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1787/f8c71341-en Valerio, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Walther, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Eilittä, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Cissé, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Muneepeerakul, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Kiker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Network analysis of regional livestock trade in West Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' PLoS ONE, 15(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='0232681 Volkova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Howey, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', Savill, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Woolhouse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Sheep Movement Networks and the Transmission of Infectious Diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' PLoS ONE, 5(6), e11185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='0011185 Wickham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' ggplot2: Elegant Graphics for Data Analysis (2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Springer International Publishing : Imprint: Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1007/978-3-319-24277-4 Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=', & Musolesi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Spatio-temporal networks: Reachability, centrality and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Royal Society Open Science, 3(6), 160196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='1098/rsos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='160196 World Bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Population Senegal [Text/HTML].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' World Bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='worldbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='org/en/country/senegal/overview Supplementary information SI Figure 1: Map of Senegal colored based on the aridity index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The inset map shows all the countries involved in livestock mobility network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=" V Mauritania Mali Niger SENEGAL MAURITANIA Gambia Burkina Faso Podor suin Nigeria Senegal Saint Louis RiverValley Matam Louga Sylvopastoral Dahra zone Kanel Pikine Tivaouane Touba Thies Mbake DAKAR Rur oue Diourbel M'bour Groundnut Kaolack basin Tambacounda Eastern MALI Senegal GAMBIA casamdne OKolda Ziguinchor GUINEA BISSAU GUINEA 0 100 200km Aridity Index Maincities(population） Agro ecologicalzones 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2 Morethan500,000 Borders 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='35 100,000 500,000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='5 Main rivers 50000 100,000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='65 o Lessthan50,000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='8 Small Ruminants Cattle Links in common Whole year 503 329 242 Months January 42 27 13 February 52 27 22 March 59 30 19 April 26 18 12 May 27 22 11 June 71 28 20 July 202 47 37 August 29 21 6 September 24 41 14 October 131 123 66 November 163 113 69 December 184 128 82 SI Table 1: Number of unique trade links in the small ruminant network and in the cattle network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The values are represented divided by month, while the “whole year” line corresponds to the number of unique links considering the network as static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The last column shows the number of links that are used by the two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Truck Others Links in common Water Walking Train Whole year 552 9 85 2 55 Months January 49 0 8 0 1 February 56 0 3 0 2 March 67 0 3 0 0 April 31 0 1 0 0 May 38 0 0 0 0 June 73 0 9 0 3 July 207 0 11 0 6 August 43 0 3 0 2 September 42 3 11 0 5 October 178 1 17 1 9 November 178 3 33 0 7 December 219 3 20 1 11 SI Table 2: Number of unique trade links according to the means of transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The values are given per month, while the “whole year” line corresponds to the number of unique links considering the network as static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The last column shows the number of links that are shared by the movements made by truck and those made by all other types of transport, including on foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' SI Figure 2: volume of livestock traded in 2020 divided by week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Cattle are shown in yellow and small ruminants are in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The black dashed line represents the day of the Tabaski festival (July 31), the black dotted line represents the day of the Grand Magal of Touba (October 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The months are indicated by the gray dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' SI Figure 3: Graphical visualization of methods chosen to assess the number of clusters: (A) Elbow method, (B) cluster dendrogram, and (C) cluster division with the HCPC (Hierarchical Clustering on Principal Components) function of the FactoMineR package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='000 B0T000 folume of animal WCK Spedes Cefle Smel rumnantOptimalnumberofclusters Factormap 5e+05 ofSqu 4e+05 2e+05 2 3 4 5 6 10 cluster Numberofclustersk ClusterDendrogram 4 Height 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dim1 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content='9%) SI Figure 4: Activity of the links over the course of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For each month, the quantity of active links is shown, colored according to their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Backbone links are those active more than nine months a year, frequent links are those active from four to nine months, intermediate links are those active two or three months, and occasional links are only active one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' The orange line represents the Tabaski festival (July 31), the violet line represents the Grand Magal of Touba festival (October 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 200 150 100 50 Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Fob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' April May Juy Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Ort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Month Berckbone I feguon I reguency category Intermeclafe Occasional SI Figure 5: Geographical representation of infection time, in the case of a disease propagated from Mali through the livestock network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been reached are in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Infectiontime Withinonemonth Withintwomonths Aftertwomonths Never 42 SI Figure 6: Geographical representation of infection time, in the case of a disease propagated from Mali through the small ruminant network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been touched are colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Infectiontime Withinonemonth Withintwomonths Aftertwomonths Never SI Figure 7: Geographical representation of infection time in the case of a disease propagated from Mali through the cattle network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been touched are colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Infectiontime Withinonemonth Withintwomonths Aftertwomonths Never SI Figure 8: Geographical representation of infection time in the case of a disease propagated from Mauritania through the livestock network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been touched are colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Infectiontime Withinonemonth Withintwomonths Aftertwomonths Never SI Figure 9: Geographical representation of infection time in the case of a disease propagated from Mauritania through the small ruminant network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been touched are colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Infectiontime Withinonemonth Withintwomonths Aftertwomonths Never 45 SI Figure 10: Geographical representation of infection time in the case of a disease propagated from Mauritania through the cattle network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Temporal network on the left, static network on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' For the static network, the colors are based on the links in the path: up to 5 in red, between 5 and 9 in orange, more than 9 in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' Nodes that have never been touched are colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} +page_content=' 37 38 39 Infectiontime 42 Withinonemonth Withintwomonths Aftertwomonths Never Gt' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFKT4oBgHgl3EQfUi2J/content/2301.11784v1.pdf'} diff --git a/btFAT4oBgHgl3EQf5R5J/content/tmp_files/2301.08732v1.pdf.txt b/btFAT4oBgHgl3EQf5R5J/content/tmp_files/2301.08732v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c61393fe2f35d20a5f3fbb84fef430f0cd06b1e --- /dev/null +++ b/btFAT4oBgHgl3EQf5R5J/content/tmp_files/2301.08732v1.pdf.txt @@ -0,0 +1,2691 @@ +Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime +Rick Bebon and Aljaˇz Godec∗ +Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 G¨ottingen, Germany +We derive general bounds on the probability that the empirical first-passage time τ n ≡ �n +i=1 τi/n +of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates +from the true mean first-passage time by more than any given amount in either direction. We +construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and +thus fill the gap between asymptotic methods and the Bayesian approach that is known to be +sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. Our +concentration-of-measure-based results allow for model-free error control and reliable error estimation +in kinetic inference, and are thus important for the analysis of experimental and simulation data in +the presence of limited sampling. +The first-passage time τ denotes the time a random pro- +cess reaches a threshold a, typically referred to as the “tar- +get”, for the first time. First-passage times [1–4] quantify +the kinetics of chemical reactions [5–10], cell signaling and +gene regulation in the low-copy [11–20] and “fastest en- +counter” limits [21–29], intracellular transport [30], RNA +biosynthesis [31], protein accumulation [32, 33] and DNA- +binding [34], emergence of drug resistance [35], virus up- +take [36], spreading of diseases [37, 38], and the foraging +behavior of bacteria and animals [39]. First-passage the- +ory was further applied to nanocluster formation [40], cell +adhesion [41–43], gating of ion channels [44], and diffusion +through interfaces [45] and across phase boundaries [46]. +In more abstract settings, first-passage times charac- +terize barrier-crossing in energy landscapes [6, 23, 47– +54], persistence properties [55–61], and the statistics of +stochastic currents [62, 63], thermodynamic entropy pro- +duction [64–67], and dynamical activity [68, 69] in non- +equilibrium systems. First-passage ideas are intimately +tied to the statistics of extremes [70–73], and were ex- +tended to quantum systems [74, 75], additive functionals +of stochastic paths [76–81], intermittent targets [82–85], +active particles [86, 87], non-Markovian dynamics [88–91], +and processes under resetting [92–101]. +Whereas theoretical studies focus on predicting first- +passage statistics, practical applications typically aim +at inferring kinetic rates—inverse mean first-passage +times—from experimental [52, 102–106] or simulation +data [51, 107–113]. The inference of empirical first-passage +times τ n ≡ �n +i=1 τi/n from data is, however, challenging +because usually only a small number of realizations n +(typically 1-10 [113–118], sometimes up to 100 [119]) is +available, which gives rise to large uncertainties and non- +Gaussian errors. Insufficient sampling is especially detri- +mental in the case of broadly distributed [51, 120, 121] +and high-dimensional data [106]. Moreover, first-passage +times are generically not exponentially distributed [8, 9, +17, 19, 23, 24, 122–127], which further complicates quan- +tification of uncertainty. A systematic understanding of +statistical deviations of the empirical from the true mean +first-passage time (see Fig. 1a), especially in the small- +sample n ≲ 100 regime, remains elusive. +Computer simulations in particular often suffer from +FIG. 1. Deviations of empirical first-passage times from the +true mean and model systems. (a) Schematic probability den- +sity of empirical first-passage time τ n inferred from a sample +of n realizations of an ergodic reversible Markov process. The +tail probability that the estimate τ n deviates from the true +mean ⟨τ⟩ by more or equal than t upwards P(τ n ≥ ⟨τ⟩ + t) +or downwards P(τ n ≤ ⟨τ⟩ − t) is shown in green and blue, +respectively. (b) Brownian molecular search process in a d- +dimensional domain (here d = 2) with outer radius R and +target radius a. Discrete-state Markov jump models of protein +folding for (c) a toy protein and (d) experimentally inferred +model of calmodulin [124]. Transitions between states are in- +dicated by arrows and obey detailed balance. For all systems +considered the absorbing target is colored red. +insufficient sampling, which leads to substantial errors in +inferred rates [128–131] and, in the worst case, erroneous +conclusions (see discussion in [113, 132]). Even extensive +computing resources may result in only a few indepen- +dent estimates spread over many orders of magnitude, +rendering uncertainty quantification challenging and not +amenable to standard error analysis [116]. +Constructing reliable confidence intervals is a fundamen- +tal challenge in statistical inference, and many prevalent +methods rely on asymptotic arguments that hold when +the number of realizations tends to infinity. However, the +applicability of asymptotic results in a finite-sample set- +ting is, by definition, problematic. In particular, Central- +Limit- and bootstrapping-based methods [133] may easily +arXiv:2301.08732v1 [cond-mat.stat-mech] 20 Jan 2023 + +2 +underestimate the uncertainty for small n and fail to guar- +antee coverage of the confidence level [116, 134–139]. +Conversely, Bayesian methods (see e.g. [140]) do not +rely on asymptotic arguments and are therefore often (in +general erroneously [141, 142]) believed to readily alle- +viate the small-sample problem. Bayesian estimates are +sensitive to, dependent on, and potentially biased by, the +specification of the prior distribution, especially in the +small-sample setting [140, 143–145]. Due to the prior +dependence of estimates and their uncertainties, Bayesian +methods must be treated with care when applied to small +samples [146, 147] (see [123, 129, 148–150] specifically for +kinetic inference) and can perform worse than asymptotic +frequentist methods [146]. +Moreover, so-called “credible intervals”—the Bayesian +analogue to confidence intervals—have a nominally differ- +ent meaning, as they treat the estimated parameter as a +random variable. Bayesian posterior intervals are similarly +affected by limited sampling [116], i.e. the constructed +uncertainty estimates and their quality are sensitive to +the choice of prior probability [141, 142] and may likely +underestimate the true uncertainty and thus fail to pro- +vide trustworthy confidence intervals [129, 151]. +On a more subtle level, the classical Bernstein-von- +Mises theorem establishes a rigorous (frequentist) justi- +fication of posterior-based Bayesian credible intervals as +asymptotically correct, prior independent confidence inter- +vals for (finite dimensional) parametric models in the large- +sample limit [152–154]. Analogous statements for semi- +parametric and (infinite dimensional) non-parametric +models are more delicate [155–158] and, despite having +received signifficant attention [159–170] (see also [171] +for misspecified and high dimensional [172] parametric +models), seem to remain—even in the asymptotic, large- +sample regime—an elusive problem. +There is thus a pressing need for understanding fluctu- +ations of inferred empirical first-passage times, a rigorous +error control, and reliable non-asymptotic error estima- +tion in the small-sample regime. These are fundamental +problems of statistical kinetics and are essential for the +analysis of experimental and simulation data. +Here, we present general bounds on fluctuations of +empirical first-passage times that allow a rigorous uncer- +tainty quantification (e.g. using confidence intervals with +guaranteed coverage probabilities for all sample sizes) +under minimal assumptions. We prove non-asymptotic +lower (L) and upper (U) bounds on the deviation proba- +bility P(τ n ≥ ⟨τ⟩ + t) and P(τ n ≤ ⟨τ⟩ − t) (see Fig. 1a), +i.e., the probability that the empirical first-passage time +inferred from a sample of n ≥ 1 realizations of an ergodic +reversible Markov process, τ n, deviates from the true +mean ⟨τ⟩ by more than t in either direction, +L± +n (t) ≤ P(±[τ n − ⟨τ⟩] ≥ t) ≤ U± +n (t) +∀t ≥ 0, +(1) +the upper bounds U± +n (t) corresponding to so-called concen- +tration inequalities [173]. The most conservative version +of the derived upper bounds is independent of any details +about the underlying dynamics. The validity and sharpness +of the bounds are demonstrated by means of spatially con- +fined Brownian molecular search processes in dimensions +1 and 3 (Fig. 1b), and discrete-state Markov jump models +of protein folding for a toy protein [24, 129, 174, 175] +(Fig. 1c) and the experimentally inferred model of calmod- +ulin [124] (Fig. 1d). We use the bounds U± +n (t) to quantify +the uncertainty of the inferred sample mean τ n in a gen- +eral setting and under minimal assumptions, for all n ≥ 1. +We conclude with a discussion of the practical implications +of the results and further research directions. +Setup.—We consider time-homogeneous Markov pro- +cesses xt on a continuous or discrete state-space Ω with +(forward) generator ˆL corresponding to a Markov rate- +matrix or an effectively one-dimensional Fokker-Planck +operator. Let the transition probability density to find +xt at x at time t given that it evolved from x0 be +pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac +or Kronecker delta for continuous and discrete state- +spaces, respectively. We assume the process to be ergodic +limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes +the equilibrium probability density and ϕ(x) the general- +ized potential in units of thermal energy kBT [176]. We +assume that ˆL obeys detailed balance [177] and is either +(i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, +or (iii) Ω is infinite but ϕ(x) sufficiently confining (see +[178]). Each of the conditions (i)-(iii) ensures that the +spectrum of ˆL is discrete [179]. +We are interested in the first-passage time to a target +a when xt=0 is drawn from a density p0(x) +τ = inf +t [ t |xt = a, p0(x0)], +(2) +and focus on p0(x) = ˜peq(x) where the tilde denotes that +the absorbing state is excluded [180]. For completeness we +also provide in [181] results for general initial conditions +p0(x) that require more precise conditions on ϕ(x) [182]. +The probability density of τ for such processes has the +generic form [23, 24] +℘a(t|x0) = +� +k>0 +µkwx0 +k e−µkt, +(3) +where µk > 0 denote first-passage rates and wx0 +k +the +(not necessarily positive) spectral “weights” normalized +according to � +k>0 wx0 +k += 1 and wx0 +1 +> 0. The m- +th moment of τ is given by ⟨τ m⟩ = m! � +k>0 wx0 +k /µm +k +and the survival probability +reads P(τ +> +t) +≡ +Sa(t|x0) += +� +k>0 wx0 +k e−µkt. +If x0 is drawn from +the equilibrium density, ˜peq(x), we have ℘a(t|˜peq) ≡ +� +Ω\a ℘a(t|x0)˜peq(x0)dx0 [183] which renders all weights +non-negative, wk ≡ +� +Ω\a wx0 +k ˜peqdx0 ≥ 0 (see proof in +[181]). We henceforth abbreviate Sa(t|˜peq) ≡ Sa(t). +To examplify the need for uncertainty bounds in Eq. (1) +we show in Fig. 2a-d that the probability that τ n − ⟨τ⟩ +lies within a desired range of say ± 10% of the longest +first-passage time scale µ−1 +1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.1, 0.1]) +is low even for n ≈ 50 for all models in Fig. 1b-d. +Lower bounds on deviation probability.—There exists +a “noise floor” for τ n for any n. Since µk ≤ µk+1 and + +3 +FIG. 2. +Deviation probabilities and corresponding bounds for a spatially confined Brownian search process in (a,e) d = 1 and +(b,f) d = 3 dimensions, and Markov-jump models of protein folding for (c,g) the experimentally inferred model of calmodulin +and (d,h) the toy protein. (a-d) Probability that δτ n = τ n − ⟨τ⟩ lies within a range of ±10% of the longest time-scale 1/µ1, +P(µ1δτ n ∈ [−0.1, 0.1]), as a function of n determined from the statistics of τ n for different fixed n for all model systems. (e-h) +Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that the sample mean τ n inferred from n realizations deviates from ⟨τ⟩ by more +than t in either direction. Right tail areas are shown for t > 0 and left for t < 0, respectively. Lower L± +n (t) and upper U± +n (t; C) +bounds are depicted as red and black lines, respectively, and the model-free upper bound U± +n (t; 2) as the dashed yellow line. +Symbols denote corresponding scaled empirical deviation probabilities as a function of t and are sampled for different n. +wk are non-negative [184] and normalized [23, 24], the +equilibrium survival probability obeys w1e−µ1t ≤ Sa(t) ≤ +e−µ1t, which directly leads to lower bounds L± +n (t) in +Eq. (1). Namely, τ n ≥ mini∈[1,n] τi ≡ τ min +n +and τ n ≤ +maxi∈[1,n] τi ≡ τ max +n +. Therefore, P(τ min +n +≥ t) ≤ P(τ n ≥ +t) ≤ P(τ max +n +≥ t) and we have P(τ min +n +≥ t) = S(t)n and +P(τ max +n +≤ t) = (1 − S(t))n, leading to lower bounds +P (τ n − ⟨τ⟩ ≥ t) ≥ +� +w1e−µ1(⟨τ⟩+t)�n +≡ L+ +n (t) +P (τ n − ⟨τ⟩ ≤ −t) ≥ +� +1 − e−µ1(⟨τ⟩−t)�n +≡ L− +n (t), +(4) +where equality is reached for n = 1 and w1 → 1. Anal- +ogous results are obtained for upper bounds (see [181]) +which, however, are much weaker than those derived be- +low with the Cram´er-Chernoff approach and concurrently +require even more information about the dynamics. +We remark that bounds on the survival probability +consequently also bound the probability density ℘(n) +a (t) +of the fastest first-passage time of n independent par- +ticles [23, 25, 26, 185] according to nw1e−µ1(n−1)t ≤ +℘(n) +a (t)/℘a(t) ≤ ne−(n−1)µ1t. We now turn to the more +challenging upper bounds. +Cram´er-Chernoff bounds.—Let δτ n ≡ |τ n−⟨τ⟩| and λ ∈ +R+. We start with the obvious inequality eλt1δτ n≥t ≤ +eλτ n, where 1b is the indicator function of the set b. Tak- +ing the expectation yields P(δτ n ≥ t) ≤ e−λt⟨eλδτ n⟩ ≡ +e−λt+ψδτn(λ), where we defined the cumulant generating +function of δτ n, ψδτ n(λ) ≡ ln⟨eλδτ n⟩. Note that τi are sta- +tistically independent. The bound can be optimized [186] +to find Chernoff’s inequality, P(δτ n ≥ t) ≤ e−nψ†δτ(t), +where ψ∗ +δτ(t) is the Cram´er transform of ψδτ(λ) [173], i.e. +ψ∗ +δτ(t) ≡ sup +λ +(λt − ψδτ(λ)), +(5) +where δτ ≡ δτ 1. On the interval λ ∈ [0, µ1) we have the +following bounds on ψδτ(λ) (see proof in [181]) +ψδτ(λ) ≤ φδτ(λ; C) ≡ +� +� +� +� +� +� +� +λ2 +2µ2 +1 +C +1 − λ/µ1 +τ ≥ ⟨τ⟩ +λ2 +2µ2 +1 +C +1 − (λ/µ1)2 +τ < ⟨τ⟩, +(6) +which are non-negative, convex, and increasing on λ ∈ +[0, µ1), and we introduced C ≡ µ2 +1⟨τ 2⟩ [187]. The bound (6) +further implies ψ∗ +δτ(t) ≥ φ∗ +δτ(t; C) ∀t ≥ 0, and may thus +be optimized according to [186] to obtain the inequalities +announced in Eq. (1) via Chernoff’s inequality: +U+ +n (t; C) = exp (−nCh+ (µ1t/C)) +0 ≤ t ≤ ∞ +U− +n (t; C) = exp (−nCh− (µ1t/C)) +0 ≤ t ≤ ⟨τ⟩ +(7) +where we defined the functions +h+(u) ≡ 1 + u − +√ +1 + 2u +(8) +h−(u) ≡ Λ(u)u − 1 +2 +Λ(u)2 +1 − Λ(u)2 +(9) +with Λ(u) ≡ 1 +2 +� +g(u) − +� +4 + 2/g(u)u − g(u)2 +� +and +g(u) ≡ +2 +√ +3 +� +1 + 2 cosh +�1 +3arcosh +� +1 + +33 +27u2 +���1/2 +. (10) + +4 +The tail behavior of δτ in Eq. (7) provides quantitative +insight into fluctuations of τ even when ⟨τ⟩ is unknown +or is an insufficient or non-representative observable [188– +190]. +Deviations are readily expressed relative to the +longest natural time scale 1/µ1 that does not need to be +known. That is, deviations are naturally parameterized +by the dimensionless variable ˜t = µ1t. Asymptotically +as n → ∞, U± +n is substantial only for ˜t/C ≪ 1 and the +tails become symmetric and sub-Gaussian [173], h+(u) = +u2/2 − O(u3) and h−(u) = u2/2 − O(u4) (see [181]). +Notably, details about the underlying dynamics only +enter the tail bounds (7) via the system-dependent con- +stant C that, however, can be bounded. In particular, for +equilibrium initial conditions we have 0 ≤ 2w1 ≤ C ≤ 2 +(see [181]). Since φδτ(λ; C) is monotonically increasing +with C ∈ (0, 2], we have φδτ(λ; C) ≤ φδτ(λ; 2) which im- +plies φ∗ +δτ(t; C) ≥ φ∗ +δτ(t; 2). Thus, we find the model-free +bounds +U± +n (t; C) ≤ U± +n (t; 2) ≡ U± +n (t) +(11) +requiring no information about the system. The non- +asymptotic bounds on deviation probabilities of τ n in +Eqs. (7) and (11) are our first main result. +Notably, analogous concentration inequalities were pre- +viously derived for time-averages of Markov processes +[191–193] (see also [194]), and were recently applied to +bound time-averaged measurement outcomes in quantum +Markov processes [195] and to derive inverse thermody- +namic uncertainty relations [196]. +Illustration of bounds.—The lower L± +n (t) and upper +U± +n (t) bounds on P(±[τ n − ⟨τ⟩] ≥ t) in Eqs. (4) and (7), +respectively, are examplified in Fig. 2e-h (see red and +black lines) for the model systems shown in Fig. 1b-d. +Note that to illustrate all bounds, for convenience in a +single panel, we formally let t → −t for the left tails +L− +n (t) and U− +n (t), such that t (as shown) has support on +[−⟨τ⟩, ∞). Deviation probabilities are in turn expressed +as P(sgn(t)δτ n ≥ |t|) where sgn(x) denotes the signum +function and δτ n = τ n − ⟨τ⟩. +To assess the quality of our bounds for several n we +further scale probabilities P1/n such that L± +n (t) and U± +n (t) +collapse onto a master curve for all n (see also inset in +Fig. 2f). Symbols denote empirical deviation probabili- +ties obtained by sampling τ n for different n (see [181] for +details), which approach the upper bound as n increases. +For n = 1 empirical right-tail deviations are close to L+ +1 (t) +even for w1 ≤ 1 [197]. As expected the model-free upper +bound U± +n (t; 2) (yellow) holds universally but is generally +more conservative, however, it is remarkably good for +C ≳ 1.3 (see e.g. Fig. 2e-g) but becomes weaker as C +approaches 0 (see e.g. Fig. 2h). +Uncertainty quantification.—The bounds (7) provide +the elusive systematic framework to rigorously quantify +the uncertainty of the estimate τ n for any, and especially +for small, sample sizes. In particular, they allow us to +construct “with high probability” guarantees such as con- +fidence intervals, which—unlike traditional confidence +intervals in statistics—are not only asymptotically correct +but hold for any n. Furthermore, these concentration- +based guarantees do not require specifying a prior belief +as in the Bayesian context. Setting U± +n (t± +α±; C) = α± for +chosen acceptable left- and right-tail error probabilities +α± (with α+ + α− < 1) we get an implicit definition of +the confidence interval [−t− +α−, t+ +α+] at confidence level (or +“coverage probability”) 1 − (α+ + α−) in the form +P(−t− +α− ≤ δτ n ≤ t+ +α+) ≥ 1 − α− − α+ ≡ 1 − α, +(12) +stating that with probability of at least 1 − α the sample +mean τ n lies within [⟨τ⟩ − t− +α−, ⟨τ⟩ + t+ +α+]. Confidence +intervals are closely related to, and can be used for, statis- +tical significance tests [198, 199]. However, they provide +more insight; instead of mere rejection/acceptance they +provide quantitative bounds on statistical uncertainty. +Two-sided intervals are not uniquely determined by +specifying a confidence level. It is customary to choose +equal tail probabilities α+ = α− = α/2 yielding so-called +central confidence intervals for which t± +α± are generally +not equidistant. Two-sided central confidence intervals +for δτ n as a function of n for a confidence level of α = 0.1 +and models systems in Fig. 1b-d are shown (rescaled to a +master scaling) in Fig. 3a. One may also choose symmetric +intervals which in turn do not necessarily imply equal tail +probabilities (i.e. α+ ̸= α−). In some situations only one- +sided confidence intervals are required P(±δτ n ≤ t± +α±) ≥ +1 − α± (for a discussion see [181]). +In particular, we may now also answer the practical +question: How many realizations are required to achieve +a desired accuracy with a specified probability? To ensure +with probability of at least 1 − α that δτ n∗ ∈ [−t− +α−, t+ +α+] +one needs n∗ realizations defined via +U+ +n∗(t+ +α+; C) + U− +n∗(t− +α−; C) = α. +(13) +The number of samples n∗ required to guarantee that τ n∗ +falls within a symmetric interval of length ∆t = 0.2/µ1, +(i.e. τ n∗ ∈ [⟨τ⟩ − 0.1/µ1, ⟨τ⟩ + 0.1/µ1]) with probability +of at least 1 − α is shown in Fig. 3b for several values of C +(intersections with the dashed line yield n∗ guaranteeing +a coverage of at least 90%). Fig. 3c depicts the comple- +mentary symmetric interval ∆t covering the range of δτ n +for a given n with probability of at least 90%. Note that +hundreds to thousands of samples may be required to +ensure an accuracy of ±0.1/µ1 with a 90% confidence, +which is seemingly not met in experiments [113–119]. +Eqs. (12) and (13) constitute our second main result as +they provide rigorous error estimates in the small-sample +regime that allow for systematic error control in kinetic +inference and can be solved for t± +α± and n∗, respectively, +using standard root-finding methods (see [181]). +Using Eq. (11) we can construct system-independent +but more conservative universal confidence intervals (see +yellow line in Fig. 3b,c). Interestingly, even when C ≈ 1 +the universal bound remains reasonably tight, only for +C ≪ 1 differences become substantial. +Conclusion.—Leveraging spectral analysis and the +framework of concentration inequalities we derived gen- +eral upper and lower bounds on the probability that the + +5 +FIG. 3. Non-asymptotic uncertainty quantification of the sam- +ple mean τ n. (a) Relative error µ1δτ n = µ1(τ n−⟨τ⟩) (symbols) +obtained from sampling of τ n for different model systems and +as a function n (re-scaled to a master scaling). The correspond- +ing two-sided central confidence interval [−µ1t− +α/2, µ1t+ +α/2] with +α = 0.1 is shown as black lines. (b) Required number of sam- +ples n∗ to ensure that the relative error δτ n∗ falls within the +symmetric interval [−0.1, 0.1] of length ∆t = 0.2/µ1 with +probability of at least 1 − α for several values of C. (c) Cor- +responding symmetric confidence interval [−µ1∆t/2, µ1∆t/2] +(only the upper limit is shown) at confidence level α = 0.1 as +a function of n for different C. +empirical first-passage time τ n inferred from n indepen- +dent realizations deviates from the true mean ⟨τ⟩ by any +given amount. We used these bounds to construct non- +asymptotic confidence intervals that hold in the elusive +small-sample regime and thus go beyond Central-Limit- +and bootstrapping-based methods, which are known to +fail for small n. The results require minimal input and +in particular do not require any prior belief as in the +Bayesian approach that is known to be problematic and +likely underestimates the uncertainty in the small-sample +setting. Our concentration-based results allow for rigor- +ous, model-free error control and reliable error estimation, +which is essential for the analysis of experimental and +simulation data. They may further be applied to popu- +lation dynamics and epidemiology, e.g. in the inference +of extinction or incubation times of diseases [200–204], +and may be extended to the concentration around the +typical instead of mean first-passage times [205] as well +as non-ergodic and irreversible dynamics. +Acknowledgments.—Financial support from Studiens- +tiftung des Deutschen Volkes (to R. B.) and the German +Research Foundation (DFG) through the Emmy Noether +Program GO 2762/1-2 (to A. G.) is gratefully acknowl- +edged. +∗ agodec@mpinat.mpg.de +[1] S. Redner, A Guide to First-Passage Processes (Cam- +bridge University Press, 2001). +[2] R. Metzler, S. Redner, and G. Oshanin, First-Passage +Phenomena and their Applications, Vol. 35 (World Scien- +tific, 2014). +[3] Y. Zhang and O. K. Dudko, Annu. Rev. Biophys. 45, 117 +(2016). +[4] S. Iyer-Biswas and A. Zilman, Adv. Chem. Phys. 160, +261 (2016). +[5] A. Szabo, K. Schulten, and Z. Schulten, J. Chem. Phys. +72, 4350–4357 (1980). +[6] P. H¨anggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. +62, 251 (1990). +[7] E. Ben-Naim, S. Redner, and F. Leyvraz, Phys. Rev. Lett. +70, 1890 (1993). +[8] D. S. Grebenkov, R. Metzler, and G. Oshanin, Phys. +Chem. Chem. Phys. 20, 16393–16401 (2018). +[9] D. S. Grebenkov, R. Metzler, and G. Oshanin, Commun. +Chem. 1, 1 (2018). +[10] D. S. Grebenkov, Phys. Rev. Lett. 117, 260201 (2016). +[11] O. G. Berg, R. B. Winter, and P. H. Von Hippel, Bio- +chemistry 20, 6929–6948 (1981). +[12] E. Koslover, M. D´ıaz de la Rosa, and A. Spakowitz, Bio- +phys. J. 101, 856 (2011). +[13] D. Holcman and Z. Schuss, J. Phys. A: Math. Theor. 47, +173001 (2014). +[14] O. B´enichou, C. Chevalier, B. Meyer, and R. Voituriez, +Phys. Rev. Lett. 106, 038102 (2011). +[15] E. G. Marklund, A. Mahmutovic, O. G. Berg, P. Hammar, +D. van der Spoel, D. Fange, and J. Elf, Proc. Natl. Acad. +Sci. 110, 19796–19801 (2013). +[16] M. Bauer and R. Metzler, PLoS ONE 8, e53956 (2013). +[17] O. B´enichou, C. Chevalier, J. Klafter, B. Meyer, and +R. Voituriez, Nat. Chem. 2, 472–477 (2010). +[18] O. B´enichou and R. Voituriez, Phys. Rep. 539, 225 (2014). +[19] A. Godec and R. Metzler, Phys. Rev. X 6, 041037 (2016). +[20] J. Newby and J. Allard, Phys. Rev. Lett. 116, 128101 +(2016). +[21] S. Redner and B. Meerson, J. Stat. Mech. 2014, P06019 +(2014). +[22] B. Meerson and S. Redner, Phys. Rev. Lett. 114, 198101 +(2015). +[23] D. Hartich and A. Godec, New J. Phys. 20, 112002 (2018). +[24] D. Hartich and A. Godec, J. Stat. Mech. 2019, 024002 +(2019). +[25] D. Hartich and A. Godec, Reaction kinetics in the few- +encounter limit, in Chemical Kinetics (World Scientific, +2019) Chap. 11, pp. 265–283. +[26] Z. Schuss, K. Basnayake, and D. Holcman, Phys. Life Rev. +28, 52–79 (2019). +[27] S. D. Lawley and J. B. Madrid, J. Chem. Phys. 150, +214113 (2019). +[28] S. D. Lawley and J. B. Madrid, J. Nonlinear Sci. 30, +1207–1227 (2020). +[29] S. D. Lawley, Phys. Rev. E 102, 062118 (2020). +[30] P. C. Bressloff and J. M. Newby, Rev. Mod. Phys. 85, +135 (2013). +[31] ´E. Rold´an, A. Lisica, D. S´anchez-Taltavull, and S. W. +Grill, Phys. Rev. E 93, 062411 (2016). +[32] K. R. Ghusinga, J. J. Dennehy, and A. Singh, Proc. Natl. +Acad. Sci. 114, 693 (2017). +[33] K. Rijal, A. Prasad, A. Singh, and D. Das, Phys. Rev. +Lett. 128, 048101 (2022). + +6 +[34] J. J. Parmar, D. Das, and R. Padinhateeri, Nucleic Acids +Res. 44, 1630 (2015). +[35] D. A. Charlebois, N. Abdennur, and M. Kaern, Phys. Rev. +Lett. 107, 218101 (2011). +[36] F. Frey, F. Ziebert, and U. S. Schwarz, Phys. Rev. Lett. +122, 088102 (2019). +[37] A. L. Lloyd and R. M. May, Science 292, 1316 (2001). +[38] L. Hufnagel, D. Brockmann, and T. Geisel, Proc. Natl. +Acad. Sci. 101, 15124 (2004). +[39] O. B´enichou, C. Loverdo, M. Moreau, and R. Voituriez, +Rev. Mod. Phys. 83, 81 (2011). +[40] F. Boccardo and O. Pierre-Louis, Phys. Rev. Lett. 128, +256102 (2022). +[41] T. Erdmann and U. S. Schwarz, Phys. Rev. Lett. 92, +108102 (2004). +[42] S. Chakrabarti, M. Hinczewski, and D. Thirumalai, Proc. +Natl. Acad. Sci. 111, 9048–9053 (2014). +[43] K. Blom and A. Godec, Phys. Rev. X 11, 031067 (2021). +[44] I. Goychuk and P. H¨anggi, Proc. Natl. Acad. Sci. 99, 3552 +(2002). +[45] T. Kay and L. Giuggioli, Phys. Rev. Res. 4, 032039 (2022). +[46] S. Bo, L. Hubatsch, J. Bauermann, C. A. Weber, and +F. J¨ulicher, Phys. Rev. Res. 3, 043150 (2021). +[47] H. Kramers, Physica 7, 284 (1940). +[48] S. Sabhapandit and S. N. Majumdar, Phys. Rev. Lett. +125, 200601 (2020). +[49] B. Trendelkamp-Schroer and F. No´e, Phys. Rev. X 6, +011009 (2016). +[50] M. Chupeau, J. Gladrow, A. Chepelianskii, U. F. Keyser, +and E. Trizac, Proc. Natl. Acad. Sci. 117, 1383 (2019). +[51] T. D. Swinburne, D. Kannan, D. J. Sharpe, and D. J. +Wales, J. Chem. Phys. 153, 134115 (2020). +[52] A. L. Thorneywork, J. Gladrow, Y. Qing, M. Rico-Pasto, +F. Ritort, H. Bayley, A. B. Kolomeisky, and U. F. Keyser, +Sci. Adv. 6, 1 (2020). +[53] A. Goychuk and E. Frey, Phys. Rev. Lett. 123, 178101 +(2019). +[54] R. Bebon and U. S. Schwarz, New J. Phys. 24, 063034 +(2022). +[55] D. B. Dougherty, I. Lyubinetsky, E. D. Williams, M. Con- +stantin, C. Dasgupta, and S. Sarma, Phys. Rev. Lett. 89, +136102 (2002). +[56] M. Constantin, S. D. Sarma, C. Dasgupta, O. Bondarchuk, +D. B. Dougherty, and E. D. Williams, Phys. Rev. Lett. +91, 086103 (2003). +[57] D. B. Dougherty, C. Tao, O. Bondarchuk, W. G. Cullen, +E. D. Williams, M. Constantin, C. Dasgupta, and S. D. +Sarma, Phys. Rev. E 71, 021602 (2005). +[58] J. Merikoski, J. Maunuksela, M. Myllys, J. Timonen, and +M. J. Alava, Phys. Rev. Lett. 90, 024501 (2003). +[59] M. Constantin, C. Dasgupta, P. P. Chatraphorn, S. N. +Majumdar, and S. D. Sarma, Phys. Rev. E 69, 061608 +(2004). +[60] C. Godr`eche, S. N. Majumdar, and G. Schehr, Phys. Rev. +Lett. 102, 240602 (2009). +[61] A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys. +62, 225–361 (2013). +[62] T. R. Gingrich and J. M. Horowitz, Phys. Rev. Lett. 119, +170601 (2017). +[63] S. Singh, P. Menczel, D. S. Golubev, I. M. Khaymovich, +J. T. Peltonen, C. Flindt, K. Saito, ´E. Rold´an, and J. P. +Pekola, Phys. Rev. Lett. 122, 230602 (2019). +[64] E. Rold´an, I. Neri, M. D¨orpinghaus, H. Meyr, and +F. J¨ulicher, Phys. Rev. Lett. 115, 250602 (2015). +[65] I. Neri, E. Rold´an, and F. J¨ulicher, Phys. Rev. X 7, 011019 +(2017). +[66] G. Falasco and M. Esposito, Phys. Rev. Lett. 125, 120604 +(2020). +[67] I. Neri, J. Phys. A: Math. Theor. 55, 304005 (2022). +[68] J. P. Garrahan, Phys. Rev. E 95, 032134 (2017). +[69] K. Hiura and S. ichi Sasa, Phys. Rev. E 103, 050103 +(2021). +[70] M. Kac, in Proceedings of the Second Berkeley Symposium +on Mathematical Statistics and Probability (University of +California Press, Berkeley, Calif., 1951) pp. 189–215. +[71] G. Schehr and S. N. Majumdar, First-Passage Phenomena +and Their Applications , 226–251 (2014). +[72] S. N. Majumdar, G. Schehr, and G. Wergen, J. Phys. A: +Math. Theor. 45, 355002 (2012). +[73] D. Hartich and A. Godec, J. Phys. A: Math. Theor. 52, +244001 (2019). +[74] H. Friedman, D. A. Kessler, and E. Barkai, Phys. Rev. E +95, 032141 (2017). +[75] F. Thiel, E. Barkai, and D. A. Kessler, Phys. Rev. Lett. +120, 040502 (2018). +[76] M. J. Kearney and S. N. Majumdar, J. Phys. A: Math. +Gen. 38, 4097 (2005). +[77] M. J. Kearney, S. N. Majumdar, and R. J. Martin, J. +Phys. A: Math. Theor. 40, F863 (2007). +[78] M. J. Kearney and S. N. Majumdar, J. Phys. A: Math. +Theor. 47, 465001 (2014). +[79] M. J. Kearney and R. J. Martin, J. Phys. A: Math. Theor. +54, 055002 (2021). +[80] S. N. Majumdar and B. Meerson, J. Stat. Mech.: Theory +Exp. 2021 (3), 039801. +[81] P. Singh and A. Pal, J. Phys. A: Math. Theor. 55, 234001 +(2022). +[82] G. Mercado-V´asquez and D. Boyer, Phys. Rev. Lett. 123, +250603 (2019). +[83] A. Kumar, A. Zodage, and M. S. Santhanam, Phys. Rev. +E 104, 052103 (2021). +[84] J. L. Spouge, A. Szabo, and G. H. Weiss, Phys. Rev. E +54, 2248 (1996). +[85] Y. Scher and S. Reuveni, Phys. Rev. Lett. 127, 018301 +(2021). +[86] E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, and J. Tailleur, +Phys. Rev. Lett. 122, 258001 (2019). +[87] F. Mori, P. L. Doussal, S. N. Majumdar, and G. Schehr, +Phys. Rev. Lett. 124, 090603 (2020). +[88] P. H¨anggi and P. Talkner, Phys. Rev. Lett. 51, 2242 +(1983). +[89] P. Hanggi and P. Talkner, Phys. Rev. A 32, 1934 (1985). +[90] T. Gu´erin, N. Levernier, O. B´enichou, and R. Voituriez, +Nature 534, 356 (2016). +[91] H. Meyer and H. Rieger, Phys. Rev. Lett. 127, 070601 +(2021). +[92] M. R. Evans and S. N. Majumdar, Phys. Rev. Lett. 106, +160601 (2011). +[93] L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and +G. Schehr, Phys. Rev. Lett. 113, 220602 (2014). +[94] S. Reuveni, Phys. Rev. Lett. 116, 170601 (2016). +[95] A. Pal and S. Reuveni, Phys. Rev. Lett. 118, 030603 +(2017). +[96] A. Pal, I. Eliazar, and S. Reuveni, Phys. Rev. Lett. 122, +020602 (2019). +[97] M. R. Evans, S. N. Majumdar, and G. Schehr, J. Phys. +A: Math. Theor. 53, 193001 (2020). + +7 +[98] B. Besga, A. Bovon, A. Petrosyan, S. N. Majumdar, and +S. Ciliberto, Phys. Rev. Res. 2, 032029 (2020). +[99] O. Tal-Friedman, A. Pal, A. Sekhon, S. Reuveni, and +Y. Roichman, J. Phys. Chem. Lett. 11, 7350 (2020). +[100] B. D. Bruyne, J. Randon-Furling, and S. Redner, Phys. +Rev. Lett. 125, 050602 (2020). +[101] B. D. Bruyne, S. N. Majumdar, and G. Schehr, Phys. +Rev. Lett. 128, 200603 (2022). +[102] D. L. Ensign and V. S. Pande, J. Phys. Chem. B 113, +12410–12423 (2009). +[103] R. Satija, A. Das, S. M¨uhle, J. Enderlein, and D. E. +Makarov, J. Phys. Chem. B 124, 3482–3493 (2020). +[104] S. Zolaktaf, F. Dannenberg, X. Rudelis, A. Condon, +J. M. Schaeffer, M. Schmidt, C. Thachuk, and E. Winfree, +Inferring Parameters for an Elementary Step Model of +DNA Structure Kinetics with Locally Context-Dependent +Arrhenius Rates (Springer International Publishing, 2017) +p. 172–187. +[105] C. Weinreb, S. Wolock, B. K. Tusi, M. Socolovsky, and +A. M. Klein, Proc. Natl. Acad. Sci. 115, E2467 (2018). +[106] P. Pearce, F. G. Woodhouse, A. Forrow, A. Kelly, +H. Kusumaatmaja, and J. Dunkel, Nat. Commun. 10, +1 (2019). +[107] Y. Zhou, C. Zhang, G. Stell, and J. Wang, J. Am. Chem. +Soc. 125, 6300–6305 (2003). +[108] J. O. Daldrop, J. Kappler, F. N. Br¨unig, and R. R. Netz, +Proc. Natl. Acad. Sci. 115, 5169–5174 (2018). +[109] D. A. Nicholson and G. C. Rutledge, J. Chem. Phys. +144, 134105 (2016). +[110] V. J. van Hijkoop, A. J. Dammers, K. Malek, and M.-O. +Coppens, J. Chem. Phys. 127, 085101 (2007). +[111] R. Belousov, +M. N. Qaisrani, +A. Hassanali, and +E. Roldan, Soft Matter 16, 9202–9216 (2020). +[112] S. Ditlevsen and O. Ditlevsen, Probabilistic Eng. Mech +23, 170–179 (2008). +[113] V. Gapsys and B. L. de Groot, eLife 9, e57589 (2020). +[114] K. Lindorff-Larsen, S. Piana, R. O. Dror, and D. E. +Shaw, Science 334, 517 (2011). +[115] J. L. Adelman and M. Grabe, J. Chem. Phys. 138, +044105 (2013). +[116] B. Mostofian and D. M. Zuckerman, J. Chem. Theory +Comput. 15, 3499 (2019). +[117] R. Mehra and K. P. Kepp, J. Chem. Phys. 151, 085101 +(2019). +[118] A. Militaru, M. Innerbichler, M. Frimmer, F. Tebbenjo- +hanns, L. Novotny, and C. Dellago, Nat. Commun. 12, 1 +(2021). +[119] L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, +and L. Novotny, Nat. Nanotechnol. 12, 1130 (2017). +[120] D. J. Sharpe and D. J. Wales, J. Chem. Phys. 153, +024121 (2020). +[121] R. M. Donovan, A. J. Sedgewick, J. R. Faeder, and D. M. +Zuckerman, J. Chem. Phys. 139, 115105 (2013). +[122] J. Sabelko, J. Ervin, and M. Gruebele, Proc. Nat. Acad. +Sci. 96, 6031 (1999). +[123] D. L. Ensign and V. S. Pande, J. Phys. Chem. B 113, +12410 (2009). +[124] J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, +and M. Rief, Science 334, 512 (2011). +[125] A. M. Berezhkovskii and A. Szabo, J. Chem. Phys. 150, +054106 (2019). +[126] I. Nayak, D. Das, and A. Nandi, Phys. Rev. Res. 2, +013114 (2020). +[127] D. J. Wales, J. Phys. Chem. Lett. 13, 6349 (2022). +[128] N. Singhal and V. S. Pande, J. Chem. Phys. 123, 204909 +(2005). +[129] G. R. Bowman, V. S. Pande, and F. No´e, An Intro- +duction to Markov State Models and their Application to +Long Timescale Molecular Simulation, Vol. 797 (Springer +Science & Business Media, 2013). +[130] A. Grossfield and D. M. Zuckerman, Annu. Rep. Comput. +Chem. 5, 23 (2009). +[131] A. Grossfield, P. N. Patrone, D. R. Roe, A. J. Schultz, +D. W. Siderius, and D. M. Zuckerman, Living J. Comp. +Mol. Sci. 1 (2018). +[132] B. Knapp, L. Ospina, and C. M. Deane, J. Chem. Theory +Comput. 14, 6127 (2018). +[133] Resampling methods like bootstrapping assume the data +to be representative of the inferred statistic, which is not +necessarily the case for small n, possibly even when n is +large but finite for broad distributions. +[134] A. C. Davison and D. V. Hinkley, Bootstrap Methods and +their Application (Cambridge University Press, 1997). +[135] J. Shao, Mathematical Statistics (Springer New York, +2003). +[136] A. Abadie and G. W. Imbens, Econometrica 76, 1537 +(2008). +[137] H. Putter and W. R. van Zwet, in Selected Works of +Willem van Zwet (Springer New York, 2011) pp. 245–266. +[138] R. V. Hogg, J. W. McKean, and A. T. Craig, Introduction +to Mathematical Statistics (Pearson, 2018). +[139] N. Schenker, J. Am. Stat. Assoc. 80, 360 (1985). +[140] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, +Bayesian Data Analysis (Chapman and Hall/CRC, 1995). +[141] A. R. Brazzale, A. C. Davison, N. Reid, et al., Applied +Asymptotics: Case Studies in Small-Sample Statistics, +Vol. 23 (Cambridge University Press, 2007). +[142] L. Lista, Statistical Methods for Data Analysis in Particle +Physics (Springer International Publishing, 2017). +[143] D. Kaplan, Bayesian Statistics for the Social Sciences +(Guilford Publications, 2014). +[144] R. McElreath, Statistical Rethinking: A Bayesian Course +with Examples in R and Stan (Chapman and Hall/CRC, +2020). +[145] M. Tavakoli, J. N. Taylor, C.-B. Li, T. Komatsuzaki, +and S. Press´e, in Advances in Chemical Physics (John +Wiley & Sons, Inc., 2017) pp. 205–305. +[146] D. McNeish, Struct. Equ. Modeling 23, 750 (2016). +[147] S. C. Smid, D. McNeish, M. Mioˇcevi´c, and R. van de +Schoot, Struct. Equ. Modeling 27, 131 (2019). +[148] S. Bacallado, J. D. Chodera, and V. Pande, J. Chem. +Phys. 131, 045106 (2009). +[149] J.-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Senne, +M. Held, J. D. Chodera, C. Sch¨utte, and F. No´e, J. Chem. +Phys. 134, 174105 (2011). +[150] B. Trendelkamp-Schroer, H. Wu, F. Paul, and F. No´e, J. +Chem. Phys. 143, 174101 (2015). +[151] J. D. Chodera and F. No´e, J. Chem. Phys. 133, 105102 +(2010). +[152] L. Le Cam, L. M. LeCam, and G. L. Yang, Asymptotics +in Statistics: Some Basic Concepts (Springer Science & +Business Media, 2000). +[153] A. W. Van der Vaart, Asymptotic Statistics, Vol. 3 (Cam- +bridge university press, 2000). +[154] L. Le Cam, Asymptotic Methods in Statistical Decision +Theory (Springer Science & Business Media, 2012). +[155] P. Diaconis and D. Freedman, Ann. Stat. 14, 1 (1986). +[156] D. D. Cox, Ann. Stat. 21, 1 (1993). + +8 +[157] P. W. Diaconis and D. Freedman, Bernoulli , 411 (1998). +[158] D. Freedman, Ann. Stat. 27, 1119 (1999). +[159] A. Barron, M. J. Schervish, and L. Wasserman, Ann. +Stat. 27, 536 (1999). +[160] S. Ghosal, J. K. Ghosh, and R. Ramamoorthi, Ann. Stat. +27, 143 (1999). +[161] J. K. Ghosh and R. V. Ramamoorthi, Bayesian Non- +parametrics (Springer-Verlag, 2003). +[162] Y. Kim and J. Lee, Ann. Stat. 32, 1 (2004). +[163] S. Boucheron and E. Gassiat, Electron. J. Stat. 3, 114 +(2009). +[164] P. J. Bickel and B. J. Kleijn, Ann. Stat. 40, 206 (2012). +[165] V. Rivoirard and J. Rousseau, Ann. Stat. 40, 1489 +(2012). +[166] I. Castillo and R. Nickl, Ann. Stat. 42, 1941 (2014). +[167] J. Rousseau, Annu. Rev. Stat. Appl. 3, 211 (2016). +[168] V. Rockov´a, in International Conference on Machine +Learning (PMLR, 2020) pp. 8137–8146. +[169] K. Ray and A. van der Vaart, Elec. J. Stat. 15, 1 (2021). +[170] S. Ghosal and A. Van der Vaart, Fundamentals of Non- +parametric Bayesian Inference, Vol. 44 (Cambridge Uni- +versity Press, 2017). +[171] B. Kleijn and A. van der Vaart, Electron. J. Stat. 6, +none (2012). +[172] I. M. Johnstone, in Institute of Mathematical Statistics +Collections (Institute of Mathematical Statistics, 2010) +pp. 87–98. +[173] S. Boucheron, G. Lugosi, and P. Massart, Concentration +Inequalities: A Nonasymptotic Theory of Independence +(Oxford University Press, 2013). +[174] J.-H. Prinz, B. Keller, and F. No´e, Phys. Chem. Chem. +Phys. 13, 16912 (2011). +[175] S. Olsson, H. Wu, F. Paul, C. Clementi, and F. No´e, +Proc. Nat. Acad. Sci. 114, 8265 (2017). +[176] G. A. Pavliotis, Stochastic Processes and Applications +(Springer New York, 2014). +[177] ˆL is self-adjoint in the left eigenspace with respect to +a scalar product weighted by e−ϕ(x) and the operator +eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat mea- +sure. +[178] Precisely, we require that ϕ(x) satisfies the Poincar´e +inequality, i.e. lim|x|→∞(|∇ϕ(x)|2/2 − ∇2ϕ(x)) = ∞. +[179] The relaxation eigenvalue problem reads −ˆLΨk(x) = +νkΨk(x) with ν0 = 0 and νk≥1 > 0 [176]. +[180] In a continuous state-space the absorbing state a has +zero measure and ˜peq(x) = peq(x); In the discrete case +˜peq(xk̸=a) ≡ peq(xk)/ � +k̸=a peq(x). +[181] See Supplemental Material at [...] for further details, +mathematical proofs, and generalizations to arbitrary +initial conditions p0(x), as well as Refs [3, 8–10, 12, 14]. +[182] When the initial condition is not sampled from ˜peq(x) +we assume that ϕ(x) is sufficiently confining to assure a +“nice” asymptotic growth of eigenvalues, limk→∞ νk = bkβ +with β > 1/2 and 0 < b < ∞. The latter condition +is automatically satisfied when Ω is finite, since regu- +lar Sturm-Liouville problems display Weyl asymptotics +with β = 2 [212]. The condition is in fact satisfied by +most physically relevant processes with discrete spectra, +incl. the Ornstein-Uhlenbeck or Rayleigh process [213] +with β = 1. +[183] When Ω is discrete the integral is replaced by a sum +over states excluding the target. +[184] wk ≥ 0 is a necessary condition for the validity of the +lower bounds. Thus, in contrast to our Cram´er-Chernoff +bounds U± +n (t) that generalize to arbitrary initial condi- +tions, L± +n (t) hold only for p0(x0) = ˜peq(x0). +[185] D. S. Grebenkov, R. Metzler, and G. Oshanin, New J. +Phys. 22, 103004 (2020). +[186] ψδτn(λ) is differentiable, convex, non-negative, and non- +decreasing and thus ψ∗ +δτ(t) = ψδτn(λ†), where λ† solves +ψ′ +δτ(λ†) = t. +[187] In case of arbitrary initial conditions ⟨τ 2⟩ becomes re- +placed by � +i wi1wi>0 < ∞ while the rest remains un- +changed. +[188] C. Mej´ıa-Monasterio, G. Oshanin, and G. Schehr, J. Stat. +Mech.: Theory Exp. 2011 (06), P06022. +[189] G. Oshanin, Y. Holovatch, and G. Schehr, Physica A +390, 4340 (2011). +[190] T. G. Mattos, C. Mej´ıa-Monasterio, R. Metzler, and +G. Oshanin, Phys. Rev. E 86, 031143 (2012). +[191] P. Lezaud, Ann. Appl. Probab. 8, 849 (1998). +[192] P. Lezaud, ESAIM Probab. Stat. 5, 183–201 (2001). +[193] F. Gao, A. Guillin, and L. Wu, Theory Probab. its Appl. +58, 358 (2014). +[194] Ref. [192] contains an error; the Proof of Lemma 2.3 is +only valid in the regime r < λ1/3||f||∞, but the Lemma +may be shown to hold in the claimed regime [214]. +[195] F. Girotti, J. P. Garrahan, and M. Gut¸˘a, Concentra- +tion inequalities for output statistics of quantum Markov +processes (2022). +[196] G. Bakewell-Smith, F. Girotti, M. Gut¸˘a, and J. P. Garra- +han, Inverse thermodynamic uncertainty relations: Gen- +eral upper bounds on the fluctuations of trajectory ob- +servables (2022). +[197] However, w1 can get arbitrary close to 0 in principle, +rendering the lower bound trivial. +[198] D. Wackerly, W. Mendenhall, and R. L. Scheaffer, Math- +ematical Statistics with Applications (Cengage Learning, +2014). +[199] W. Q. Meeker, G. J. Hahn, and L. A. Escobar, Statistical +Intervals: A Guide for Practitioners and Researchers, Vol. +541 (John Wiley & Sons, 2017). +[200] M. Dykman, I. Schwartz, and A. Landsman, Phys. Rev. +Lett. 101, 078101 (2008). +[201] J. A. Gilbert, L. A. Meyers, A. P. Galvani, and J. P. +Townsend, Epidemics 6, 37 (2014). +[202] B. Ottino-Loffler, J. G. Scott, and S. H. Strogatz, eLife +6, e30212 (2017). +[203] M. Aliee, K. S. Rock, and M. J. Keeling, J. R. Soc. +Interface 17, 20200540 (2020). +[204] D. Hathcock and S. H. Strogatz, Phys. Rev. Lett. 128, +218301 (2022). +[205] S. Belan, Phys. Rev. Res. 2, 013243 (2020). +[14] R. L. Burden, J. D. Faires, and A. M. Burden, Numerical +Analysis (Cengage Learning, 2015). +[10] E. Barkai, E. Aghion, and D. Kessler, Phys. Rev. X 4, +021036 (2014). +[9] J. W. Pitman, Adv. Appl. Probab. 7, 511 (1975). +[3] A. J. F. Siegert, Phys. Rev. 81, 617 (1951). +[8] U. Seifert, Annu. Rev. Condens. Matter Phys. 10, 171 +(2019). +[12] G. Cowan, Statistical Data Analysis (Oxford University +Press, 1998). +[212] G. Teschl, Ordinary Differential Equations and Dynami- +cal Systems (American Mathematical Society, 2012). +[213] C. W. Gardiner, Handbook of Stochastic Methods for +Physics, Chemistry and the Natural Sciences, 3rd ed., + +9 +Springer Series in Synergetics, Vol. 13 (Springer-Verlag, +Berlin, 2004). +[214] S. C. Ibanez, Concentration inequalities for Markov jump +processes (2022). + +1 +Supplementary Material for: +Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime +Rick Bebon and Aljaˇz Godec +Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 G¨ottingen +In this Supplementary Material (SM) we present additional background and details of the calculations, auxiliary +results, numerical methods, and mathematical proofs of the claims made in the Letter. The sections are organized in +the order as they appear in the Letter. +CONTENTS +References +5 +S1. Spectral representation and preparatory Lemmas +2 +A. Spectral representation +2 +B. Lemma 1: All weights are non-negative for equilibrium initial conditions +3 +C. Lemma 2: Sum of positive weights is bounded from above +3 +S2. Extreme value bounds and comparison with Cram´er-Chernoff bounds +4 +A. Extreme value bounds +4 +B. Comparison of Cram´er-Chernoff vs Extreme value Bounds +4 +S3. Complete proof of concentration inequalities and their asymptotics +5 +A. Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩ +5 +B. Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ +7 +C. Behavior of upper bounds U± +n (t) for large sample sizes +9 +D. Proof of bounds on C and model-free concentration inequalities +9 +S4. Model systems and details on numerical methods +10 +A. Continuous-time discrete-state Markov jump process +10 +1. Transitions rates of the 8-state toy protein model +11 +2. Transitions rates of the calmodulin protein model +11 +B. Spatially confined Brownian molecular search process +11 +C. Statistics of first-passage times ⟨τ⟩ and the sample mean τ n +12 +S5. Uncertainty quantification with confidence intervals +13 +References +15 + +2 +S1. +SPECTRAL REPRESENTATION AND PREPARATORY LEMMAS +In this section we provide additional background on the spectral analysis of first-passage problems and some auxiliary +Lemmas. In particular, we prove that for equilibrium initial conditions all spectral first-passage weights wk(˜peq) are +non-negative and that general initial conditions p0(x) the sum of positive spectral weights is always bounded. +A. +Spectral representation +First, we recall some general results using the spectral representation of first-passage processes (for more details on +see e.g. [1, 2]). As stated in the Letter, we consider time-homogeneous Markov processes xt on a continuous or discrete +state-space Ω with (forward) generator ˆL corresponding to a Markov rate-matrix or an effectively one-dimensional +Fokker-Planck operator. Let the transition probability density to find xt at x at time t given that it evolved from x0 +be pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac or Kronecker delta for continuous and discrete state-spaces, +respectively. We assume the process to be ergodic limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes the +equilibrium probability density and ϕ(x) the corresponding generalized potential in units of thermal energy kBT. We +assume that ˆL obeys detailed balance, such that it is self-adjoint in the left eigenspace with respect to a scalar product +weighted by e−ϕ(x) and the operator eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat measure. +We assume that ˆL is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or that (iii) Ω is infinite but ϕ(x) +is sufficiently confining (precisely, we require that ϕ(x) satisfies the Poincar´e inequality, i.e. lim|x|→∞(|∇ϕ(x)|2/2 − +∇2ϕ(x)) = ∞.). Each of the conditions (i)-(iii) ensures that the eigenvalue spectrum of ˆL is discrete. The relaxation +eigenvalue problem (for the inner product (·|·) defined with respect to a flat Lebesgue measure) reads −ˆLΨR +k (x) = +νkΨR +k (x) with ΨL +k(x) = ΨR +k (x)eϕ(x), ν0 = 0 and νk≥1 > 0. +The first-passage time to a target a for xt=0 drawn from a density p0(x) is defined as τ = inft[ t |xt = a, p0(x0)]. +We will use ⟨·⟩ to denote an average over all first-passage paths {xt′}0≤t′≤τ, i.e. those that hit a only once. The +first-passage time density to a, ℘a(t|x0) = ⟨δ(t − τ[{xt′}])⟩ to reach the absorbing target at x = a, starting initially +from x0, has the general spectral representation +℘a(t|x0) = +� +k≥1 +wk(x0)µke−µkt, +(S1) +where µk is the k-th first-passage rate and wk(x0) its corresponding first-passage weight. In similar fashion the survival +probability is expressed as +Sa(t|x0) ≡ +� ∞ +t +℘a(t′|x0)dt′ = +� +k≥1 +wk(x0)e−µkt. +(S2) +We note that in contrast to the relaxation eigenvalues νk, the first-passage rates µk = µk(a) depend in the location of +the absorbing target. Moreover, for any target location a the interlacing theorem holds [1, 2] : +νk−1 ≤ µk(a) ≤ νk +∀k, a +(S3) +where equality occurs iff wk(x0) = 0, i.e. for a where ΨR +k (a) = 0. +Laplace transforming the spectral expansion of the first-passage time density (S1)—according to ˜f(s) ≡ +� +e−stf(t) dt +with f being a generic function locally integrable on t ∈ [0, ∞)—yields +˜℘a(s) = +� +k≥1 +wk(x0)µk +s + µk +. +(S4) +The first-passage weights are then obtained by using the residue theorem to invert the Laplace transformed renewal +theorem [1–3] +wk(x0) = ˜p(a, −µk|x0) +µk ˙˜p(a, −µk|a) = +� +l≥0(1 − νl/µk)−1ΨR +l (a)ΨL +l (x0) +� +l≥0(1 − νl/µk)−2ΨR +l (a)ΨL +l (a) < ∞, +(S5) +where ˙˜p(a, s|a) = ∂s˜p(a, s|a) is taken at s = −µk and {νl, ΨR +l , ΨL +l } are the corresponding relaxation eigenmodes [1, 2]. +The weights satisfy � +k≥1 wk(x0) = 1 and the first non-zero weight is strictly positive w1(x0) > 0. Moreover, the +relaxation eigenvalues ν0 = 0 and all νk>0 ≥ 0 are real as a result of detailed balance. + +3 +B. +Lemma 1: All weights are non-negative for equilibrium initial conditions +In the Letter we focus on equilibrium initial conditions, that is we assume that x0 is drawn from the invariant +measure, peq(x0), which in the particular case of diffusion processes is assumed to have a reflecting boundary at a +(i.e. we focus on the one-sided first-passage process). We further introduce the non-negative modified spectral weights +¯wk(x0) ≡ wk(x0)θ(sgn[wk(x0)]) and now prove that for a normalized equilibrium probability density of initial conditions +p0(x0) that excludes the target—i.e. ˜peq(x0) ≡ peq(x0)[1 − δa(x0)]/(1|peq(x0)[1 − δa(x0)]) where δa(x0) is the Dirac +measure (note that (1|˜peq) = 1)—all weights wk are rendered non-negative. We thus have ¯wk(˜peq) = wk(˜peq) ≥ 0, ∀k. +Namely, because ΨL +l (a) = eβU(a)ΨR +l (a) we have ΨR +l (a)ΨL +l (a) ≥ 0, ∀l, and from bi-orthogonality (ΨL +l |peq) = δl,0 it +follows that +˜wk ≡ (wk|˜peq) = ˜peq(a) +1 − � +l≥0(1 − νl/µk)−1ΨR +l (a)ΨL +l (a) +� +l≥0(1 − νl/µk)−2ΨR +l (a)ΨL +l (a) += +˜peq(a) +� +l≥0(1 − νl/µk)−2ΨR +l (a)ΨL +l (a) ≥ 0 +(S6) +because by definition µk > 0, ∀k ≥ 1 denotes the zeros of ˜p(a, s|a), i.e. ˜p(a, −µk|a) = � +l≥0(νl − µk)−1ΨR +l (a)ΨL +l (a) = +µ−1 +k +� +l≥0(1 − νl/µk)−1ΨR +l (a)ΨL +l (a) = 0 which completes the proof of the Lemma. +C. +Lemma 2: Sum of positive weights is bounded from above +For the sake of completeness we here additionally present results for general initial conditions p0(x0). Recall from +the Letter that we require some additional conditions on ϕ(x) or Ω in this more general setting. +In particular, we assume that ϕ(x) is sufficiently confining to assure a “nice” asymptotic growth of eigenvalues, +limk→∞ νk = bkβ with β > 1/2 and 0 < b < ∞. The latter condition is automatically satisfied when Ω is finite, since +regular Sturm-Liouville problems display Weyl asymptotics with β = 2 [4]. The condition is in fact satisfied by most +physically relevant processes with discrete spectra, incl. the (Sturm-Liouville irregular) Ornstein-Uhlenbeck or Rayleigh +process [5] with β = 1. This implies, by the interlacing theorem (S3) that b(k − 1)β ≤ µk ≤ bkβ and therefore there +exists a real constant C ∈ (0, ∞) such that limk→∞ µk diverges as Ckβ. +Recall further that the m-th moment of τ is given by ⟨τ m⟩ = m! � +k≥1 wk(p0)/µm +k . By construction we obtain +2 � +k≥1 ¯wk(p0)/µ2 +k ≡ ⟨¯τ 2 +p0⟩ ≥ 2 � +k≥1 wk(p0)/µ2 +k ≡ ⟨τ 2 +p0⟩, where equality holds when p0 = ˜peq (since in this case all +wk ≥ 0, i.e., ¯wk(˜peq) = wk(˜peq) as discussed before). +Moreover, because we only consider Markov jump processes on finite state-spaces as well as processes for which +limk→∞ µk = Ckα with 0 < C < ∞ and α > 1/2 (this includes confined Markov jump processes on infinite state-spaces +and all regular Sturm-Liouville problems) convergence is ensured, i.e. 2 � +k≥1 ¯wk(p0)/µ2+n +k +< ∞, ∀n ≥ 0. +To prove this consider wmax ≡ maxk≥k∗ ¯wk(p0) such that wmax/µ2+n +k +≥ wk(p0)/µ2+n +k +, ∀k. Let the smallest k for +which the asymptotic scaling holds be k∗ then we may split the summation as � +k≥1 = �k∗−1 +k=1 + � +k≥k∗ such that +� +k≥1 +¯wk(p0) +µ2+n +k +≤ +k∗−1 +� +k=1 +¯wk(p0) +µ2+n +k ++ +� +k≥k∗ +wmax +µ2+n +k +. +Because the first term is nominally finite we only need to prove convergence of the second sum, which we do by +means of the integral test. We define a function f(k) ≡ wmax/µ2+n +k +that is monotonically decaying in k. This +implies f(x) ≤ f(k), ∀x ∈ [k, ∞) and f(x) ≥ f(k), ∀x ∈ [k∗, k]. +We then have for every integer k ≥ k∗ that +� k+1 +k +f(x)dx ≤ +� k+1 +k +f(k)dx = f(k) and conversely, for every integer k ≥ k∗+1 that +� k +k−1 f(x)dx ≥ +� k +k−1 f(k)dx = f(k). +We now sum over all k ≥ k∗ to obtain, using µk = Ckα∀k ≥ k∗ +� ∞ +k∗ +wmax +(Cxα)(2+n) dx ≤ +� +k +wmax +µ2+n +k +≤ +wmax +(Ckα∗ )2+n + +� ∞ +k∗ +wmax +(Cxα)2+n dx → +wmaxC−(2+n)k1−α(2+n) +∗ +α(2 + n) − 1 +≤ +� +k +wmax +µ2+n +k +≤ wmax(Ckα +∗ )−(2+n) + wmaxC−(2+n)k1−α(2+n) +∗ +α(2 + n) − 1 +< ∞ +where the last integral converges because 1−α(2+n) < 0, ∀n ≥ 0, which in turn proves convergence of � +k≥1 ¯wk(p0)/µ2 +k. + +4 +S2. +EXTREME VALUE BOUNDS AND COMPARISON WITH CRAM´ER-CHERNOFF BOUNDS +In the Letter we derive lower bounds L± +n (t) on the deviation probability P(τ n − ⟨τ⟩ ≥ t) and P(⟨τ⟩ − τ n ≥ t) +by utilizing extremal events, i.e., we consider the maximal and minimal first-passage time in a sample of n ≥ 1 +i.i.d. realizations. In this section we derive analogous upper bounds building on the same ideas. +A. +Extreme value bounds +Recall that for the reversible Markov dynamics considered the equilibrium survival probability Sa(t|˜peq) ≡ Sa(t) in +its spectral representation (S2) obeys +w1e−µ1t ≤ Sa(t) ≤ e−µ1t. +(S7) +For the upper bound we use µk ≤ µk+1 and that � +k>0 wk = 1 are normalized, whereas the lower bound follows since +wk ≥ 0, ∀k, as we consider equilibrium initial conditions throughout. Moreover, from extreme value theory it follows +P(τ min +n +≥ t) = Sa(t)n +⇔ +P(τ min +n +≤ t) = 1 − Sa(t)n, +P(τ max +n +≤ t) = (1 − Sa(t))n +⇔ +P(τ max +n +≥ t) = 1 − (1 − Sa(t))n , +(S8) +where we introduce τ max +n +≡ maxi∈[1,n] τi and τ min +n +≡ mini∈[1,n] τi, respectively. Clearly, since τ min +n +≤ τ n ≤ τ max +n +we +can write P(τ min +n +≥ t) ≤ P(τ n ≥ t) ≤ P(τ max +n +≥ t) and analogously P(τ min +n +≤ t) ≥ P(τ n ≤ t) ≥ P(τ max +n +≤ t). Using +Eq. (S8) in combination with Eq. (S7) we directly arrive at the lower bounds L± +n (t) (see Eq. (4) in the Letter) +P(τ n ≥ ⟨τ⟩ + t) ≥ P(τ min +n +≥ ⟨τ⟩ + t) = Sa(t + ⟨τ⟩)n ≥ +� +w1e−µ1(⟨τ⟩+t)�n +(S9) +P(τ n ≤ ⟨τ⟩ − t) ≥ P(τ max +n +≤ ⟨τ⟩ − t) = (1 − Sa(⟨τ⟩ − t))n ≥ +� +1 − e−µ1(⟨τ⟩−t)�n +. +(S10) +Introduced considerations are, however, not restricted to only lower bounds such that we can further leverage bounds +on the equilibrium survival probability (S7) to analogously obtain corresponding upper bounds as +P(τ n ≥ ⟨τ⟩ + t) ≤ P(τ max +n +≥ t) = 1 − (1 − Sa(⟨τ⟩ + t))n ≤ 1 − +� +1 − eµ1(⟨τ⟩+t)�n +, +P(τ n ≤ ⟨τ⟩ − t) ≤ P(τ min +n +≤ t) = 1 − Sa(t)n ≤ 1 − +� +w1e−µ1(⟨τ⟩−t)�n +. +(S11) +As we will illustrate next, the upper bounds (S11) are much weaker than those derived with the Cram´er-Chernoff +approach (Eq. (7) in the Letter) and require more information about the dynamics. +B. +Comparison of Cram´er-Chernoff vs Extreme value Bounds +In this section we directly compare the concentration-based upper bounds U± +n (t) (see Eq. (7) in the Letter) that +are obtained with the Cram´er-Chernoff approach, with the upper bounds (S11) which are based on extreme value +considerations in analogy to the lower bounds L± +n (t). Similar to Fig. 2e-h of the Letter we now exemplify and compare +both upper bounds in Fig. S1 for the model systems shown in Fig. 1b-d. +In Fig. S1a-d we equivalently express re-scaled deviation probabilities P1/n(sgn(t)δτ n ≥ |t|) in a single panel, i.e., +for the left tail we formally let t → −t such that t as shown now has support in [−⟨τ⟩, ∞) and sgn(x) = ±1 for +±x > 0 and sgn(0) = 0 denotes the signum function. Empirical deviation probabilities (symbols) as a function of t are +computed from statistics obtained by sampling τ n for different fixed n values. Extreme value lower bounds L± +n (t) (S10) +(or Eq. (4)) for both tails are depicted in red. Here we now focus on comparing the upper bounds. Concentration +inequalities U± +n (t; C) (Eq.(7)) are again depicted as black lines whereas the corresponding extreme value upper bounds +are represented as dashed/dotted lines where the respective coloring indicates the number of realizations n. Note, that +the concentration bounds (and the lower bounds) collapse onto a single master curve due to the employed scaling P1/n, +whereas the extreme value upper bounds do not due to their different functional form (compare Eq. (S11)). Evidently, +while for n = 1 the extreme value bounds remains close to the actual deviation probability, already for n = 3 they +become considerably less tight and overshoot heavily for all considered models. Moreover, extreme value upper bounds +become increasingly weak (even trivial at times) as n increases, therefore highlighting that Cram´er-Chernoff-type +bounds are vastly more suitable. + +5 +Motivated by the discussion above we next want to gain more quantitative insights for which sample sizes n the +Cram´er-Chernoff approach becomes more favorable. For this purpose we introduce a quality factor Q ∈ [0, ∞) that is +informally defined as +Q ≡ +Extreme value upper bound +Cram´er-Chernoff-type upper bound. +(S12) +A value Q > 1 therefore indicates that the Cram´er-Chernoff bound is tighter and Q < 1 suggests that the extreme +value bound should be favored, respectively. In Fig. S1e-h we illustrate the quality factor Q as a function of sample +size n for different fixed dimensionless deviation values µ1t (star symbols in Fig. S1a-d). Remarkably for all model +systems considered—which span a large range of possible C values—the Cram´er-Chernoff approach is already superior +even in the small-sample regime n ≲ 4. Moreover, we can further study the particular n∗, for which one would reach +Q = 1, as a function of some desired deviation µ1t relative to the longest time scale 1/µ1. Note, that again for the +left tail we let t → −t (see discussion above). As depicted in Fig. S1i-l for our model systems, n∗ (blue) generally is +found to be well below n = 8, i.e., even for most small sample sizes the derived Cram´er-Chernoff-type bounds can be +considered to be the better choice, especially when considering large µ1t (i.e. large deviations). +Lastly, one could ask the question why the extreme value upper bound is so “weak” when n increases even just +slightly. To answer this question we recall that—since we are interested in deviations of the sample mean τ n around +⟨τ⟩—we bound the sample mean with the minimal and maximal first-passage time according to τ min +n +≤ τ n ≤ τ max +n +which is further used, in combination with bounds on the survival probability (S7), to derive corresponding upper +bounds (S11). Clearly, as n increases we expect this bound to become increasingly loose as by larger sample sizes we +increase the chances of sampling rare first-passage times, i.e., maximal and minimal first-passage time that strongly +deviate from the (sample) mean—this also explain why bounds (S10) and (S11) are only particularly tight for n = 1 +as here τ min +n += τ 1 = τ max +n +. In contrast, the Cram´er-Chernoff method requires a much more delicate mathematical +analysis involving bounds of the moment generating function. The Cram´er-Chernoff-type bound has the additional +advantage that it can be further used to universally bound deviation probabilities where no specific information about +the underlying system is required (see Eq. (11) in the Letter). Moreover, even the version of Cram´er-Chernoff bounds +U± +n (t; C) that require input of one system-dependent constant C still require less information about the dynamics since +extreme value upper bounds (S11) partly also require knowledge about the first-passage weight w1 and ⟨τ⟩ itself. +S3. +COMPLETE PROOF OF CONCENTRATION INEQUALITIES AND THEIR ASYMPTOTICS +In this section we provide various additional details on the upper bounds U± +n (t; C) (Eq. (7) of the Letter). In +particular, we prove the required bounds on the cumulant generating function, compute their corresponding Cram´er +transform, and give further information about the large-sample limit n → ∞, as well as the model-free version of the +bounds. +A. +Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩ +We begin with the right tail, i.e. upwards deviations such that τ ≥ ⟨τ⟩, and start by proving a bound for the +moment generating function of the deviation of the first-passage time τ from the mean ⟨τ⟩. Using the spectral +representation (S1) and the inequality x ≤ ex−1, ∀x ∈ R, we find +⟨eλ(τ−⟨τ⟩)⟩ = e−λ⟨τ⟩ � +k>0 +wk +1 − λ/µk +≤ exp +� +−λ⟨τ⟩ + +� +k>0 +wk +1 − λ/µk +− 1 +� +(S13) +for all λ < µk. Moreover, for |λ| < µ1 we may further expand the sum � +k>0 +wk +1−λ/µk = � +m≥0 λm � +k>0 wk/µm +k using +the geometric series. Recall that the moments are given by ⟨τ m⟩ = m! � +k>0 wk/µm +k , such that we obtain +⟨eλ(τ−⟨τ⟩)⟩ ≤ exp +� +�� +m≥2 +λm � +k>0 +wk +µm +k +� +� = exp +� +λ2 ⟨τ 2⟩ +2 ++ +� +m>2 +λm � +k>0 +wk +µm +k +� +. +(S14) + +6 +FIG. S1. Comparison between Cram´er-Chernoff-type upper bounds U± +n (t; C) and extreme value upper bounds for a spatially +confined Brownian search process in dimensions (a,e,i) d = 1 and (b,f,j) d = 3, and discrete-state Markov jump processes for +(c,d,k) the inferred model of calmodulin and (d,h,l) a 8-state toy protein. (a-d) Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that +the sample mean τ n inferred from n ≥ 1 realizations deviations from ⟨τ⟩ by more than t in either direction. Right tail areas +are shown for t > 0 and left for t < 0, respectively. Cram´er-Chernoff upper bounds U± +n (t; C) as black and extreme value upper +bounds as dashed lines, respectively. Corresponding lower bounds L± +n (t) are depicted as red lines and symbols denoted scaled +empirical deviation probabilities obtained from the statistics of τ n for different n. (e-h) Quality factor Q as a function of n for +different fixed relative deviations µ1t (see star symbols (a-d)). (i-l) Sample size n∗ (blue) for which both upper bounds are +equal, i.e., Q = 1, as a function of re-scaled deviations. +Since µ1 ≤ µk>1 and all first-passage weights wk are positive (due to equilibrium initial conditions) we find +⟨eλ(τ−⟨τ⟩)⟩ ≤ exp +� +λ2 ⟨τ 2⟩ +2 ++ +� +m>2 +λm � +k>0 +wk +µm +k +� +≤ exp +� +λ2 ⟨τ 2⟩ +2 ++ +� +m>2 +λm +µm−2 +1 +� +k>0 +wk +µ2 +k +� +≤ exp +� +λ2 ⟨τ 2⟩ +2 +� +1 + +λ +µ1 − λ +�� += exp +� +λ2 +⟨τ 2⟩/2 +(1 − λ/µ1) +� +. +Introducing ψδτ(λ) ≡ ln⟨eλδτ⟩, with δτ = τ − ⟨τ⟩ for the right tail, we immediately identify the upper bound +ψδτ(λ) ≤ λ2 +2 +⟨τ 2⟩ +1 − λ/µ1 += +˜λ2 +2 +C +1 − ˜λ ≡ φδτ(˜λ; C) +τ ≥ ⟨τ⟩, +(S15) +which concludes the derivation of the upper expression in Eq. (6) of the Letter. Note that we further have introduced +the dimensionless quantities ˜t ≡ µ1t, C = µ2 +1⟨τ 2⟩, and ˜λ = λ/µ1 in the last step. In the case of general initial conditions +p0(x0) ̸= peq(x0) we must simply replace µ2 +1⟨τ 2⟩ → C from Lemma 2. +Next, we find the optimizing value of ˜λ, i.e., we compute the Cram´er transform of Eq. (S15) defined as +φ∗ +δτ(˜t; C) ≡ +sup +˜λ∈[0,1) +[˜λ˜t − φδτ(˜λ; C)] = +sup +˜λ∈[0,1) +� +˜λ˜t − +˜λ2 +2 +C +1 − ˜λ +� +. +(S16) + +7 +φδτ(˜λ; C) is differentiable, non-negative, convex, and increasing on +˜ +lambda ∈ [0, 1), which implies that Eq. (S16) can be +obtained by differentiation of ˜λ˜t − φδτ(˜λ; C) with respect to ˜λ, hence φ∗ +δτ(˜t; C) = ˜λ†˜t − φδτ(˜λ†; C) where the optimum ˜λ† +solves φ′ +δτ(˜λ†; C) = t. Accordingly, we find the supremum to be attained at ˜λ†(˜t) = 1 − 1/ +� +1 + 2˜t/C. For convenience +we further introduce the auxiliary function h+(u) ≡ 1 + u − √1 + 2u such that we finally arrive at +φ∗ +δτ(˜t; C) = Ch+(˜t/C) = Ch+(µ1t/C), +0 ≤ t ≤ ⟨τ⟩. +(S17) +By using Chernoff’s inequality we subsequently obtain the upper bound P(δτ n ≥ t) ≤ e−nφ∗ +δτ (t;C) ≡ U+ +n (t; C) for +0 ≤ t ≤ ∞ which completes the proof of Theorem 1 and thus the first announced inequality (7) in the Letter. +0 +0.005 +0.010 (a) +˜t = 0.1 +˜λ˜t +φ∗ +δτ (˜t; 2) +φ∗ +δτ (˜t; C) +φδτ (˜λ; C) +φδτ (˜λ; 2) +0 +0.005 +0.010 (b) +0 +0.005 +0.010 +(c) +0 +0.01 +0.02 (d) +0 +0.05 +0.10 +˜λ +0 +0.002 +0.004 (e) +˜λ†(˜t; C) +˜λ†(˜t; 2) +φ∗ +δτ (˜t; C) +φ∗ +δτ (˜t; 2) +˜λ˜t − φδτ (˜λ; C) +˜λ˜t − φδτ (˜λ; 2) +0 +0.05 +0.10 +˜λ +0 +0.001 +0.002 +0.003 (f) +0 +0.05 +0.10 +˜λ +0 +0.002 +0.004 (g) +0 +0.1 +0.2 +˜λ +0 +0.004 +0.008 (h) +FIG. S2. Illustration of the Cram´er-Chernoff bounding method for the right tail with ˜t = 0.1 and parameters for spatially +confined Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for +the model of calmodulin (c,d) and a 8-state toy protein (d,h). Top row depicts bounds of the cumulant generating function +φδτ(˜λ; C) (black) and φδτ(˜λ; 2) (yellow) as a function of ˜λ, respectively. Bottom row shows the differences ˜λ˜t − φδτ(˜λ; C) (red) +and ˜λ˜t − φδτ(˜λ; 2) (green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue). The corresponding suprema are +obtained at ˜λ†(˜t; C) and ˜λ†(˜t; 2) (dotted lines) and define the Cram´er transforms φ∗ +δτ(˜t; C) and φ∗ +δτ(˜t; 2) (compare top row). For +all considered models we demonstrate φδτ(˜λ; C) ≤ φδτ(˜λ; 2) and φ∗ +δτ(˜t; C) ≥ φ∗ +δτ(˜t; 2) as derived in the maintext. Note for the +panels (b,f) we have φδτ(˜λ; C) ⪅ φδτ(˜λ; 2) and φ∗ +δτ(˜t; C) ⪆ φ∗ +δτ(˜t; 2) since C = 1.99 ≈ 2. +B. +Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ +Next we turn to the left tail, τ < ⟨τ⟩, where the corresponding moment generating function analogously reads +⟨eλ(⟨τ⟩−τ)⟩ = eλ⟨τ⟩ � +k>0 +wk +1 + λ/µk +≤ exp +� +λ⟨τ⟩ + +� +k>0 +wk +1 + λ/µk +− 1 +� +for λ < µk. Using equivalent arguments as for the right tail above we may further write +eλ(⟨τ⟩−τ)⟩ ≤ exp +� +�� +m≥2 +(−λ)m � +k>0 +wk +µm +k +� +� += exp +� +λ2 ⟨τ 2⟩ +2 ++ +� +m>2 +(−λ)m � +k>0 +wk +µm +k +� +≤ exp +� +λ2 ⟨τ 2⟩ +2 ++ +� +m>0 +λ2m +µ2m−2 +1 +� +k>0 +wk +µ2 +k +� += exp +� +λ2 +⟨τ 2⟩/2 +1 − (λ/µ1)2 +� +. +(S18) +Recall the definition of the cumulant generating function, ψδτ(λ) ≡ ln⟨eλδτ⟩, such that Eq. (S18) directly yields +ψδτ(λ) ≤ λ2 +2 +⟨τ 2⟩ +1 − (λ/µ1)2 = +˜λ2 +2 +C +1 − ˜λ2 ≡ φδτ(˜λ; C) +(S19) + +8 +which completes the derivation of the lower expression in Eq. (6) of the Letter. Note that for the left tail we +have δτ = ⟨τ⟩ − τ and we again let ˜t ≡ µ1t, ˜λ ≡ λ/µ1, and C ≡ µ2 +1⟨τ 2⟩. In the case of general initial conditions +p0(x0) ̸= peq(x0) we must simply replace µ2 +1⟨τ 2⟩ → C from Lemma 2. +Analogous to the right tail we next compute the Cram´er transform of Eq. (S19), i.e., +φ∗ +δτ(˜t; C) ≡ +sup +˜λ∈[0,1) +[˜λ˜t − φδτ(˜λ; C)] = +sup +˜λ∈[0,1) +� +˜λ˜t − +˜λ2 +2 +C +1 − ˜λ2 +� +, +(S20) +where we find the optimal value ˜λ†(˜t; C) to be determined by the transcendental quartic, ˜λ†(˜t) : (1− ˜λ2)2 −C˜t˜λ = 0 with +C˜t ≡ C/˜t, which we solve according to the method of Descartes. First, we re-arrange the quartic as ˜λ4−2˜λ2−C˜t˜λ+1 = 0 +and make the factorization ansatz +(˜λ2 − y˜t˜λ2 + w˜t)(˜λ2 + y˜t˜λ2 + z˜t) = 0 +w˜t + z˜t − y2 +˜t = −2 +y˜t(w˜t − z˜t) = −C˜t +z˜tw˜t = 1. +(S21) +The system of equations (S21) is solved by +w˜t(y˜t) = (y2 +˜t − 2 − C˜t/y˜t)/2, +z˜t(y˜t) = (y2 +˜t − 2 + C˜t/y˜t)/2, +(S22) +where y2 +˜t ≡ Y˜t is the solution of the cubic Y 3 +˜t − 4Y 2 +˜t − C2 +˜t = 0. Moreover, since the discriminant D is strictly +negative, i.e. D = −28C2 +˜t − 33C4 +˜t < 0, the qubic has only one real root. +The corresponding depressed qubic +reads ˜t3 − 24/3˜t − (27/33 + C2 +˜t ) = 0 with ˜tY˜t − 4/3. +Let p = −24/3 < 0 and q = −(27/33 + C2 +˜t ) < 0 then +22p3 + 33q2 = −212/33 + 33(27/33 + C2 +˜t )2 > 0 for any ˜t ≥ 0. We can express the unique real root as +y2 +˜t = 4 +3 +� +1 + 2 cosh +�1 +3arcosh +� +1 + 33C2 +˜t +27 +��� +(S23) +and y = ± +� +y2 with y2 from Eq. (S23) can now be plugged into Eqs. (S22) to obtain w˜t(y) and z˜t(y) that are required +to solve the pair of quadratic equations (S21). The four roots of the transcendental quartic are hence given by +˜λ1(˜t) = y˜t +2 +� +1 + +� +1 − 4w˜t(y˜t)/y2 +˜t +� +, +˜λ2(˜t) = y˜t +2 +� +1 − +� +1 − 4w˜t(y˜t)/y2 +˜t +� +, +˜λ3(˜t) = −y˜t +2 +� +1 − +� +1 − 4z˜t(y˜t)/y2 +˜t +� +, +˜λ4(˜t) = −y˜t +2 +� +1 + +� +1 − 4z˜t(y˜t)/y2 +˜t +� +. +(S24) +Moreover, we find w˜t(y˜t)/y2 +˜t = (1 − 2/y2 +˜t − C˜t/y3 +˜t )/2 and z˜t(y˜t)/y2 +˜t = (1 − 2/y2 +˜t + C˜t/y3 +˜t )/2. Since y˜t > 0 while +˜λ ∈ [0, 1), ˜λ2, ˜λ3 in Eq. (S24) are excluded automatically (note also that the square root in ˜λ2, ˜λ3 becomes complex for +˜t → ∞). We also have lim˜t→∞ y2 +˜t = 4 and lim˜t→∞ w˜t(y˜t) = 1 such that lim˜t→∞ ˜λ1 = ˜λ2 = 1. Conversely, we find that +lim˜t→0 ˜t2/3y2 +˜t = C2/3 = lim˜t→0 ˜t2/3C˜t/y˜t such that lim˜t→0 w˜t(y˜t) = −1 while lim˜t→0 w˜t(y˜t)y2 +˜t = −C2/3 = 0. Therefore, +lim˜t→0 ˜λ1 = y˜t → ∞ whereas lim˜t→0 ˜λ2(˜t) = y˜t × 0/2 ↘ 0. We recall that ˜λ ∈ [0, 1) which therefore excludes ˜λ1(˜t) +and identifies ˜λ†(˜t) = ˜λ2(˜t) as the supremum. Finally, we introduce the auxiliary functions +g(u) ≡ +2 +√ +3 +� +1 + 2 cosh +�1 +3arcosh +� +1 + +33 +27u2 +���1/2 +and +Λ(u) ≡ 1 +2 +� +g(u) − +� +4 + 2/g(u)u − g(u)2 +� +, +(S25) +as well as +h−(u) ≡ Λ(u)u − 1 +2 +Λ(u)2 +1 − Λ(u)2 +(S26) +which allows us to obtain and write the Cram´er transform as +φ∗ +δτ(˜t; C) = Ch−(˜t/C) = Ch−(µ1t/C), +0 ≤ t ≤ ⟨τ⟩. +(S27) +In the last step we use Chernoff’s inequality to obtain the bound P(δτ n ≥ t) ≤ e−nφ∗ +δτ (t;C) ≡ U− +n (t; C) for 0 ≤ t ≤ ⟨τ⟩ +which completes the proof of Theorem 2 and hence the derivation of the lower expression in Eq. (7) of the Letter. + +9 +0 +0.005 +0.010 (a) +˜t = 0.1 +˜λ˜t +φ∗ +δτ (˜t; 2) +φ∗ +δτ (˜t; C) +φδτ (˜λ; C) +φδτ (˜λ; 2) +0 +0.005 +0.010 (b) +0 +0.005 +0.010 +(c) +0 +0.01 +0.02 (d) +0 +0.05 +0.10 +˜λ +0 +0.002 +0.004 (e) +˜λ†(˜t; C) +˜λ†(˜t; 2) +φ∗ +δτ (˜t; C) +φ∗ +δτ (˜t; 2) +˜λ˜t − φδτ (˜λ; C) +˜λ˜t − φδτ (˜λ; 2) +0 +0.05 +0.10 +˜λ +0 +0.001 +0.002 +0.003 (f) +0 +0.05 +0.10 +˜λ +0 +0.002 +0.004 (g) +0 +0.1 +0.2 +˜λ +0 +0.004 +0.008 (h) +FIG. S3. Illustration of the Cram´er-Chernoff bounding method for the left tail with ˜t = 0.1 and parameters for spatially confined +Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for the model of +calmodulin (c,d) and a 8-state toy protein (d,h). Top row depicts bounds of the cumulant generating function φδτ(˜λ; C) (black) +and φδτ(˜λ; 2) (yellow) as a function of ˜λ, respectively. Bottom row shows the differences ˜λ˜t − φδτ(˜λ; C) (red) and ˜λ˜t − φδτ(˜λ; 2) +(green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue). The corresponding suprema are obtained at ˜λ†(˜t; C) +and ˜λ†(˜t; 2) (dotted lines) and define the Cram´er transforms φ∗ +δτ(˜t; C) and φ∗ +δτ(˜t; 2) (compare top row). For all considered models +we demonstrate φδτ(˜λ; C) ≤ φδτ(˜λ; 2) and φ∗ +δτ(˜t; C) ≥ φ∗ +δτ(˜t; 2) as derived in the maintext. Note for the panels (b,f) we have +φδτ(˜λ; C) ⪅ φδτ(˜λ; 2) and φ∗ +δτ(˜t; C) ⪆ φ∗ +δτ(˜t; 2) since C = 1.99 ≈ 2. +C. +Behavior of upper bounds U± +n (t) for large sample sizes +Here, we present some further remarks about the limit of large sample sizes. Asymptotically as n → ∞, U± +n (t) is +substantial only for ˜t/C ≪ 1. For the right tail bound h+(u) we immediately find that for u ≪ 1 we can Taylor expand +√1 + 2u = 1 + u − u2/2 + O(u3). Consequently we directly obtain h+(u) = −u2/2 + O(u3), i.e., the upper tail is +sub-Gaussian for small deviations and will converge to a Gaussian as n → ∞. For the left tail we furthermore have +arcosh(1 + x) = ln(1 + x + +� +x(x + 2)) and thus limx→∞ arcosh(1 + x) = ln(2x) − 1/(2x)2. As a result it follows that +1 +3 limu→0 arcosh(1 + 33/27u2) ≃ 1 +3 ln(33/26u2) = ln(3/4u2/3) − u4212/37 and thus +lim +u→0 g(u) ≃ +2 +√ +3{1 + 2 cosh ln(3/4u2/3)}1/2 = +2 +√ +3[1 + 3/4u2/3]1/2 +(S28) += u−1/3[1 + 4u2/3/3]1/2 = u−1/3[1 + 2u2/3/3 + O(u4/3)] = u−1/3 + 2 +3u1/3 + O(u). +(S29) +A lengthy but straightforward calculation subsequently reveals that limu→0 Λ(u) = u − O(u3) such that +lim +u→0 +Λ(u)2 +1 − Λ(u)2 ≃ +u2 +1 − u2 = u2 + O(u4). +(S30) +We therefore have that limu→0 h−(u) = u2/2 − O(u4), i.e., both tails are sub-Gaussian for ˜t/C ≪ 1 with C ≡ µ2 +1⟨τ 2⟩. +D. +Proof of bounds on C and model-free concentration inequalities +Notably, system details only enter the Cram´er transforms (S17) and (S27) (and consequently upper bounds on the +deviation probability due to Chernoff’s inequality) in the form of a system-specific constant C ≡ µ2 +1⟨τ 2⟩. Note that here +we only allow for equilibrium initial conditions. Recalling that the moments of the first-passage time τ are expressed +as ⟨τ m⟩ = m! � +k>0 wk/µm +k allows us to write +0 ≤ 2w1 +µ2 +1 +≤ ⟨τ 2⟩ = 2 +� +k>0 +wk +µ2 +k +≤ 2 +� +k>0 +wk +µ2 +1 += 2 +µ2 +1 +(S31) + +10 +for equilibrium initial conditions where we have used that wk are non-negative, normalized, and µ1 ≤ µk>1. Conse- +quently, by Eq. (S31), we immediately find that the system-constant itself is bounded 0 ≤ 2w1 ≤ C ≤ 2. Note that +analogous considerations can be used to more generally obtain 0 ≤ m!w1 ≤ µm +1 ⟨τ m⟩ ≤ m! for the m-th moment. +The fact that C ∈ (0, 2] can now be further leveraged to arrive at the model-free bounds (Eq. (11) in the Letter) +which require no information about the underlying system. Recall the upper bounds of the cumulant generating +function φδτ(˜λ; C) and their corresponding Cram´er transform φ∗ +δτ(˜t; C), i.e., +φδτ(˜λ; C) = +� +� +� +� +� +� +� +˜λ2 +2 +C +1 − ˜λ +τ ≥ ⟨τ⟩ +˜λ2 +2 +C +1 − ˜λ2 +τ < ⟨τ⟩, +and +φ∗ +δτ(˜t; C) = +� +Ch+(˜t/C) +τ ≥ ⟨τ⟩ +Ch−(˜t/C) +τ < ⟨τ⟩. +(S32) +Since φδτ(˜λ; C) is monotonically increasing in C it follows that φδτ(˜λ; C) ≤ φδτ(˜λ; 2), ∀˜λ ∈ [0, 1) (see Figs S2 and S3 top +row). By definition of φ∗ +δτ(˜t; C) ≡ sup˜λ∈[0,1)(˜λ˜t−φδτ(˜λ; C)) this bound in turn implies that φ∗ +δτ(˜t; C) ≥ φ∗ +δτ(˜t; 2) (compare +Figs. S2 and S3 bottom row). With Chernoff’s inequality we moreover arrive at P(δτ n ≥ t) ≤ e−nφ∗ +δτ (t;C) ≤ e−nφ∗ +δτ (t;2) +and hence U± +n (t; C) ≤ U± +n (t; 2) which completes the derivation of Eq. (11) in the Letter. +S4. +MODEL SYSTEMS AND DETAILS ON NUMERICAL METHODS +In the Letter we exemplify our results by considering a Brownian molecular search process in dimensions d = 1 and +d = 3, as well as discrete-state Markov-jump models of protein folding for a 8-state toy protein and the experimentally +inferred model of calmodulin (compare Fig. 1b-d). In this section we present further details on the model systems and +their numerical treatment. +A. +Continuous-time discrete-state Markov jump process +As illustrative discrete-state continuous-time Markov-jump models of protein folding we consider a simple 8-state +toy protein [2, 6] and further use the experimentally inferred folding network of the cellular calcium sensor protein +calmodulin [7]. Since we consider equilibrium initial conditions, proteins start from an initial state drawn from the +equilibrium density ˜peq(x)—note that the tilde denotes that the absorbing target is excluded—from which they search +the native state a (here a = (1, 1, 1) for the 8-state model and a = F1234 for calmodulin; cf. Fig. 1b-d). Arrows in the +networks denote possible transitions, e.g. a transition from state i to state j that occurs with the corresponding rate +Lji. We consider reversible dynamics, i.e., the resulting transition matrix ˆL of the relaxation process satisfies detailed +balance peq,j/peq,i = Lji/Lij = exp(Fi − Fj) and transitions rates are connected to the free energy of the states Fi [8]. +We recall that the first-passage time density ℘a(t) can be evaluated by using the spectral representation (S1). To this +end we set up the modified transition matrix, adopting in this section the Dirac bra-ket notation, ˆLa = ˆL−|a⟩⟨a| where +|a⟩ ≡ (0, . . . , 0, 1, 0, . . .)⊺ defines a vector with all entries zero expect at the a-th position of the absorbing state where +it equals one. This effectively removes all transitions that correspond to jumps leaving the absorbing state a. Next, we +carry out an eigendecomposition of ˆLa and determine the eigenvalues µk, right eigenvectors |φR⟩, and left eigenvectors +⟨φL|. We subsequently use obtained eigenmodes to compute the first-passage weights wk(x0) = −⟨a|φR +k ⟩⟨φL +k|x0⟩ +(see [1, 2]), and recall that µk and wk determine the moments according to ⟨τ m⟩ = m! � +k>0 wk/µm +k . Corresponding +relevant parameters of the Markov jump models are listed in Tab. I. Next we give further details on how corresponding +transition rates are constructed. +TABLE I. Parameters for the Markov jump models for the 8-state toy protein and the inferred model of calmodulin. Listed are +values for the first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩. +Model +µ1 +w1 +µ2 +w2 +µ3 +w3 +µ4 +w4 +µ5 +w5 +µ6 +w6 +µ7 +w7 +⟨τ⟩ +⟨τ 2⟩ +Toy protein 0.976 0.337 6.148 0.009 1.551 +0.583 +4.203 +0.001 +4.396 +0.0001 6.233 0.060 12.834 0.010 0.385 0.713 +Calmodulin 0.469 0.651 3.763 0.349 19.097 9.98E-5 143.749 2.42E-9 1581.629 1.52E-6 +– +– +– +– +1.479 5.958 + +11 +1. +Transitions rates of the 8-state toy protein model +For the 8-state toy protein model we randomly generate a free energy level Fi for each state i ∈ {1, 2, 3, 4, 5, 6, 7, 8} +with Fi uniformly distributed within the interval 0 ≤ Fi ≤ 10. Transition rates that satisfy detailed balance are then +obtained using the ansatz +ki→j ≡ Lji = exp(∆Fi/2) +and +kj→i ≡ Lij = exp(−∆Fi/2), +(S33) +where ∆Fi ≡ Fi − Fj and thus ln(Lji/Lij) = ∆Fi = Fi − Fj. Obtained individual transition rates are listed in Tab. II. +TABLE II. Transition rates for the 8-state toy protein model obtained via the ansatz described in the main text. +transition rate ki→j transition rate ki→j transition rate ki→j transition rate ki→j +1 → 2 +1.878 +2 → 5 +2.648 +3 → 7 +4.549 +5 → 8 +0.106 +2 → 1 +5.327 +5 → 2 +3.421 +7 → 3 +1.00994 +8 → 5 +124.477 +1 → 3 +0.00463 +2 → 6 +0.527 +4 → 6 +0.358 +6 → 8 +0.712 +3 → 1 +0.507 +6 → 2 +36.0577 +6 → 4 +36.457 +8 → 6 +15.794 +1 → 4 +0.326 +3 → 5 +1.109 +4 → 7 +0.523 +7 → 8 +0.322 +4 → 1 +0.623 +5 → 3 +0.0371 +7 → 4 +6.670 +8 → 7 +56.998 +2. +Transitions rates of the calmodulin protein model +In the experimental setup a constant external force f, a so-called pretension, is applied to the calmodulin protein +via optical tweezers. Folding and unfolding processes are observed at different pretensions ranging from 6pN to 13 pN +and corresponding force-dependent transition rates ki→j(f) = Lji(f) between two conformational states i and j are +measured. Note that i, j ∈ {Unfold, F12, F123, F23, F34, F1234} and we further map states according to Unfold ↔ 1, +F12 ↔ 2, F123 ↔ 3, F23 ↔ 4, F34 ↔ 5, and F1234 ↔ 6 for convenience. For our purposes we choose, without loss of +generality, a pretension of f = 9 pN and obtain the corresponding measured transitions rates from Fig. S8 in the +Supplementary Material of [7]. Clearly, experimental transitions rates are accompanied with measurement uncertainties +which is reflected in slight “deviations” from a mathematically precise definition of detailed balance. To mitigate this +issue, and to ensure that transition rates precisely obey detailed balance ki→jpeq,i = kj→ipeq,j, we further have to +slightly adjust the rates. +First, we compute the invariant density peq from the experimental rates and obtain a corresponding free energy +level Fi = − ln(peq,i). Next, we use the ansatz (S33), i.e., Lji = Ai exp(∆Fi/2) and Lij = Ai exp(−∆Fi/2) where we +introduce a constant Ai. Finally, Ai’s are chosen such that resulting transition rates fall within experimental error +bars in Ref. [7]. Obtained transition rates are listed in Table III. +TABLE III. Transition rates of the Markov jump model for the calmodulin protein. Rates are extracted from the Supplemental +Material of Ref. [7] and modified such that they obey detailed balance precisely according to the maintext. +transition rate ki→j transition rate ki→j transition rate ki→j +1 → 2 +5.997 +1 → 4 +13.439 +1 → 5 +15.330 +2 → 1 +0.774 +4 → 1 +127.968 +5 → 1 +0.121 +5 → 6 +3.749 +2 → 3 +1514.820 +2 → 6 +13.441 +6 → 5 +13.326 +3 → 2 +53.0661 +6 → 2 +2.922 +B. +Spatially confined Brownian molecular search process +We also test our theory for Markov processes on a continuous state-space. More precisely, we consider the spatially +confined diffusive search of a Brownian particle in a d-dimensional unit sphere with a reflecting boundary at R = 1 and +a perfectly absorbing spherical target of radius 0 < a < 1, here a = 0.1, in the center (compare Fig. 1b). The closest +distance of the particle to the surface of the absorbing sphere at time t is a confined Bessel process (see e.g. [2, 9, 10]) +which time evolution obeys the Itˆo equation +dxt = (d − 1)x−1 +t dt + +√ +2dWt, +(S34) + +12 +where dWt is the increment of a Wiener process (i.e. Gaussian white noise) with ⟨dWt⟩ = 0 and ⟨dWtdWt′⟩ = δ(t−t′)dt, +and we have set, without loss of generality, D = 1. The general case with any 0 < D < ∞ and a sphere of radius R is +covered by expressing time in units of R2/D. +For d = 1 Eq. (S34) reduces to a 1 dimensional Brownian motion which has the equilibrium first-passage weights +weq +k = 2 +π2 +1 − sin [(k − 1)π] +(k − 1/2)2 +(S35) +and matching first-passage eigenvalues are obtained as µk = π2(k − 1/2)2. Moreover, for d = 3 the first-passage time +probability density of the Bessel process can be evaluated exactly and has the equilibrium weights +weq +k = 2 +µk +3a2 +1 − a3 +tan[(1 − a)√µk] + +1 +√µk +(1 − a) tan[(1 − a)√µk] − +a +√µk +, +(S36) +with the first-passage eigenvalues µk being the solutions of the transcendental equation √µk = tan([1 − a]√µk) that +can be solved analytically using Newton’s series [2]. Relevant parameters for the spatially confined Brownian search +process with a = 0.1 are listed in Tab. IV. +TABLE IV. Parameters for the spatially confined Brownian molecular search process in dimensions 1 and 3. Listed are values +for the first 5 first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩, respectively a. +Model +µ1 +w1 +µ2 +w2 +µ3 +w3 +µ4 +w4 +µ5 +w5 +⟨τ⟩ +⟨τ 2⟩ +1D Brownian motion 2.467 0.811 22.207 0.0901 61.685 +0.0324 +120.903 +0.0165 +199.859 0.01001 0.333 0.267 +3D Bessel process +0.363 0.994 25.174 0.00277 73.926 9.163E-4 147.037 4.573E-4 244.516 2.742E-4 2.739 1.509 +a For the numerical evaluation of ⟨τ⟩ and ⟨τ 2⟩ as listed we truncate the sum after M = 1000 terms. +C. +Statistics of first-passage times ⟨τ⟩ and the sample mean τ n +Here we provide some further details on the sampling method used to obtain the statistics of (i) the first-passage +time τ and (ii) the sample-mean τ n ≡ � +i τi/n at some fixed value n for our considered models. +We recall that after determining the first-passage eigenvalues µk and first-passage weights wk, the first-passage time +density ℘a(t) (S1) and survival probability Sa(t) (S2) are fully characterized. To now sample the random variable τ, +i.e. individual realizations of the first-passage process, we employ the so-called inversion sampling method [11]. This +method allows us to generate independent samples of τ from ℘a(t) given its cumulative distribution function (CDF) +which is directly related to the survival probability according to 1 − Sa(t). Note that for discrete-state dynamics the +number of states M is finite, i.e. k = 1, . . . , M and therefore Eq. (S2) (and hence the CDF) is a finite sum. In contrast, +for continuous-state dynamics we formally have M = ∞, meaning that sums are here not finite. For the following +numerical evaluation of the spatially confined Brownian search process we therefore truncate the sum after M = 1000 +terms. The first-passage time densities ℘a(t) obtained via inversion sampling (symbols) for all considered models are +shown in Fig. S4a-d and corroborated by the corresponding analytical result (S1) (dashed black line). +For Fig. 2a-d in the Letter empirical probabilities that τ n − ⟨τ⟩ lies within a desired range of ± 10% of the +longest first-passage time scale µ−1 +1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.1, 0.1]), are computed using statistics of the sample +mean τ n by fixing n, i.e., the number of individual realizations the average is taken over. In particular, we have +n ∈ {1, 2, 3, 5, 10, 20, 30, 40, 50, 75, 100, 150, 200, 300, 400, 500}. Subsequently, for each individual fixed n the sample +mean τ n itself is sampled a total of N = 106 times. That is, we first draw n first-passage times τ, compute τ n by +averaging over the drawn n realizations, and finally repeat this step N = 106 times to obtain statistics of τ n for all n +values introduced above. Probability densities of the sample mean are shown in Fig. S4e-h for n ∈ {3, 5, 10, 20} and all +model systems. Corresponding true mean first-passage times ⟨τ⟩ are highlighted in grey. +In Fig. 2e-h of the Letter the probabilities to deviate more than t in either direction, P(±[τ n − ⟨τ⟩] ≥ t), are +computed from analogous statistics of the sample mean τ n. Since we also consider empirical probabilities for rare +events with large deviations (i.e. large µ1t) we however require substantially more statistics of τ n. To this end we now +have N = 107 for n ∈ {1, 3} and N = 1011 for n ∈ {5, 10, 20}. In addition it should be further noted that we re-scale +obtained probabilities according to P1/n. To compute an empirical deviation probability where e.g. P1/20 = 0.1 one +would be thus required to sample rare events that occur with a probability of ≃ 10−20. + +13 +In Fig. 3a of the Letter each data point corresponds to the relative error µ1(τ n − ⟨τ⟩) (note that µ1 and ⟨τ⟩ are +different for each model) where the sample mean τ n is again obtained by first fixing n and then sampling n first-passage +times τ according to the inversion sampling method and subsequently taking the average. +10−9 +10−5 +10−1 +t +10−2 +101 +104 +℘a(t) +(a) +10−5 +10−1 +t +10−2 +100 +102 (b) +10−4 +10−2 +100 +t +10−2 +10−1 +100 +101 +(c) +10−5 +10−2 +t +100 +102 +(d) +0 +0.5 +1 +τ n +0 +1 +2 +3 +4 +5 +p(τ n) +⟨τ⟩ +(e) +n = 3 +n = 5 +n = 10 +n = 20 +0 +2 +4 +6 +τ n +0 +0.2 +0.4 +0.6 +0.8 (f) +0 +2 +4 +τ n +0 +0.5 +1 (g) +0 +0.5 +1 +τ n +0 +1 +2 +3 +4 +5 (h) +FIG. S4. Inversion sampling of first-passage statistics for a spatially confined Brownian search process in dimensions (a,e) d = 1 +and (b,f) d = 3, and discrete-state Markov jump processes for (c,d) the inferred model of calmodulin and (d,h) a 8-state toy +protein. (a-d) First-passage time density ℘a(t) obtained using inversion sampling (symbols) and analytical result as black dashed +lines. (e-h) Empirical probability density of the sample mean τ n for different n values. True mean first-passage times ⟨τ⟩ are +shown in grey. +S5. +UNCERTAINTY QUANTIFICATION WITH CONFIDENCE INTERVALS +In this section we extend the discussion and present some further details on the confidence intervals introduced in +the Letter. Our derived upper bounds U± +n (t) can be applied to construct non-asymptotic performance guarantees such +as confidence intervals. In particular, they can be employed to bound the probability that δτ n ≡ τ n − ⟨τ⟩ is found to +be in some interval [−t− +α−, t+ +α+], i.e., +P(δτ n ∈ [−t− +α−, t+ +α+]) = P(−t− +α− ≤ δτ n ≤ t+ +α+) += P(δτ n ≥ −t− +α− ∩ δτ n ≤ t+ +α+) +≥ 1 − P(δτ n ≤ −t− +α−) − P(δτ n ≥ t+ +α+) +≥ 1 − U− +n (t− +α−) +� +�� +� +≡α− +− U+ +n (t+ +α+) +� +�� +� +≡α+ +. +(S37) +In passing from the second to the third line we have applied Boole’s second inequality, and from the third to forth +line we use bounds (7) of the Letter. In the last line we additionally introduced acceptable right and left tail error +probabilities α±. The implicit interval [−t− +α−, t+ +α+] therefore defines a confidence interval at a confidence level of 1 − α +with α ≡ α+ + α−, and α+ + α− < 1. In general the choice of the confidence interval for a fixed probability 1 − α is +not unique. Some common options in the literature (see e.g. [12, 13]), all having the same confidence level, are listed +below. +• One common choice are so-called central intervals (blue lines in Fig. S5) which correspond to equal tail probabilities +α+ = α− = α/2 for the complementary intervals [−⟨τ⟩, −t− +α−] and [t+ +α+, ∞). Notably, we remark that central +confidence intervals do not generally imply that t+ +α+ and t− +α− are equidistant from another, i.e., t+ +α+ ̸= t− +α−. + +14 +• As an alternative one could likewise choose t+ +α+ = t− +α− ≡ ∆t/2, which subsequently leads to the symmetric +interval [−∆t/2, ∆t/2] with total length ∆t (see red lines in Fig. S5) . Analogously, a symmetric interval does +not necessarily imply that the corresponding tail probabilities are equal, i.e., in general α+ ̸= α−. +• Both considerations above lead to two-sided intervals. However, another possible choice includes the fully +asymmetric intervals [−⟨τ⟩, t+ +α+] and [−t− +α−, ∞), i.e., one-sided intervals with a corresponding confidence level +1 − α+ (for the upper limit t+ +α+) and 1 − α− (for the lower limit t− +α−), respectively, +P(±δτ n ≤ t± +α±) ≥ 1 − α±. +(S38) +0 +0.5 +1 +µ1t+ +α+ +0 +0.5 +1 +µ1t− +α− +n = 15 +(a) +0.5 +0.7 +0.9 +central int. +symm. int. +0 +0.5 +1 +µ1t+ +α+ +n = 20 +(b) +0.5 +0.7 +0.9 +0 +0.5 +1 +µ1t+ +α+ +n = 30 +(c) +0.5 +0.7 +0.9 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1 − α +FIG. S5. +Contour plot of different choices of possible two-sided confidence intervals [−µ1t− +α−, µ1t+ +α+] for a fixed confidence level +α and (a) n = 15, (b) n = 20, (c) n = 30. Contour lines for α ∈ {0.1, 0.3, 0.5} are depicted in white. Specific choices of central +and symmetric are shown in blue and red, respectively, and we let C = 1 for all panels. +Confidence intervals are practically useful as they answer questions such as e.g.: +How many realizations are required to achieve a desired accuracy with a specified probability? +Or: For a given number of realizations a desired accuracy is achieved with at least what probability? +In the case of symmetric confidence intervals t+ +α+ = t− +α− (see Fig. S5 red lines) the interval endpoints are implicitly +defined via the last line of Eq. (S37) which is easily solved using standard root-finding procedures like the bi-section +method [14]. The same holds true for other interval choices, however, when specifying the error probabilities α± +directly—as done for e.g. two-sided central intervals (α± = α/2) or one-sided intervals—it suffices to solve Eq. (S38) with +the respective α±. Hereby, the lower confidence limit t− +α− is again easily obtained using standard root-finding methods. +Notably, the upper confidence limit t+ +α+ can now be solved analytically. To show this we consider U+ +n (t+ +α+; C) = α+, +i.e., we identify the t+ +α+ that solves +0 = −nCh+(µ1t+ +α+/C) − ln(α+). +(S39) +The roots are identified as +t1 = −ln (α+) +µ1n +− +√ +2 +� +− ln (α+) +µ1 +� +n/C +and +t2 = −ln (α+) +µ1n ++ +√ +2 +� +− ln (α+) +µ1 +� +n/C +, +(S40) +and we identify t+ +α+ = t2 as the relevant solution. Having obtained an explicit expression for t+ +α+ further allows us to +re-insert it into the left-hand side of Eq. (S38), i.e., we find that with a probability of at least 1 − α+ +δτ n ≤ −ln (α+) +µ1n ++ +√ +2 +� +− ln (α+) +µ1 +� +n/C +. +(S41) +The required number of realizations n∗ to ensure with a probability of at least 1 − α that δτ n is found within some +interval [−t− +α−, t+ +α+] (e.g. symmetric interval in Fig. 3b) is analogous identified according to Eq. (14) in the Letter +Un∗(t+ +α+; C) + Un∗(t− +α−; C) = α, +(S42) + +15 +which once again is readily solved via e.g. the bisection method. Moreover, in the case of one-sided intervals one +immediately finds the corresponding analytical expression +n∗ ≥ − +ln(α±) +Ch±(µ1t/C), +(S43) +where n∗ denotes the required number to ensure that ±δτ n ≤ t with at least 1 − α±. +∗ agodec@mpinat.mpg.de +[1] D. Hartich and A. Godec, New J. Phys. 20, 112002 (2018). +[2] D. Hartich and A. Godec, J. Phys. A: Math. Theor. 52, 244001 (2019). +[3] A. J. F. Siegert, Phys. Rev. 81, 617 (1951). +[4] G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, 2012). +[5] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed., Springer Series +in Synergetics, Vol. 13 (Springer-Verlag, Berlin, 2004). +[6] G. R. Bowman, V. S. Pande, and F. No´e, An Introduction to Markov State Models and their Application to Long Timescale +Molecular Simulation, Vol. 797 (Springer Science & Business Media, 2013). +[7] J. Stigler, F. Ziegler, A. Gieseke, J. C. M. Gebhardt, and M. Rief, Science 334, 512 (2011). +[8] U. Seifert, Annu. Rev. Condens. Matter Phys. 10, 171 (2019). +[9] J. W. Pitman, Adv. Appl. Probab. 7, 511 (1975). +[10] E. Barkai, E. Aghion, and D. Kessler, Phys. Rev. X 4, 021036 (2014). +[11] L. Devroye, Non-Uniform Random Variate Generation (Springer New York, 1986). +[12] G. Cowan, Statistical Data Analysis (Oxford University Press, 1998). +[13] L. Lista, Statistical Methods for Data Analysis in Particle Physics (Springer International Publishing, 2017). +[14] R. L. Burden, J. D. Faires, and A. M. Burden, Numerical Analysis (Cengage Learning, 2015). + diff --git a/btFAT4oBgHgl3EQf5R5J/content/tmp_files/load_file.txt b/btFAT4oBgHgl3EQf5R5J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..01807664e3ee7d6fd98a995c0d0c6caf8995da8c --- /dev/null +++ b/btFAT4oBgHgl3EQf5R5J/content/tmp_files/load_file.txt @@ -0,0 +1,2463 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf,len=2462 +page_content='Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime Rick Bebon and Aljaˇz Godec∗ Mathematical bioPhysics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Max Planck Institute for Multidisciplinary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 37077 G¨ottingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Germany We derive general bounds on the probability that the empirical first-passage time τ n ≡ �n i=1 τi/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The first-passage time τ denotes the time a random pro- cess reaches a threshold a, typically referred to as the “tar- get”, for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' First-passage times [1–4] quantify the kinetics of chemical reactions [5–10], cell signaling and gene regulation in the low-copy [11–20] and “fastest en- counter” limits [21–29], intracellular transport [30], RNA biosynthesis [31], protein accumulation [32, 33] and DNA- binding [34], emergence of drug resistance [35], virus up- take [36], spreading of diseases [37, 38], and the foraging behavior of bacteria and animals [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' First-passage the- ory was further applied to nanocluster formation [40], cell adhesion [41–43], gating of ion channels [44], and diffusion through interfaces [45] and across phase boundaries [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In more abstract settings, first-passage times charac- terize barrier-crossing in energy landscapes [6, 23, 47– 54], persistence properties [55–61], and the statistics of stochastic currents [62, 63], thermodynamic entropy pro- duction [64–67], and dynamical activity [68, 69] in non- equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' First-passage ideas are intimately tied to the statistics of extremes [70–73], and were ex- tended to quantum systems [74, 75], additive functionals of stochastic paths [76–81], intermittent targets [82–85], active particles [86, 87], non-Markovian dynamics [88–91], and processes under resetting [92–101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Whereas theoretical studies focus on predicting first- passage statistics, practical applications typically aim at inferring kinetic rates—inverse mean first-passage times—from experimental [52, 102–106] or simulation data [51, 107–113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The inference of empirical first-passage times τ n ≡ �n i=1 τi/n from data is, however, challenging because usually only a small number of realizations n (typically 1-10 [113–118], sometimes up to 100 [119]) is available, which gives rise to large uncertainties and non- Gaussian errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Insufficient sampling is especially detri- mental in the case of broadly distributed [51, 120, 121] and high-dimensional data [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, first-passage times are generically not exponentially distributed [8, 9, 17, 19, 23, 24, 122–127], which further complicates quan- tification of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A systematic understanding of statistical deviations of the empirical from the true mean first-passage time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1a), especially in the small- sample n ≲ 100 regime, remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Computer simulations in particular often suffer from FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Deviations of empirical first-passage times from the true mean and model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (a) Schematic probability den- sity of empirical first-passage time τ n inferred from a sample of n realizations of an ergodic reversible Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The tail probability that the estimate τ n deviates from the true mean ⟨τ⟩ by more or equal than t upwards P(τ n ≥ ⟨τ⟩ + t) or downwards P(τ n ≤ ⟨τ⟩ − t) is shown in green and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (b) Brownian molecular search process in a d- dimensional domain (here d = 2) with outer radius R and target radius a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Discrete-state Markov jump models of protein folding for (c) a toy protein and (d) experimentally inferred model of calmodulin [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transitions between states are in- dicated by arrows and obey detailed balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For all systems considered the absorbing target is colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' insufficient sampling, which leads to substantial errors in inferred rates [128–131] and, in the worst case, erroneous conclusions (see discussion in [113, 132]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Even extensive computing resources may result in only a few indepen- dent estimates spread over many orders of magnitude, rendering uncertainty quantification challenging and not amenable to standard error analysis [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Constructing reliable confidence intervals is a fundamen- tal challenge in statistical inference, and many prevalent methods rely on asymptotic arguments that hold when the number of realizations tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' However, the applicability of asymptotic results in a finite-sample set- ting is, by definition, problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, Central- Limit- and bootstrapping-based methods [133] may easily arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='08732v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='stat-mech] 20 Jan 2023 2 underestimate the uncertainty for small n and fail to guar- antee coverage of the confidence level [116, 134–139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Conversely, Bayesian methods (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [140]) do not rely on asymptotic arguments and are therefore often (in general erroneously [141, 142]) believed to readily alle- viate the small-sample problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bayesian estimates are sensitive to, dependent on, and potentially biased by, the specification of the prior distribution, especially in the small-sample setting [140, 143–145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Due to the prior dependence of estimates and their uncertainties, Bayesian methods must be treated with care when applied to small samples [146, 147] (see [123, 129, 148–150] specifically for kinetic inference) and can perform worse than asymptotic frequentist methods [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, so-called “credible intervals”—the Bayesian analogue to confidence intervals—have a nominally differ- ent meaning, as they treat the estimated parameter as a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bayesian posterior intervals are similarly affected by limited sampling [116], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' the constructed uncertainty estimates and their quality are sensitive to the choice of prior probability [141, 142] and may likely underestimate the true uncertainty and thus fail to pro- vide trustworthy confidence intervals [129, 151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' On a more subtle level, the classical Bernstein-von- Mises theorem establishes a rigorous (frequentist) justi- fication of posterior-based Bayesian credible intervals as asymptotically correct, prior independent confidence inter- vals for (finite dimensional) parametric models in the large- sample limit [152–154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Analogous statements for semi- parametric and (infinite dimensional) non-parametric models are more delicate [155–158] and, despite having received signifficant attention [159–170] (see also [171] for misspecified and high dimensional [172] parametric models), seem to remain—even in the asymptotic, large- sample regime—an elusive problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' There is thus a pressing need for understanding fluctu- ations of inferred empirical first-passage times, a rigorous error control, and reliable non-asymptotic error estima- tion in the small-sample regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' These are fundamental problems of statistical kinetics and are essential for the analysis of experimental and simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Here, we present general bounds on fluctuations of empirical first-passage times that allow a rigorous uncer- tainty quantification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' using confidence intervals with guaranteed coverage probabilities for all sample sizes) under minimal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We prove non-asymptotic lower (L) and upper (U) bounds on the deviation proba- bility P(τ n ≥ ⟨τ⟩ + t) and P(τ n ≤ ⟨τ⟩ − t) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', the probability that the empirical first-passage time inferred from a sample of n ≥ 1 realizations of an ergodic reversible Markov process, τ n, deviates from the true mean ⟨τ⟩ by more than t in either direction, L± n (t) ≤ P(±[τ n − ⟨τ⟩] ≥ t) ≤ U± n (t) ∀t ≥ 0, (1) the upper bounds U± n (t) corresponding to so-called concen- tration inequalities [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The most conservative version of the derived upper bounds is independent of any details about the underlying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The validity and sharpness of the bounds are demonstrated by means of spatially con- fined Brownian molecular search processes in dimensions 1 and 3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b), and discrete-state Markov jump models of protein folding for a toy protein [24, 129, 174, 175] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1c) and the experimentally inferred model of calmod- ulin [124] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We use the bounds U± n (t) to quantify the uncertainty of the inferred sample mean τ n in a gen- eral setting and under minimal assumptions, for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We conclude with a discussion of the practical implications of the results and further research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—We consider time-homogeneous Markov pro- cesses xt on a continuous or discrete state-space Ω with (forward) generator ˆL corresponding to a Markov rate- matrix or an effectively one-dimensional Fokker-Planck operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Let the transition probability density to find xt at x at time t given that it evolved from x0 be pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac or Kronecker delta for continuous and discrete state- spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We assume the process to be ergodic limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes the equilibrium probability density and ϕ(x) the general- ized potential in units of thermal energy kBT [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We assume that ˆL obeys detailed balance [177] and is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or (iii) Ω is infinite but ϕ(x) sufficiently confining (see [178]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Each of the conditions (i)-(iii) ensures that the spectrum of ˆL is discrete [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We are interested in the first-passage time to a target a when xt=0 is drawn from a density p0(x) τ = inf t [ t |xt = a, p0(x0)], (2) and focus on p0(x) = ˜peq(x) where the tilde denotes that the absorbing state is excluded [180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For completeness we also provide in [181] results for general initial conditions p0(x) that require more precise conditions on ϕ(x) [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The probability density of τ for such processes has the generic form [23, 24] ℘a(t|x0) = � k>0 µkwx0 k e−µkt, (3) where µk > 0 denote first-passage rates and wx0 k the (not necessarily positive) spectral “weights” normalized according to � k>0 wx0 k = 1 and wx0 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The m- th moment of τ is given by ⟨τ m⟩ = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' � k>0 wx0 k /µm k and the survival probability reads P(τ > t) ≡ Sa(t|x0) = � k>0 wx0 k e−µkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' If x0 is drawn from the equilibrium density, ˜peq(x), we have ℘a(t|˜peq) ≡ � Ω\\a ℘a(t|x0)˜peq(x0)dx0 [183] which renders all weights non-negative, wk ≡ � Ω\\a wx0 k ˜peqdx0 ≥ 0 (see proof in [181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We henceforth abbreviate Sa(t|˜peq) ≡ Sa(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To examplify the need for uncertainty bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (1) we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2a-d that the probability that τ n − ⟨τ⟩ lies within a desired range of say ± 10% of the longest first-passage time scale µ−1 1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1]) is low even for n ≈ 50 for all models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lower bounds on deviation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—There exists a “noise floor” for τ n for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since µk ≤ µk+1 and 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Deviation probabilities and corresponding bounds for a spatially confined Brownian search process in (a,e) d = 1 and (b,f) d = 3 dimensions, and Markov-jump models of protein folding for (c,g) the experimentally inferred model of calmodulin and (d,h) the toy protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (a-d) Probability that δτ n = τ n − ⟨τ⟩ lies within a range of ±10% of the longest time-scale 1/µ1, P(µ1δτ n ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1]), as a function of n determined from the statistics of τ n for different fixed n for all model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (e-h) Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that the sample mean τ n inferred from n realizations deviates from ⟨τ⟩ by more than t in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Right tail areas are shown for t > 0 and left for t < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lower L± n (t) and upper U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) bounds are depicted as red and black lines, respectively, and the model-free upper bound U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) as the dashed yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Symbols denote corresponding scaled empirical deviation probabilities as a function of t and are sampled for different n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' wk are non-negative [184] and normalized [23, 24], the equilibrium survival probability obeys w1e−µ1t ≤ Sa(t) ≤ e−µ1t, which directly leads to lower bounds L± n (t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Namely, τ n ≥ mini∈[1,n] τi ≡ τ min n and τ n ≤ maxi∈[1,n] τi ≡ τ max n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Therefore, P(τ min n ≥ t) ≤ P(τ n ≥ t) ≤ P(τ max n ≥ t) and we have P(τ min n ≥ t) = S(t)n and P(τ max n ≤ t) = (1 − S(t))n, leading to lower bounds P (τ n − ⟨τ⟩ ≥ t) ≥ � w1e−µ1(⟨τ⟩+t)�n ≡ L+ n (t) P (τ n − ⟨τ⟩ ≤ −t) ≥ � 1 − e−µ1(⟨τ⟩−t)�n ≡ L− n (t), (4) where equality is reached for n = 1 and w1 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Anal- ogous results are obtained for upper bounds (see [181]) which, however, are much weaker than those derived be- low with the Cram´er-Chernoff approach and concurrently require even more information about the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We remark that bounds on the survival probability consequently also bound the probability density ℘(n) a (t) of the fastest first-passage time of n independent par- ticles [23, 25, 26, 185] according to nw1e−µ1(n−1)t ≤ ℘(n) a (t)/℘a(t) ≤ ne−(n−1)µ1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We now turn to the more challenging upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cram´er-Chernoff bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—Let δτ n ≡ |τ n−⟨τ⟩| and λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We start with the obvious inequality eλt1δτ n≥t ≤ eλτ n, where 1b is the indicator function of the set b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tak- ing the expectation yields P(δτ n ≥ t) ≤ e−λt⟨eλδτ n⟩ ≡ e−λt+ψδτn(λ), where we defined the cumulant generating function of δτ n, ψδτ n(λ) ≡ ln⟨eλδτ n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that τi are sta- tistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The bound can be optimized [186] to find Chernoff’s inequality, P(δτ n ≥ t) ≤ e−nψ†δτ(t), where ψ∗ δτ(t) is the Cram´er transform of ψδτ(λ) [173], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ψ∗ δτ(t) ≡ sup λ (λt − ψδτ(λ)), (5) where δτ ≡ δτ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' On the interval λ ∈ [0, µ1) we have the following bounds on ψδτ(λ) (see proof in [181]) ψδτ(λ) ≤ φδτ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≡ � � � � � � � λ2 2µ2 1 C 1 − λ/µ1 τ ≥ ⟨τ⟩ λ2 2µ2 1 C 1 − (λ/µ1)2 τ < ⟨τ⟩, (6) which are non-negative, convex, and increasing on λ ∈ [0, µ1), and we introduced C ≡ µ2 1⟨τ 2⟩ [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The bound (6) further implies ψ∗ δτ(t) ≥ φ∗ δτ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ∀t ≥ 0, and may thus be optimized according to [186] to obtain the inequalities announced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (1) via Chernoff’s inequality: U+ n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = exp (−nCh+ (µ1t/C)) 0 ≤ t ≤ ∞ U− n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = exp (−nCh− (µ1t/C)) 0 ≤ t ≤ ⟨τ⟩ (7) where we defined the functions h+(u) ≡ 1 + u − √ 1 + 2u (8) h−(u) ≡ Λ(u)u − 1 2 Λ(u)2 1 − Λ(u)2 (9) with Λ(u) ≡ 1 2 � g(u) − � 4 + 2/g(u)u − g(u)2 � and g(u) ≡ 2 √ 3 � 1 + 2 cosh �1 3arcosh � 1 + 33 27u2 ���1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (10) 4 The tail behavior of δτ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) provides quantitative insight into fluctuations of τ even when ⟨τ⟩ is unknown or is an insufficient or non-representative observable [188– 190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Deviations are readily expressed relative to the longest natural time scale 1/µ1 that does not need to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' That is, deviations are naturally parameterized by the dimensionless variable ˜t = µ1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Asymptotically as n → ∞, U± n is substantial only for ˜t/C ≪ 1 and the tails become symmetric and sub-Gaussian [173], h+(u) = u2/2 − O(u3) and h−(u) = u2/2 − O(u4) (see [181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Notably, details about the underlying dynamics only enter the tail bounds (7) via the system-dependent con- stant C that, however, can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, for equilibrium initial conditions we have 0 ≤ 2w1 ≤ C ≤ 2 (see [181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since φδτ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) is monotonically increasing with C ∈ (0, 2], we have φδτ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ φδτ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) which im- plies φ∗ δτ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≥ φ∗ δτ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thus, we find the model-free bounds U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) ≡ U± n (t) (11) requiring no information about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The non- asymptotic bounds on deviation probabilities of τ n in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) and (11) are our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Notably, analogous concentration inequalities were pre- viously derived for time-averages of Markov processes [191–193] (see also [194]), and were recently applied to bound time-averaged measurement outcomes in quantum Markov processes [195] and to derive inverse thermody- namic uncertainty relations [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Illustration of bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—The lower L± n (t) and upper U± n (t) bounds on P(±[τ n − ⟨τ⟩] ≥ t) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (4) and (7), respectively, are examplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2e-h (see red and black lines) for the model systems shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that to illustrate all bounds, for convenience in a single panel, we formally let t → −t for the left tails L− n (t) and U− n (t), such that t (as shown) has support on [−⟨τ⟩, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Deviation probabilities are in turn expressed as P(sgn(t)δτ n ≥ |t|) where sgn(x) denotes the signum function and δτ n = τ n − ⟨τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To assess the quality of our bounds for several n we further scale probabilities P1/n such that L± n (t) and U± n (t) collapse onto a master curve for all n (see also inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Symbols denote empirical deviation probabili- ties obtained by sampling τ n for different n (see [181] for details), which approach the upper bound as n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For n = 1 empirical right-tail deviations are close to L+ 1 (t) even for w1 ≤ 1 [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' As expected the model-free upper bound U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (yellow) holds universally but is generally more conservative, however, it is remarkably good for C ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='3 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2e-g) but becomes weaker as C approaches 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—The bounds (7) provide the elusive systematic framework to rigorously quantify the uncertainty of the estimate τ n for any, and especially for small, sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, they allow us to construct “with high probability” guarantees such as con- fidence intervals, which—unlike traditional confidence intervals in statistics—are not only asymptotically correct but hold for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Furthermore, these concentration- based guarantees do not require specifying a prior belief as in the Bayesian context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Setting U± n (t± α±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = α± for chosen acceptable left- and right-tail error probabilities α± (with α+ + α− < 1) we get an implicit definition of the confidence interval [−t− α−, t+ α+] at confidence level (or “coverage probability”) 1 − (α+ + α−) in the form P(−t− α− ≤ δτ n ≤ t+ α+) ≥ 1 − α− − α+ ≡ 1 − α, (12) stating that with probability of at least 1 − α the sample mean τ n lies within [⟨τ⟩ − t− α−, ⟨τ⟩ + t+ α+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Confidence intervals are closely related to, and can be used for, statis- tical significance tests [198, 199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' However, they provide more insight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' instead of mere rejection/acceptance they provide quantitative bounds on statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Two-sided intervals are not uniquely determined by specifying a confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' It is customary to choose equal tail probabilities α+ = α− = α/2 yielding so-called central confidence intervals for which t± α± are generally not equidistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Two-sided central confidence intervals for δτ n as a function of n for a confidence level of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 and models systems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d are shown (rescaled to a master scaling) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' One may also choose symmetric intervals which in turn do not necessarily imply equal tail probabilities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' α+ ̸= α−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In some situations only one- sided confidence intervals are required P(±δτ n ≤ t± α±) ≥ 1 − α± (for a discussion see [181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, we may now also answer the practical question: How many realizations are required to achieve a desired accuracy with a specified probability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To ensure with probability of at least 1 − α that δτ n∗ ∈ [−t− α−, t+ α+] one needs n∗ realizations defined via U+ n∗(t+ α+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) + U− n∗(t− α−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (13) The number of samples n∗ required to guarantee that τ n∗ falls within a symmetric interval of length ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2/µ1, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' τ n∗ ∈ [⟨τ⟩ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1/µ1, ⟨τ⟩ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1/µ1]) with probability of at least 1 − α is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3b for several values of C (intersections with the dashed line yield n∗ guaranteeing a coverage of at least 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3c depicts the comple- mentary symmetric interval ∆t covering the range of δτ n for a given n with probability of at least 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that hundreds to thousands of samples may be required to ensure an accuracy of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1/µ1 with a 90% confidence, which is seemingly not met in experiments [113–119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (12) and (13) constitute our second main result as they provide rigorous error estimates in the small-sample regime that allow for systematic error control in kinetic inference and can be solved for t± α± and n∗, respectively, using standard root-finding methods (see [181]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (11) we can construct system-independent but more conservative universal confidence intervals (see yellow line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Interestingly, even when C ≈ 1 the universal bound remains reasonably tight, only for C ≪ 1 differences become substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—Leveraging spectral analysis and the framework of concentration inequalities we derived gen- eral upper and lower bounds on the probability that the 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Non-asymptotic uncertainty quantification of the sam- ple mean τ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (a) Relative error µ1δτ n = µ1(τ n−⟨τ⟩) (symbols) obtained from sampling of τ n for different model systems and as a function n (re-scaled to a master scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The correspond- ing two-sided central confidence interval [−µ1t− α/2, µ1t+ α/2] with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 is shown as black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (b) Required number of sam- ples n∗ to ensure that the relative error δτ n∗ falls within the symmetric interval [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1] of length ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2/µ1 with probability of at least 1 − α for several values of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (c) Cor- responding symmetric confidence interval [−µ1∆t/2, µ1∆t/2] (only the upper limit is shown) at confidence level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 as a function of n for different C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' empirical first-passage time τ n inferred from n indepen- dent realizations deviates from the true mean ⟨τ⟩ by any given amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We used these bounds to construct non- asymptotic confidence intervals that hold in the elusive small-sample regime and thus go beyond Central-Limit- and bootstrapping-based methods, which are known to fail for small n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The results require minimal input and in particular do not require any prior belief as in the Bayesian approach that is known to be problematic and likely underestimates the uncertainty in the small-sample setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Our concentration-based results allow for rigor- ous, model-free error control and reliable error estimation, which is essential for the analysis of experimental and simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' They may further be applied to popu- lation dynamics and epidemiology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' in the inference of extinction or incubation times of diseases [200–204], and may be extended to the concentration around the typical instead of mean first-passage times [205] as well as non-ergodic and irreversible dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='—Financial support from Studiens- tiftung des Deutschen Volkes (to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=') and the German Research Foundation (DFG) through the Emmy Noether Program GO 2762/1-2 (to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=') is gratefully acknowl- edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ∗ agodec@mpinat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='de [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner, A Guide to First-Passage Processes (Cam- bridge University Press, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, First-Passage Phenomena and their Applications, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 35 (World Scien- tific, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zhang and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dudko, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 45, 117 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Iyer-Biswas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zilman, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 160, 261 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Szabo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schulten, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schulten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 72, 4350–4357 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H¨anggi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Talkner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Borkovec, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 62, 251 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ben-Naim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Leyvraz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 70, 1890 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grebenkov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 20, 16393–16401 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grebenkov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grebenkov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 117, 260201 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [11] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Berg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Winter, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Von Hippel, Bio- chemistry 20, 6929–6948 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Koslover, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D´ıaz de la Rosa, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spakowitz, Bio- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 101, 856 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Holcman and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schuss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 47, 173001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [14] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B´enichou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chevalier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meyer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Voituriez, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 106, 038102 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Marklund, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mahmutovic, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Berg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hammar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van der Spoel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Fange, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Elf, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 110, 19796–19801 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bauer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, PLoS ONE 8, e53956 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B´enichou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chevalier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Klafter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meyer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Voituriez, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2, 472–477 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [18] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B´enichou and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Voituriez, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 539, 225 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 6, 041037 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Newby and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Allard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 116, 128101 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meerson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2014, P06019 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meerson and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 114, 198101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 20, 112002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2019, 024002 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, Reaction kinetics in the few- encounter limit, in Chemical Kinetics (World Scientific, 2019) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 265–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [26] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schuss, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Basnayake, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Holcman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Life Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 28, 52–79 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lawley and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Madrid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 150, 214113 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lawley and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Madrid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 30, 1207–1227 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lawley, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 102, 062118 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bressloff and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Newby, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 85, 135 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [31] ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rold´an, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lisica, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S´anchez-Taltavull, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grill, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 93, 062411 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ghusinga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dennehy, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Singh, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 114, 693 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rijal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Prasad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Singh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Das, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 128, 048101 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 6 [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Parmar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Das, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Padinhateeri, Nucleic Acids Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 44, 1630 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Charlebois, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Abdennur, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kaern, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 107, 218101 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Frey, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ziebert, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schwarz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 122, 088102 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lloyd and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' May, Science 292, 1316 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [38] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hufnagel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Brockmann, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Geisel, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 101, 15124 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [39] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B´enichou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Loverdo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreau, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Voituriez, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 83, 81 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [40] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Boccardo and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pierre-Louis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 128, 256102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [41] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Erdmann and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schwarz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 92, 108102 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chakrabarti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hinczewski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thirumalai, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 111, 9048–9053 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [43] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Blom and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 11, 031067 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [44] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Goychuk and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H¨anggi, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 99, 3552 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kay and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Giuggioli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 4, 032039 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hubatsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bauermann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Weber, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J¨ulicher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3, 043150 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [47] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kramers, Physica 7, 284 (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sabhapandit and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 125, 200601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [49] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Trendelkamp-Schroer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 6, 011009 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chupeau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gladrow, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chepelianskii, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Keyser, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Trizac, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 117, 1383 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [51] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Swinburne, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kannan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sharpe, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wales, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 153, 134115 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thorneywork, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gladrow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Qing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rico-Pasto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ritort, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bayley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kolomeisky, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Keyser, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 6, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Goychuk and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Frey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 123, 178101 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [54] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bebon and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schwarz, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 24, 063034 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [55] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dougherty, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lyubinetsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Con- stantin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dasgupta, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sarma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 89, 136102 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Constantin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sarma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dasgupta, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bondarchuk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dougherty, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Williams, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 91, 086103 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dougherty, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tao, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bondarchuk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cullen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Constantin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dasgupta, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sarma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 71, 021602 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [58] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Merikoski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Maunuksela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Myllys, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Timonen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Alava, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 90, 024501 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Constantin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dasgupta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chatraphorn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sarma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 69, 061608 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [60] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godr`eche, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 102, 240602 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [61] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 62, 225–361 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [62] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gingrich and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Horowitz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 119, 170601 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [63] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Singh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Menczel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Golubev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Khaymovich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Peltonen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Flindt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Saito, ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rold´an, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pekola, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 122, 230602 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [64] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rold´an, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Neri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D¨orpinghaus, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meyr, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J¨ulicher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 115, 250602 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [65] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Neri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rold´an, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J¨ulicher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 7, 011019 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [66] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Falasco and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Esposito, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 125, 120604 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [67] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Neri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 55, 304005 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [68] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Garrahan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 95, 032134 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [69] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hiura and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ichi Sasa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 103, 050103 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [70] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kac, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, Calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', 1951) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 189–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [71] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, First-Passage Phenomena and Their Applications , 226–251 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [72] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wergen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 45, 355002 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [73] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 52, 244001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [74] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Friedman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kessler, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Barkai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 95, 032141 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [75] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thiel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Barkai, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kessler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 120, 040502 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [76] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kearney and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 38, 4097 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [77] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kearney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 40, F863 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [78] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kearney and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 47, 465001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kearney and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 54, 055002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [80] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meerson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' : Theory Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2021 (3), 039801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [81] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Singh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 55, 234001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [82] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mercado-V´asquez and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Boyer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 123, 250603 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zodage, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Santhanam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 104, 052103 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [84] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spouge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Szabo, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Weiss, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 54, 2248 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [85] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Scher and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reuveni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 127, 018301 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [86] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Woillez, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kafri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lecomte, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tailleur, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 122, 258001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [87] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mori, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Doussal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 124, 090603 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [88] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H¨anggi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Talkner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 51, 2242 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [89] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hanggi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Talkner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A 32, 1934 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [90] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gu´erin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Levernier, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B´enichou, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Voituriez, Nature 534, 356 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [91] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meyer and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rieger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 127, 070601 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [92] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Evans and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 106, 160601 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [93] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kusmierz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sabhapandit, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 113, 220602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [94] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reuveni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 116, 170601 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [95] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reuveni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 118, 030603 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [96] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pal, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Eliazar, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reuveni, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 122, 020602 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [97] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Evans, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 53, 193001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 7 [98] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Besga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bovon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Petrosyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ciliberto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2, 032029 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [99] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tal-Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sekhon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reuveni, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Roichman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 11, 7350 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [100] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bruyne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Randon-Furling, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Redner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 125, 050602 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [101] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bruyne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Majumdar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 128, 200603 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [102] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ensign and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B 113, 12410–12423 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [103] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Satija, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M¨uhle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Enderlein, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Makarov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B 124, 3482–3493 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [104] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zolaktaf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dannenberg, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rudelis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Condon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schaeffer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thachuk, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Winfree, Inferring Parameters for an Elementary Step Model of DNA Structure Kinetics with Locally Context-Dependent Arrhenius Rates (Springer International Publishing, 2017) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 172–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [105] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Weinreb, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wolock, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tusi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Socolovsky, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Klein, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 115, E2467 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [106] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pearce, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Woodhouse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Forrow, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kelly, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kusumaatmaja, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dunkel, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 10, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [107] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zhou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stell, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 125, 6300–6305 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [108] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Daldrop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kappler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Br¨unig, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Netz, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 115, 5169–5174 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [109] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nicholson and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rutledge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 144, 134105 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [110] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van Hijkoop, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dammers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Malek, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Coppens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 127, 085101 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [111] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Belousov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Qaisrani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hassanali, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Roldan, Soft Matter 16, 9202–9216 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [112] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ditlevsen and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ditlevsen, Probabilistic Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mech 23, 170–179 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [113] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gapsys and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' de Groot, eLife 9, e57589 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [114] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lindorff-Larsen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Piana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dror, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Shaw, Science 334, 517 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [115] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Adelman and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 138, 044105 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [116] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mostofian and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zuckerman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 15, 3499 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [117] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mehra and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kepp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 151, 085101 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [118] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Militaru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Innerbichler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Frimmer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tebbenjo- hanns, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Novotny, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dellago, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 12, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [119] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gieseler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ricci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Quidant, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dellago, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Novotny, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 12, 1130 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [120] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sharpe and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wales, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 153, 024121 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [121] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Donovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sedgewick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Faeder, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zuckerman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 139, 115105 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [122] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sabelko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ervin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gruebele, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 96, 6031 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [123] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ensign and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B 113, 12410 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [124] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stigler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ziegler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gieseke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gebhardt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rief, Science 334, 512 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [125] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Berezhkovskii and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Szabo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 150, 054106 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [126] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nayak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Das, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nandi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2, 013114 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [127] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wales, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 13, 6349 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [128] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Singhal and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 123, 204909 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [129] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bowman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, An Intro- duction to Markov State Models and their Application to Long Timescale Molecular Simulation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 797 (Springer Science & Business Media, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [130] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grossfield and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zuckerman, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 5, 23 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [131] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grossfield, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Patrone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Roe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schultz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Siderius, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Zuckerman, Living J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [132] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Knapp, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ospina, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Deane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 14, 6127 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [133] Resampling methods like bootstrapping assume the data to be representative of the inferred statistic, which is not necessarily the case for small n, possibly even when n is large but finite for broad distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [134] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Davison and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hinkley, Bootstrap Methods and their Application (Cambridge University Press, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [135] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Shao, Mathematical Statistics (Springer New York, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [136] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Abadie and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Imbens, Econometrica 76, 1537 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [137] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Putter and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van Zwet, in Selected Works of Willem van Zwet (Springer New York, 2011) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 245–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [138] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hogg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' McKean, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Craig, Introduction to Mathematical Statistics (Pearson, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [139] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schenker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 80, 360 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [140] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gelman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Carlin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stern, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rubin, Bayesian Data Analysis (Chapman and Hall/CRC, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [141] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Brazzale, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Davison, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Reid, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', Applied Asymptotics: Case Studies in Small-Sample Statistics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 23 (Cambridge University Press, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [142] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lista, Statistical Methods for Data Analysis in Particle Physics (Springer International Publishing, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [143] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kaplan, Bayesian Statistics for the Social Sciences (Guilford Publications, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [144] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' McElreath, Statistical Rethinking: A Bayesian Course with Examples in R and Stan (Chapman and Hall/CRC, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [145] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Tavakoli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Taylor, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Komatsuzaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Press´e, in Advances in Chemical Physics (John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', 2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 205–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [146] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' McNeish, Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Modeling 23, 750 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [147] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Smid, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' McNeish, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mioˇcevi´c, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van de Schoot, Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Modeling 27, 131 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [148] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bacallado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chodera, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 131, 045106 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [149] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Prinz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sarich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Keller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Senne, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Held, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chodera, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sch¨utte, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 134, 174105 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [150] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Trendelkamp-Schroer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Paul, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 143, 174101 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [151] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chodera and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 133, 105102 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [152] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Le Cam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' LeCam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Yang, Asymptotics in Statistics: Some Basic Concepts (Springer Science & Business Media, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [153] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Van der Vaart, Asymptotic Statistics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3 (Cam- bridge university press, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [154] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Le Cam, Asymptotic Methods in Statistical Decision Theory (Springer Science & Business Media, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [155] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Diaconis and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Freedman, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 14, 1 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [156] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cox, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 21, 1 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 8 [157] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Diaconis and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Freedman, Bernoulli , 411 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [158] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Freedman, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 27, 1119 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [159] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Barron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schervish, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wasserman, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 27, 536 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [160] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ghosal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ghosh, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ramamoorthi, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 27, 143 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [161] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ghosh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ramamoorthi, Bayesian Non- parametrics (Springer-Verlag, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [162] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kim and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lee, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 32, 1 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [163] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Boucheron and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gassiat, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3, 114 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [164] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bickel and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kleijn, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 40, 206 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [165] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rivoirard and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rousseau, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 40, 1489 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [166] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Castillo and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nickl, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 42, 1941 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [167] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rousseau, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3, 211 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [168] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rockov´a, in International Conference on Machine Learning (PMLR, 2020) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 8137–8146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [169] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ray and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van der Vaart, Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 15, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [170] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ghosal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Van der Vaart, Fundamentals of Non- parametric Bayesian Inference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 44 (Cambridge Uni- versity Press, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [171] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kleijn and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' van der Vaart, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 6, none (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [172] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Johnstone, in Institute of Mathematical Statistics Collections (Institute of Mathematical Statistics, 2010) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 87–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [173] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Boucheron, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lugosi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence (Oxford University Press, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [174] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Prinz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Keller, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 13, 16912 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [175] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Olsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Paul, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Clementi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 114, 8265 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [176] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pavliotis, Stochastic Processes and Applications (Springer New York, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [177] ˆL is self-adjoint in the left eigenspace with respect to a scalar product weighted by e−ϕ(x) and the operator eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat mea- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [178] Precisely, we require that ϕ(x) satisfies the Poincar´e inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' lim|x|→∞(|∇ϕ(x)|2/2 − ∇2ϕ(x)) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [179] The relaxation eigenvalue problem reads −ˆLΨk(x) = νkΨk(x) with ν0 = 0 and νk≥1 > 0 [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [180] In a continuous state-space the absorbing state a has zero measure and ˜peq(x) = peq(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In the discrete case ˜peq(xk̸=a) ≡ peq(xk)/ � k̸=a peq(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [181] See Supplemental Material at [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='] for further details, mathematical proofs, and generalizations to arbitrary initial conditions p0(x), as well as Refs [3, 8–10, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [182] When the initial condition is not sampled from ˜peq(x) we assume that ϕ(x) is sufficiently confining to assure a “nice” asymptotic growth of eigenvalues, limk→∞ νk = bkβ with β > 1/2 and 0 < b < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The latter condition is automatically satisfied when Ω is finite, since regu- lar Sturm-Liouville problems display Weyl asymptotics with β = 2 [212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The condition is in fact satisfied by most physically relevant processes with discrete spectra, incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' the Ornstein-Uhlenbeck or Rayleigh process [213] with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [183] When Ω is discrete the integral is replaced by a sum over states excluding the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [184] wk ≥ 0 is a necessary condition for the validity of the lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Thus, in contrast to our Cram´er-Chernoff bounds U± n (t) that generalize to arbitrary initial condi- tions, L± n (t) hold only for p0(x0) = ˜peq(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [185] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Grebenkov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 22, 103004 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [186] ψδτn(λ) is differentiable, convex, non-negative, and non- decreasing and thus ψ∗ δτ(t) = ψδτn(λ†), where λ† solves ψ′ δτ(λ†) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [187] In case of arbitrary initial conditions ⟨τ 2⟩ becomes re- placed by � i wi1wi>0 < ∞ while the rest remains un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [188] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mej´ıa-Monasterio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' : Theory Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2011 (06), P06022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [189] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Holovatch, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schehr, Physica A 390, 4340 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [190] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mattos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mej´ıa-Monasterio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Metzler, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Oshanin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' E 86, 031143 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [191] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lezaud, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 8, 849 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [192] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lezaud, ESAIM Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 5, 183–201 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [193] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Guillin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wu, Theory Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' its Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 58, 358 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [194] Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [192] contains an error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' the Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='3 is only valid in the regime r < λ1/3||f||∞, but the Lemma may be shown to hold in the claimed regime [214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [195] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Girotti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Garrahan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gut¸˘a, Concentra- tion inequalities for output statistics of quantum Markov processes (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [196] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bakewell-Smith, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Girotti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gut¸˘a, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Garra- han, Inverse thermodynamic uncertainty relations: Gen- eral upper bounds on the fluctuations of trajectory ob- servables (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [197] However, w1 can get arbitrary close to 0 in principle, rendering the lower bound trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [198] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Wackerly, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Mendenhall, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Scheaffer, Math- ematical Statistics with Applications (Cengage Learning, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [199] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meeker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hahn, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Escobar, Statistical Intervals: A Guide for Practitioners and Researchers, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 541 (John Wiley & Sons, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [200] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Dykman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Schwartz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Landsman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 101, 078101 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [201] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gilbert, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Meyers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Galvani, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Townsend, Epidemics 6, 37 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [202] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ottino-Loffler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Scott, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Strogatz, eLife 6, e30212 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [203] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Aliee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rock, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Keeling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Interface 17, 20200540 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [204] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hathcock and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Strogatz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 128, 218301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [205] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Belan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2, 013243 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Burden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Faires, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Burden, Numerical Analysis (Cengage Learning, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Barkai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Aghion, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kessler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 4, 021036 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pitman, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 7, 511 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Siegert, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 81, 617 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [8] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Seifert, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 10, 171 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cowan, Statistical Data Analysis (Oxford University Press, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [212] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Teschl, Ordinary Differential Equations and Dynami- cal Systems (American Mathematical Society, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [213] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', 9 Springer Series in Synergetics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 13 (Springer-Verlag, Berlin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [214] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ibanez, Concentration inequalities for Markov jump processes (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1 Supplementary Material for: Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime Rick Bebon and Aljaˇz Godec Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 G¨ottingen In this Supplementary Material (SM) we present additional background and details of the calculations, auxiliary results, numerical methods, and mathematical proofs of the claims made in the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The sections are organized in the order as they appear in the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' CONTENTS References 5 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spectral representation and preparatory Lemmas 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spectral representation 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lemma 1: All weights are non-negative for equilibrium initial conditions 3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lemma 2: Sum of positive weights is bounded from above 3 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Extreme value bounds and comparison with Cram´er-Chernoff bounds 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Extreme value bounds 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Comparison of Cram´er-Chernoff vs Extreme value Bounds 4 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Complete proof of concentration inequalities and their asymptotics 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩ 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Behavior of upper bounds U± n (t) for large sample sizes 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Proof of bounds on C and model-free concentration inequalities 9 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Model systems and details on numerical methods 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Continuous-time discrete-state Markov jump process 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transitions rates of the 8-state toy protein model 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transitions rates of the calmodulin protein model 11 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spatially confined Brownian molecular search process 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Statistics of first-passage times ⟨τ⟩ and the sample mean τ n 12 S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Uncertainty quantification with confidence intervals 13 References 15 2 S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' SPECTRAL REPRESENTATION AND PREPARATORY LEMMAS In this section we provide additional background on the spectral analysis of first-passage problems and some auxiliary Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, we prove that for equilibrium initial conditions all spectral first-passage weights wk(˜peq) are non-negative and that general initial conditions p0(x) the sum of positive spectral weights is always bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spectral representation First, we recall some general results using the spectral representation of first-passage processes (for more details on see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' As stated in the Letter, we consider time-homogeneous Markov processes xt on a continuous or discrete state-space Ω with (forward) generator ˆL corresponding to a Markov rate-matrix or an effectively one-dimensional Fokker-Planck operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Let the transition probability density to find xt at x at time t given that it evolved from x0 be pt(x|x0) ≡ eˆLtδx0(x) where δx0(x) denotes the Dirac or Kronecker delta for continuous and discrete state-spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We assume the process to be ergodic limt→∞ pt(x|x0) = peq(x), where peq(x) ≡ e−ϕ(x) denotes the equilibrium probability density and ϕ(x) the corresponding generalized potential in units of thermal energy kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We assume that ˆL obeys detailed balance, such that it is self-adjoint in the left eigenspace with respect to a scalar product weighted by e−ϕ(x) and the operator eϕ(x)/2 ˆLe−ϕ(x)/2 is self-adjoint with respect to a flat measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We assume that ˆL is either (i) bounded, (ii) Ω is finite with reflecting boundary ∂Ω, or that (iii) Ω is infinite but ϕ(x) is sufficiently confining (precisely, we require that ϕ(x) satisfies the Poincar´e inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' lim|x|→∞(|∇ϕ(x)|2/2 − ∇2ϕ(x)) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Each of the conditions (i)-(iii) ensures that the eigenvalue spectrum of ˆL is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The relaxation eigenvalue problem (for the inner product (·|·) defined with respect to a flat Lebesgue measure) reads −ˆLΨR k (x) = νkΨR k (x) with ΨL k(x) = ΨR k (x)eϕ(x), ν0 = 0 and νk≥1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The first-passage time to a target a for xt=0 drawn from a density p0(x) is defined as τ = inft[ t |xt = a, p0(x0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We will use ⟨·⟩ to denote an average over all first-passage paths {xt′}0≤t′≤τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' those that hit a only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The first-passage time density to a, ℘a(t|x0) = ⟨δ(t − τ[{xt′}])⟩ to reach the absorbing target at x = a, starting initially from x0, has the general spectral representation ℘a(t|x0) = � k≥1 wk(x0)µke−µkt, (S1) where µk is the k-th first-passage rate and wk(x0) its corresponding first-passage weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In similar fashion the survival probability is expressed as Sa(t|x0) ≡ � ∞ t ℘a(t′|x0)dt′ = � k≥1 wk(x0)e−µkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S2) We note that in contrast to the relaxation eigenvalues νk, the first-passage rates µk = µk(a) depend in the location of the absorbing target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, for any target location a the interlacing theorem holds [1, 2] : νk−1 ≤ µk(a) ≤ νk ∀k, a (S3) where equality occurs iff wk(x0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' for a where ΨR k (a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Laplace transforming the spectral expansion of the first-passage time density (S1)—according to ˜f(s) ≡ � e−stf(t) dt with f being a generic function locally integrable on t ∈ [0, ∞)—yields ˜℘a(s) = � k≥1 wk(x0)µk s + µk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S4) The first-passage weights are then obtained by using the residue theorem to invert the Laplace transformed renewal theorem [1–3] wk(x0) = ˜p(a, −µk|x0) µk ˙˜p(a, −µk|a) = � l≥0(1 − νl/µk)−1ΨR l (a)ΨL l (x0) � l≥0(1 − νl/µk)−2ΨR l (a)ΨL l (a) < ∞, (S5) where ˙˜p(a, s|a) = ∂s˜p(a, s|a) is taken at s = −µk and {νl, ΨR l , ΨL l } are the corresponding relaxation eigenmodes [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The weights satisfy � k≥1 wk(x0) = 1 and the first non-zero weight is strictly positive w1(x0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, the relaxation eigenvalues ν0 = 0 and all νk>0 ≥ 0 are real as a result of detailed balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lemma 1: All weights are non-negative for equilibrium initial conditions In the Letter we focus on equilibrium initial conditions, that is we assume that x0 is drawn from the invariant measure, peq(x0), which in the particular case of diffusion processes is assumed to have a reflecting boundary at a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' we focus on the one-sided first-passage process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We further introduce the non-negative modified spectral weights ¯wk(x0) ≡ wk(x0)θ(sgn[wk(x0)]) and now prove that for a normalized equilibrium probability density of initial conditions p0(x0) that excludes the target—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ˜peq(x0) ≡ peq(x0)[1 − δa(x0)]/(1|peq(x0)[1 − δa(x0)]) where δa(x0) is the Dirac measure (note that (1|˜peq) = 1)—all weights wk are rendered non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We thus have ¯wk(˜peq) = wk(˜peq) ≥ 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Namely, because ΨL l (a) = eβU(a)ΨR l (a) we have ΨR l (a)ΨL l (a) ≥ 0, ∀l, and from bi-orthogonality (ΨL l |peq) = δl,0 it follows that ˜wk ≡ (wk|˜peq) = ˜peq(a) 1 − � l≥0(1 − νl/µk)−1ΨR l (a)ΨL l (a) � l≥0(1 − νl/µk)−2ΨR l (a)ΨL l (a) = ˜peq(a) � l≥0(1 − νl/µk)−2ΨR l (a)ΨL l (a) ≥ 0 (S6) because by definition µk > 0, ∀k ≥ 1 denotes the zeros of ˜p(a, s|a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ˜p(a, −µk|a) = � l≥0(νl − µk)−1ΨR l (a)ΨL l (a) = µ−1 k � l≥0(1 − νl/µk)−1ΨR l (a)ΨL l (a) = 0 which completes the proof of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lemma 2: Sum of positive weights is bounded from above For the sake of completeness we here additionally present results for general initial conditions p0(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Recall from the Letter that we require some additional conditions on ϕ(x) or Ω in this more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, we assume that ϕ(x) is sufficiently confining to assure a “nice” asymptotic growth of eigenvalues, limk→∞ νk = bkβ with β > 1/2 and 0 < b < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The latter condition is automatically satisfied when Ω is finite, since regular Sturm-Liouville problems display Weyl asymptotics with β = 2 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The condition is in fact satisfied by most physically relevant processes with discrete spectra, incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' the (Sturm-Liouville irregular) Ornstein-Uhlenbeck or Rayleigh process [5] with β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' This implies, by the interlacing theorem (S3) that b(k − 1)β ≤ µk ≤ bkβ and therefore there exists a real constant C ∈ (0, ∞) such that limk→∞ µk diverges as Ckβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Recall further that the m-th moment of τ is given by ⟨τ m⟩ = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' � k≥1 wk(p0)/µm k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' By construction we obtain 2 � k≥1 ¯wk(p0)/µ2 k ≡ ⟨¯τ 2 p0⟩ ≥ 2 � k≥1 wk(p0)/µ2 k ≡ ⟨τ 2 p0⟩, where equality holds when p0 = ˜peq (since in this case all wk ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', ¯wk(˜peq) = wk(˜peq) as discussed before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, because we only consider Markov jump processes on finite state-spaces as well as processes for which limk→∞ µk = Ckα with 0 < C < ∞ and α > 1/2 (this includes confined Markov jump processes on infinite state-spaces and all regular Sturm-Liouville problems) convergence is ensured, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2 � k≥1 ¯wk(p0)/µ2+n k < ∞, ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To prove this consider wmax ≡ maxk≥k∗ ¯wk(p0) such that wmax/µ2+n k ≥ wk(p0)/µ2+n k , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Let the smallest k for which the asymptotic scaling holds be k∗ then we may split the summation as � k≥1 = �k∗−1 k=1 + � k≥k∗ such that � k≥1 ¯wk(p0) µ2+n k ≤ k∗−1 � k=1 ¯wk(p0) µ2+n k + � k≥k∗ wmax µ2+n k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Because the first term is nominally finite we only need to prove convergence of the second sum, which we do by means of the integral test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We define a function f(k) ≡ wmax/µ2+n k that is monotonically decaying in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' This implies f(x) ≤ f(k), ∀x ∈ [k, ∞) and f(x) ≥ f(k), ∀x ∈ [k∗, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We then have for every integer k ≥ k∗ that � k+1 k f(x)dx ≤ � k+1 k f(k)dx = f(k) and conversely, for every integer k ≥ k∗+1 that � k k−1 f(x)dx ≥ � k k−1 f(k)dx = f(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We now sum over all k ≥ k∗ to obtain, using µk = Ckα∀k ≥ k∗ � ∞ k∗ wmax (Cxα)(2+n) dx ≤ � k wmax µ2+n k ≤ wmax (Ckα∗ )2+n + � ∞ k∗ wmax (Cxα)2+n dx → wmaxC−(2+n)k1−α(2+n) ∗ α(2 + n) − 1 ≤ � k wmax µ2+n k ≤ wmax(Ckα ∗ )−(2+n) + wmaxC−(2+n)k1−α(2+n) ∗ α(2 + n) − 1 < ∞ where the last integral converges because 1−α(2+n) < 0, ∀n ≥ 0, which in turn proves convergence of � k≥1 ¯wk(p0)/µ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 4 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' EXTREME VALUE BOUNDS AND COMPARISON WITH CRAM´ER-CHERNOFF BOUNDS In the Letter we derive lower bounds L± n (t) on the deviation probability P(τ n − ⟨τ⟩ ≥ t) and P(⟨τ⟩ − τ n ≥ t) by utilizing extremal events, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', we consider the maximal and minimal first-passage time in a sample of n ≥ 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In this section we derive analogous upper bounds building on the same ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Extreme value bounds Recall that for the reversible Markov dynamics considered the equilibrium survival probability Sa(t|˜peq) ≡ Sa(t) in its spectral representation (S2) obeys w1e−µ1t ≤ Sa(t) ≤ e−µ1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S7) For the upper bound we use µk ≤ µk+1 and that � k>0 wk = 1 are normalized, whereas the lower bound follows since wk ≥ 0, ∀k, as we consider equilibrium initial conditions throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, from extreme value theory it follows P(τ min n ≥ t) = Sa(t)n ⇔ P(τ min n ≤ t) = 1 − Sa(t)n, P(τ max n ≤ t) = (1 − Sa(t))n ⇔ P(τ max n ≥ t) = 1 − (1 − Sa(t))n , (S8) where we introduce τ max n ≡ maxi∈[1,n] τi and τ min n ≡ mini∈[1,n] τi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Clearly, since τ min n ≤ τ n ≤ τ max n we can write P(τ min n ≥ t) ≤ P(τ n ≥ t) ≤ P(τ max n ≥ t) and analogously P(τ min n ≤ t) ≥ P(τ n ≤ t) ≥ P(τ max n ≤ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S8) in combination with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S7) we directly arrive at the lower bounds L± n (t) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (4) in the Letter) P(τ n ≥ ⟨τ⟩ + t) ≥ P(τ min n ≥ ⟨τ⟩ + t) = Sa(t + ⟨τ⟩)n ≥ � w1e−µ1(⟨τ⟩+t)�n (S9) P(τ n ≤ ⟨τ⟩ − t) ≥ P(τ max n ≤ ⟨τ⟩ − t) = (1 − Sa(⟨τ⟩ − t))n ≥ � 1 − e−µ1(⟨τ⟩−t)�n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S10) Introduced considerations are, however, not restricted to only lower bounds such that we can further leverage bounds on the equilibrium survival probability (S7) to analogously obtain corresponding upper bounds as P(τ n ≥ ⟨τ⟩ + t) ≤ P(τ max n ≥ t) = 1 − (1 − Sa(⟨τ⟩ + t))n ≤ 1 − � 1 − eµ1(⟨τ⟩+t)�n , P(τ n ≤ ⟨τ⟩ − t) ≤ P(τ min n ≤ t) = 1 − Sa(t)n ≤ 1 − � w1e−µ1(⟨τ⟩−t)�n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S11) As we will illustrate next, the upper bounds (S11) are much weaker than those derived with the Cram´er-Chernoff approach (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) in the Letter) and require more information about the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Comparison of Cram´er-Chernoff vs Extreme value Bounds In this section we directly compare the concentration-based upper bounds U± n (t) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) in the Letter) that are obtained with the Cram´er-Chernoff approach, with the upper bounds (S11) which are based on extreme value considerations in analogy to the lower bounds L± n (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2e-h of the Letter we now exemplify and compare both upper bounds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1 for the model systems shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1a-d we equivalently express re-scaled deviation probabilities P1/n(sgn(t)δτ n ≥ |t|) in a single panel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', for the left tail we formally let t → −t such that t as shown now has support in [−⟨τ⟩, ∞) and sgn(x) = ±1 for ±x > 0 and sgn(0) = 0 denotes the signum function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Empirical deviation probabilities (symbols) as a function of t are computed from statistics obtained by sampling τ n for different fixed n values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Extreme value lower bounds L± n (t) (S10) (or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (4)) for both tails are depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Here we now focus on comparing the upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Concentration inequalities U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7)) are again depicted as black lines whereas the corresponding extreme value upper bounds are represented as dashed/dotted lines where the respective coloring indicates the number of realizations n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note, that the concentration bounds (and the lower bounds) collapse onto a single master curve due to the employed scaling P1/n, whereas the extreme value upper bounds do not due to their different functional form (compare Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Evidently, while for n = 1 the extreme value bounds remains close to the actual deviation probability, already for n = 3 they become considerably less tight and overshoot heavily for all considered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, extreme value upper bounds become increasingly weak (even trivial at times) as n increases, therefore highlighting that Cram´er-Chernoff-type bounds are vastly more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 5 Motivated by the discussion above we next want to gain more quantitative insights for which sample sizes n the Cram´er-Chernoff approach becomes more favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For this purpose we introduce a quality factor Q ∈ [0, ∞) that is informally defined as Q ≡ Extreme value upper bound Cram´er-Chernoff-type upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S12) A value Q > 1 therefore indicates that the Cram´er-Chernoff bound is tighter and Q < 1 suggests that the extreme value bound should be favored, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1e-h we illustrate the quality factor Q as a function of sample size n for different fixed dimensionless deviation values µ1t (star symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Remarkably for all model systems considered—which span a large range of possible C values—the Cram´er-Chernoff approach is already superior even in the small-sample regime n ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, we can further study the particular n∗, for which one would reach Q = 1, as a function of some desired deviation µ1t relative to the longest time scale 1/µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note, that again for the left tail we let t → −t (see discussion above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1i-l for our model systems, n∗ (blue) generally is found to be well below n = 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', even for most small sample sizes the derived Cram´er-Chernoff-type bounds can be considered to be the better choice, especially when considering large µ1t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' large deviations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lastly, one could ask the question why the extreme value upper bound is so “weak” when n increases even just slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To answer this question we recall that—since we are interested in deviations of the sample mean τ n around ⟨τ⟩—we bound the sample mean with the minimal and maximal first-passage time according to τ min n ≤ τ n ≤ τ max n which is further used, in combination with bounds on the survival probability (S7), to derive corresponding upper bounds (S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Clearly, as n increases we expect this bound to become increasingly loose as by larger sample sizes we increase the chances of sampling rare first-passage times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', maximal and minimal first-passage time that strongly deviate from the (sample) mean—this also explain why bounds (S10) and (S11) are only particularly tight for n = 1 as here τ min n = τ 1 = τ max n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In contrast, the Cram´er-Chernoff method requires a much more delicate mathematical analysis involving bounds of the moment generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The Cram´er-Chernoff-type bound has the additional advantage that it can be further used to universally bound deviation probabilities where no specific information about the underlying system is required (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (11) in the Letter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, even the version of Cram´er-Chernoff bounds U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) that require input of one system-dependent constant C still require less information about the dynamics since extreme value upper bounds (S11) partly also require knowledge about the first-passage weight w1 and ⟨τ⟩ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' COMPLETE PROOF OF CONCENTRATION INEQUALITIES AND THEIR ASYMPTOTICS In this section we provide various additional details on the upper bounds U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) of the Letter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, we prove the required bounds on the cumulant generating function, compute their corresponding Cram´er transform, and give further information about the large-sample limit n → ∞, as well as the model-free version of the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theorem 1: Cram´er-Chernoff bound for the right tail τ ≥ ⟨τ⟩ We begin with the right tail, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' upwards deviations such that τ ≥ ⟨τ⟩, and start by proving a bound for the moment generating function of the deviation of the first-passage time τ from the mean ⟨τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Using the spectral representation (S1) and the inequality x ≤ ex−1, ∀x ∈ R, we find ⟨eλ(τ−⟨τ⟩)⟩ = e−λ⟨τ⟩ � k>0 wk 1 − λ/µk ≤ exp � −λ⟨τ⟩ + � k>0 wk 1 − λ/µk − 1 � (S13) for all λ < µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, for |λ| < µ1 we may further expand the sum � k>0 wk 1−λ/µk = � m≥0 λm � k>0 wk/µm k using the geometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Recall that the moments are given by ⟨τ m⟩ = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' � k>0 wk/µm k , such that we obtain ⟨eλ(τ−⟨τ⟩)⟩ ≤ exp � �� m≥2 λm � k>0 wk µm k � � = exp � λ2 ⟨τ 2⟩ 2 + � m>2 λm � k>0 wk µm k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S14) 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Comparison between Cram´er-Chernoff-type upper bounds U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and extreme value upper bounds for a spatially confined Brownian search process in dimensions (a,e,i) d = 1 and (b,f,j) d = 3, and discrete-state Markov jump processes for (c,d,k) the inferred model of calmodulin and (d,h,l) a 8-state toy protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (a-d) Scaled probabilities P1/n(sgn(t)δτ n ≥ |t|) that the sample mean τ n inferred from n ≥ 1 realizations deviations from ⟨τ⟩ by more than t in either direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Right tail areas are shown for t > 0 and left for t < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cram´er-Chernoff upper bounds U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) as black and extreme value upper bounds as dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Corresponding lower bounds L± n (t) are depicted as red lines and symbols denoted scaled empirical deviation probabilities obtained from the statistics of τ n for different n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (e-h) Quality factor Q as a function of n for different fixed relative deviations µ1t (see star symbols (a-d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (i-l) Sample size n∗ (blue) for which both upper bounds are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', Q = 1, as a function of re-scaled deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since µ1 ≤ µk>1 and all first-passage weights wk are positive (due to equilibrium initial conditions) we find ⟨eλ(τ−⟨τ⟩)⟩ ≤ exp � λ2 ⟨τ 2⟩ 2 + � m>2 λm � k>0 wk µm k � ≤ exp � λ2 ⟨τ 2⟩ 2 + � m>2 λm µm−2 1 � k>0 wk µ2 k � ≤ exp � λ2 ⟨τ 2⟩ 2 � 1 + λ µ1 − λ �� = exp � λ2 ⟨τ 2⟩/2 (1 − λ/µ1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Introducing ψδτ(λ) ≡ ln⟨eλδτ⟩, with δτ = τ − ⟨τ⟩ for the right tail, we immediately identify the upper bound ψδτ(λ) ≤ λ2 2 ⟨τ 2⟩ 1 − λ/µ1 = ˜λ2 2 C 1 − ˜λ ≡ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) τ ≥ ⟨τ⟩, (S15) which concludes the derivation of the upper expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (6) of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that we further have introduced the dimensionless quantities ˜t ≡ µ1t, C = µ2 1⟨τ 2⟩, and ˜λ = λ/µ1 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In the case of general initial conditions p0(x0) ̸= peq(x0) we must simply replace µ2 1⟨τ 2⟩ → C from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Next, we find the optimizing value of ˜λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', we compute the Cram´er transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S15) defined as φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≡ sup ˜λ∈[0,1) [˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C)] = sup ˜λ∈[0,1) � ˜λ˜t − ˜λ2 2 C 1 − ˜λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S16) 7 φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) is differentiable, non-negative, convex, and increasing on ˜ lambda ∈ [0, 1), which implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S16) can be obtained by differentiation of ˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) with respect to ˜λ, hence φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = ˜λ†˜t − φδτ(˜λ†;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) where the optimum ˜λ† solves φ′ δτ(˜λ†;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Accordingly, we find the supremum to be attained at ˜λ†(˜t) = 1 − 1/ � 1 + 2˜t/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For convenience we further introduce the auxiliary function h+(u) ≡ 1 + u − √1 + 2u such that we finally arrive at φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = Ch+(˜t/C) = Ch+(µ1t/C), 0 ≤ t ≤ ⟨τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S17) By using Chernoff’s inequality we subsequently obtain the upper bound P(δτ n ≥ t) ≤ e−nφ∗ δτ (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='C) ≡ U+ n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) for 0 ≤ t ≤ ∞ which completes the proof of Theorem 1 and thus the first announced inequality (7) in the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (a) ˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 ˜λ˜t φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='02 (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 (e) ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) ˜λ˜t − φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ˜λ˜t − φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='003 (f) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 (g) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='008 (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Illustration of the Cram´er-Chernoff bounding method for the right tail with ˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 and parameters for spatially confined Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for the model of calmodulin (c,d) and a 8-state toy protein (d,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Top row depicts bounds of the cumulant generating function φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (black) and φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (yellow) as a function of ˜λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bottom row shows the differences ˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (red) and ˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The corresponding suprema are obtained at ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (dotted lines) and define the Cram´er transforms φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (compare top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For all considered models we demonstrate φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≥ φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) as derived in the maintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note for the panels (b,f) we have φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ⪅ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ⪆ φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) since C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='99 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theorem 2: Cram´er-Chernoff bound for the left tail ⟨τ⟩ < τ Next we turn to the left tail, τ < ⟨τ⟩, where the corresponding moment generating function analogously reads ⟨eλ(⟨τ⟩−τ)⟩ = eλ⟨τ⟩ � k>0 wk 1 + λ/µk ≤ exp � λ⟨τ⟩ + � k>0 wk 1 + λ/µk − 1 � for λ < µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Using equivalent arguments as for the right tail above we may further write eλ(⟨τ⟩−τ)⟩ ≤ exp � �� m≥2 (−λ)m � k>0 wk µm k � � = exp � λ2 ⟨τ 2⟩ 2 + � m>2 (−λ)m � k>0 wk µm k � ≤ exp � λ2 ⟨τ 2⟩ 2 + � m>0 λ2m µ2m−2 1 � k>0 wk µ2 k � = exp � λ2 ⟨τ 2⟩/2 1 − (λ/µ1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S18) Recall the definition of the cumulant generating function, ψδτ(λ) ≡ ln⟨eλδτ⟩, such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S18) directly yields ψδτ(λ) ≤ λ2 2 ⟨τ 2⟩ 1 − (λ/µ1)2 = ˜λ2 2 C 1 − ˜λ2 ≡ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (S19) 8 which completes the derivation of the lower expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (6) of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that for the left tail we have δτ = ⟨τ⟩ − τ and we again let ˜t ≡ µ1t, ˜λ ≡ λ/µ1, and C ≡ µ2 1⟨τ 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In the case of general initial conditions p0(x0) ̸= peq(x0) we must simply replace µ2 1⟨τ 2⟩ → C from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Analogous to the right tail we next compute the Cram´er transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S19), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≡ sup ˜λ∈[0,1) [˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C)] = sup ˜λ∈[0,1) � ˜λ˜t − ˜λ2 2 C 1 − ˜λ2 � , (S20) where we find the optimal value ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) to be determined by the transcendental quartic, ˜λ†(˜t) : (1− ˜λ2)2 −C˜t˜λ = 0 with C˜t ≡ C/˜t, which we solve according to the method of Descartes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' First, we re-arrange the quartic as ˜λ4−2˜λ2−C˜t˜λ+1 = 0 and make the factorization ansatz (˜λ2 − y˜t˜λ2 + w˜t)(˜λ2 + y˜t˜λ2 + z˜t) = 0 w˜t + z˜t − y2 ˜t = −2 y˜t(w˜t − z˜t) = −C˜t z˜tw˜t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S21) The system of equations (S21) is solved by w˜t(y˜t) = (y2 ˜t − 2 − C˜t/y˜t)/2, z˜t(y˜t) = (y2 ˜t − 2 + C˜t/y˜t)/2, (S22) where y2 ˜t ≡ Y˜t is the solution of the cubic Y 3 ˜t − 4Y 2 ˜t − C2 ˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, since the discriminant D is strictly negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D = −28C2 ˜t − 33C4 ˜t < 0, the qubic has only one real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The corresponding depressed qubic reads ˜t3 − 24/3˜t − (27/33 + C2 ˜t ) = 0 with ˜tY˜t − 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Let p = −24/3 < 0 and q = −(27/33 + C2 ˜t ) < 0 then 22p3 + 33q2 = −212/33 + 33(27/33 + C2 ˜t )2 > 0 for any ˜t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We can express the unique real root as y2 ˜t = 4 3 � 1 + 2 cosh �1 3arcosh � 1 + 33C2 ˜t 27 ��� (S23) and y = ± � y2 with y2 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S23) can now be plugged into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S22) to obtain w˜t(y) and z˜t(y) that are required to solve the pair of quadratic equations (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The four roots of the transcendental quartic are hence given by ˜λ1(˜t) = y˜t 2 � 1 + � 1 − 4w˜t(y˜t)/y2 ˜t � , ˜λ2(˜t) = y˜t 2 � 1 − � 1 − 4w˜t(y˜t)/y2 ˜t � , ˜λ3(˜t) = −y˜t 2 � 1 − � 1 − 4z˜t(y˜t)/y2 ˜t � , ˜λ4(˜t) = −y˜t 2 � 1 + � 1 − 4z˜t(y˜t)/y2 ˜t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S24) Moreover, we find w˜t(y˜t)/y2 ˜t = (1 − 2/y2 ˜t − C˜t/y3 ˜t )/2 and z˜t(y˜t)/y2 ˜t = (1 − 2/y2 ˜t + C˜t/y3 ˜t )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since y˜t > 0 while ˜λ ∈ [0, 1), ˜λ2, ˜λ3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S24) are excluded automatically (note also that the square root in ˜λ2, ˜λ3 becomes complex for ˜t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We also have lim˜t→∞ y2 ˜t = 4 and lim˜t→∞ w˜t(y˜t) = 1 such that lim˜t→∞ ˜λ1 = ˜λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Conversely, we find that lim˜t→0 ˜t2/3y2 ˜t = C2/3 = lim˜t→0 ˜t2/3C˜t/y˜t such that lim˜t→0 w˜t(y˜t) = −1 while lim˜t→0 w˜t(y˜t)y2 ˜t = −C2/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Therefore, lim˜t→0 ˜λ1 = y˜t → ∞ whereas lim˜t→0 ˜λ2(˜t) = y˜t × 0/2 ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We recall that ˜λ ∈ [0, 1) which therefore excludes ˜λ1(˜t) and identifies ˜λ†(˜t) = ˜λ2(˜t) as the supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Finally, we introduce the auxiliary functions g(u) ≡ 2 √ 3 � 1 + 2 cosh �1 3arcosh � 1 + 33 27u2 ���1/2 and Λ(u) ≡ 1 2 � g(u) − � 4 + 2/g(u)u − g(u)2 � , (S25) as well as h−(u) ≡ Λ(u)u − 1 2 Λ(u)2 1 − Λ(u)2 (S26) which allows us to obtain and write the Cram´er transform as φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = Ch−(˜t/C) = Ch−(µ1t/C), 0 ≤ t ≤ ⟨τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S27) In the last step we use Chernoff’s inequality to obtain the bound P(δτ n ≥ t) ≤ e−nφ∗ δτ (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='C) ≡ U− n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) for 0 ≤ t ≤ ⟨τ⟩ which completes the proof of Theorem 2 and hence the derivation of the lower expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (7) of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (a) ˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 ˜λ˜t φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='02 (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 (e) ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) φ∗ δτ (˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) ˜λ˜t − φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ˜λ˜t − φδτ (˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='003 (f) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='10 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 (g) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2 ˜λ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='008 (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Illustration of the Cram´er-Chernoff bounding method for the left tail with ˜t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 and parameters for spatially confined Brownian search process in dimensions d = 1 (a,e) or d = 3 (b,f), and discrete-state Markov jump processes for the model of calmodulin (c,d) and a 8-state toy protein (d,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Top row depicts bounds of the cumulant generating function φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (black) and φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (yellow) as a function of ˜λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bottom row shows the differences ˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) (red) and ˜λ˜t − φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (green) as a function of ˜λ, respectively (see also top row with ˜λ˜t in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The corresponding suprema are obtained at ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and ˜λ†(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (dotted lines) and define the Cram´er transforms φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (compare top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For all considered models we demonstrate φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≥ φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) as derived in the maintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note for the panels (b,f) we have φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ⪅ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ⪆ φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) since C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='99 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Behavior of upper bounds U± n (t) for large sample sizes Here, we present some further remarks about the limit of large sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Asymptotically as n → ∞, U± n (t) is substantial only for ˜t/C ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For the right tail bound h+(u) we immediately find that for u ≪ 1 we can Taylor expand √1 + 2u = 1 + u − u2/2 + O(u3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Consequently we directly obtain h+(u) = −u2/2 + O(u3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', the upper tail is sub-Gaussian for small deviations and will converge to a Gaussian as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For the left tail we furthermore have arcosh(1 + x) = ln(1 + x + � x(x + 2)) and thus limx→∞ arcosh(1 + x) = ln(2x) − 1/(2x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' As a result it follows that 1 3 limu→0 arcosh(1 + 33/27u2) ≃ 1 3 ln(33/26u2) = ln(3/4u2/3) − u4212/37 and thus lim u→0 g(u) ≃ 2 √ 3{1 + 2 cosh ln(3/4u2/3)}1/2 = 2 √ 3[1 + 3/4u2/3]1/2 (S28) = u−1/3[1 + 4u2/3/3]1/2 = u−1/3[1 + 2u2/3/3 + O(u4/3)] = u−1/3 + 2 3u1/3 + O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S29) A lengthy but straightforward calculation subsequently reveals that limu→0 Λ(u) = u − O(u3) such that lim u→0 Λ(u)2 1 − Λ(u)2 ≃ u2 1 − u2 = u2 + O(u4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S30) We therefore have that limu→0 h−(u) = u2/2 − O(u4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', both tails are sub-Gaussian for ˜t/C ≪ 1 with C ≡ µ2 1⟨τ 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Proof of bounds on C and model-free concentration inequalities Notably, system details only enter the Cram´er transforms (S17) and (S27) (and consequently upper bounds on the deviation probability due to Chernoff’s inequality) in the form of a system-specific constant C ≡ µ2 1⟨τ 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that here we only allow for equilibrium initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Recalling that the moments of the first-passage time τ are expressed as ⟨τ m⟩ = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' � k>0 wk/µm k allows us to write 0 ≤ 2w1 µ2 1 ≤ ⟨τ 2⟩ = 2 � k>0 wk µ2 k ≤ 2 � k>0 wk µ2 1 = 2 µ2 1 (S31) 10 for equilibrium initial conditions where we have used that wk are non-negative, normalized, and µ1 ≤ µk>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Conse- quently, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S31), we immediately find that the system-constant itself is bounded 0 ≤ 2w1 ≤ C ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that analogous considerations can be used to more generally obtain 0 ≤ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='w1 ≤ µm 1 ⟨τ m⟩ ≤ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' for the m-th moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The fact that C ∈ (0, 2] can now be further leveraged to arrive at the model-free bounds (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (11) in the Letter) which require no information about the underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Recall the upper bounds of the cumulant generating function φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) and their corresponding Cram´er transform φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = � � � � � � � ˜λ2 2 C 1 − ˜λ τ ≥ ⟨τ⟩ ˜λ2 2 C 1 − ˜λ2 τ < ⟨τ⟩, and φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = � Ch+(˜t/C) τ ≥ ⟨τ⟩ Ch−(˜t/C) τ < ⟨τ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S32) Since φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) is monotonically increasing in C it follows that φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2), ∀˜λ ∈ [0, 1) (see Figs S2 and S3 top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' By definition of φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≡ sup˜λ∈[0,1)(˜λ˜t−φδτ(˜λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C)) this bound in turn implies that φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≥ φ∗ δτ(˜t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) (compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S2 and S3 bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' With Chernoff’s inequality we moreover arrive at P(δτ n ≥ t) ≤ e−nφ∗ δτ (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='C) ≤ e−nφ∗ δτ (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2) and hence U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) ≤ U± n (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2) which completes the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (11) in the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' MODEL SYSTEMS AND DETAILS ON NUMERICAL METHODS In the Letter we exemplify our results by considering a Brownian molecular search process in dimensions d = 1 and d = 3, as well as discrete-state Markov-jump models of protein folding for a 8-state toy protein and the experimentally inferred model of calmodulin (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In this section we present further details on the model systems and their numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Continuous-time discrete-state Markov jump process As illustrative discrete-state continuous-time Markov-jump models of protein folding we consider a simple 8-state toy protein [2, 6] and further use the experimentally inferred folding network of the cellular calcium sensor protein calmodulin [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since we consider equilibrium initial conditions, proteins start from an initial state drawn from the equilibrium density ˜peq(x)—note that the tilde denotes that the absorbing target is excluded—from which they search the native state a (here a = (1, 1, 1) for the 8-state model and a = F1234 for calmodulin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Arrows in the networks denote possible transitions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' a transition from state i to state j that occurs with the corresponding rate Lji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We consider reversible dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', the resulting transition matrix ˆL of the relaxation process satisfies detailed balance peq,j/peq,i = Lji/Lij = exp(Fi − Fj) and transitions rates are connected to the free energy of the states Fi [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We recall that the first-passage time density ℘a(t) can be evaluated by using the spectral representation (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To this end we set up the modified transition matrix, adopting in this section the Dirac bra-ket notation, ˆLa = ˆL−|a⟩⟨a| where |a⟩ ≡ (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' , 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' )⊺ defines a vector with all entries zero expect at the a-th position of the absorbing state where it equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' This effectively removes all transitions that correspond to jumps leaving the absorbing state a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Next, we carry out an eigendecomposition of ˆLa and determine the eigenvalues µk, right eigenvectors |φR⟩, and left eigenvectors ⟨φL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We subsequently use obtained eigenmodes to compute the first-passage weights wk(x0) = −⟨a|φR k ⟩⟨φL k|x0⟩ (see [1, 2]), and recall that µk and wk determine the moments according to ⟨τ m⟩ = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' � k>0 wk/µm k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Corresponding relevant parameters of the Markov jump models are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Next we give further details on how corresponding transition rates are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Parameters for the Markov jump models for the 8-state toy protein and the inferred model of calmodulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Listed are values for the first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Model µ1 w1 µ2 w2 µ3 w3 µ4 w4 µ5 w5 µ6 w6 µ7 w7 ⟨τ⟩ ⟨τ 2⟩ Toy protein 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='337 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='583 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='001 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0001 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='060 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='713 Calmodulin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='651 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='349 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='097 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='98E-5 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='749 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='42E-9 1581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='629 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='52E-6 – – – – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='479 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='958 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transitions rates of the 8-state toy protein model For the 8-state toy protein model we randomly generate a free energy level Fi for each state i ∈ {1, 2, 3, 4, 5, 6, 7, 8} with Fi uniformly distributed within the interval 0 ≤ Fi ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transition rates that satisfy detailed balance are then obtained using the ansatz ki→j ≡ Lji = exp(∆Fi/2) and kj→i ≡ Lij = exp(−∆Fi/2), (S33) where ∆Fi ≡ Fi − Fj and thus ln(Lji/Lij) = ∆Fi = Fi − Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Obtained individual transition rates are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transition rates for the 8-state toy protein model obtained via the ansatz described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' transition rate ki→j transition rate ki→j transition rate ki→j transition rate ki→j 1 → 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='878 2 → 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='648 3 → 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='549 5 → 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='106 2 → 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='327 5 → 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='421 7 → 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='00994 8 → 5 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='477 1 → 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='00463 2 → 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='527 4 → 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='358 6 → 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='712 3 → 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='507 6 → 2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0577 6 → 4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='457 8 → 6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='794 1 → 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='326 3 → 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='109 4 → 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='523 7 → 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='322 4 → 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='623 5 → 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0371 7 → 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='670 8 → 7 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='998 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transitions rates of the calmodulin protein model In the experimental setup a constant external force f, a so-called pretension, is applied to the calmodulin protein via optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Folding and unfolding processes are observed at different pretensions ranging from 6pN to 13 pN and corresponding force-dependent transition rates ki→j(f) = Lji(f) between two conformational states i and j are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that i, j ∈ {Unfold, F12, F123, F23, F34, F1234} and we further map states according to Unfold ↔ 1, F12 ↔ 2, F123 ↔ 3, F23 ↔ 4, F34 ↔ 5, and F1234 ↔ 6 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For our purposes we choose, without loss of generality, a pretension of f = 9 pN and obtain the corresponding measured transitions rates from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S8 in the Supplementary Material of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Clearly, experimental transitions rates are accompanied with measurement uncertainties which is reflected in slight “deviations” from a mathematically precise definition of detailed balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To mitigate this issue, and to ensure that transition rates precisely obey detailed balance ki→jpeq,i = kj→ipeq,j, we further have to slightly adjust the rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' First, we compute the invariant density peq from the experimental rates and obtain a corresponding free energy level Fi = − ln(peq,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Next, we use the ansatz (S33), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', Lji = Ai exp(∆Fi/2) and Lij = Ai exp(−∆Fi/2) where we introduce a constant Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Finally, Ai’s are chosen such that resulting transition rates fall within experimental error bars in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Obtained transition rates are listed in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Transition rates of the Markov jump model for the calmodulin protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rates are extracted from the Supplemental Material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [7] and modified such that they obey detailed balance precisely according to the maintext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' transition rate ki→j transition rate ki→j transition rate ki→j 1 → 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='997 1 → 4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='439 1 → 5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='330 2 → 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='774 4 → 1 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='968 5 → 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='121 5 → 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='749 2 → 3 1514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='820 2 → 6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='441 6 → 5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='326 3 → 2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0661 6 → 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='922 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Spatially confined Brownian molecular search process We also test our theory for Markov processes on a continuous state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' More precisely, we consider the spatially confined diffusive search of a Brownian particle in a d-dimensional unit sphere with a reflecting boundary at R = 1 and a perfectly absorbing spherical target of radius 0 < a < 1, here a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, in the center (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The closest distance of the particle to the surface of the absorbing sphere at time t is a confined Bessel process (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [2, 9, 10]) which time evolution obeys the Itˆo equation dxt = (d − 1)x−1 t dt + √ 2dWt, (S34) 12 where dWt is the increment of a Wiener process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gaussian white noise) with ⟨dWt⟩ = 0 and ⟨dWtdWt′⟩ = δ(t−t′)dt, and we have set, without loss of generality, D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The general case with any 0 < D < ∞ and a sphere of radius R is covered by expressing time in units of R2/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For d = 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S34) reduces to a 1 dimensional Brownian motion which has the equilibrium first-passage weights weq k = 2 π2 1 − sin [(k − 1)π] (k − 1/2)2 (S35) and matching first-passage eigenvalues are obtained as µk = π2(k − 1/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, for d = 3 the first-passage time probability density of the Bessel process can be evaluated exactly and has the equilibrium weights weq k = 2 µk 3a2 1 − a3 tan[(1 − a)√µk] + 1 √µk (1 − a) tan[(1 − a)√µk] − a √µk , (S36) with the first-passage eigenvalues µk being the solutions of the transcendental equation √µk = tan([1 − a]√µk) that can be solved analytically using Newton’s series [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Relevant parameters for the spatially confined Brownian search process with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Parameters for the spatially confined Brownian molecular search process in dimensions 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Listed are values for the first 5 first-passage eigenvalues µk, first-passage weights wk, and the first ⟨τ⟩ and second moment ⟨τ 2⟩, respectively a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Model µ1 w1 µ2 w2 µ3 w3 µ4 w4 µ5 w5 ⟨τ⟩ ⟨τ 2⟩ 1D Brownian motion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='811 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0901 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0324 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0165 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='01001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='267 3D Bessel process 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='994 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='00277 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='926 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='163E-4 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='037 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='573E-4 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='516 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='742E-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='739 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='509 a For the numerical evaluation of ⟨τ⟩ and ⟨τ 2⟩ as listed we truncate the sum after M = 1000 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Statistics of first-passage times ⟨τ⟩ and the sample mean τ n Here we provide some further details on the sampling method used to obtain the statistics of (i) the first-passage time τ and (ii) the sample-mean τ n ≡ � i τi/n at some fixed value n for our considered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' We recall that after determining the first-passage eigenvalues µk and first-passage weights wk, the first-passage time density ℘a(t) (S1) and survival probability Sa(t) (S2) are fully characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To now sample the random variable τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' individual realizations of the first-passage process, we employ the so-called inversion sampling method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' This method allows us to generate independent samples of τ from ℘a(t) given its cumulative distribution function (CDF) which is directly related to the survival probability according to 1 − Sa(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Note that for discrete-state dynamics the number of states M is finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' , M and therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S2) (and hence the CDF) is a finite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In contrast, for continuous-state dynamics we formally have M = ∞, meaning that sums are here not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For the following numerical evaluation of the spatially confined Brownian search process we therefore truncate the sum after M = 1000 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The first-passage time densities ℘a(t) obtained via inversion sampling (symbols) for all considered models are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S4a-d and corroborated by the corresponding analytical result (S1) (dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2a-d in the Letter empirical probabilities that τ n − ⟨τ⟩ lies within a desired range of ± 10% of the longest first-passage time scale µ−1 1 , P(µ1[τ n − ⟨τ⟩] ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1]), are computed using statistics of the sample mean τ n by fixing n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', the number of individual realizations the average is taken over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, we have n ∈ {1, 2, 3, 5, 10, 20, 30, 40, 50, 75, 100, 150, 200, 300, 400, 500}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Subsequently, for each individual fixed n the sample mean τ n itself is sampled a total of N = 106 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' That is, we first draw n first-passage times τ, compute τ n by averaging over the drawn n realizations, and finally repeat this step N = 106 times to obtain statistics of τ n for all n values introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Probability densities of the sample mean are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S4e-h for n ∈ {3, 5, 10, 20} and all model systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Corresponding true mean first-passage times ⟨τ⟩ are highlighted in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 2e-h of the Letter the probabilities to deviate more than t in either direction, P(±[τ n − ⟨τ⟩] ≥ t), are computed from analogous statistics of the sample mean τ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Since we also consider empirical probabilities for rare events with large deviations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' large µ1t) we however require substantially more statistics of τ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To this end we now have N = 107 for n ∈ {1, 3} and N = 1011 for n ∈ {5, 10, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In addition it should be further noted that we re-scale obtained probabilities according to P1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To compute an empirical deviation probability where e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' P1/20 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1 one would be thus required to sample rare events that occur with a probability of ≃ 10−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 13 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3a of the Letter each data point corresponds to the relative error µ1(τ n − ⟨τ⟩) (note that µ1 and ⟨τ⟩ are different for each model) where the sample mean τ n is again obtained by first fixing n and then sampling n first-passage times τ according to the inversion sampling method and subsequently taking the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 10−9 10−5 10−1 t 10−2 101 104 ℘a(t) (a) 10−5 10−1 t 10−2 100 102 (b) 10−4 10−2 100 t 10−2 10−1 100 101 (c) 10−5 10−2 t 100 102 (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 τ n 0 1 2 3 4 5 p(τ n) ⟨τ⟩ (e) n = 3 n = 5 n = 10 n = 20 0 2 4 6 τ n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='8 (f) 0 2 4 τ n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 (g) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 τ n 0 1 2 3 4 5 (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Inversion sampling of first-passage statistics for a spatially confined Brownian search process in dimensions (a,e) d = 1 and (b,f) d = 3, and discrete-state Markov jump processes for (c,d) the inferred model of calmodulin and (d,h) a 8-state toy protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (a-d) First-passage time density ℘a(t) obtained using inversion sampling (symbols) and analytical result as black dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (e-h) Empirical probability density of the sample mean τ n for different n values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' True mean first-passage times ⟨τ⟩ are shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' UNCERTAINTY QUANTIFICATION WITH CONFIDENCE INTERVALS In this section we extend the discussion and present some further details on the confidence intervals introduced in the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Our derived upper bounds U± n (t) can be applied to construct non-asymptotic performance guarantees such as confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In particular, they can be employed to bound the probability that δτ n ≡ τ n − ⟨τ⟩ is found to be in some interval [−t− α−, t+ α+], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', P(δτ n ∈ [−t− α−, t+ α+]) = P(−t− α− ≤ δτ n ≤ t+ α+) = P(δτ n ≥ −t− α− ∩ δτ n ≤ t+ α+) ≥ 1 − P(δτ n ≤ −t− α−) − P(δτ n ≥ t+ α+) ≥ 1 − U− n (t− α−) � �� � ≡α− − U+ n (t+ α+) � �� � ≡α+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S37) In passing from the second to the third line we have applied Boole’s second inequality, and from the third to forth line we use bounds (7) of the Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In the last line we additionally introduced acceptable right and left tail error probabilities α±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The implicit interval [−t− α−, t+ α+] therefore defines a confidence interval at a confidence level of 1 − α with α ≡ α+ + α−, and α+ + α− < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In general the choice of the confidence interval for a fixed probability 1 − α is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Some common options in the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [12, 13]), all having the same confidence level, are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' One common choice are so-called central intervals (blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S5) which correspond to equal tail probabilities α+ = α− = α/2 for the complementary intervals [−⟨τ⟩, −t− α−] and [t+ α+, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Notably, we remark that central confidence intervals do not generally imply that t+ α+ and t− α− are equidistant from another, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', t+ α+ ̸= t− α−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 14 As an alternative one could likewise choose t+ α+ = t− α− ≡ ∆t/2, which subsequently leads to the symmetric interval [−∆t/2, ∆t/2] with total length ∆t (see red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Analogously, a symmetric interval does not necessarily imply that the corresponding tail probabilities are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', in general α+ ̸= α−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Both considerations above lead to two-sided intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' However, another possible choice includes the fully asymmetric intervals [−⟨τ⟩, t+ α+] and [−t− α−, ∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', one-sided intervals with a corresponding confidence level 1 − α+ (for the upper limit t+ α+) and 1 − α− (for the lower limit t− α−), respectively, P(±δτ n ≤ t± α±) ≥ 1 − α±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S38) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 µ1t+ α+ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 µ1t− α− n = 15 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='9 central int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 µ1t+ α+ n = 20 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 1 µ1t+ α+ n = 30 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='0 1 − α FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Contour plot of different choices of possible two-sided confidence intervals [−µ1t− α−, µ1t+ α+] for a fixed confidence level α and (a) n = 15, (b) n = 20, (c) n = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Contour lines for α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='5} are depicted in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Specific choices of central and symmetric are shown in blue and red, respectively, and we let C = 1 for all panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Confidence intervals are practically useful as they answer questions such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' : How many realizations are required to achieve a desired accuracy with a specified probability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Or: For a given number of realizations a desired accuracy is achieved with at least what probability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' In the case of symmetric confidence intervals t+ α+ = t− α− (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S5 red lines) the interval endpoints are implicitly defined via the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S37) which is easily solved using standard root-finding procedures like the bi-section method [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' The same holds true for other interval choices, however, when specifying the error probabilities α± directly—as done for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' two-sided central intervals (α± = α/2) or one-sided intervals—it suffices to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S38) with the respective α±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hereby, the lower confidence limit t− α− is again easily obtained using standard root-finding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Notably, the upper confidence limit t+ α+ can now be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' To show this we consider U+ n (t+ α+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = α+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', we identify the t+ α+ that solves 0 = −nCh+(µ1t+ α+/C) − ln(α+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S39) The roots are identified as t1 = −ln (α+) µ1n − √ 2 � − ln (α+) µ1 � n/C and t2 = −ln (α+) µ1n + √ 2 � − ln (α+) µ1 � n/C , (S40) and we identify t+ α+ = t2 as the relevant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Having obtained an explicit expression for t+ α+ further allows us to re-insert it into the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S38), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', we find that with a probability of at least 1 − α+ δτ n ≤ −ln (α+) µ1n + √ 2 � − ln (α+) µ1 � n/C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (S41) The required number of realizations n∗ to ensure with a probability of at least 1 − α that δτ n is found within some interval [−t− α−, t+ α+] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' symmetric interval in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 3b) is analogous identified according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' (14) in the Letter Un∗(t+ α+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) + Un∗(t− α−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C) = α, (S42) 15 which once again is readily solved via e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' the bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Moreover, in the case of one-sided intervals one immediately finds the corresponding analytical expression n∗ ≥ − ln(α±) Ch±(µ1t/C), (S43) where n∗ denotes the required number to ensure that ±δτ n ≤ t with at least 1 − α±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' ∗ agodec@mpinat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content='de [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 20, 112002 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Hartich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Godec, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 52, 244001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Siegert, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 81, 617 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=', Springer Series in Synergetics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 13 (Springer-Verlag, Berlin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Bowman, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pande, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' No´e, An Introduction to Markov State Models and their Application to Long Timescale Molecular Simulation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 797 (Springer Science & Business Media, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Stigler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Ziegler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gieseke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Gebhardt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rief, Science 334, 512 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [8] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Seifert, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 10, 171 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Pitman, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' 7, 511 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Barkai, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Aghion, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Kessler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' X 4, 021036 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Devroye, Non-Uniform Random Variate Generation (Springer New York, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Cowan, Statistical Data Analysis (Oxford University Press, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Lista, Statistical Methods for Data Analysis in Particle Physics (Springer International Publishing, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Burden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Faires, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} +page_content=' Burden, Numerical Analysis (Cengage Learning, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFAT4oBgHgl3EQf5R5J/content/2301.08732v1.pdf'} diff --git a/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/2301.05626v1.pdf.txt b/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/2301.05626v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..963f9f3fe44c86f8a1d155d6d11c8afbda6dbf0a --- /dev/null +++ b/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/2301.05626v1.pdf.txt @@ -0,0 +1,1078 @@ +Channel Measurement for Holographic MIMO: +Benefits and Challenges of Spatial Oversampling +Tengjiao Wang∗, Yongxi Liu†, Ming Zhang†, Wei E. I. Sha‡, Cen Ling∗, Chao Li∗, Shaobo Wang∗ +∗Wireless Network RAN Research Department, Huawei Technologies CO., Ltd, Shanghai, China +†School of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi, China +‡College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China +Emails: ∗{wangtengjiao6, lingcen, lichao18, shaobo.wang}@huawei.com, +†liuyongxi@stu.xjtu.edu.cn, ming20.zhang@xjtu.edu.cn, ‡ weisha@zju.edu.cn +Abstract—In this paper, the channel of an indoor holographic +multiple-input multiple-output (MIMO) system is measured. It +is demonstrated through experiments for the first time that +the spatial oversampling of holographic MIMO systems is able +to increase the capacity of a wireless communication system +significantly. However, the antenna efficiency is the most crucial +challenge preventing us from getting the capacity improvement. +An extended EM-compliant channel model is also proposed +for holographic MIMO systems, which is able to take the +non-isotropic characteristics of the propagation environment, +the antenna pattern distortion, the antenna efficiency, and the +polarization characteristics into consideration. +Index Terms—Holographic MIMO, massive MIMO, spatial +oversampling, channel measurement, electromagnetic informa- +tion theory. +I. INTRODUCTION +In order to increase the spectral efficiency and energy +efficiency for the future 5G-Advanced and 6G wireless com- +munication systems, the concept of holographic multiple-input +multiple-output (MIMO) is proposed recently [1]. By inte- +grating an infinite number of antennas into a limited surface, +holographic MIMO is expected to fully exploit the propagation +characteristics offered by the electromagnetic channel and ap- +proach the fundamental performance limit [2]. For holographic +MIMO systems, both the accurate channel modeling and the +realistic performance evaluation are key problems. +In the literature, many efforts have been devoted to accu- +rately model the channel of holographic MIMO systems [3]– +[7]. In [3], a Fourier plane-wave series expansion-based chan- +nel model is proposed for holographic MIMO systems. Based +on the Fourier spectral representation, it provides a physically +meaningful model capturing the propagation characteristics of +the electromagnetic (EM) wave. In [4], the authors extend the +Fourier plane-wave channel model to a multi-user scenario. +In [5], the non-isotropic scattering environment is further con- +sidered by using the von Mises-Fisher (VMF) distributions [8]. +Then in our previous works [6], [7], an EM-compliant channel +model is proposed. By combining the VMF distributions [8] +and the 3GPP TR 38.901 channel model [9], a realistic angular +power spectrum is modeled. The non-ideal factors caused by +mutual coupling at the transceivers [10], including the antenna +pattern distortion and the decrease of antenna efficiency are +also accounted for. However, in the state-of-the-art channel +models [3]–[7], the polarization of EM wave is not taken into +consideration. The polarization is an inherent characteristic of +EM waves and will have significant impact on the performance +of holographic MIMO systems. +Another key problem for holographic MIMO is the accurate +performance evaluation. In [5], the ergodic capacity of a +single-user holographic MIMO system is analyzed, which +shows the capacity improvement of holographic MIMO over +conventional MIMO. In [11], the capacity of a single-user +holographic MIMO system is investigated from a continuous +point of view, the capacity enhancement is also demonstrated. +In our previous work [6], both the single-user and the multi- +user downlink channel capacities of holographic MIMO sys- +tems are investigated. However, the performance evaluations +in the state-of-the-art researches [5], [6], [11] are all numerical +simulations based on theoretical models. No experiments have +been done to verify the accuracy of the performance evalua- +tions. +In this paper, we try to solve the above two problems +for holographic MIMO systems. Firstly, an extended EM- +compliant channel model is proposed based on the channel +model in our previous work [6]. Secondly, the channel of holo- +graphic MIMO is measured through real-world experiments, +and the performance is evaluated based on the measurement +results. The contributions of this paper can be summarized as +follows: +• An extended EM-compliant channel model is proposed +for holographic MIMO systems. In the extended model, +not only the non-isotropic characteristics of the propaga- +tion environment, the antenna pattern distortion, the an- +tenna efficiency, but also the polarization of the antennas +and the propagation environment can be modeled. +• Based on the extended channel model, the real-world +channel for holographic MIMO systems is measured for +the first time in an indoor environment. An experiment +is carefully designed, in which an electrically controlled +virtual dense array is used to realize arbitrary element +spacings of holographic MIMO. +• The channel capacity of holographic MIMO systems is +evaluated according to the measurement results. It is +arXiv:2301.05626v1 [cs.IT] 13 Jan 2023 + +demonstrated that the spatial oversampling of holographic +MIMO is able to provide a two to three times capacity +enhancement, without considering the antenna efficiency +loss. The antenna efficiency is the most crucial challenge +for holographic MIMO. +The rest of this paper is organized as follows. In Section II, +the proposed extended EM-compliant channel model for holo- +graphic MIMO is explained in details. Then, the setup for the +channel measurement is given in Section III. The measurement +results and the corresponding performance evaluations are +provided in Section IV. Finally, this paper is concluded in +Section V. +II. EXTENDED EM-COMPLIANT CHANNEL MODEL +A. EM-Compliant Channel Model +In this subsection, the EM-compliant channel model for +holographic MIMO in our previous work [6] is briefly in- +troduced. The central frequency and wavelength are denoted +by fc and λ. Planar antenna arrays with size {Lx +R, Ly +R} and +{Lx +S, Ly +S} are equipped at the receiver and the transmitter, +respectively. The numbers of antenna elements are denoted +by NR and NS. The spacings between antenna elements are +denoted by {∆x +R, ∆y +R} and {∆x +S, ∆y +S}. The coordinates of the +antenna elements are represented by rq = (rx +q , ry +q, rz +q), q = +1, 2, · · · , NR and sp = (sx +p, sy +p, sz +p), p = 1, 2, · · · , NS, respec- +tively. +According to [6], the channel matrix H ∈ CNR×NS can be +expressed as +H = ΓRΨRHaΨH +S ΓS, +(1) +where Ha ∈ CnR×nS denotes the wavenumber-domain chan- +nel matrix, which has nR × nS elements. Here, nR = |ER| +and nS = |ES| are the cardinalities of the sets ER and +ES, with ER = +� +(lx, ly) ∈ Z2 : +� +lxλ +Lx +R +�2 ++ +� +lyλ +Ly +R +�2 +≤ 1 +� +and +ES = +� +(mx, my) ∈ Z2 : +� +mxλ +Lx +S +�2 ++ +� +myλ +Ly +S +�2 +≤ 1 +� +. Each el- +ement [Ha]β,α of Ha is a random Fourier coefficient fol- +lowing the complex Gaussian distribution CN(0, σ2 +β,α), β = +1, 2, · · · , nR, α = 1, 2, · · · , nS. The variance can be further +given by +σ2 +β,α = +�� +ΩR(lx +β,ly +β) +A2(θR, φR) sin θRdθRdφR× +�� +ΩS(mxα,my +α) +A2(θS, φS) sin θSdθSdφS, +(2) +where A2(θR, φR) and A2(θS, φS) denote the angular power +spectrum at the receiver and the transmitter, respectively. The +angular power spectrum can be further modeled by a mixture +of VMF distributions [8] +A2 +R(θR, φR) = +Nc +� +i=1 +wR,ipR,i(θR, φR), +(3) +and +A2 +S(θS, φS) = +Nc +� +i=1 +wS,ipS,i(θS, φS), +(4) +where Nc denotes the number of clusters of the scatters in the +propagation environment. wR,i and wS,i denote the normal- +ization factor with �Nc +i +wR,i = �Nc +i +wS,i = 1. pR,i(θR, φR) +and pS,i(θS, φS) denote the probability functions of the VMF +distribution, which can be further expressed as [8] +pR,i(θR, φR) = +αR,i +4πsinh(αR,i)× +eαR,i(sin θR sin ¯θR,i cos(φR− ¯φR,i)+cos θR cos ¯θR,i), +(5) +and +pS,i(θS, φS) = +αS,i +4πsinh(αS,i)× +eαS,i(sin θS sin ¯θS,i cos(φS− ¯φS,i)+cos θS cos ¯θS,i), +(6) +where {¯φR,i, ¯θR,i} and {¯φS,i, ¯θS,i} denote the elevation and +azimuth angles of the i-th cluster at the receiver and the +transmitter. αS,i and αR,i denote the concentration parameters +for the i-th cluster. These angles can be derived from the 3GPP +TR 38.901 channel model [9] according to the relationship +defined in [6]. +In (1), ΨR ∈ CNR×nR and ΨS ∈ CNS×nS denote the +modified Fourier harmonics, which take the antenna pattern +distortion into consideration. Each element of ΨR and ΨS +can be further derived as +[ΨR]q,β = +1 +√NR +e +j +� +2πlx +β +Lx +R rx +q + +2πly +β +Ly +R +ry +q +γR(lx +β,ly +β)rz +q +� +× FR,q +� +ˆθR(lx +β, ly +β), ˆφR(lx +β, ly +β) +� +, +(7) +and +[ΨS]p,α = +1 +√NS +e +j +� +2πmx +α +Lx +S +sx +p+ 2πmy +α +Ly +S +sy +p+γS(mx +α,my +α)sz +p +� +× FS,p +� +ˆθS(mx +α, my +α), ˆφS(mx +α, my +α) +� +, +(8) +where +γR(lx, ly) += +� +( 2π +λ )2 − ( 2πlx +Lx +R )2 − ( 2πly +Ly +R )2 +and +γS(mx, my) += +� +( 2π +λ )2 − ( 2πmx +Lx +S )2 − ( 2πmy +Ly +S )2. +FR,q (θR, φR) and FS,p (θS, φS) represent the embedded +element directivity pattern of the q-th antenna at the receiver +and the p-th antenna at the transmitter. The corresponding +elevation and azimuth angles {ˆφR(lx, ly), ˆθR(lx, ly)} and +{ˆφS(mx, my), ˆθS(mx, my)} for the Fourier harmonics (lx, ly) +and (mx, my) can be calculated by a transformation from the +wavenumber domain to the angular domain [6]. +In the end, ΓR ∈ RNR×NR and ΓS ∈ RNS×NS are diagonal +matrices representing the efficiency of the antenna element +at the receiver and the transmitter, respectively. More details +of the channel model can be found in [6]. However, the +polarization characteristics of the antenna and the environment +are not taken into consideration. +B. Extended Channel Model with Polarization +In this subsection, we extend the EM-compliant channel +model to account for the polarization characteristics of the +antennas and the propagation environment. + +From (1), the channel between the p-th antenna at the trans- +mitter and the q-th antenna at the receiver can be expressed +as +[H]q,p = +nR +� +β=1 +nS +� +α=1 +ηR,q[ΨR]q,β[Ha]β,α[ΨS]∗ +p,αηS,p, +(9) +where ηR,q and ηS,p denote the diagonal elements of ΓR and +ΓS, representing the antenna efficiency of the corresponding +antenna. According to [10], the polarization pattern of an EM +wave can be decomposed into two components orthogonal to +the propagation direction, i.e., the vertical polarization and the +horizontal polarization. Therefore, the channel considering the +polarization characteristics can be written as +[H]pol +q,p = +nR +� +β=1 +nS +� +α=1 +ηR,q[Ψθ +R]q,β[Hθθ +a ]β,α[Ψθ +S]∗ +p,αηS,p ++ +nR +� +β=1 +nS +� +α=1 +ηR,q[Ψθ +R]q,β[Hθφ +a ]β,α[Ψφ +S]∗ +p,αηS,p ++ +nR +� +β=1 +nS +� +α=1 +ηR,q[Ψφ +R]q,β[Hφθ +a ]β,α[Ψθ +S]∗ +p,αηS,p ++ +nR +� +β=1 +nS +� +α=1 +ηR,q[Ψφ +R]q,β[Hφφ +a ]β,α[Ψφ +S]∗ +p,αηS,p, +(10) +where Ψθ +R +∈ CNR×nR and Ψθ +S +∈ CNS×nS denote the +modified Fourier harmonics with the horizontal polarization, +while Ψφ +R ∈ CNR×nR and Ψφ +S ∈ CNS×nS denote the modified +Fourier harmonics with vertical polarization. Hθθ +a ∈ CnR×nS, +Hφφ +a +∈ CnR×nS, Hθφ +a +∈ CnR×nS, and Hφθ +a +∈ CnR×nS +denote the co-polarization and cross-polarization wavenumber- +domain channel matrices. The details of these parameters are +explained in the following paragraphs. +Polarization of Antennas: Firstly, the polarization charac- +teristics of the antennas are modeled. The polarization of a +specific antenna can be described by its antenna pattern [10]. +Therefore, we further modify the Fourier harmonics to take the +polarization of the antennas into consideration. The modified +Fourier harmonics with polarization at the receiver can be +expressed as +[Ψθ +R]q,β = +1 +√NR +e +j +� +2πlx +β +Lx +R rx +q + +2πly +β +Ly +R +ry +q +γR(lx +β,ly +β)rz +q +� +× F θ +R,q +� +ˆθR(lx +β, ly +β), ˆφR(lx +β, ly +β) +� +, +(11) +and +[Ψφ +R]q,β = +1 +√NR +e +j +� +2πlx +β +Lx +R rx +q + +2πly +β +Ly +R +ry +q +γR(lx +β,ly +β)rz +q +� +× F φ +R,q +� +ˆθR(lx +β, ly +β), ˆφR(lx +β, ly +β) +� +, +(12) +where F θ +R,q (θR, φR) and F φ +R,q (θR, φR) denote the embedded +element directivity patterns in the horizontal and the vertical +polarization. Similarly, the modified Fourier harmonics at the +transmitter can be expressed as +[Ψθ +S]p,α = +1 +√NS +e +j +� +2πmx +α +Lx +S +sx +p+ 2πmy +α +Ly +S +sy +p+γS(mx +α,my +α)sz +p +� +× F θ +S,p +� +ˆθS(mx +α, my +α), ˆφS(mx +α, my +α) +� +, +(13) +and +[Ψφ +S]p,α = +1 +√NS +e +j +� +2πmx +α +Lx +S +sx +p+ 2πmy +α +Ly +S +sy +p+γS(mx +α,my +α)sz +p +� +× F φ +S,p +� +ˆθS(mx +α, my +α), ˆφS(mx +α, my +α) +� +, +(14) +where F θ +S,p (θS, φS) and F φ +S,p (θS, φS) denote the embedded +element directivity patterns in the horizontal and the vertical +polarization for the p-th antenna at the transmitter. +Polarization of Propagation Environment: Secondly, the +polarization characteristics of the propagation environment is +modeled. Similar to [9], we involve random phase shifts and +cross polarization power ratios (XPR) to model the polariza- +tion characteristics of the propagation environment. For the +co-polarization wavenumber-domain channels, a phase shift +is added to account for the polarization distortion by the +propagation environment, which can be expressed as +[Hθθ +a ]β,α = [Ha]β,α × ejΦθθ +β,α, +(15) +and +[Hφφ +a ]β,α = [Ha]β,α × ejΦφφ +β,α. +(16) +Otherwise, the cross-polarization wavenumber-domain chan- +nels can be expressed as +[Hθφ +a ]β,α = [Ha]β,α × ejΦθφ +β,α × +� +κ−1 +β,α, +(17) +and +[Hφθ +a ]β,α = [Ha]β,α × ejΦφθ +β,α × +� +κ−1 +β,α, +(18) +where Φθθ +β,α, Φφφ +β,α, Φφθ +β,α, and Φθφ +β,α are random phase shifts +following the uniform distribution within [−π, π]. κβ,α de- +notes the XPR of the propagation environment, which follows +the log-normal distribution κβ,α = 10Xβ,α/10 with Xβ,α ∼ +N(µXPR, σ2 +XPR). +Matrix Formulation: Finally, we transform the item-wise +channel model into a matrix form. The item-wise channel +model in (10) can be further expressed as +Hpol =ΓRΨθ +RHθθ +a Ψθ +S +HΓS + ΓRΨθ +RHθφ +a Ψφ +S +HΓS ++ ΓRΨφ +RHφθ +a Ψθ +S +HΓS + ΓRΨφ +RHφφ +a Ψφ +S +HΓS +=ΓR +� +Ψθ +R, Ψφ +R +� �Hθθ +a +Hθφ +a +Hφθ +a +Hφφ +a +� � +Ψθ +S, Ψφ +S +�H ΓS. +(19) +Therefore, the final channel matrix can be derived as +Hpol = ΓRΨpol +R Hpol +a Ψpol +S +HΓS, +(20) +where Ψpol +R += [Ψθ +R, Ψφ +R], Ψpol +S += [Ψθ +S, Ψφ +S], and Hpol +a += +�Hθθ +a +Hθφ +a +Hφθ +a +Hφφ +a +� +. + +Network Analyzer +Computer +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 13 14 15 16 +17 +33 +49 +65 +81 +97 +113 +129 +145 +161 +177 +193 +209 +225 +241 +120 121 +128 +1 +2 +3 +4 +13 +16 +Receiver +Transmitter +Virtual +Antenna +Array +Position +Control +Data +Collect +Antenna +Array +Channel +Signal +Generate +Signal +Detect +ൗ +𝜆 2 +ൗ +𝜆 2 +ൗ +𝜆 4 +ൗ +𝜆 8 +9 +5 +6 +7 +8 +10 +11 +12 +13 +14 +15 +16 +(120,6) +(121,6) +(113,6) +(128,6) +Fig. 1. Schematic diagram of the measurement equipment. +As a result, not only the non-isotropic characteristics of the +propagation environment, the antenna pattern distortion, the +antenna efficiency, but also the polarization characteristics of +the antennas and the propagation environment are all taken +into consideration in the extended channel model. +III. MEASUREMENT SETUP +In order to evaluate the extended EM-compliant channel +model and implement a realistic performance evaluation for +holographic MIMO systems, an experiment is conducted to +measure the real-world channel of holographic MIMO sys- +tems. To the best of our knowledge, it is the first attempt to +measure the channel of a holographic MIMO system. +A schematic diagram of the measurement equipment is +shown in Fig. 1. In the experiment, the dense antenna array of +holographic MIMO is realized by a virtual antenna array. A +discone antenna is used to achieve an omnidirectional pattern +and the position of it is controlled by an electrical machine. +Through computer programming, the antenna can be moved +to different positions to construct a virtual dense array with +arbitrary element spacings. In the experiment, the virtual an- +tenna array is equipped at the receiver to realize a holographic +MIMO array with spacing ∆x +R = ∆y +R ∈ {λ/8, λ/4, λ/2}. At +the transmitter, a conventional antenna array with NS = 16 +antennas and spacings ∆x +S = ∆y +S = λ/2 is used. It is +composed of patch antennas whose half power beam width +is 70◦. A calibrated network analyzer is used to measure the +channel and a computer is utilized to collect the measurement +results. The center frequency is fc = 4.7 GHz and the +bandwidth is 200 MHz from 4.6 GHz to 4.8 GHz with 1023 +samples. +The experiment is performed in an indoor environment +where the line-of-sight path is blocked by a metal object. Many +scatters are present to create a rich scattering environment. The +schematic diagram of the measurement environment is shown +in Fig. 2. We consider two scenarios. In the first scenario, the +virtual receive array plane is perpendicular to the transmitter, +Receiver +(Scenario2) +Receiver +(Scenario1) +Transmitter +Work Bench +Storage Rack +Medal Object +1.55m +5.65m +1.70m +2.18m +2.00m +Fig. 2. Schematic diagram of the measurement environment. +while in the second scenario, the virtual receive array plane is +parallel to the transmit array plane. +IV. MEASUREMENT RESULTS AND EVALUATIONS +In this section, we use the measurement results to evaluate +the performance of an indoor holographic MIMO system. +The measurement results are shown in Section IV-A and +corresponding performance evaluations are provided in Sec- +tion IV-B. Because the dense array of holographic MIMO +is implemented virtually, the antenna efficiency loss is not +accounted for in the measurement. Finally, the performance +evaluations with antenna efficiency loss are provided in Sec- +tion IV-C. +A. Channel Measurement Results +After measurement, a channel matrix Hpol with size NR × +NS can be derived. Here we plot several channel measurement +results to show the correlation of antennas at different position. +We use the pair (q, p) to represent the q-th antenna in the +virtual receive array and the p-th antenna in the transmit array. +The channel responses corresponding to (120, 6) and +(121, 6) transceiver pairs are shown in Fig. 3a. In these two +pairs, the transmit antennas are the same and the receive +antennas are adjacent, we can observe that the channel re- +sponses are quite similar. Instead, if we choose transceiver +pairs whose receiver elements are not adjacent, e.g., (113, 6) +and (128, 6), the results are shown in Fig. 3b. Since their +receiver elements are separated by 2λ, we can find that the +red line differs from the blue line in the spectrum, showing +a low correlation compared with the results in Fig. 3a. The +difference between these two figures shows the effect of spatial +coherence. In an array, adjacent elements are more likely +to sense the channel inside the same cluster, and thus the +correlation of their channel responses are stronger. +B. Performance Evaluation without Antenna Efficiency Loss +Once the channel matrix Hpol is obtained, we can evaluate +the channel capacity based on the measurement results. The +variation of channel capacity with different element spacings +in both scenarios are shown in Fig 4. In these figures, the +antenna spacings are ∆x +R = ∆y +R ∈ {λ/8, λ/4, λ/2} and the +corresponding numbers of antennas are NR ∈ {256, 64, 16} +at the receiver. The signal to noise ratio (SNR) is set to 0 dB. + +4.6 +4.62 +4.64 +4.66 +4.68 +4.7 +4.72 +4.74 +4.76 +4.78 +4.8 +Frequency [GHz] +-100 +-95 +-90 +-85 +-80 +-75 +-70 +-65 +-60 +S parrameter [dB] +(a) +4.6 +4.62 +4.64 +4.66 +4.68 +4.7 +4.72 +4.74 +4.76 +4.78 +4.8 +Frequency [GHz] +-100 +-95 +-90 +-85 +-80 +-75 +-70 +-65 +-60 +S parrameter [dB] +(b) +Fig. 3. +Channel response between q-th antenna at the receiver and p-th +antenna at the transmitter in Scenario 2. (a) q = 120, 121, p = 6; (b) +q = 113, 128, p = 6. +Both the water filling and the equal power allocation strategies +are adopted to evaluate the channel capacity performance. The +blue lines correspond to the channel capacity, while the red +lines correspond to the relative capacity with respect to the +case with ∆x +R = ∆y +R = λ/2 spacing. +From the results in Fig. 4, we can see that the spatial +oversampling of holographic MIMO is able to increase the +channel capacity. Using equal power allocation strategy, a four +times spatial oversampling with ∆x +R = ∆y +R = λ/4 can offer +about 120% capacity gain, and a 16 times oversampling with +∆x +R = ∆y +R = λ/8 provides more than 300% capacity gain. +While using the water filling strategy, the corresponding ca- +pacity gains are about 80% and 200%. Therefore, the capacity +enhancement capability of holographic MIMO stated in the +previous research works [5], [6], [11] is verified by practical +experiment. It is worth noting that the antenna efficiency +loss at the receiver is not taken into consideration in the +measurement because the dense array is realized virtually, +Antenna spacing +Capacity [bit/s/Hz] +Water filling +Equal power +Water filling +Equal power +(a) +Antenna spacing +Capacity [bit/s/Hz] +Water filling +Equal power +Water filling +Equal power +(b) +Fig. 4. Channel capacity and relative capacity with different antenna spacings. +(a) Scenario 1; (b) Scenario 2. +which means ηR,q = 1. In the next subsection, the antenna +efficiency loss is further considered in analyzing the capacity +of a holographic MIMO system. +C. Performance Evaluation with Antenna Efficiency Loss +In a practical dense antenna array with small element +spacing, the efficiency of the antenna elements will decrease +because of the mutual coupling among them. In [12], a +relationship between the antenna efficiency and the element +spacing is established for a dense array, which is called +Hannan’s element efficiency. According to [12], for a practical +dense array at the receiver, the efficiency of the antenna +element can be estimated as +ηR,q ≈ π∆x +R∆y +R +λ2 +, +(21) +which means that the element efficiency is proportional to the +area allocated to the element. It implies that when the spacing +of antenna element is small (∆x +R∆y +R < λ2/π), the element + +Antenna spacing +Capacity [bit/s/Hz] +Water filling +Equal power +Water filling +Equal power +(a) +Antenna spacing +Capacity [bit/s/Hz] +Water filling +Equal power +Water filling +Equal power +(b) +Fig. 5. Channel capacity and relative capacity with antenna efficiency loss. (a) Scenario 1; (b) Scenario 2. +efficiency cannot reach 1, and it will decrease as the antenna +elements are placed closer. +Using the efficiency estimation in (21), we modify the chan- +nel measurement results and evaluate the channel capacity of +holographic MIMO systems. The results are shown in Fig. 5. +It can be seen that in both scenarios, the channel capacities +will not keep increasing with more antenna elements and +smaller element spacings. Using the equal power allocation +strategy, a 16 times oversampling with ∆x +R = ∆y +R = λ/8 +can only provide 4% capacity gain. While using the water +filling strategy, the channel capacities even slightly decrease. +The reason behind this is that the array gain and multiplexing +gain by deploying more antenna elements are reduced by the +decrease of the antenna efficiency. +From the above analyses, we can find that although the +channel correlation increases with smaller antenna spacings, +the spatial oversampling of holographic MIMO is able to +offer an obvious capacity enhancement. However, the antenna +efficiency loss due to mutual coupling will greatly decrease the +capacity gain, which is one of the most important challenges +for a practical holographic MIMO system. Therefore, design- +ing a dense antenna array with element efficiency above the +Hannan’s efficiency scaling law will be the promising ways to +exploit the benefit of spatial oversampling for the holographic +MIMO systems. +V. CONCLUSION +In this paper, an extended EM-compliant channel model +is proposed for holographic MIMO systems, which takes the +non-isotropic characteristics of the propagation environment, +the antenna pattern distortion, the antenna efficiency, and the +polarization into account. An experiment is also conducted to +measure the channel of an indoor holographic MIMO system +through virtual antenna arrays. It is demonstrated through +experiments for the first time that the spatial oversampling +of holographic MIMO is able to increase the capacity of a +wireless communication system significantly. However, the +antenna efficiency is the most crucial challenge preventing us +from getting the capacity improvement. +REFERENCES +[1] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen, +R. Zhang, M. D. Renzo, and M. Debbah, “Holographic MIMO surfaces +for 6G wireless networks: Opportunities, challenges, and trends,” IEEE +Wireless Commun., vol. 27, no. 5, pp. 118–125, Oct. 2020. +[2] D. Dardari and N. Decarli, “Holographic communication using intelli- +gent surfaces,” IEEE Commun. Mag., vol. 59, no. 6, pp. 35–41, Jun. +2021. +[3] A. Pizzo, T. L. Marzetta, and L. Sanguinetti, “Spatially-stationary +model for holographic MIMO small-scale fading,” IEEE J. Sel. Areas +Commun., vol. 38, no. 9, pp. 1964–1979, Sep. 2020. +[4] L. Wei, C. Huang, G. Alexandropoulus, W. E. I. Sha, Z. Zhang, +M. Debbah, and C. Yuen, “Multi-user holographic MIMO surfaces: +Channel modeling and spectral efficiency analysis,” IEEE J. Sel. Topics +Signal Process., vol. 16, no. 5, pp. 1112–1124, Aug. 2022. +[5] A. Pizzo, L. Sanguinetti, and T. L. Marzetta, “Fourier plane-wave +series expansion for holographic MIMO communications,” IEEE Trans. +Wireless Commun., vol. 21, no. 9, pp. 6890–6905, Sep. 2022. +[6] T. Wang, W. Han, Z. Zhong, J. Pang, G. Zhou, S. Wang, and +Q. Li, “Electromagnetic-compliant channel modeling and performance +evaluation for holographic MIMO,” in IEEE Global Communications +Conference (GLOBECOM), Rio de Janeiro, Brazil, Dec. 2022. +[7] Y. Liu, M. Zhang, and T. Wang, “Effect of antenna pattern on electro- +magnetic MIMO communication,” in IEEE International Conference on +Communication Technology (ICCT), Nanjing, China, Nov. 2022. +[8] K. Mammasis, R. W. Stewart, and J. S. Thompson, “Spatial fading +correlation model using mixtures of von Mises Fisher distributions,” +IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 2046–2055, Apr. 2009. +[9] 3GPP TR 38.901, “Study on channel model for frequencies from 0.5 to +100 GHz,” 2020. +[10] C. A. Balanis, Antenna Theory: Analysis and Design. +New York: John +Wiley & Sons Ltd, 2015. +[11] Z. Zhang and L. Dai, “Continuous-aperture MIMO for electromagnetic +information theory,” arXiv, 2021. [Online]. Available: https://arxiv.org/ +abs/2111.08630 +[12] P. Hannan, “The element-gain paradox for a phased-array antenna,” +IEEE Trans. Antenna Propag., vol. 12, no. 4, pp. 423–433, Jul. 1964. + diff --git a/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/load_file.txt b/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b13b0f7909634ac057b77001299f33776be968b2 --- /dev/null +++ b/cNE5T4oBgHgl3EQffQ_r/content/tmp_files/load_file.txt @@ -0,0 +1,418 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf,len=417 +page_content='Channel Measurement for Holographic MIMO: Benefits and Challenges of Spatial Oversampling Tengjiao Wang∗, Yongxi Liu†, Ming Zhang†, Wei E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sha‡, Cen Ling∗, Chao Li∗, Shaobo Wang∗ ∗Wireless Network RAN Research Department, Huawei Technologies CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', Ltd, Shanghai, China †School of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi, China ‡College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China Emails: ∗{wangtengjiao6, lingcen, lichao18, shaobo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='wang}@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='com, †liuyongxi@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='cn, ming20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='zhang@xjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='cn, ‡ weisha@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='cn Abstract—In this paper, the channel of an indoor holographic multiple-input multiple-output (MIMO) system is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It is demonstrated through experiments for the first time that the spatial oversampling of holographic MIMO systems is able to increase the capacity of a wireless communication system significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, the antenna efficiency is the most crucial challenge preventing us from getting the capacity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' An extended EM-compliant channel model is also proposed for holographic MIMO systems, which is able to take the non-isotropic characteristics of the propagation environment, the antenna pattern distortion, the antenna efficiency, and the polarization characteristics into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Index Terms—Holographic MIMO, massive MIMO, spatial oversampling, channel measurement, electromagnetic informa- tion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' INTRODUCTION In order to increase the spectral efficiency and energy efficiency for the future 5G-Advanced and 6G wireless com- munication systems, the concept of holographic multiple-input multiple-output (MIMO) is proposed recently [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' By inte- grating an infinite number of antennas into a limited surface, holographic MIMO is expected to fully exploit the propagation characteristics offered by the electromagnetic channel and ap- proach the fundamental performance limit [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' For holographic MIMO systems, both the accurate channel modeling and the realistic performance evaluation are key problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the literature, many efforts have been devoted to accu- rately model the channel of holographic MIMO systems [3]– [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [3], a Fourier plane-wave series expansion-based chan- nel model is proposed for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Based on the Fourier spectral representation, it provides a physically meaningful model capturing the propagation characteristics of the electromagnetic (EM) wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [4], the authors extend the Fourier plane-wave channel model to a multi-user scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [5], the non-isotropic scattering environment is further con- sidered by using the von Mises-Fisher (VMF) distributions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Then in our previous works [6], [7], an EM-compliant channel model is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' By combining the VMF distributions [8] and the 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='901 channel model [9], a realistic angular power spectrum is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The non-ideal factors caused by mutual coupling at the transceivers [10], including the antenna pattern distortion and the decrease of antenna efficiency are also accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, in the state-of-the-art channel models [3]–[7], the polarization of EM wave is not taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The polarization is an inherent characteristic of EM waves and will have significant impact on the performance of holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Another key problem for holographic MIMO is the accurate performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [5], the ergodic capacity of a single-user holographic MIMO system is analyzed, which shows the capacity improvement of holographic MIMO over conventional MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [11], the capacity of a single-user holographic MIMO system is investigated from a continuous point of view, the capacity enhancement is also demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In our previous work [6], both the single-user and the multi- user downlink channel capacities of holographic MIMO sys- tems are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, the performance evaluations in the state-of-the-art researches [5], [6], [11] are all numerical simulations based on theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' No experiments have been done to verify the accuracy of the performance evalua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In this paper, we try to solve the above two problems for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Firstly, an extended EM- compliant channel model is proposed based on the channel model in our previous work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Secondly, the channel of holo- graphic MIMO is measured through real-world experiments, and the performance is evaluated based on the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The contributions of this paper can be summarized as follows: An extended EM-compliant channel model is proposed for holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the extended model, not only the non-isotropic characteristics of the propaga- tion environment, the antenna pattern distortion, the an- tenna efficiency, but also the polarization of the antennas and the propagation environment can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Based on the extended channel model, the real-world channel for holographic MIMO systems is measured for the first time in an indoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' An experiment is carefully designed, in which an electrically controlled virtual dense array is used to realize arbitrary element spacings of holographic MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The channel capacity of holographic MIMO systems is evaluated according to the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='05626v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='IT] 13 Jan 2023 demonstrated that the spatial oversampling of holographic MIMO is able to provide a two to three times capacity enhancement, without considering the antenna efficiency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The antenna efficiency is the most crucial challenge for holographic MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In Section II, the proposed extended EM-compliant channel model for holo- graphic MIMO is explained in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Then, the setup for the channel measurement is given in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The measurement results and the corresponding performance evaluations are provided in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Finally, this paper is concluded in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' EXTENDED EM-COMPLIANT CHANNEL MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' EM-Compliant Channel Model In this subsection, the EM-compliant channel model for holographic MIMO in our previous work [6] is briefly in- troduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The central frequency and wavelength are denoted by fc and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Planar antenna arrays with size {Lx R, Ly R} and {Lx S, Ly S} are equipped at the receiver and the transmitter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The numbers of antenna elements are denoted by NR and NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The spacings between antenna elements are denoted by {∆x R, ∆y R} and {∆x S, ∆y S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The coordinates of the antenna elements are represented by rq = (rx q , ry q, rz q), q = 1, 2, · · · , NR and sp = (sx p, sy p, sz p), p = 1, 2, · · · , NS, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' According to [6], the channel matrix H ∈ CNR×NS can be expressed as H = ΓRΨRHaΨH S ΓS, (1) where Ha ∈ CnR×nS denotes the wavenumber-domain chan- nel matrix, which has nR × nS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Here, nR = |ER| and nS = |ES| are the cardinalities of the sets ER and ES, with ER = � (lx, ly) ∈ Z2 : � lxλ Lx R �2 + � lyλ Ly R �2 ≤ 1 � and ES = � (mx, my) ∈ Z2 : � mxλ Lx S �2 + � myλ Ly S �2 ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Each el- ement [Ha]β,α of Ha is a random Fourier coefficient fol- lowing the complex Gaussian distribution CN(0, σ2 β,α), β = 1, 2, · · · , nR, α = 1, 2, · · · , nS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The variance can be further given by σ2 β,α = �� ΩR(lx β,ly β) A2(θR, φR) sin θRdθRdφR× �� ΩS(mxα,my α) A2(θS, φS) sin θSdθSdφS, (2) where A2(θR, φR) and A2(θS, φS) denote the angular power spectrum at the receiver and the transmitter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The angular power spectrum can be further modeled by a mixture of VMF distributions [8] A2 R(θR, φR) = Nc � i=1 wR,ipR,i(θR, φR), (3) and A2 S(θS, φS) = Nc � i=1 wS,ipS,i(θS, φS), (4) where Nc denotes the number of clusters of the scatters in the propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' wR,i and wS,i denote the normal- ization factor with �Nc i wR,i = �Nc i wS,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' pR,i(θR, φR) and pS,i(θS, φS) denote the probability functions of the VMF distribution, which can be further expressed as [8] pR,i(θR, φR) = αR,i 4πsinh(αR,i)× eαR,i(sin θR sin ¯θR,i cos(φR− ¯φR,i)+cos θR cos ¯θR,i), (5) and pS,i(θS, φS) = αS,i 4πsinh(αS,i)× eαS,i(sin θS sin ¯θS,i cos(φS− ¯φS,i)+cos θS cos ¯θS,i), (6) where {¯φR,i, ¯θR,i} and {¯φS,i, ¯θS,i} denote the elevation and azimuth angles of the i-th cluster at the receiver and the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' αS,i and αR,i denote the concentration parameters for the i-th cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' These angles can be derived from the 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='901 channel model [9] according to the relationship defined in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In (1), ΨR ∈ CNR×nR and ΨS ∈ CNS×nS denote the modified Fourier harmonics, which take the antenna pattern distortion into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Each element of ΨR and ΨS can be further derived as [ΨR]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β = 1 √NR e j � 2πlx β Lx R rx q + 2πly β Ly R ry q +γR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='ly β)rz q � × FR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q � ˆθR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (7) and [ΨS]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α = 1 √NS e j � 2πmx α Lx S sx p+ 2πmy α Ly S sy p+γS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='my α)sz p � × FS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p � ˆθS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (8) where γR(lx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly) = � ( 2π λ )2 − ( 2πlx Lx R )2 − ( 2πly Ly R )2 and γS(mx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my) = � ( 2π λ )2 − ( 2πmx Lx S )2 − ( 2πmy Ly S )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' FR,q (θR, φR) and FS,p (θS, φS) represent the embedded element directivity pattern of the q-th antenna at the receiver and the p-th antenna at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The corresponding elevation and azimuth angles {ˆφR(lx, ly), ˆθR(lx, ly)} and {ˆφS(mx, my), ˆθS(mx, my)} for the Fourier harmonics (lx, ly) and (mx, my) can be calculated by a transformation from the wavenumber domain to the angular domain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the end, ΓR ∈ RNR×NR and ΓS ∈ RNS×NS are diagonal matrices representing the efficiency of the antenna element at the receiver and the transmitter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' More details of the channel model can be found in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, the polarization characteristics of the antenna and the environment are not taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Extended Channel Model with Polarization In this subsection, we extend the EM-compliant channel model to account for the polarization characteristics of the antennas and the propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' From (1), the channel between the p-th antenna at the trans- mitter and the q-th antenna at the receiver can be expressed as [H]q,p = nR � β=1 nS � α=1 ηR,q[ΨR]q,β[Ha]β,α[ΨS]∗ p,αηS,p, (9) where ηR,q and ηS,p denote the diagonal elements of ΓR and ΓS, representing the antenna efficiency of the corresponding antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' According to [10], the polarization pattern of an EM wave can be decomposed into two components orthogonal to the propagation direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', the vertical polarization and the horizontal polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' the channel considering the polarization characteristics can be written as [H]pol q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p = nR � β=1 nS � α=1 ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q[Ψθ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β[Hθθ a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α[Ψθ S]∗ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p + nR � β=1 nS � α=1 ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q[Ψθ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β[Hθφ a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α[Ψφ S]∗ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p + nR � β=1 nS � α=1 ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q[Ψφ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β[Hφθ a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α[Ψθ S]∗ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p + nR � β=1 nS � α=1 ηR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q[Ψφ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β[Hφφ a ]β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α[Ψφ S]∗ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='αηS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (10) where Ψθ R ∈ CNR×nR and Ψθ S ∈ CNS×nS denote the modified Fourier harmonics with the horizontal polarization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' while Ψφ R ∈ CNR×nR and Ψφ S ∈ CNS×nS denote the modified Fourier harmonics with vertical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Hθθ a ∈ CnR×nS, Hφφ a ∈ CnR×nS, Hθφ a ∈ CnR×nS, and Hφθ a ∈ CnR×nS denote the co-polarization and cross-polarization wavenumber- domain channel matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The details of these parameters are explained in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Polarization of Antennas: Firstly, the polarization charac- teristics of the antennas are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The polarization of a specific antenna can be described by its antenna pattern [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Therefore, we further modify the Fourier harmonics to take the polarization of the antennas into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The modified Fourier harmonics with polarization at the receiver can be expressed as [Ψθ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β = 1 √NR e j � 2πlx β Lx R rx q + 2πly β Ly R ry q +γR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='ly β)rz q � × F θ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q � ˆθR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (11) and [Ψφ R]q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='β = 1 √NR e j � 2πlx β Lx R rx q + 2πly β Ly R ry q +γR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='ly β)rz q � × F φ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q � ˆθR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφR(lx β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ly β) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (12) where F θ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q (θR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' φR) and F φ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='q (θR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' φR) denote the embedded element directivity patterns in the horizontal and the vertical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' the modified Fourier harmonics at the transmitter can be expressed as [Ψθ S]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α = 1 √NS e j � 2πmx α Lx S sx p+ 2πmy α Ly S sy p+γS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='my α)sz p � × F θ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p � ˆθS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (13) and [Ψφ S]p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='α = 1 √NS e j � 2πmx α Lx S sx p+ 2πmy α Ly S sy p+γS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='my α)sz p � × F φ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p � ˆθS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' ˆφS(mx α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' my α) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (14) where F θ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p (θS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' φS) and F φ S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='p (θS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' φS) denote the embedded element directivity patterns in the horizontal and the vertical polarization for the p-th antenna at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Polarization of Propagation Environment: Secondly, the polarization characteristics of the propagation environment is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Similar to [9], we involve random phase shifts and cross polarization power ratios (XPR) to model the polariza- tion characteristics of the propagation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' For the co-polarization wavenumber-domain channels, a phase shift is added to account for the polarization distortion by the propagation environment, which can be expressed as [Hθθ a ]β,α = [Ha]β,α × ejΦθθ β,α, (15) and [Hφφ a ]β,α = [Ha]β,α × ejΦφφ β,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (16) Otherwise, the cross-polarization wavenumber-domain chan- nels can be expressed as [Hθφ a ]β,α = [Ha]β,α × ejΦθφ β,α × � κ−1 β,α, (17) and [Hφθ a ]β,α = [Ha]β,α × ejΦφθ β,α × � κ−1 β,α, (18) where Φθθ β,α, Φφφ β,α, Φφθ β,α, and Φθφ β,α are random phase shifts following the uniform distribution within [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' κβ,α de- notes the XPR of the propagation environment, which follows the log-normal distribution κβ,α = 10Xβ,α/10 with Xβ,α ∼ N(µXPR, σ2 XPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Matrix Formulation: Finally, we transform the item-wise channel model into a matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The item-wise channel model in (10) can be further expressed as Hpol =ΓRΨθ RHθθ a Ψθ S HΓS + ΓRΨθ RHθφ a Ψφ S HΓS + ΓRΨφ RHφθ a Ψθ S HΓS + ΓRΨφ RHφφ a Ψφ S HΓS =ΓR � Ψθ R, Ψφ R � �Hθθ a Hθφ a Hφθ a Hφφ a � � Ψθ S, Ψφ S �H ΓS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (19) Therefore, the final channel matrix can be derived as Hpol = ΓRΨpol R Hpol a Ψpol S HΓS, (20) where Ψpol R = [Ψθ R, Ψφ R], Ψpol S = [Ψθ S, Ψφ S], and Hpol a = �Hθθ a Hθφ a Hφθ a Hφφ a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Network Analyzer Computer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 120 121 128 1 2 3 4 13 16 Receiver Transmitter Virtual Antenna Array Position Control Data Collect Antenna Array Channel Signal Generate Signal Detect ൗ 𝜆 2 ൗ 𝜆 2 ൗ 𝜆 4 ൗ 𝜆 8 9 5 6 7 8 10 11 12 13 14 15 16 (120,6) (121,6) (113,6) (128,6) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Schematic diagram of the measurement equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' As a result, not only the non-isotropic characteristics of the propagation environment, the antenna pattern distortion, the antenna efficiency, but also the polarization characteristics of the antennas and the propagation environment are all taken into consideration in the extended channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' MEASUREMENT SETUP In order to evaluate the extended EM-compliant channel model and implement a realistic performance evaluation for holographic MIMO systems, an experiment is conducted to measure the real-world channel of holographic MIMO sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' To the best of our knowledge, it is the first attempt to measure the channel of a holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' A schematic diagram of the measurement equipment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the experiment, the dense antenna array of holographic MIMO is realized by a virtual antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' A discone antenna is used to achieve an omnidirectional pattern and the position of it is controlled by an electrical machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Through computer programming, the antenna can be moved to different positions to construct a virtual dense array with arbitrary element spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the experiment, the virtual an- tenna array is equipped at the receiver to realize a holographic MIMO array with spacing ∆x R = ∆y R ∈ {λ/8, λ/4, λ/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' At the transmitter, a conventional antenna array with NS = 16 antennas and spacings ∆x S = ∆y S = λ/2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It is composed of patch antennas whose half power beam width is 70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' A calibrated network analyzer is used to measure the channel and a computer is utilized to collect the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The center frequency is fc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='7 GHz and the bandwidth is 200 MHz from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='6 GHz to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='8 GHz with 1023 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The experiment is performed in an indoor environment where the line-of-sight path is blocked by a metal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Many scatters are present to create a rich scattering environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The schematic diagram of the measurement environment is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' We consider two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the first scenario, the virtual receive array plane is perpendicular to the transmitter, Receiver (Scenario2) Receiver (Scenario1) Transmitter Work Bench Storage Rack Medal Object 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='55m 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='65m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='70m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='18m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='00m Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Schematic diagram of the measurement environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' while in the second scenario, the virtual receive array plane is parallel to the transmit array plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' MEASUREMENT RESULTS AND EVALUATIONS In this section, we use the measurement results to evaluate the performance of an indoor holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The measurement results are shown in Section IV-A and corresponding performance evaluations are provided in Sec- tion IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Because the dense array of holographic MIMO is implemented virtually, the antenna efficiency loss is not accounted for in the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Finally, the performance evaluations with antenna efficiency loss are provided in Sec- tion IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Channel Measurement Results After measurement, a channel matrix Hpol with size NR × NS can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Here we plot several channel measurement results to show the correlation of antennas at different position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' We use the pair (q, p) to represent the q-th antenna in the virtual receive array and the p-th antenna in the transmit array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The channel responses corresponding to (120, 6) and (121, 6) transceiver pairs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In these two pairs, the transmit antennas are the same and the receive antennas are adjacent, we can observe that the channel re- sponses are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Instead, if we choose transceiver pairs whose receiver elements are not adjacent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', (113, 6) and (128, 6), the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Since their receiver elements are separated by 2λ, we can find that the red line differs from the blue line in the spectrum, showing a low correlation compared with the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The difference between these two figures shows the effect of spatial coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In an array, adjacent elements are more likely to sense the channel inside the same cluster, and thus the correlation of their channel responses are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Performance Evaluation without Antenna Efficiency Loss Once the channel matrix Hpol is obtained, we can evaluate the channel capacity based on the measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The variation of channel capacity with different element spacings in both scenarios are shown in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In these figures, the antenna spacings are ∆x R = ∆y R ∈ {λ/8, λ/4, λ/2} and the corresponding numbers of antennas are NR ∈ {256, 64, 16} at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The signal to noise ratio (SNR) is set to 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='72 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='8 Frequency [GHz] 100 95 90 85 80 75 70 65 60 S parrameter [dB] (a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='72 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='8 Frequency [GHz] 100 95 90 85 80 75 70 65 60 S parrameter [dB] (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Channel response between q-th antenna at the receiver and p-th antenna at the transmitter in Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (a) q = 120, 121, p = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (b) q = 113, 128, p = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Both the water filling and the equal power allocation strategies are adopted to evaluate the channel capacity performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The blue lines correspond to the channel capacity, while the red lines correspond to the relative capacity with respect to the case with ∆x R = ∆y R = λ/2 spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' From the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 4, we can see that the spatial oversampling of holographic MIMO is able to increase the channel capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Using equal power allocation strategy, a four times spatial oversampling with ∆x R = ∆y R = λ/4 can offer about 120% capacity gain, and a 16 times oversampling with ∆x R = ∆y R = λ/8 provides more than 300% capacity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' While using the water filling strategy, the corresponding ca- pacity gains are about 80% and 200%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Therefore, the capacity enhancement capability of holographic MIMO stated in the previous research works [5], [6], [11] is verified by practical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It is worth noting that the antenna efficiency loss at the receiver is not taken into consideration in the measurement because the dense array is realized virtually, Antenna spacing Capacity [bit/s/Hz] Water filling Equal power Water filling Equal power (a) Antenna spacing Capacity [bit/s/Hz] Water filling Equal power Water filling Equal power (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Channel capacity and relative capacity with different antenna spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (a) Scenario 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (b) Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' which means ηR,q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In the next subsection, the antenna efficiency loss is further considered in analyzing the capacity of a holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Performance Evaluation with Antenna Efficiency Loss In a practical dense antenna array with small element spacing, the efficiency of the antenna elements will decrease because of the mutual coupling among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' In [12], a relationship between the antenna efficiency and the element spacing is established for a dense array, which is called Hannan’s element efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' According to [12], for a practical dense array at the receiver, the efficiency of the antenna element can be estimated as ηR,q ≈ π∆x R∆y R λ2 , (21) which means that the element efficiency is proportional to the area allocated to the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It implies that when the spacing of antenna element is small (∆x R∆y R < λ2/π), the element Antenna spacing Capacity [bit/s/Hz] Water filling Equal power Water filling Equal power (a) Antenna spacing Capacity [bit/s/Hz] Water filling Equal power Water filling Equal power (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Channel capacity and relative capacity with antenna efficiency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (a) Scenario 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' (b) Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' efficiency cannot reach 1, and it will decrease as the antenna elements are placed closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Using the efficiency estimation in (21), we modify the chan- nel measurement results and evaluate the channel capacity of holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It can be seen that in both scenarios, the channel capacities will not keep increasing with more antenna elements and smaller element spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Using the equal power allocation strategy, a 16 times oversampling with ∆x R = ∆y R = λ/8 can only provide 4% capacity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' While using the water filling strategy, the channel capacities even slightly decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' The reason behind this is that the array gain and multiplexing gain by deploying more antenna elements are reduced by the decrease of the antenna efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' From the above analyses, we can find that although the channel correlation increases with smaller antenna spacings, the spatial oversampling of holographic MIMO is able to offer an obvious capacity enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, the antenna efficiency loss due to mutual coupling will greatly decrease the capacity gain, which is one of the most important challenges for a practical holographic MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Therefore, design- ing a dense antenna array with element efficiency above the Hannan’s efficiency scaling law will be the promising ways to exploit the benefit of spatial oversampling for the holographic MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' CONCLUSION In this paper, an extended EM-compliant channel model is proposed for holographic MIMO systems, which takes the non-isotropic characteristics of the propagation environment, the antenna pattern distortion, the antenna efficiency, and the polarization into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' An experiment is also conducted to measure the channel of an indoor holographic MIMO system through virtual antenna arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' It is demonstrated through experiments for the first time that the spatial oversampling of holographic MIMO is able to increase the capacity of a wireless communication system significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' However, the antenna efficiency is the most crucial challenge preventing us from getting the capacity improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Hu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Alexandropoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zappone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Yuen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Renzo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Debbah, “Holographic MIMO surfaces for 6G wireless networks: Opportunities, challenges, and trends,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 118–125, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Dardari and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Decarli, “Holographic communication using intelli- gent surfaces,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 35–41, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Pizzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Marzetta, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sanguinetti, “Spatially-stationary model for holographic MIMO small-scale fading,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 1964–1979, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Alexandropoulus, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sha, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Debbah, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Yuen, “Multi-user holographic MIMO surfaces: Channel modeling and spectral efficiency analysis,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Topics Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 1112–1124, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Pizzo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Sanguinetti, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Marzetta, “Fourier plane-wave series expansion for holographic MIMO communications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 6890–6905, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Pang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Li, “Electromagnetic-compliant channel modeling and performance evaluation for holographic MIMO,” in IEEE Global Communications Conference (GLOBECOM), Rio de Janeiro, Brazil, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wang, “Effect of antenna pattern on electro- magnetic MIMO communication,” in IEEE International Conference on Communication Technology (ICCT), Nanjing, China, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Mammasis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Stewart, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Thompson, “Spatial fading correlation model using mixtures of von Mises Fisher distributions,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2046–2055, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [9] 3GPP TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='901, “Study on channel model for frequencies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='5 to 100 GHz,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Balanis, Antenna Theory: Analysis and Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' New York: John Wiley & Sons Ltd, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [11] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Zhang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Dai, “Continuous-aperture MIMO for electromagnetic information theory,” arXiv, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='org/ abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content='08630 [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Hannan, “The element-gain paradox for a phased-array antenna,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' Antenna Propag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 423–433, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} +page_content=' 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE5T4oBgHgl3EQffQ_r/content/2301.05626v1.pdf'} diff --git a/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/2301.00414v1.pdf.txt b/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/2301.00414v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b662e93c468e854af60dbd1c77fca12aa54014d --- /dev/null +++ b/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/2301.00414v1.pdf.txt @@ -0,0 +1,2547 @@ +21 +DaeMon: Architectural Support for Efficient Data Movement +in Disaggregated Systems +CHRISTINA GIANNOULA, University of Toronto, Canada and National Technical University of Athens, +Greece +KAILONG HUANG∗, University of Toronto, Canada +JONATHAN TANG∗, University of Toronto, Canada +NECTARIOS KOZIRIS, National Technical University of Athens, Greece +GEORGIOS GOUMAS, National Technical University of Athens, Greece +ZESHAN CHISHTI, Intel Corporation, USA +NANDITA VIJAYKUMAR, University of Toronto, Canada +Resource disaggregation offers a cost effective solution to resource scaling, utilization, and failure-handling +in data centers by physically separating hardware devices in a server. Servers are architected as pools of +processor, memory, and storage devices, organized as independent failure-isolated components interconnected +by a high-bandwidth network. A critical challenge, however, is the high performance penalty of accessing +data from a remote memory module over the network. Addressing this challenge is difficult as disaggregated +systems have high runtime variability in network latencies/bandwidth, and page migration can significantly +delay critical path cache line accesses in other pages. +This paper conducts a characterization analysis on different data movement strategies in fully disaggregated +systems, evaluates their performance overheads in a variety of workloads, and introduces DaeMon, the first +software-transparent mechanism to significantly alleviate data movement overheads in fully disaggregated +systems. First, to enable scalability to multiple hardware components in the system, we enhance each compute +and memory unit with specialized engines that transparently handle data migrations. Second, to achieve high +performance and provide robustness across various network, architecture and application characteristics, we +implement a synergistic approach of bandwidth partitioning, link compression, decoupled data movement of +multiple granularities, and adaptive granularity selection in data movements. We evaluate DaeMon in a wide +variety of workloads at different network and architecture configurations using a state-of-the-art accurate +simulator. DaeMon improves system performance and data access costs by 2.39× and 3.06×, respectively, over +the widely-adopted approach of moving data at page granularity. +CCS Concepts: • General and reference → Performance; Design; Evaluation; Experimentation; • Com- +puter systems organization → Architectures; • Hardware; +Additional Key Words and Phrases: data movement, data access, memory access, hardware support, hard- +ware mechanism, high performance, memory systems, memory disaggregation, resource disaggregation, +disaggregated systems, workload characterization, benchmarking, performance characterization +∗Equal contribution to this research work. +Authors’ addresses: Christina Giannoula, christina.giann@gmail.com, University of Toronto, Canada, National Technical +University of Athens, Greece; Kailong Huang, University of Toronto, Canada; Jonathan Tang, University of Toronto, Canada; +Nectarios Koziris, National Technical University of Athens, Greece; Georgios Goumas, National Technical University of +Athens, Greece; Zeshan Chishti, Intel Corporation, USA; Nandita Vijaykumar, University of Toronto, Canada. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the +full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. +Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2022 Copyright held by the owner/author(s). Publication rights licensed to ACM. +2476-1249/2022/3-ART21 $15.00 +https://doi.org/10.1145/3508041 +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. +arXiv:2301.00414v1 [cs.AR] 1 Jan 2023 + +21:2 +Christina Giannoula, et al. +ACM Reference Format: +Christina Giannoula, Kailong Huang, Jonathan Tang, Nectarios Koziris, Georgios Goumas, Zeshan Chishti, +and Nandita Vijaykumar. 2022. DaeMon: Architectural Support for Efficient Data Movement in Disaggregated +Systems. Proc. ACM Meas. Anal. Comput. Syst. 6, 1, Article 21 (March 2022), 34 pages. https://doi.org/10.1145/ +3508041 +1 +Introduction +With recent advances in network technologies [33, 41, 63, 79, 80] that enable high bandwidth +networks, resource disaggregation [33, 87] has emerged as a promising technology for data cen- +ters [16, 33, 37, 54, 87, 94]. Resource disaggregation proposes the physical separation of hardware +devices (CPU, accelerator, memory, and disk) in a server as independent and failure-isolated com- +ponents connected over a high-bandwidth network such as RDMA [63] and Gen-Z [80]. Compared +to monolithic servers that tightly integrate these components (Figure 1a), disaggregated systems +can greatly improve resource utilization, as memory/storage components can be shared across +applications; resource scaling, as hardware components can be flexibly added, removed, or upgraded; +and failure handling, as the entire server does not need to be replaced in the event of a fault in a +device. Thus, resource disaggregation can significantly decrease data center costs. +Disaggregated systems comprise multiple compute, memory and storage components, intercon- +nected over a high-bandwidth network (Figure 1b), each independently managed by a specialized +kernel module (monitor). Typically, each compute component includes a small amount (a few +GBs) of main memory (henceforth referred to as local memory) to improve memory performance. +However, almost all the memory in the data center is separated as network-attached disaggregated +memory components to maximize resource sharing and independence in failure handling (different +from typical hybrid memory architectuers). Thus, the majority of the application working sets is +accessed from the disaggregated memory components (henceforth referred to as remote memory). +Each memory component includes its own controller and can be flexibly shared by many compute +components. Thus, disaggregated systems can provide high memory capacity for applications +with large working sets (e.g., bioinformatics, graph processing and neural networks) at lower cost. +Fine-grain microsecond-latency networking technologies [36, 41, 63, 79, 80] that interconnect all +hardware components have made fully disaggregated systems feasible, being only 2-8× slower +than DRAM bus bandwidth. However, since a large fraction of the application’s data (typically +∼80%) [33, 54, 87] is located and accessed from remote memory, the higher latencies of remotely +accessing data over the network can cause large performance penalties. +Alleviating data access overheads is challenging in disaggregated systems for the following +reasons. First, disaggregated systems are not monolithic and comprise independently managed +entities: each component has its own hardware controller, its resource allocation is transparent from +other components and a specialized kernel monitor uses its own functionality/implementation to +manage the component it runs on (only communicates with other monitors via network messaging +if there is a need to access remote resources). This characteristic necessitates a distributed and +disaggregated solution that can scale to a large number of independent components in the system. +Second, there is high variability in data access latencies as they depend on the location of the +remote memory component and the contention with other compute components that share the +same memory components and network. Data placements can also vary during runtime or between +multiple executions, since data is dynamically allocated in one or more remote memory components +and hardware updates can flexibly change the architecture of the memory component and the +network topology. Third, data is typically migrated at page granularity [5, 6, 10, 35, 54, 57, 87, +103, 109] as it enables: (i) transparency to avoid modifications to existing OS/applications; (ii) +low metadata overheads for address translation; and (iii) leveraging spatial locality within pages. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:3 +network across components +(a) +(b) +Local +Memory +CPU +Compute +Component +processor +monitor +VA -> PA Table +Remote +Memory +Controller +memory +monitor +Memory +Component +VA -> PA Table +DRAM Cache (fast) +CPU +Monolithic Server +Monolithic OS +Main Memory +(slow) +VA -> PA Table +Remote +Memory +Controller +memory +monitor +Memory +Component +VA -> PA Table +Fig. 1. (a) A monolithic system versus (b) a disaggregated system. +However, we observe in §2.2 that moving memory pages in disaggregated systems, i.e., moving +data at a large granularity over the network, can significantly increase bandwidth consumption +and slow down accesses to cache lines in other concurrently accessed pages. +Recent works on hybrid memory systems [3, 4, 23, 26–29, 42, 45, 47, 52, 58–61, 64, 82, 83, 103], for +example, those that integrate die-stacked DRAM [44] caches aim to address the high page movement +costs between main memory and the DRAM cache [23, 42, 60, 61, 83] with mechanisms to move data +at smaller granularities [23, 43, 60, 61, 76, 96, 97], e.g., cache line, or by using page placement/hot +page selection mechanisms [3, 4, 26–29, 45, 47, 52, 58, 59, 64, 82, 103]. However, these prior works +are tailored for a monolithic tightly-integrated architecture (Figure 1a), and are not suitable for +disaggregated systems (See § 7). These works assume centralized data management/allocation +(unlike in disaggregated systems). For instance, software runtimes [3, 4, 26–29, 45, 47, 52, 58, 59, 64, +82, 103] running on CPUs in hybrid systems leverage TLBs/page tables to track page hotness and +move pages across different memory devices (Figure 1a). Instead, in fully disaggregated systems +all hardware memory functionalities (e.g., TLBs, page tables) of remote pages are moved to the +memory components themselves [37, 87] (Figure 1b). Thus they cannot be used to track page +hotness at the CPU side to implement intelligent page placement/movement in local memory. +Similarly, hardware-based approaches [43, 55, 76, 96] add centralized hardware units in the CPU to +track metadata for pages in second-tier memory. This however would incur high hardware costs in +disaggregated systems that enable large amounts (e.g., TBs) of remote memory [87]. Requiring each +compute component to control/track a large number of pages in remote memory components would +impose significant hardware costs and scalability challenges, and thus might annihilate the benefits +of resource disaggregation. Moreover, disaggregated systems incur significant variations in access +latencies and bandwidth based on the current network architecture and concurrent jobs sharing the +memory components/network, which are not addressed by prior work. This necessitates a solution +primarily designed for robustness to this variability. +In this work, we analyze different data movement strategies in fully disaggregated systems, +and introduce DaeMon, an efficient software-transparent mechanism to alleviate data movement +overheads in disaggregated systems. DaeMon provides (i) high performance on dynamic workload +demands, (ii) robustness to variations in architectures, network characteristics and application +behavior, and (iii) independence and scalability to multiple compute components and memory +components that are managed transparently to each other and are flexibly added/removed in the +system. +DaeMon consists of two key ideas. First, we offload data migrations to dedicated hardware +engines, named DaeMon compute and memory engine, that are added at each compute component +and memory component, respectively. This key idea enables independence and scalability to a large +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:4 +Christina Giannoula, et al. +number of compute components and memory components of disaggregated systems. Compared to +a centralized design, DaeMon’s distributed management of data movement enables simultaneous +processing of data movement across multiple components and decreases the processing costs and +queuing delays to serve data requests. Second, we leverage the synergy of three key techniques to +provide robustness to the high variability in network latencies/bandwidth. 1) We use a bandwidth +partitioning approach to enable the decoupled movement of data at two granularities, i.e., page and +cache line, and prioritize cache line granularity data moves over page moves. This design enables +low access latencies to remote memory for the cache line requests on the critical path, while the +associated pages can be still be moved independently at slower rates to retain the benefits of spatial +locality. 2) We design an adaptive approach to decide on-the-fly if a request should be served +by a cache line, page or both. Via selective granularity data movement, we provide robustness +to variations in network, architectures and application characteristics. 3) We leverage hardware +link compression when migrating pages to reduce network bandwidth consumption and alleviate +queuing delays. +The synergy of the afforementioned key techniques provides a robust solution for disaggregated +systems: decoupled multiple granularity data movement effectively prioritizes cache line requests +on the critical path, and migrates pages at a slower rate leveraging compression to reduce bandwidth +consumption. The adaptive granularity selection mechanism effectively adapts to the characteristics +of the application data, e.g., by favoring moving more pages if application data is highly compressible. +The decoupled cache line granularity movement also enables the use of more sophisticated and +effective compression algorithms (with relatively high compression latency) for page migrations. +We evaluate DaeMon using a range of capacity intensive workloads with different memory access +patterns from machine learning, high-performance-computing, graph processing, and bioinfor- +matics domains. Over the widely-adopted approach of moving data at page granularity, DaeMon +decreases memory access latencies by 3.06× on average, and improves system performance by +2.39× on average. We demonstrate that DaeMon provides (i) robustness and significant performance +benefits on various network/architecture configurations and application behavior (Figures 8 and 13), +(ii) scalability to multiple hardware components and networks, (Figure 17), and (iii) adaptivity to +dynamic workload demands, even when multiple heterogeneous jobs are concurrently executed in +the disaggregated system (Figure 18). +This paper makes the following contributions: +• We heavily modify a state-of-the-art simulator to develop and evaluate the overheads of different +data movement strategies in fully disaggregated systems, analyze the challenges of providing +efficient data movement in such systems, and develop DaeMon, an adaptive distributed data +movement mechanism for fully disaggregated systems. +• We enable decoupled data movement at two granularities, and migrate the requested critical +data quickly at cache line granularity and the corresponding pages opportunistically without +stalling critical cache line requests. We dynamically control the data movement granularity to +effectively adapt to the current system load and application behavior. We employ a high-latency +compression scheme to further reduce bandwidth consumption during page migrations. +• We evaluate DaeMon using a wide range of capacity intensive workloads, various architecture/net- +work configurations, and in multi-workload executions of concurrent heterogeneous jobs. We +demonstrate that DaeMon significantly outperforms the state-of-the-art data movement strategy, +and constitutes a robust and scalable approach for data movement in fully disaggregated systems. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:5 +2 +Background and Motivation +2.1 +Baseline Disaggregated System +Figure 2 shows the baseline organization of the disaggregated system, which includes several +compute components and memory components as network-attached components. To improve +performance, each compute component tightly includes a few GBs of main memory, referred to as +local memory, which can typically host 20-25% of the application’s memory footprint [33, 84, 87]. +Each memory component includes its own controller and connects multiple DIMM modules, +referred to as remote memory. +Data +CPU +Cores +On-Chip +Caches +Coherent +Interconnect +FPGA +Control Logic +Translation +Unit +Local Memory +VA -> PA Table +(a) +Controller +Translation Unit +Remote Memory +VA -> PA Table +(b) +Remote +Memory +Controller +Memory +Component +memory +monitor +Local +Memory +CPU +Compute +Component +processor +monitor +Remote +Memory +Controller +Memory +Component +memory +monitor +Remote +Memory +Controller +Memory +Component +memory +monitor +Local +Memory +CPU +Compute +Component +processor +monitor +Fig. 2. High-level organization of a disaggregated system. +We assume distributed OS modules that coordinate and communicate with each other via network +messaging when needed, similar to [37, 54, 87]: processor and memory kernel monitors run at +compute components and memory components, respectively. The memory allocation/management +of remote memory is performed at the memory component itself [37, 87], transparently to compute +components, enabling the different components to be independent. The on-chip caches and the local +memory of compute components are typically indexed by virtual addresses [87], and remote data +is requested from memory components using virtual addresses [16, 37, 54, 87] (unlike in hybrid +memory systems). The data management is typically performed at page granularity [16, 54, 87] (e.g., +4KB). The local memory of the compute component can be treated as a cache with tags [87] or a +local virtual to physical translation mapping [54] can be used (either approach works with DaeMon, +however we assume the second approach in our evaluation). The physical memory addresses of +the local memory can be found by accessing and traversing metadata (tags or local page tables) +kept in a dedicated (pre-reserved) DRAM memory space (kernel metadata is directly indexed via +physical addresses). When an local memory miss happens, either (i) the processor kernel module of +the compute component triggers a page fault and fetches the requested page from remote memory +components [87], or (ii) dedicated software runtimes co-designed with hardware primitives [16] +(e.g., supported in FPGA-based controllers as shown in Figure 2a) handle remote data requests on +demand completely eliminating expensive page faults. Either approach works with DaeMon. We +assume that the controllers of memory components implement hardware-based address translation +(Figure 2b) to access pages in remote memory as proposed in [37]. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:6 +Christina Giannoula, et al. +Jobs running at different compute components can share read-only pages located at multiple +memory components. Similarly to prior state-of-the-art works [16, 37, 87], we assume that the +system does not support writable shared pages across compute components, since they are rare +across datacenter jobs [37, 87]. +2.2 +Data Movement Overheads in Fully Disaggregated Systems +Prior state-of-the-art works [14, 35, 37, 46, 54, 74, 75, 99, 109, 112] typically enable data management +at page granularity for three compelling reasons. First, the memory allocation and management +is transparent, i.e., requires little to no modification to OS or application code. Second, the coarse +granularity enables low metadata overheads for address translation in local memory and remote +memory. Managing local memory as a cache at cache line granularity would incur prohibitively +high metadata overheads [87]. Third, page movements enable exploiting spatial locality in common +memory access patterns [43, 56, 102], and increase the number of accesses served from the lower +cost local memory instead of remote memory. +Figure 3 compares the performance of different data movement strategies in disaggregated +systems across various workloads (See Table 3). We evaluate one memory component and one +compute component having local memory to fit ∼20% of the application’s working set. We use +100ns/400ns latency [33, 54] to model the propagation and switching delays on the network +(referred to as switch latency), and configure the network bandwidth between the compute +component and the memory component (referred to as bandwidth factor) to be 1/4× the DRAM +bus bandwidth [33, 87] of the local memory or remote memory. We compare six configurations: (i) +Local: all accesses are served from the local memory; (ii) cache-line: accessing data from remote +memory at cache line granularity, and directly writing data to the Last Level Cache (LLC) of the +compute component (local memory is not used), (iii) Remote: accessing data from remote memory +at page granularity (moving pages to local memory) accounting for all network-related overheads, +(iv) page-free: remote accesses incur the latency of one cache line granularity remote access and +the corresponding page is transferred to local memory at zero cost (spatial locality is leveraged), (v) +cache-line+page: requesting data from remote memory at both cache line (moved to LLC) and page +granularity (moved to local memory) and servicing data requests using the latency of the packet +that arrives earlier to compute component (accounting for all network-related overheads), and (vi) +DaeMon: accessing data from remote memory using DaeMon (accounting for all network-related +overheads). +We make four observations. First, Remote, i.e., the typically-used approach of moving data at +page granularity, incurs significant performance slowdowns, on average 3.86×, over the monolithic +Local configuration due to transferring large amounts of data over the network. In addition to the +large network bandwidth consumption, migrating pages can slow down critical path accesses to +data in other concurrently accessed pages. Second, page-free achieves almost the same performance +as the Local scheme. A small penalty is incurred as the first access to a page in remote memory +incurs cache line granularity latency access cost. However, since the whole page is migrated to +local memory for free, performance significantly improves thanks to spatial locality benefits of +migrating pages in addition to the requested cache line. Thus, migrating pages to local memory +is critical to achieving high performance. Third, cache-line outperforms Remote in some latency- +bound workloads with poor spatial locality, however its performance benefits depend on network +characteristics. For example, in tr, cache-line outperforms Remote by 1.42× when the switch latency +is 100ns, while it incurs 1.82× performance slowdown over Remote with 400ns switch latency. +Fourth, the cache-line+page scheme, i.e., naively moving data at both granularities, is still inefficient +(only 1.11× better than Remote), since critical cache lines are still queued behind large pages. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:7 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +2 +4 +Slowdown +stch-lat=100 bw-fact=1/4 +10 11 +39 39 +cache-line +Remote +page-free +cache-line+page +DaeMon +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +2 +4 +6 +8 +Slowdown +stch-lat=400 bw-fact=1/4 +10 11 +39 39 +12 +10 +cache-line +Remote +page-free +cache-line+page +DaeMon +Fig. 3. Data movement overheads in disaggregated systems. +Overall, we draw two conclusions. (i) Page migrations incur high performance penalties and +can significantly slow down the critical path cache line requests to other concurrently accessed +pages. However, if the overheads of migrating pages can be mitigated, moving data at page granu- +larity offers a critical opportunity to alleviate remote access costs. (ii) There is no one-size-fits-all +granularity in data movements to always perform best across all network configurations and appli- +cations. Depending on the spatial locality and the network load, some applications benefit from +cache line-only accesses that avoids unnecessary congestion of pages in the network, while some +applications significantly benefit from page movements that leverage spatial locality. To this end, +we design DaeMon to significantly reduce data movement costs across various application, network +and architecture characteristics. Figure 3 demonstrates that DaeMon significantly outperforms the +Remote and cache-line+page schemes by on average 2.38× and 2.14×, respectively. +3 +DaeMon: Our Approach +DaeMon is an adaptive and scalable data movement mechanism for fully disaggregated systems that +supports low-overhead page migration, enables software transparency, and provides robustness +to variations in memory component placements, network architectures and application behavior. +DaeMon comprises two key ideas: +(1) Disaggregated Hardware Support for Data Movement Acceleration. We enhance each +compute component and memory component with specialized engines, i.e., DaeMon compute +and memory engine (Figure 4), respectively, to manage data movements across the network of +disaggregated systems. DaeMon engines enable independence and high scalability to a large number +of compute components/memory components that are flexibly added/removed in disaggregated +systems. Moreover, distributed management of data migrations at multiple DaeMon engines in- +creases the execution parallelism and decreases the processing costs and queuing delays to serve +data requests. +(2) Synergy of Three Key Techniques. DaeMon incorporates three synergistic key techniques +shown in Figure 4: +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:8 +Christina Giannoula, et al. +Remote +Memory +Controller +Compute +Component +Memory +Component +Local +Memory +CPU + LLC +Page +Queue +Sub-block +Queue +(De) +Compr. +Unit +Cachelines +Compressed +Pages +Network +DaeMon Compute Engine +DaeMon Memory Engine +Page +Queue +Sub-block +Queue +Selection +Granularity Unit +(De) +Compr. +Unit +Fig. 4. High-level overview of DaeMon. +(I) Decoupled Multiple Granularity Data Movement. First, we integrate two separate hardware +queues to manage and serve data requests from remote memory at two granularities, i.e., cache line +(via the sub-block queue) and page (via the page queue) granularity. Cache line requests are directly +moved to Last Level Cache (LLC) of the compute component to avoid additional metadata overheads +and eliminate memory latency. Page requests are moved to local memory of the compute component. +Second, we prioritize moving cache lines over moving pages via a bandwidth partitioning approach: +a queue controller serves cache line and page requests with a predefined fixed ratio to ensure +that at any given time a certain fraction of the bandwidth resources is always allocated to serve +cache line requests quickly. DaeMon implements both network and remote memory bus bandwidth +partitioning. +This technique provides two benefits. First, retaining page migrations in DaeMon (i) enables +software-transparency, (ii) allows maintaining metadata for DRAM at page granularity (thus +incurring low metadata overheads), and (iii) exploits the performance benefits of data (spatial) +locality within pages. Second, cache line data movements that are on the critical path are quickly +served, and have fewer slowdowns from expensive page movements that may have been previously +triggered, since DaeMon effectively prioritizes cache line movements. +(II) Selection Granularity Data Movement. To handle network, architecture and application +variability in disaggregated systems, we design an dynamic approach to decide whether a request +should be served by a cache line, page, or both, depending on application and network characteristics. +At DaeMon’s engine of each compute component, we include two separate hardware buffers to +track pending data migrations for both cache line and page granularity, and a selection granularity +unit to control the granularity of upcoming data requests based on the utilization of the above +buffers. The utilization of these buffers allows us to capture dynamic information regarding the +current traffic in the system and the application behavior (i.e., locality). Our proposed selection +granularity data movement enables robustness against fluctuations in network, architecture and +application characteristics (we explain how this is implemented in §4.2). +(III) Link Compression on Page Movements. We leverage the decoupled page movement to use +a high-latency link compression scheme (with high compression ratio), when moving pages across +the network. We integrate hardware compression units at both the compute components and +memory components to highly compress pages moved over the network: the page is compressed +before it is being transferred over the network, and decompressed when it arrives at the destination +(before it is written in memory modules). Link compression on page movements reduces the network +bandwidth consumption and alleviates network bottlenecks. +Overall, DaeMon cooperatively integrates all three key techniques, the synergy of which provides +a versatile solution: +(1) Prioritizing requested cache lines helps DaeMon to tolerate high (de)compression latencies in +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:9 +page migrations over the network, while also leveraging benefits of page migrations (low metadata +overheads, spatial locality). +(2) Moving compressed pages consumes less network bandwidth, helping DaeMon to reserve part +of the bandwidth to effectively prioritize critical path cache line accesses. +(3) Selection granularity movement helps DaeMon to adapt to the application data compressibility: +if the pages are highly compressible, the number of pending page migrations is relatively low, +thus DaeMon favors moving data more at page granularity instead of cache line granularity (and +vice-versa). +4 +DaeMon: Detailed Design +We design DaeMon to be a disaggregated solution: a DaeMon compute engine is added at each +compute component of the system to handle data requests to remote memory, and a DaeMon +memory engine is integrated at the controller of each memory component of the system. Figure 5 +shows our proposed architecture. +Network +Compute Component +Memory Component +CPU +Cores +LLC +Local Memory +Coherent +Interconnect +Remote Memory +FPGA +DaeMon +Compute +Engine +Control +Logic +Controller +DaeMon Memory +Engine +Fig. 5. Proposed architecture for compute component and memory component. +The baseline architecture of each compute component includes a chiplet-based CPU+FPGA +architecture (this CPU+FPGA integrated design has also been proposed to prior state-of-the-art +work [16, 40]), which is expected to have small cost [16] compared to the overall cost savings +enabled by disaggregated systems, while it is also socket compatible to current systems [25, 40]. The +FPGA has three communication paths: i) a coherent path, i.e., CPU-FPGA coherent links, to access +the CPU on-chip cache hierarchy, ii) an interface (channel-based connection) to access the local +memory, and iii) an external connection to the network controller to move data to/from remote +memory. We propose extending the FPGA by adding a new lightweight hardware component to +handle data requests, i.e., the DaeMon compute engine. +Each memory component includes its own controller [37, 54, 87], that has two communication +paths: a channel-based connection to DIMM modules of remote memory, and an external connection +to the network, which is used to move data from/to compute components. We propose extending +the controller of each memory component by adding a new hardware component to handle data +movements, i.e., the DaeMon memory engine. +In our study, we assume that the local memory is an inclusive cache for the remote memory, +which contains all application data. The local memory implements an approximate LRU replacement +policy, similar to prior state-of-the-art work [87]. +4.1 +Enabling Decoupled Multiple Granularity Data Movement +Figure 6 shows the detailed design of the DaeMon compute engine and DaeMon memory engine. +DaeMon engine includes two queues to handle requests at each granularity: cache line granularity +via the sub-block queue 1 , and page granularity via the page queue 2 . It also includes a queue +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:10 +Christina Giannoula, et al. +controller 7 to serve requests from both queues, and a packet buffer 6 to temporarily keep +arrived packets, while they are being processed. +CPU +Interface +LLC +LLC +Sub-block +Queue +Queue +Controller +Packet Unit +Compression +Unit +Decompression +Unit +Network +Interface +DaeMon Compute Engine +Packet +Buffer +Inflight +Sub-block +Buffer +Inflight +Page Buffer +Selection Granularity Unit +Page +Queue +Dirty Unit +Dirty Data +Buffer +Controller +Memory Interface +1 +2 +5 +7 +3 +4 +6 +9 +8 +10 +Network +Interface +Sub-block +Queue +Queue Controller +Memory +Interface +DaeMon Memory Engine +Page +Queue +Packet Unit +Decompression +Unit +Compression +Unit +Packet +Buffer +1 +2 +7 +6 +9 +10 +Fig. 6. Detailed design of DaeMon engines for the compute (left) and memory (right) components. +Approximate Bandwidth Partitioning. To prioritize cache line data movements while also +ensuring that page movements are not aggressively stalled, we design an approximate bandwidth +partitioning approach between the cache line and page movements, and configure the queue +controller to serve cache line and page requests with a predefined fixed ratio. Assuming that +cache line and page requests transfer 64B and 4KB of data, respectively, and having a bandwidth +partitioning ratio of 25% (Figure 11 presents a sensitivity study on this ratio), 25% of the bandwidth +is reserved for cache lines as follows: for each page request issued through the network, which +results in transferring 4KB data, the queue controller needs to serve 4096/64 ∗ 0.25/(1 − 0.25) ≈ 21 +cache line requests, each transferring 64B of data. To ensure this approximate partitioning is always +maintained, we retain this alternate serving of page and cache line requests even if either queue is +empty (i.e., requests may not issued in all cycles). DaeMon implements an approximate bandwidth +partitioning both in the network across components of the system and when accessing data from +remote memory modules. +4.2 +Selecting the Data Movement Granularity +DaeMon compute engine additionally includes two separate hardware buffers to track data requests +which are scheduled to be moved or in the process of being migrated (henceforth referred to as +inflight): (i) the inflight sub-block buffer for the cache line granularity requests 3 , and (ii) the +inflight page buffer for the page granularity requests 4 . Both buffers are used to track pending data +migrations and avoid requesting the same data multiple times. DaeMon compute engine includes +a selection granularity unit 5 which throttles data requests to avoid requesting the same data +multiple times, and decides at which granularity the request should be served (cache line, page, or +both granularities). +Scheduling Page Granularity Data Movements. When DaeMon compute engine receives a data +request, the selection granularity units checks (i) the utilization of the inflight page buffer, and +(ii) if the corresponding page has already been scheduled to be moved. If the page has already +been requested or the inflight page buffer is full, the selection granularity unit does not request the +page. Thus at any given time, the number of pages scheduled to be moved is automatically limited +by the selection granularity unit, also limiting storage/area overheads to track the pending page +migrations. +If the inflight page buffer is not full, the selection granularity units schedules the page migration +by adding a new entry in the page queue and the inflight page buffer, marking the page as scheduled. +When the queue controller issues the movement, the corresponding entry is released in the page +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:11 +queue, and the page entry in the inflight page buffer is marked as moved. When the requested page +arrives, the corresponding entry is released (invalid state) in the inflight page buffer. The page +is written to local memory and all pending requests are serviced via local memory. Any entries +in the inflight sub-block buffer with requests to cache lines in the same page are removed and +thus, any data packets that arrive in the future with cache lines from the same page are simply +ignored. In DaeMon, we retain existing data management and address translation mechanisms at +page granularity. Local page table updates at the compute component are only performed in page +migrations. +Scheduling Cache Line Granularity Data Movements. To decide whether a cache line gran- +ularity movement should be made, the selection granularity unit checks (i) the utilization of the +inflight sub-block buffer and (ii) if the corresponding page was already scheduled to be moved +(by a previous request). There are two cases. First, if the corresponding page is not scheduled to +be moved according to the inflight page buffer, the selection granularity unit always schedules a +cache line granularity data movement. Second, if the corresponding page is already scheduled to +be moved, the selection granularity unit sends the cache line only if: (i) the sub-block buffer has +lower utilization than the page buffer and (ii) the page is not already in the process of migration +(i.e., the page is in the page queue). Otherwise, it drops the request as the page has already been +requested. This avoids unnecessarily sending cache lines when the corresponding page is likely to +arrive faster and when the sub-block queue is likely to be slow due to oversaturation. +If a cache line is scheduled, a new entry is added both in the sub-block queue and the inflight +sub-block buffer. When the queue controller issues the movement, the corresponding entry is +released in the sub-block queue. When the requested cache line arrives at the compute component, +the corresponding entry is released in the sub-block buffer, and the data is directly written to LLC +through the FPGA-based coherent interconnect. +The above mechanism enables an adaptive approach for the data movement granularity based +on the dynamic network/architecture and application characteristics: +(1) If there is high locality within pages, there are fewer pages requested, and the sub-block buffer +fills up faster than the page buffer. Thus, DaeMon favors issuing pages and throttles cache line +requests. If there is low locality within pages, the page buffer fills up faster than the sub-block buffer, +since cache line requests are served with a higher rate than page requests (e.g., 21:1 cache lines +versus pages requests for 25% bandwidth ratio). Thus, DaeMon favors issuing cache line movements +and throttles page migrations. +(2) If both the page and sub-block buffers are fully utilized, DaeMon detects bandwidth constrained +scenarios. In bandwidth constrained scenarios, DaeMon favors issuing more cache line movements +to alleviate bandwidth bottlenecks. When the bottleneck is mitigated, (inflight buffers are not fully +utilized), DaeMon schedules more page movements to obtain locality benefits. +(3) Additionally, when using link compression to transfer pages, DaeMon is able to adapt to the +compressibility of the application data: if the pages are highly compressible, the inflight page buffer +empties at a faster rate and thus DaeMon favors sending more page migrations (and vice versa). +4.3 +Handling Dirty Data +Dirty data (cache lines/pages) is always directly written to remote memory. Data (cache line or +page granularity) can be in one of the three states: (i) local: when data is cached in on-chip caches +(for cache lines) or local memory (for pages), (ii) remote: when data is only in remote memory, and +(iii) inflight: when data is being migrated. With DaeMon, data can be present simultaneously in two +states: for example, local as a cache line (in the cache hierarchy of the compute component) and +inflight as a page or vice versa. This poses coherence issues if the processor writes to data in the +above state. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:12 +Christina Giannoula, et al. +There are two scenarios: (i) if a page arrives to compute component before a prioritized cache +line, any modifications to the page may be overwritten by the stale cache line that arrives later, +and (ii) if a dirty cache line is evicted from the LLC while the corresponding page is in transit, the +modifications would be lost when the page arrives to compute component. As explained, in the +(i) scenario, when a page arrives, the corresponding entries in the inflight sub-block buffer with +requests to cache lines in the same page are removed and thus, any data packets that arrive in the +future with cache lines from the same page are simply ignored. In the (ii) scenario, for every dirty +cache line that gets evicted by the LLC and also misses in the local memory, its corresponding page +can be either inflight or in remote memory. To ensure correctness, DaeMon compute engine first +checks if there is an inflight page request in the inflight page buffer. If there is no inflight page +request (according to the inflight page buffer), the evicted dirty cache line is directly migrated to +remote memory. In the other case, we need to retain the dirty cache line until the page arrives. We +include a dirty unit 8 in the DaeMon compute engine with a dirty data buffer that temporarily +stores these dirty cache lines. When the corresponding inflight page arrives, the DaeMon compute +engine flushes the dirty cache line(s) from the dirty buffer to local memory. +Prior works [2, 16] observe that typically a few cache lines (1-8 cache lines) or all cache lines of +a page are accessed. Thus, when the evicted dirty cache lines of the same page increase beyond a +predefined threshold (e.g., 8 cache lines), the DaeMon compute engine flushes all dirty cache lines +to remote memory, and marks the corresponding entry for that page in the inflight page buffer as +throttled. When the inflight page arrives, the DaeMon compute engine ignores it, since its entry is +in the throttled state, and sends a new request for that page to receive the up-to-date data. This +enables lower area/storage overheads for the dirty data buffer. +4.4 +Link Compression in Page Migrations +Approaches for data compression are typically of two types: (i) latency-optimized compression +schemes [8, 21, 73, 104, 105], which optimize/minimize the (de)compression latencies, and (ii) +ratio-optimized compression schemes [1, 49, 93, 111], which provide higher compression ratios +while incurring relatively high (de)compression latencies. We select a ratio-optimized compression +scheme in DaeMon based on two observations (§6): (i) in disaggregated systems, queueing delays and +network latencies can be significant, thus compression benefits outweigh the high (de)compression +latencies, and (ii) DaeMon prioritizes cache lines that are on the critical path, thus we can tolerate +relatively high (de)compression latencies for page migrations. +DaeMon engines include (de)compression units 9 +10 that compress pages transferred through +the network. We implement a hardware design similar to IBM MXT [1, 93], using the LZ77 compres- +sion algorithm [111], and operating at 1KB granularity at a time. (De)Compression units include 4 +engines, each of which operates on 256B of data and uses a 256B shared dictionary, incurring in +total a 64-cycle latency according to [1, 93]. +4.5 +DaeMon’s Hardware Structures +We estimate the overheads of DaeMon’s hardware structures for each compute component assuming +a 64-core CPU, using CACTI [66]. The sizes of the DaeMon sub-block and page queues and the +sub-block and page buffers have been selected based on the maximum possible number of pending +data migrations at a time, which is determined by the number of the available LLC MSHRs (Miss +Status Holding Registers) in a typical CPU system, and is independent of the workloads’ patterns +and the mix of workloads that are running at each time. For the hardware structures at each +memory component, we scale the sizes of the DaeMon sub-block and page queues, assuming that +each memory component can concurrently serve up to 4 compute components. Table 1 presents +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:13 +the hardware overheads of DaeMon compute engine (C) and DaeMon memory engine (M). Figure 7 +shows an inflight sub-block buffer entry, an inflight page buffer entry, and a dirty data buffer entry. +Hardware +Entries +Size +Access +Area +Energy +Structure +(KB) +Cost (ns) +Cost (mm2) +Cost (nJ) +Sub-block Queue (C) +128 +0.5 +0.34 +0.084 +0.038 +Sub-block Queue (M) +512 +2 +0.38 +0.093 +0.039 +Page Queue (C) +256 +1 +0.35 +0.087 +0.038 +Page Queue (M) +1024 +4 +0.40 +0.105 +0.041 +Inflight Sub-block Buffer (C) +128 +1.625 +0.56 +0.041 +0.046 +Inflight Page Buffer (C) +256 +3.25 +0.77 +0.089 +0.096 +Dirty Data Buffer (C) +256 +17 +0.62 +0.168 +0.046 +Packet Buffer (C) +- +8 +0.538 +0.137 +0.044 +Packet Buffer (M) +- +32 +1.032 +0.263 +0.047 +2 × Dictionary Table (C,M) +1024 +1 +0.28 +0.015 +0.020 +Table 1. DaeMon’s hardware overheads for C: compute engine and M: memory engine. +Address State +Dirty Cache +Line Offset +Cache Line +Offset +Address State +Data +Address +00: scheduled +01: moved +10: throttled +11: invalid +0: scheduled +1: invalid +0010...11010 +32 bits 2 bits +64 bits +64 bits +32 bits +1 bit +512 bits +32 bits +b) Inflight Page Buffer Entry +a) Inflight Sub-block Buffer Entry +c) Dirty Data Buffer Entry +0010...11010 +Fig. 7. An inflight sub-block buffer entry, an inflight page buffer entry, and a dirty data buffer entry. +Sub-block Queue (SRAM), 128 entries: The sub-block queue size is limited by the available LLC +MSHRs of the compute component. +Page Queue (SRAM) - 256 entries: The page queue has 256 entries, since DaeMon serves requests +from the page queue at a smaller rate than the sub-block queue. +Inflight Sub-block Buffer (CAM) - 128 entries: Similar to the sub-block queue, this buffer has +128 entries. We design this hardware structure to be indexed using the corresponding page address +to achieve smaller area costs, since at a given time there may be multiple inflight cache line requests +to the same page. Each entry (Figure 7a) includes the page address, the state (scheduled or invalid), +and a 64-bit queue that is used to indicate the offsets within the page of the inflight cache requests +by (re)setting the corresponding bits. +Inflight Page Buffer (CAM) - 256 entries: An inflight page buffer entry (Figure 7b) includes +the page address, the state that can be scheduled, moved, throttled (when the page needs to be +re-requested) or invalid, and a 64-bit queue to indicate the offsets of the dirty cache lines of the +inflight page that are temporarily kept in the dirty data buffer. +Dirty Data Buffer (SRAM) - 256 entries: A dirty data buffer entry (Figure 7c) includes the evicted +cache line and its address. +Packet Buffer (SRAM) - 8KB: We use an 8KB buffer to temporarily store arrived data packets +until they are processed. +Dictionary Tables for (De)Compression (CAM) - 2KB: DaeMon proposes 4 engines at each +(de)compression unit, each of them has 256B CAM [1, 93]. In total, we estimate each dictionary +table as 1KB CAM. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:14 +Christina Giannoula, et al. +Overall, DaeMon’s hardware overheads are due to the cache memories corresponding to the +sub-block and page queues, the sub-block and page buffers, and the dictionary tables used for data +compression. The total sizes of the DaeMon cache memories are ∼34KB and 40KB for the DaeMon +compute and memory engine, respectively. Therefore, DaeMon’s hardware overheads are similar to +that of the small L1 cache memory of a modern state-of-the-art processor (e.g., Intel Xeon). We +conclude that our proposed hardware structures incur very modest hardware and financial costs to +be integrated into the compute components and memory components of disaggregated systems. +4.6 +Handling Failures +DaeMon handles compute component, memory component and network failures using fault- +tolerance approaches of prior works [16, 54, 87]. If the compute component fails (CPU or DaeMon +compute engine), the application needs to be restarted potentially on a different compute com- +ponent of the system. Network failures are handled using timeouts: DaeMon engines can trigger +timeouts when pending page or cache line requests have not arrived after a long time, or when +ACK messages have not been received for migrations of dirty data. The exploration of the timeout +period value is left for is future work. Finally, memory component failures are handled via data +replication, similarly to prior work [87]: DaeMon can send the evicted dirty data to more than one +memory component, and wait to receive ACK messages from all of them. +4.7 +DaeMon Extensions +Prefetching. DaeMon can flexibly support hardware/software-based prefetchers. Existing CPU +prefetchers might generate data requests, which DaeMon can normally serve by migrating the +prefetched data at a cache line granularity, page granularity or both granularities, via on our +proposed selection granularity scheme. Page prefetchers [62] might generate page-granularity +data requests, which DaeMon can serve by migrating the prefetched data at page granularity or +throttling the page request based on our selection granularity scheme. +Large Pages. DaeMon can be easily extended to support large granularity pages (e.g., 2MB). To +effectively prioritize cache line requests over page requests, DaeMon’s predefined ratio for the +approximate bandwidth partitioning needs to be properly configured based on the size of the large +page. To enable multiple page sizes (e.g., both 4KB and 2MB), we could enhance DaeMon to split +large pages (e.g., 2MB) to consecutive page requests of smaller sizes (e.g., 4KB) issued in the page +queue. +5 +Methodology +Simulation Methodology. We use Sniper [17, 18], a state-of-the-art accurate simulator, and we +heavily modified it to model a disaggregated system with one compute component and multiple +memory components interconnected across the network. We present detailed evaluation results +using one memory component and provide a characterization study of multiple memory components +with various network configurations in Figure 17. For the network across components, we use +both (i) a fixed latency of 100ns/400ns [33, 54] to model propagation and switching delays inside +network (referred to as switch latency), and (ii) a variable latency of modeling the current bandwidth +utilization at each simulation interval (100K ns) when configuring the network bandwidth to be 2-8× +less than DRAM bandwidth [33, 87] (referred to as bandwidth factor). For the compute component, +we configure a state-of-the-art CPU server with on-chip cache memories of typical sizes and +x86 OoO cores of 3.6GHz frequency. The local memory size is configured to fit ∼20% of each +application’s working set, and we evaluate LRU replacement policy [87] in local memory, unless +otherwise stated. The aforementioned configuration is consistent with prior state-of-the-art works +in disaggregated systems [33, 54, 87]. For both the local memory and remote memory, we evaluate +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:15 +a DDR4 memory model with 17GB/s bus bandwidth, and we simulate hardware-based address +translation for memory pages, having overhead as one DRAM access cost per lookup, as explained +in prior state-of-the-art work [37]. We evaluate access overheads in DaeMon queues/buffers using +CACTI [66] (See Table 1). Table 2 lists the parameters of our simulated system. +CPU +3.6 GHz, 4-way OoO x86 cores, 224-entry ROB; +L1 Instr. Cache +32 KB, 4-way associativity, LRU; +L1 Data Cache +32 KB, 8-way associativity, 4-cycle access latency, LRU; +L2 Cache +256 KB, 8-way associativity, 8-cycle access latency, LRU; +LLC +4MB, 16-way associativity, 30-cycle access latency, LRU; +Local Memory +2400MHz, 15ns process. latency, 17GB/s bus bandwidth [13]; +Network +2-8× less than bus bandwidth, 100-400ns switching latency [33, 87]; +Remote Memory +2400MHz, 15ns process. latency, 17GB/s bus bandwidth [13]; +Table 2. Configuration of simulated system. +Workloads. We evaluate various workloads with different memory access patterns from various +application domains including graph processing, machine learning, bioinformatics, linear algebra, +data analytics, and HPC domains, shown in Table 3. The dynamic working sets at any given point at +runtime range from 43.2MB to 1.32GB. In a fully disaggregated system, the application working set +(irrespective of the size) is primarily housed in remote memory to provide the benefits of improved +elasticity, heterogeneity, and failure isolation. Therefore, we configure the local memory size to fit +∼20% of each application’s working set (similar to prior state-of-the-art work [33, 54, 87]). All data +is initially located at remote memory. We simulate most workloads to full execution and for slower +long running workloads, we simulate 1B instructions. +Workload +Domain +Input Data +K-Core Decomposition (kc) [88] +Graph Processing +1M vertices x 10M edges +Triangle Counting (tr) [88] +Graph Processing +1M vertices x 10M edges +Page Rank (pr) [88] +Graph Processing +1M vertices x 10M edges +Needle Wunsch (nw) [20] +Bioinformatics +4096 base pairs per sequence +Breath First Search (bf) [88] +Graph Processing +1M vertices x 10M edges +Betweenness Centrality (bc) [88] +Graph Processing +1M vertices x 10M edges +Timeseries (ts) [106] +Data Analytics +262144 elements in sequence +Sparse Matrix Vector Multipl. (sp) [51] +Linear Algebra +pkustk14 matrix +Sparse Lengths Sum (sl) [67] +Machine Learning +Kaggle Criteo 10GB Dataset +High Perf. Conjugate Gradient (hp) [39] +HPC +104 x 104 x 104 +Particle Filter (pf) [20] +HPC +4096 x 4096, 30000 particles +Darknet19 (dr) [81] +Machine Learning +dog.jpg (768 x 576 pixels) +Resnet50 (rs) [81] +Machine Learning +dog.jpg (768 x 576 pixels) +Table 3. Summary of workloads. +6 +Evaluation +We evaluate six schemes: (i) Remote: the typically-used approach [16, 54, 87] of moving data to/from +remote memory at page granularity; (ii) LC: DaeMon’s link compression for page movement without +enabling cache line granularity data movement, i.e., moving data at page granularity with LZ-based +link compression enabled; (iii) BP: enabling only DaeMon’s decoupled multiple granularity data +movement with 25% bandwidth partitioning ratio for cache line movements, i.e., moving data always +at both granularities; (iv) PQ: enabling DaeMon’s both decoupled multiple granularity and selection +granularity data movement with 25% bandwidth partitioning ratio for cache line movements +(without enabling data compression in page migrations); (v) DaeMon: DaeMon’s complete design +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:16 +Christina Giannoula, et al. +enabling all its three techniques (25% bandwidth partitioning ratio); and (vi) Local: the monolithic +approach where all the data fits in local memory of the compute component. +Performance. Figure 8 compares all schemes with different network configurations. Our evaluated +workloads exhibit three patterns and we make the following observations. +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=100 bw-fact=1/2 +11 +14 +20 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=100 bw-fact=1/4 +7 +9 +10 +15 +23 +39 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +6 +7 +Speedup +stch-lat=100 bw-fact=1/8 +20 +173797 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=400 bw-fact=1/2 +10 +9 +20 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=400 bw-fact=1/4 +7 810 +142239 +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +6 +7 +Speedup +stch-lat=400 bw-fact=1/8 +20 +193797 +Fig. 8. Speedup in all workloads normalized to Remote using various network configurations. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +LC +BP +PQ +DaeMon +LocalDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:17 +First, kc, tr, pr, and nw exhibit relatively poor spatial locality within pages. In such workloads, BP +effectively prioritizes critical cache line requests. However, PQ provides significant benefits thanks +to dynamically selecting the data movement granularity: the page buffer saturates faster than the +sub-block buffer given the poor locality and the higher servicing rate of the cache line requests in +the queue controller, thus the selection granularity unit enables the movement of more cache lines +and fewer pages. This results to reduced access latencies as critical path cache line requests are no +longer stalled behind many page migrations. +Second, bf, bc, and ts exhibit medium spatial locality within pages. In such workloads, both LC +and PQ decrease data access costs using different approaches: LC enables exploiting more spatial +locality by moving more pages, while PQ accelerates accesses to the critical path cache line requests, +both of which benefit these workloads. +Third, the remaining workloads exhibit high spatial locality within pages, thus page migration is +critical to leverage data locality. In these workloads, BP incurs high performance slowdowns, since +it is oblivious to application behavior. Instead, PQ effectively enables more page movements and +throttles cache line movements by tracking pending data requests, thus achieving similar system +performance to Remote. LC performs better for sp, sl, hp, and pf, since these workloads have higher +data compressibility than dr and rs. +Fourth, when network bandwidth is more constrained, LC provides even higher performance over +Remote, while PQ is unaffected by bandwidth as the bandwidth partitioning approach prioritizes +cache line movements even with low available bandwidth. +Fifth, PQ is slightly affected by the switch latency (Please also see Figure 20 in Appendix § A): +PQ outperforms Remote by 1.60× and 1.51× for 100ns and 400ns switch latency, respectively. The +slightly lower benefits are due to PQ’s inability to hide network switch latencies in critical cache +line movements. Instead, LC is unaffected by switch latency, as page movement incurs much higher +overheads (due to very high network processing and queueing delays) over the smaller switch +latency, which link compression is able to alleviate. +Finally, DaeMon provides high performance benefits for all three classes of workloads with +different locality characteristics thanks to synergistically integrating both LC and PQ: (i) PQ helps +hide the (de)compression latencies in LC and migrate fewer pages in order to prioritize critical path +cache line movements, and (ii) LC releases network bandwidth resources and helps recover the lost +spatial locality in pages by moving more pages with the available network bandwidth. dr and rs +show only 1.05× speedup over Remote as neither LC nor PQ is able to provide speedups due to the +poor application data compressibility and high spatial locality within pages (which favors moving +pages rather than cache lines). DaeMon’s adaptive approach also provides high performance benefits +across all network configurations: (i) when the switch latencies are high, cache lines movements are +slowed down and the sub-block queue fills up faster, thus DaeMon favors moving more pages, which +is more effective at high network switch latencies; and (ii) the approximate bandwidth partitioning +approach effectively prioritizes cache line over page movements even when network bandwidth is +constrained. Therefore, DaeMon significantly outperforms the state-of-the-art Remote scheme by +1.85×, 2.36×, 2.97× for 1/2, 1/4, and 1/8 bandwidth factor, respectively. +Overall, we conclude that DaeMon’s cooperative techniques provide a robust approach to alleviate +data movement overheads across various network characteristics and application behavior. +Memory Access Costs. Figure 9 compares the average data access costs (latencies) achieved by +various schemes normalized to Remote. Due to space limitations, in the remaining plots, we present +a representative subset of our evaluated workloads, but we report geometric mean values across +all evaluated workloads. Please also see Figure 19 in Appendix § A, which compares the network +bandwidth utilization achieved by the various data movement schemes. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:18 +Christina Giannoula, et al. +pr +nw +bf +bc +sp +hp +dr +rs +GM +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Access Costs +stch-lat=100 bw-fact=1/2 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +pr +nw +bf +bc +sp +hp +dr +rs +GM +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Access Costs +stch-lat=400 bw-fact=1/4 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +1.0 +Fig. 9. Data access costs achieved by various schemes normalized to Remote. +We make three observations. First, LC improves data access costs over Remote by 2.12× across +all network configurations (not graphed), because it reduces the network processing costs and +queueing delays by sending fewer bytes through the network. PQ improves access costs (2.06× +over Remote across all configurations) by prioritizing critical path cache line movements. Second, +PQ significantly reduces data access costs in workloads with poor page locality (e.g., pr, nw), since +critical path cache line movements are not stalled by migrating pages. However, in applications +with high data locality (e.g., dr, rs), although PQ reduces data access costs by 1.43× over Remote, +it improves performance by only 1.05×, because the selection granularity unit favors sending +pages for workloads with high locality and a few requests are served at cache line granularity. +Third, DaeMon significantly reduces data access costs by 3.06× over Remote. DaeMon employs link +compression to migrate more pages with lower network overhead over PQ, thus exploiting more +data locality, while also leveraging the ability to prioritize critical cache line requests. In pr, DaeMon +can achieve lower access latency than Local, since serving requests from both local memory and +remote memory increases the effective aggregate memory bandwidth. +Hit Ratio in Local Memory. Figure 10 presents the hit ratio in local memory, and is thus a measure +of the page movement benefits. To prioritize cache lines, PQ throttles some page migrations, thus +reducing the local memory hit rate as a tradeoff for reduced access latencies to critical path cache +line requests. However, DaeMon enables moving more pages over PQ thanks to link compression, +while still retaining the cache line prioritization benefits of PQ. The numbers shown over each +bar for DaeMon present the additional pages that were moved in DaeMon as a percentage over +PQ, thanks to the reduced bandwidth consumption provided by link compression. A zero value +indicates that neither PQ nor DaeMon has throttled any page movement. +We draw three findings. First, Remote has on average 97.7% hit ratio in local memory. Thanks +to high spatial locality, all workloads benefit from page migration, leading to high hit rates: even +workloads with relatively poor spatial locality (e.g., nw) have 90% hit ratio in local memory. Second, +PQ decreases the hit ratio in local memory by up to 18.4% over Remote, because PQ throttles page +movements in some workloads to prioritize cache line requests, thus increasing the number of +accesses to remote memory. Third, DaeMon recovers most of the lost local memory hits, achieving +on average only 0.4% worse hit ratio over Remote. Leveraging link compression in DaeMon reduces +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +Remote +LC +PQ +DaeMon +LocalDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:19 +pr +nw +bf +bc +sp +hp +dr +rs +GM +80.0 +82.5 +85.0 +87.5 +90.0 +92.5 +95.0 +97.5 +100.0 +Hit Ratio (%) +stch-lat=100 bw-fact=1/2 +46 +45 +100 +100 +0 +0 +0 +0 +72 +pr +nw +bf +bc +sp +hp +dr +rs +GM +80.0 +82.5 +85.0 +87.5 +90.0 +92.5 +95.0 +97.5 +100.0 +Hit Ratio (%) +stch-lat=400 bw-fact=1/4 +25 +70 +91 +82 +0 +100 +100 +100 +70 +Fig. 10. Hit ratio in local memory achieved by various schemes. +network bandwidth consumption and significantly increases the number of pages that can be +migrated over PQ. Across all configurations (not graphed) DaeMon migrates 68.9% of the pages +throttled by PQ via leveraging link compression. We conclude that DaeMon enables both leveraging +the benefits of data locality within pages and the prioritization of critical path cache line requests. +Bandwidth Partitioning Ratio. Figure 11 presents a sensitivity study on the bandwidth parti- +tioning ratio between the cache line and page movements. +pr +nw +bf +bc +sp +hp +dr +rs +GM +1 +2 +3 +4 +5 +6 +7 +Speedup +stch-lat=100 bw-fact=1/2 +10 +11 +12 +11 +14 +16 +16 +10% PQ +25% PQ +50% PQ +80% PQ +10% DaeMon +25% DaeMon +50% DaeMon +80% DaeMon +pr +nw +bf +bc +sp +hp +dr +rs +GM +1 +2 +3 +4 +5 +Speedup +stch-lat=400 bw-fact=1/2 +10 +10 +11 +9 +9 +11 +13 +10% PQ +25% PQ +50% PQ +80% PQ +10% DaeMon +25% DaeMon +50% DaeMon +80% DaeMon +Fig. 11. Performance of PQ and DaeMon normalized to Remote varying the bandwidth partitioning ratio. +We draw three findings. First, a higher bandwidth partitioning ratio (e.g., 50%) than DaeMon’s +default 25% ratio, incurs slowdowns in workloads of medium and high spatial locality, and only +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +Remote +LC +PQ +DaeMon21:20 +Christina Giannoula, et al. +improves performance in workloads with very low locality within pages (e.g., pr, nw). This is because +high bandwidth partitioning ratios favor cache line movements and throttle a higher number +of page movements. Second, since cache line data movements are affected more by the switch +latency compared to page movements, the performance benefits of higher bandwidth partitioning +ratios reduce at higher switch latencies. For example, in pr, the 50% bandwidth partitioning ratio +outperforms 25% ratio by 1.19× and 1.08× using DaeMon at 100ns and 400ns switch latency, +respectively. Finally, across all different bandwidth factors (not graphed), DaeMon’s default 25% +ratio outperforms the 50% ratio by 1.02× and 1.04× for 100ns and 400ns switch latency, respectively, +and the 80% ratio by 1.07× and 1.33× for 100ns and 400ns switch latency, respectively. We conclude +that DaeMon’s default 25% ratio on average performs best across all various network and application +characteristics. +Compression Scheme. Figure 12 compares the performance of LC normalized to Remote with +three compression schemes: (i) fpcbdi: a latency-optimized hybrid scheme of BDI [73] and FPC [8] +with 4-cycle (de)compression latency per cache line [49]; (ii) fve: the latency-optimized FVE [91] +scheme using a 256B dictionary table and having 6-cycle (de)compression latency per cache line [91]; +and (iii) LZ: DaeMon’s compression ratio-optimized LZ-based scheme [49, 93] (See details on § 4.4). +pr +nw +bf +bc +sp +hp +dr +rs +GM +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Speedup +stch-lat=100 bw-fact=1/2 +5.0 +fpcbdi +fve +LZ +pr +nw +bf +bc +sp +hp +dr +rs +GM +0 +1 +2 +3 +4 +Speedup +stch-lat=100 bw-fact=1/8 +7.6 +fpcbdi +fve +LZ +Fig. 12. Performance of LC varying the compression scheme. +We observe that LZ always outperforms Remote, despite the high (de)compression latencies, +because the network overheads are significantly higher, indicating that link compression is a highly +effective solution for disaggregated systems. dr and rs show little performance improvement with +LZ, because the application data is less compressible (their compression ratio is 1.42× versus 4.47× +on average across all evaluated workloads). Moreover, LZ outperforms fpcbdi and fve across all +network configurations (not graphed) by 1.54× and 1.44× on average, respectively, since it achieves +higher compression ratios (on average 2.92× and 2.73× higher compression ratio than fpcbdi and +fve respectively). The benefits of LZ over fpcbdi and fve are even higher in the more bandwidth +limited configurations (e.g., with 1/8 bandwidth factor). Therefore, we conclude that the high +network overheads in disaggregated systems favor compression algorithms that provide higher +compression ratios, since the benefits of the reduced bandwidth consumption outweigh the higher +(de)compression latencies. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:21 +Network Disturbance. Figures 13 and 14 compare the IPC and the hit ratio in local memory +respectively, of LC, PQ and DaeMon, when the network traffic varies during runtime: we simulate +contention from other compute components that share the same network, by artificially injecting +packets inside the network. We evaluate pr and nw, as they incur the highest data movement costs. +0.0 +0.5 +1.0 +1.5 +2.0 +Executed Instructions +1e8 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +IPC +nw stch-lat=100 bw-fact=1/2 +LC +PQ +DaeMon +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Executed Instructions +1e9 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +IPC +pr stch-lat=100 bw-fact=1/2 +LC +PQ +DaeMon +Fig. 13. Performance of LC, PQ, DaeMon, when creating artificial disturbance in the network during runtime. +0.0 +0.5 +1.0 +1.5 +2.0 +Executed Instructions +1e8 +86 +88 +90 +92 +94 +96 +98 +100 +Hit Ratio (%) +nw net-lat=100 bw-fact=1/2 +LC +PQ +DaeMon +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Executed Instructions +1e9 +86 +88 +90 +92 +94 +96 +98 +Hit Ratio (%) +pr net-lat=100 bw-fact=1/2 +LC +PQ +DaeMon +Fig. 14. Hit ratio in local memory of LC, PQ and DaeMon, when creating artificial disturbance in the network +during runtime. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:22 +Christina Giannoula, et al. +DaeMon outperforms both LC and PQ by 2.85× and 1.19×, respectively, even when network traffic +varies during runtime. DaeMon effectively adapts to varying application behavior and network +conditions at runtime. For example, in nw, in the first 50M instructions, DaeMon benefits more from +LC as the application has high bandwidth consumption and higher locality within pages. In the +next 100M instructions, the workload exhibits less data locality within pages, and DaeMon benefits +more from PQ, which provides significant performance benefits over LC by effectively prioritizing +critical path cache line requests. In the last part of execution, DaeMon again leverages the benefits +of LC. Therefore, we conclude that DaeMon provides a versatile approach to dynamic and variable +runtime application and network characteristics. +Multithreaded Performance. Figure 15 shows DaeMon’s performance benefits for multithreaded +workloads on 8 OoO cores, thus evaluating more bandwidth-limited executions compared to that of +Figure 8. Please also see Figure 21 in Appendix § A which evaluates even more bandwidth-limited +executions. Across all workloads and network configurations (not graphed), DaeMon outperforms +the typically-used Remote scheme by 2.73× on average. When network bandwidth is very limited, +e.g., 1/16 bandwidth factor (Figure 21), DaeMon’s benefits are even higher, by 3.95× over Remote. +kc +tr +pr +nw +bf +bc +ts +sp +sl +GM +0 +2 +4 +6 +8 +10 +Speedup +stch-lat=100 bw-fact=1/2 +10.4 +LC +PQ +DaeMon +Local +kc +tr +pr +nw +bf +bc +ts +sp +sl +GM +0 +2 +4 +6 +8 +10 +Speedup +stch-lat=400 bw-fact=1/2 +10.5 +LC +PQ +DaeMon +Local +kc +tr +pr +nw +bf +bc +ts +sp +sl +GM +02468 +10 +12 +14 +Speedup +stch-lat=400 bw-fact=1/4 +21.1 +LC +PQ +DaeMon +Local +Fig. 15. Speedup achieved by various schemes in multithreaded workloads normalized to Remote. +FIFO Replacement Policy in Local Memory. Figure 16 compares DaeMon and Local normalized +to Remote, when using First-In-First-Out (FIFO) replacement policy in local memory. Across all +workloads and network configurations (not graphed), DaeMon outperforms the widely-adopted +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:23 +Remote scheme by 2.63×, when using a FIFO replacement policy in local memory. DaeMon is +orthogonal to the replacement policy used in local memory, and can be used synergistically with +any arbitrary replacement policy in local memory to even further reduce data access costs. Overall, +DaeMon can significantly mitigate the data movement overheads in fully disaggregated systems +independently on the number of data migrations happens during runtime: even when a small +number of data migrations happens during runtime (e.g., thanks to sophisticated approaches such +as intelligent replacement policies in local memory, hot page placement/selection techniques, +page prefetchers), DaeMon can even further alleviate the data movement costs by dynamically +selecting the granularity of data movements, prioritizing the critical cache line requests, and +opportunistically moving compressed pages at slower rates. Therefore, we conclude that DaeMon +can work synergistically with sophisticated replacement policies in local memory, page prefetchers +and intelligent page placement/movement techniques to even further improve system performance. +pr +nw +bf +bc +sp +hp +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=100 bw-fact=1/2 +8.88.9 +14.7 +19.2 +Remote +DaeMon +Local +pr +nw +bf +bc +sp +hp +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +stch-lat=100 bw-fact=1/4 +12.5 +19.2 +27.6 +46.2 +Remote +DaeMon +Local +Fig. 16. Performance of Local and DaeMon over Remote, when using FIFO replacement policy in local memory. +Multiple Memory Components. Figure 17 compares Remote and DaeMon normalized to Local, +when varying the number of memory components and having a different network configuration +for each memory component. Please also see Figure 22 in Appendix § A. We evaluated distributing +memory pages with either a round-robin way or randomly across remote memory components, +and draw the same key observation for both distributions. When adding more memory components +using the same network configuration with that of when having one memory component (e.g., +having 100ns switch latency and 1/4 bandwidth factor for each memory component), performance +of both Remote and DaeMon improves over Local: memory pages are distributed across multiple +memory components and the system provides larger aggregate network and memory bandwidth, +thus data migrations incur smaller overheads. Finally, DaeMon significantly outperforms Remote +by 3.25× across all workload-architecture combinations, and constitutes a scalable solution for +large-scale disaggregated systems with multiple hardware components and various architectures. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:24 +Christina Giannoula, et al. +MC1.1 +MC2.1 +MC2.2 +MC2.3 +MC4.1 +MC4.2 +MC4.3 +MC4.4 +MC1.1 +MC2.1 +MC2.2 +MC2.3 +MC4.1 +MC4.2 +MC4.3 +MC4.4 +MC1.1 +MC2.1 +MC2.2 +MC2.3 +MC4.1 +MC4.2 +MC4.3 +MC4.4 +MC1.1 +MC2.1 +MC2.2 +MC2.3 +MC4.1 +MC4.2 +MC4.3 +MC4.4 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Slowdown +39 21 40 42 21 50 42 78 +nw +ts +sp +dr +Remote +DaeMon +#memory components +stch-lat +bw-fact +MC1.1 +1 +100 +1/4 +MC2.1 +2 +100-100 +1/4-1/4 +MC2.2 +2 +400-400 +1/4-1/8 +MC2.3 +2 +100-100 +1/8-1/8 +MC4.1 +4 +100-100-100-100 +1/4-1/4-1/4-1/4 +MC4.2 +4 +100-400-100-400 +1/4-1/8-1/4-1/8 +MC4.3 +4 +400-400-400-400 +1/8-1/8-1/8-1/8 +MC4.4 +4 +100-100-100-100 +1/8-1/16-1/8-1/16 +Fig. 17. Performance of Remote and DaeMon over Local when using multiple memory components. +Multiple Concurrent Workloads. Figure 18 shows DaeMon’s performance benefits when concur- +rently running multiple workloads on a compute component with 4 OoO cores. The performance +of each core is normalized to that of the same core using Remote. The local memory hosts ∼15% +and ∼9% of each application’s working set, when running 2 and 4 workloads, respectively. DaeMon +outperforms Remote by 1.96× across all multiple-workload experiments, thus being highly efficient +and performant when multiple heterogeneous jobs concurrently run in the disaggregated system. +bf-nw +bf-ts +pr-nw +nw-sp +tr-nw +bc-ts +nw-ts +ts-dr +ts-sp +bf-dr-ts-nw +bf-dr-ts-sp +pr-dr-sp-nw +pr-nw-ts-sp +bc-dr-ts-sp +tr-dr-ts-sp +bc-nw-ts-sp +tr-nw-ts-sp +dr-ts-sp-nw +0 +1 +2 +3 +4 +5 +6 +7 +Speedup +stch-lat=100 bw-fact=1/2 +Core 1 +Core 2 +Core 3 +Core 4 +Fig. 18. Performance of DaeMon over Remote when running multiple concurrent workloads in a 4-CPU +compute component and a memory component. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:25 +6.1 +Key Takeaways and Recommendations +This section summarizes our key takeaways and recommendations extracted from our evaluations. +Key Takeaway #1. There is no one-size-fits-all granularity in data movements: the best-performing +granularity at each time depends on the network/system load and the application data access patterns, +which can significantly vary across applications and within application during runtime. Figure 8 +demonstrates that some applications significantly benefit from the prioritization of critical cache +line data movements (e.g., pr, nw), and some applications only benefit from page migrations that +leverage data locality (e.g., dr, rs). Figure 12 shows that some applications have highly compressible +data, and thus greatly benefit from compressed page granularity data movements. Finally, Figure 13 +proves that the application behavior and network traffic can highly vary during runtime, and thus +the best-performing data movement granularity needs to adapt to the application characteristics +and network/system conditions. Therefore, we recommend that system and hardware designers of +disaggregated systems implement system-level solutions and hardware mechanisms that dynami- +cally change and adapt their configurations and selection methods to the availability of the system +resources and the runtime behavior of the heterogeneous applications. +Key Takeaway #2. Typical datacenter applications exhibit high data locality within memory pages +(e.g., 4KB). Figure 10 shows that Remote achieves high data locality, i.e., always has at least 90% hit +ratio in local memory, across a wide variety of datacenter workloads with diverse access patterns. +Therefore, migrating data at a large granularity, e.g., page granularity, is very effective and critical +to achieving high system performance in fully disaggregated systems. To this end, we suggest that +hardware and system designers of disaggregated systems retain coarse-grained data migration +(i.e., page granularity data migration), since it both enables high performance and maintains low +metadata overheads for address translation in local memory and remote memory. +Key Takeaway #3. Aggressively prioritizing the cache line granularity data movements that are +on the critical path might hurt performance. Figure 11 shows that a high bandwidth partitioning +ratio, e.g., 50% or 80% bandwidth partitioning ratio, which significantly prioritizes the cache +line granularity data movements over the page granularity data movements, incurs significant +performance slowdowns in workloads with medium and high spatial locality. As a result, we suggest +that hardware and system designers of data movement solutions tailored for disaggregated systems +always ensure that page migrations are not aggressively stalled. +Key Takeaway #4. Distributed and disaggregated data movements solutions are highly effective and +efficient in fully disaggregated systems. disaggregated systems are distributed architectures and +comprise multiple hardware devices, each of them is independently and transparently managed +from other hardware components in the system. Our evaluations in Figures 17 and 22 show that +distributed and disaggregated solutions for data movement (i.e., DaeMon) better leverage the +available aggregate network and memory bandwidth in the system, and enable high scalability to +large-scale disaggregated systems with multiple hardware components. To this end, we recommend +that hardware architects design distributed hardware mechanisms for fully disaggregated systems. +7 +Related Work +To our knowledge, this is first work to (i) analyze and alleviate the data movement problem in +fully disaggregated systems; (ii) enable prioritized and decoupled movement of data at multiple +granularities simultaneously to reduce access latencies; (iii) propose a dynamic selection granularity +mechanism with approximate bandwidth partitioning to effectively leverage both cache line and +page movement depending on application and network characteristics; and (iv) implement a +synergistic solution of link compression, bandwidth partitioning, and adaptive granularity selection +in data movements. We discuss prior work. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:26 +Christina Giannoula, et al. +Disaggregated Systems. Several prior works [5, 6, 10, 14–16, 33–35, 37, 38, 46, 54, 57, 70, 74, 75, +78, 87, 99, 108–110, 112] propose OS modules, system-level solutions, programming frameworks, +software management systems, architectures and emulators for disaggregated systems. These +works do not tackle the data movement problem in disaggregated systems, and thus DaeMon is +orthogonal to these proposals. MIND [54] proposes memory sharing among compute components +by implementing coherence and address translation in network switches. Kona [16] is a software +runtime to track cache line granularity accesses to remote memory, and eliminate page faults by +decoupling the application memory access tracking from the virtual memory page size. However, +Kona and MIND do not mitigate data movement overheads in disaggregated systems, as data is +always moved at page granularity. Thus, DaeMon is largely orthogonal to these works and could be +used to further improve performance. Clio [37] proposes a disaggregated system that virtualizes +and manages remote memory at the hardware level (independently to compute components), +and eliminates expensive page faults in memory components. Clio accesses remote data at a byte +granularity via dedicated API, however not being transparent to programmers. As explained in +§ 2.2, moving data always at a small granularity can cause significant performance penalties in +many applications, and does not provide robustness against fluctuations in network characteristics. +Instead, DaeMon is software-transparent, robust and significantly alleviates data movement costs +via decoupled and selective data movement at multiple granularities. Lim et al. [56] propose a +disaggregated architecture and characterize moving data only at cache line or page granularity. +The authors show that the page-based configuration outperforms the cache line configuration at +most common patterns (as observed in § 2.2), however it does not address the high performance +penalties of page migrations. Maruf and Chowdhury [62] propose a page prefetching scheme for +disaggregated systems, which however can only help applications with high locality within pages, +and does not capture the significant variability in data access costs of fully disaggregated systems. +DaeMon is orthogonal to page prefetchers and can work synergistically with them to even further +improve performance, as described in Section 4.7. We leave the experimentation of their synergy +for future work. +Hybrid Memory Systems. Numerous works for hybrid memory systems propose data placement +schemes [3, 19, 22, 28, 29, 32, 45, 47, 59, 82, 100], or selection methods [4, 26, 27, 43, 52, 53, 58, +64, 76, 89, 96, 103] to identify hot memory pages that are migrated to die-stacked DRAM, that is +organized as a cache of a larger main memory. Compared to these approaches, first, intelligent page +placement/movement is orthogonal to DaeMon, and cannot by itself address the high overheads +caused by remote page migrations across the network, that can be significantly slower than that +within the server and more latency/bandwidth-constrained in the context of fully disaggregated +systems. Second, these prior works assume a monolithic centralized system where TLBs/page tables +can be leveraged to track page hotness of remote pages (e.g., [4, 26, 27, 52, 58, 64, 103]) or that +memory allocation/placement is handled by the server itself (e.g., [3, 28, 29, 32, 45, 47, 55, 59, 82]). +However, in disaggregated systems, address translation and memory management are distributed +across memory components and cannot be used to track pages at the CPU server side, while compute +components and memory components are managed by independent kernel monitors that have no +visibility/control of other components or data management/placement across components. Similarly, +hardware-based approaches [43, 55, 76, 96] for hybrid systems add centralized hardware units at +the server side to store page tracking metadata for the second-tier main memory. For example, +Chop [43] adds 4MB of metadata to track 16GB of second-tier memory. These schemes would incur +significant area overheads (in the order of GBs) to track large amounts of remote memory (in the +order of TBs) enabled by disaggregated systems [87]. Requiring each compute component to track +a large number of pages enabled by multiple remote memory components would cause scalability +issues and significantly limit the benefits of resource disaggregation. Thus, designing an effective +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:27 +scalable hot page selection scheme for fully disaggregated systems is an open challenge, and DaeMon +could work in conjunction with such schemes to further improve performance. Third, all these +prior works do not handle variability in data access costs of disaggregated systems. disaggregated +systems necessitate an adaptive mechanism given the significant variations in access latencies and +bandwidth. Fourth, applying/adapting the design of prior schemes tailored for tightly-integrated +hybrid systems in disaggregated systems might incur significantly higher overheads and require +important modifications than that described in the original papers. +A few recent works design hardware schemes for commodity servers to enable moving data only +at cache line granularity [23, 60, 61, 97] or a larger sub-block granularity (a few cache lines) [42, 83]. +Ekman et al. [30] evaluate a critical-block first approach, where each 8KB page is split in blocks of +2KB data, and the requested (critical) 2KB block of data is transferred first, and written in DRAM +cache. As we show in § 2.2, moving data at a single granularity (page or cache line) can incur +high performance costs and does not provide robustness towards significant variations in network +bandwidth and latencies. +Hardware Compression. Prior works propose compression schemes [1, 8, 12, 21, 24, 31, 48, 50, +68, 69, 71–73, 77, 86, 91, 92, 101, 104, 105, 107] for cache memory, main memory and memory +bus links in CPUs/GPUs [65, 85, 90, 98], and selection methods to dynamically enable/disable +compression [7, 9, 95], or find the best-performing compression scheme [11, 49]. These works +integrate ratio-optimized or latency-optimized compression schemes depending on the particular +context and system’s characteristics they target. Our work enables link compression in page +movements synergistically with decoupled multiple granularity data movement, which allows us to +tolerate the high compression latencies of ratio-optimized compression schemes such as LZ [111]. +8 +Conclusion +DaeMon is the first adaptive data movement solution for fully disaggregated systems. DaeMon +supports low-cost page migration, scales elastically to multiple hardware components, enables +software transparency, and provides robustness across various architecture/network characteristics +and the application behavior by effectively monitoring pending cache line and page movements. Our +evaluations using a state-of-the-art accurate simulator show that DaeMon significantly improves +system performance and data access costs for a wide range of applications under various architecture +and network configurations, and when multiple jobs are simultaneously running in the system. +We conclude that DaeMon is an efficient, scalable and robust solution to alleviate data movement +overheads in disaggregated systems, and hope that this work encourages further studies of the data +movement problem in disaggregated systems. +Acknowledgments +We thank the anonymous reviewers from SIGMETRICS 2023, and our shepherd, Abhishek Chandra, +for their comments and suggestions. We also thank Konstantinos Kanellopoulos and Ivan Fernandez +for their help on technical aspects of this work. The final version of our paper is also available on +arXiv . +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:28 +Christina Giannoula, et al. +References +[1] B. Abali, H. Franke, D. E. Poff, R. A. Saccone, C. O. Schulz, L. M. Herger, and T. B. Smith. 2001. Memory Expansion +Technology (MXT): Software Support and Performance. IBM Journal of Research and Development (2001). +[2] Atul Adya, Robert Grandl, Daniel Myers, and Henry Qin. 2019. Fast Key-Value Stores: An Idea Whose Time Has +Come and Gone. In HotOS. +[3] Neha Agarwal, David Nellans, Mark Stephenson, Mike O’Connor, and Stephen W. Keckler. 2015. Page Placement +Strategies for GPUs within Heterogeneous Memory Systems. In ASPLOS. +[4] Neha Agarwal and Thomas F. Wenisch. 2017. Thermostat: Application-Transparent Page Management for Two-Tiered +Main Memory. In ASPLOS. +[5] Marcos K. Aguilera, Nadav Amit, Irina Calciu, Xavier Deguillard, Jayneel Gandhi, Stanko Novaković, Arun Ra- +manathan, Pratap Subrahmanyam, Lalith Suresh, Kiran Tati, Rajesh Venkatasubramanian, and Michael Wei. 2018. +Remote Regions: A Simple Abstraction for Remote Memory. In ATC. +[6] Marcos K. Aguilera, Nadav Amit, Irina Calciu, Xavier Deguillard, Jayneel Gandhi, Pratap Subrahmanyam, Lalith +Suresh, Kiran Tati, Rajesh Venkatasubramanian, and Michael Wei. 2017. Remote Memory in the Age of Fast Networks. +In SoCC. +[7] A.R. Alameldeen and D.A. Wood. 2004. Adaptive Cache Compression for High-Performance Processors. In ISCA. +[8] Alaa Alameldeen and David Wood. 2004. Frequent Pattern Compression: A Significance-Based Compression Scheme +for L2 Caches. (2004). +[9] Alaa R. Alameldeen and David A. Wood. 2007. Interactions Between Compression and Prefetching in Chip Multipro- +cessors. In HPCA. +[10] Sebastian Angel, Mihir Nanavati, and Siddhartha Sen. 2020. Disaggregation and the Application. In HotCloud. +[11] Angelos Arelakis, Fredrik Dahlgren, and Per Stenstrom. 2015. HyComp: A Hybrid Cache Compression Method for +Selection of Data-Type-Specific Compression Methods. In MICRO. +[12] Angelos Arelakis and Per Stenstrom. 2014. SC2: A Statistical Compression Cache Scheme. In ISCA. +[13] JEDEC Solid State Technology Assn. 2017. JESD79-4B: DDR4 SDRAM Standard. +[14] Laurent Bindschaedler, Ashvin Goel, and Willy Zwaenepoel. 2020. Hailstorm: Disaggregated Compute and Storage +for Distributed LSM-Based Databases. In ASPLOS. +[15] Dhantu Buragohain, Abhishek Ghogare, Trishal Patel, Mythili Vutukuru, and Purushottam Kulkarni. 2017. DiME: A +Performance Emulator for Disaggregated Memory Architectures. In APSys. +[16] Irina Calciu, M. Talha Imran, Ivan Puddu, Sanidhya Kashyap, Hasan Al Maruf, Onur Mutlu, and Aasheesh Kolli. 2021. +Rethinking Software Runtimes for Disaggregated Memory. In ASPLOS. +[17] Trevor E. Carlson, Wim Heirman, and Lieven Eeckhout. 2011. Sniper: Exploring the Level of Abstraction for Scalable +and Accurate Parallel Multi-Core Simulations. In SC. +[18] Trevor E. Carlson, Wim Heirman, Stijn Eyerman, Ibrahim Hur, and Lieven Eeckhout. 2014. An Evaluation of +High-Level Mechanistic Core Models. TACO (2014). +[19] Chia-Hao Chang, Adithya Kumar, and Anand Sivasubramaniam. 2021. To Move or Not to Move? Page Migration for +Irregular Applications in over-Subscribed GPU Memory Systems with DynaMap. In SYSTOR. +[20] Shuai Che, Michael Boyer, Jiayuan Meng, David Tarjan, Jeremy W. Sheaffer, Sang-Ha Lee, and Kevin Skadron. 2009. +Rodinia: A Benchmark Suite for Heterogeneous Computing. In IISWC. +[21] Xi Chen, Lei Yang, Robert P. Dick, Li Shang, and Haris Lekatsas. 2010. C-Pack: A High-Performance Microprocessor +Cache Compression Algorithm. VLSI (2010). +[22] Chiachen Chou, Aamer Jaleel, and Moinuddin Qureshi. 2017. BATMAN: Techniques for Maximizing System Bandwidth +of Memory Systems with Stacked-DRAM. In MEMSYS. +[23] Chia Chen Chou, Aamer Jaleel, and Moinuddin K. Qureshi. 2014. CAMEO: A Two-Level Memory Organization with +Capacity of Main Memory and Flexibility of Hardware-Managed Cache. In MICRO. +[24] Esha Choukse, Mattan Erez, and Alaa R. Alameldeen. 2018. Compresso: Pragmatic Main Memory Compression. In +MICRO. +[25] David Cock, Abishek Ramdas, Daniel Schwyn, Michael Giardino, Adam Turowski, Zhenhao He, Nora Hossle, Dario +Korolija, Melissa Licciardello, Kristina Martsenko, Reto Achermann, Gustavo Alonso, and Timothy Roscoe. 2022. +Enzian: An Open, General, CPU/FPGA Platform for Systems Software Research. In ASPLOS. +[26] Xiangyu Dong, Yuan Xie, Naveen Muralimanohar, and Norman P. Jouppi. 2010. Simple but Effective Heterogeneous +Main Memory with On-Chip Memory Controller Support. In SC. +[27] Thaleia Dimitra Doudali, Sergey Blagodurov, Abhinav Vishnu, Sudhanva Gurumurthi, and Ada Gavrilovska. 2019. +Kleio: A Hybrid Memory Page Scheduler with Machine Intelligence. In HPDC. +[28] Thaleia Dimitra Doudali, Daniel Zahka, and Ada Gavrilovska. 2021. Cori: Dancing to the Right Beat of Periodic Data +Movements over Hybrid Memory Systems. In IPDPS. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:29 +[29] Subramanya R. Dulloor, Amitabha Roy, Zheguang Zhao, Narayanan Sundaram, Nadathur Satish, Rajesh Sankaran, +Jeff Jackson, and Karsten Schwan. 2016. Data Tiering in Heterogeneous Memory Systems. In EuroSys. +[30] Magnus Ekman and Per Stenstrom. 2005. A Cost-Effective Main Memory Organization for Future Servers. In IPDPS. +[31] M. Ekman and P. Stenstrom. 2005. A Robust Main-Memory Compression Scheme. In ISCA. +[32] M. J. Feeley, W. E. Morgan, E. P. Pighin, A. R. Karlin, H. M. Levy, and C. A. Thekkath. 1995. Implementing Global +Memory Management in a Workstation Cluster. In SOSP. +[33] Peter X. Gao, Akshay Narayan, Sagar Karandikar, Joao Carreira, Sangjin Han, Rachit Agarwal, Sylvia Ratnasamy, and +Scott Shenker. 2016. Network Requirements for Resource Disaggregation. In OSDI. +[34] Donghyun Gouk, Sangwon Lee, Miryeong Kwon, and Myoungsoo Jung. 2022. Direct Access, High-Performance +Memory Disaggregation with DirectCXL. In ATC. +[35] Juncheng Gu, Youngmoon Lee, Yiwen Zhang, Mosharaf Chowdhury, and Kang G. Shin. 2017. Efficient Memory +Disaggregation with Infiniswap. In NSDI. +[36] Chuanxiong Guo, Haitao Wu, Zhong Deng, Gaurav Soni, Jianxi Ye, Jitu Padhye, and Marina Lipshteyn. 2016. RDMA +over Commodity Ethernet at Scale. In SIGCOMM. +[37] Zhiyuan Guo, Yizhou Shan, Xuhao Luo, Yutong Huang, and Yiying Zhang. 2022. Clio: A Hardware-Software +Co-Designed Disaggregated Memory System. In ASPLOS. +[38] Sangjin Han, Norbert Egi, Aurojit Panda, Sylvia Ratnasamy, Guangyu Shi, and Scott Shenker. 2013. Network Support +for Resource Disaggregation in Next-Generation Datacenters. In HotNets. +[39] HPCG. 2019. High Performance Conjugate Gradient Benchmark. https://github.com/hpcg-benchmark/hpcg +[40] Ranggi Hwang, Taehun Kim, Youngeun Kwon, and Minsoo Rhu. 2020. Centaur: A Chiplet-Based, Hybrid Sparse-Dense +Accelerator for Personalized Recommendations. In ISCA. +[41] Intel. 2021. Intel Omni-Path Architecture. https://www.intel.com/content/www/us/en/high-performance-computing- +fabrics/omni-path-driving-exascale-computing.html +[42] Djordje Jevdjic, Stavros Volos, and Babak Falsafi. 2013. Die-Stacked DRAM Caches for Servers: Hit Ratio, Latency, or +Bandwidth? Have It All with Footprint Cache. In ISCA. +[43] Xiaowei Jiang, Niti Madan, Li Zhao, Mike Upton, Ravishankar Iyer, Srihari Makineni, Donald Newell, Yan Solihin, and +Rajeev Balasubramonian. 2010. CHOP: Adaptive Filter-Based DRAM Caching for CMP Server Platforms. In HPCA. +[44] Hongshin Jun, Jinhee Cho, Kangseol Lee, Ho-Young Son, Kwiwook Kim, Hanho Jin, and Keith Kim. 2017. HBM DRAM +Technology and Architecture. In IMW. +[45] Sudarsun Kannan, Ada Gavrilovska, Vishal Gupta, and Karsten Schwan. 2017. HeteroOS: OS Design for Heterogeneous +Memory Management in Datacenter. In ISCA. +[46] K. Katrinis, D. Syrivelis, D. Pnevmatikatos, G. Zervas, D. Theodoropoulos, I. Koutsopoulos, K. Hasharoni, D. Raho, +C. Pinto, F. Espina, S. Lopez-Buedo, Q. Chen, M. Nemirovsky, D. Roca, H. Klos, and T. Berends. 2016. Rack-Scale +Disaggregated Cloud Data Centers: The dReDBox Project Vision. In DATE. +[47] Jonghyeon Kim, Wonkyo Choe, and Jeongseob Ahn. 2021. Exploring the Design Space of Page Management for +Multi-Tiered Memory Systems. In ATC. +[48] Jungrae Kim, Michael Sullivan, Esha Choukse, and Mattan Erez. 2016. Bit-Plane Compression: Transforming Data for +Better Compression in Many-Core Architectures. In ISCA. +[49] Seikwon Kim, Seonyoung Lee, Taehoon Kim, and Jaehyuk Huh. 2017. Transparent Dual Memory Compression +Architecture. In PACT. +[50] M. Kjelso, M. Gooch, and S. Jones. 1996. Design and Performance of a Main Memory Hardware Data Compressor. In +EUROMICRO. +[51] Fredrik Kjolstad, Stephen Chou, David Lugato, Shoaib Kamil, and Saman Amarasinghe. 2017. Taco: A Tool to Generate +Tensor Algebra Kernels. In ASE. +[52] Jagadish B. Kotra, Haibo Zhang, Alaa R. Alameldeen, Chris Wilkerson, and Mahmut T. Kandemir. 2018. CHAMELEON: +A Dynamically Reconfigurable Heterogeneous Memory System. In MICRO. +[53] Andres Lagar-Cavilla, Junwhan Ahn, Suleiman Souhlal, Neha Agarwal, Radoslaw Burny, Shakeel Butt, Jichuan Chang, +Ashwin Chaugule, Nan Deng, Junaid Shahid, Greg Thelen, Kamil Adam Yurtsever, Yu Zhao, and Parthasarathy +Ranganathan. 2019. Software-Defined Far Memory in Warehouse-Scale Computers. In ASPLOS. +[54] Seung-seob Lee, Yanpeng Yu, Yupeng Tang, Anurag Khandelwal, Lin Zhong, and Abhishek Bhattacharjee. 2021. +MIND: In-Network Memory Management for Disaggregated Data Centers. In SOSP. +[55] Yang Li, Saugata Ghose, Jongmoo Choi, Jin Sun, Hui Wang, and Onur Mutlu. 2017. Utility-Based Hybrid Memory +Management. In CLUSTER. +[56] Kevin Lim, Jichuan Chang, Trevor Mudge, Parthasarathy Ranganathan, Steven K. Reinhardt, and Thomas F. Wenisch. +2009. Disaggregated Memory for Expansion and Sharing in Blade Servers. In ISCA. +[57] Kevin Lim, Yoshio Turner, Jose Renato Santos, Alvin AuYoung, Jichuan Chang, Parthasarathy Ranganathan, and +Thomas F. Wenisch. 2012. System-Level Implications of Disaggregated Memory. In HPCA. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:30 +Christina Giannoula, et al. +[58] Haikun Liu, Yujie Chen, Xiaofei Liao, Hai Jin, Bingsheng He, Long Zheng, and Rentong Guo. 2017. Hardware/Software +Cooperative Caching for Hybrid DRAM/NVM Memory Architectures. In ICS. +[59] Lei Liu, Shengjie Yang, Lu Peng, and Xinyu Li. 2019. Hierarchical Hybrid Memory Management in OS for Tiered +Memory Systems. TPDS (2019). +[60] Gabriel Loh and Mark D. Hill. 2012. Supporting Very Large DRAM Caches with Compound-Access Scheduling and +MissMap. IEEE Micro (2012). +[61] Gabriel H. Loh and Mark D. Hill. 2011. Efficiently Enabling Conventional Block Sizes for Very Large Die-Stacked +DRAM Caches. In MICRO. +[62] Hasan Al Maruf and Mosharaf Chowdhury. 2020. Effectively Prefetching Remote Memory with Leap. In ATC. +[63] Mellanox. 2020. Mellanox Innova Adapters. https://www.nvidia.com/en-us/networking/products/data-processing- +unit/?mtag=programmable_adapter_cards +[64] Mitesh R. Meswani, Sergey Blagodurov, David Roberts, John Slice, Mike Ignatowski, and Gabriel H. Loh. 2015. +Heterogeneous Memory Architectures: A HW/SW Approach for Mixing Die-Stacked and Off-Package Memories. In +HPCA. +[65] Sparsh Mittal and Jeffrey S. Vetter. 2016. A Survey Of Architectural Approaches for Data Compression in Cache and +Main Memory Systems. TPDS (2016). +[66] Naveen Muralimanohar, Rajeev Balasubramonian, and Norm Jouppi. 2007. Optimizing NUCA Organizations and +Wiring Alternatives for Large Caches with CACTI 6.0. In MICRO. +[67] Maxim Naumov, Dheevatsa Mudigere, Hao-Jun Michael Shi, Jianyu Huang, Narayanan Sundaraman, Jongsoo Park, +Xiaodong Wang, Udit Gupta, Carole-Jean Wu, Alisson G. Azzolini, Dmytro Dzhulgakov, Andrey Mallevich, Ilia +Cherniavskii, Yinghai Lu, Raghuraman Krishnamoorthi, Ansha Yu, Volodymyr Kondratenko, Stephanie Pereira, Xianjie +Chen, Wenlin Chen, Vijay Rao, Bill Jia, Liang Xiong, and Misha Smelyanskiy. 2019. Deep Learning Recommendation +Model for Personalization and Recommendation Systems. In arXiv. +[68] Tri M. Nguyen, Adi Fuchs, and David Wentzlaff. 2018. CABLE: A CAche-Based Link Encoder for Bandwidth-Starved +Manycores. In MICRO. +[69] Tri M. Nguyen and David Wentzlaff. 2015. MORC: A Manycore-Oriented Compressed Cache. In MICRO. +[70] Vlad Nitu, Boris Teabe, Alain Tchana, Canturk Isci, and Daniel Hagimont. 2018. Welcome to Zombieland: Practical +and Energy-Efficient Memory Disaggregation in a Datacenter. In EuroSys. +[71] Sungbo Park, Ingab Kang, Yaebin Moon, Jung Ho Ahn, and G. Edward Suh. 2021. BCD Deduplication: Effective +Memory Compression Using Partial Cache-Line Deduplication. In ASPLOS. +[72] Gennady Pekhimenko, Vivek Seshadri, Yoongu Kim, Hongyi Xin, Onur Mutlu, Phillip B. Gibbons, Michael A. Kozuch, +and Todd C. Mowry. 2013. Linearly Compressed Pages: A Low-Complexity, Low-Latency Main Memory Compression +Framework. In MICRO. +[73] Gennady Pekhimenko, Vivek Seshadri, Onur Mutlu, Phillip B. Gibbons, Michael A. Kozuch, and Todd C. Mowry. 2012. +Base-Delta-Immediate Compression: Practical Data Compression for on-Chip Caches. In PACT. +[74] Ivy Peng, Roger Pearce, and Maya Gokhale. 2020. On the Memory Underutilization: Exploring Disaggregated Memory +on HPC Systems. In SBAC-PAD. +[75] Christian Pinto, Dimitris Syrivelis, Michele Gazzetti, Panos Koutsovasilis, Andrea Reale, Kostas Katrinis, and H. Peter +Hofstee. 2020. ThymesisFlow: A Software-Defined, HW/SW co-Designed Interconnect Stack for Rack-Scale Memory +Disaggregation. In MICRO. +[76] Andreas Prodromou, Mitesh Meswani, Nuwan Jayasena, Gabriel Loh, and Dean M. Tullsen. 2017. MemPod: A +Clustered Architecture for Efficient and Scalable Migration in Flat Address Space Multi-level Memories. In HPCA. +[77] Cheng Qian, Libo Huang, Qi Yu, Zhiying Wang, and Bruce Childers. 2018. CMH: Compression Management for +Improving Capacity in the Hybrid Memory Cube. In CF. +[78] Pramod Subba Rao and George Porter. 2016. Is Memory Disaggregation Feasible? A Case Study with Spark SQL. In +ANCS. +[79] RDMA. 2019. RDMA Consortium. http://www.rdmaconsortium.org/ +[80] RDMA. 2022. Gen-Z Core Specification. https://genzconsortium.org/ +[81] Joseph Redmon. 2013–2016. Darknet: Open Source Neural Networks in C. http://pjreddie.com/darknet/ +[82] Zhenyuan Ruan, Malte Schwarzkopf, Marcos K. Aguilera, and Adam Belay. 2020. +AIFM: High-Performance, +Application-Integrated Far Memory. In OSDI). +[83] Jee Ho Ryoo, Mitesh R. Meswani, Andreas Prodromou, and Lizy K. John. 2017. SILC-FM: Subblocked InterLeaved +Cache-Like Flat Memory Organization. In HPCA. +[84] Amedeo Sapio, Ibrahim Abdelaziz, Abdulla Aldilaijan, Marco Canini, and Panos Kalnis. 2017. In-Network Computation +is a Dumb Idea Whose Time Has Come. In HotNets. +[85] Vijay Sathish, Michael J. Schulte, and Nam Sung Kim. 2012. Lossless and Lossy Memory I/O Link Compression for +Improving Performance of GPGPU Workloads. In PACT. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:31 +[86] Ali Shafiee, Meysam Taassori, Rajeev Balasubramonian, and Al Davis. 2014. MemZip: Exploring Unconventional +Benefits from Memory Compression. In HPCA. +[87] Yizhou Shan, Yutong Huang, Yilun Chen, and Yiying Zhang. 2018. LegoOS: A Disseminated, Distributed OS for +Hardware Resource Disaggregation. In OSDI. +[88] Julian Shun and Guy E. Blelloch. 2013. Ligra: A Lightweight Graph Processing Framework for Shared Memory. In +PpopP. +[89] Gagandeep Singh, Rakesh Nadig, Jisung Park, Rahul Bera, Nastaran Hajinazar, David Novo, Juan Gómez-Luna, Sander +Stuijk, Henk Corporaal, and Onur Mutlu. 2022. Sibyl: Adaptive and Extensible Data Placement in Hybrid Storage +Systems Using Online Reinforcement Learning. In ISCA. +[90] Martin Thuresson, Lawrence Spracklen, and Per Stenstrom. 2008. Memory-Link Compression Schemes: A Value +Locality Perspective. IEEE Trans. Comput. (2008). +[91] Martin Thuresson and Per Stenström. 2008. Accommodation of the Bandwidth of Large Cache Blocks Using +Cache/Memory Link Compression. In ICPP. +[92] Yingying Tian, Samira M. Khan, Daniel A. Jiménez, and Gabriel H. Loh. 2014. Last-Level Cache Deduplication. In ICS. +[93] R.B. Tremaine, T.B. Smith, M. Wazlowski, D. Har, Kwok-Ken Mak, and S. Arramreddy. 2001. Pinnacle: IBM MXT in a +Memory Controller Chip. IEEE Micro (2001). +[94] Shin-Yeh Tsai and Yiying Zhang. 2017. LITE Kernel RDMA Support for Datacenter Applications. In SOSP. +[95] Irina Chihaia Tuduce and Thomas Gross. 2005. Adaptive Main Memory Compression. In ATC. +[96] Evangelos Vasilakis, Vassilis Papaefstathiou, Pedro Trancoso, and Ioannis Sourdis. 2019. LLC-Guided Data Migration +in Hybrid Memory Systems. In IPDPS. +[97] Vasilakis, Evangelos and Papaefstathiou, Vassilis and Trancoso, Pedro and Sourdis, Ioannis. 2020. Hybrid2: Combining +Caching and Migration in Hybrid Memory Systems. In HPCA. +[98] Nandita Vijaykumar, Gennady Pekhimenko, Adwait Jog, Abhishek Bhowmick, Rachata Ausavarungnirun, Chita Das, +Mahmut Kandemir, Todd C. Mowry, and Onur Mutlu. 2015. A Case for Core-Assisted Bottleneck Acceleration in +GPUs: Enabling Flexible Data Compression with Assist Warps. In ISCA. +[99] Chenxi Wang, Haoran Ma, Shi Liu, Yuanqi Li, Zhenyuan Ruan, Khanh Nguyen, Michael D. Bond, Ravi Netravali, +Miryung Kim, and Guoqing Harry Xu. 2020. Semeru: A Memory-Disaggregated Managed Runtime. In OSDI. +[100] Johannes Weiner, Niket Agarwal, Dan Schatzberg, Leon Yang, Hao Wang, Blaise Sanouillet, Bikash Sharma, Tejun Heo, +Mayank Jain, Chunqiang Tang, and Dimitrios Skarlatos. 2022. TMO: Transparent Memory Offloading in Datacenters. +In ASPLOS. +[101] P. Wilson, Scott F. Kaplan, and Y. Smaragdakis. 1999. The Case for Compressed Caching in Virtual Memory Systems. +In ATC. +[102] Dong Hyuk Woo, Nak Hee Seong, Dean L. Lewis, and Hsien-Hsin Sean Lee. 2010. An Optimized 3D-stacked Memory +Architecture by Exploiting Excessive, High-Density TSV Bandwidth. HPCA (2010). +[103] Zi Yan, Daniel Lustig, David Nellans, and Abhishek Bhattacharjee. 2019. Nimble Page Management for Tiered Memory +Systems. In ASPLOS. +[104] Jun Yang, Rajiv Gupta, and Chuanjun Zhang. 2004. Frequent Value Encoding for Low Power Data Buses. TODAES +(2004). +[105] Jun Yang, Youtao Zhang, and R. Gupta. 2000. Frequent Value Compression in Data Caches. In MICRO. +[106] Chin-Chia Michael Yeh, Yan Zhu, Liudmila Ulanova, Nurjahan Begum, Yifei Ding, Hoang Anh Dau, Diego Furtado +Silva, Abdullah Mueen, and Eamonn Keogh. 2016. Matrix Profile I: All Pairs Similarity Joins for Time Series: A +Unifying View that Includes Motifs, Discords and Shapelets. In ICDM. +[107] Vinson Young, Sanjay Kariyappa, and Moinuddin K. Qureshi. 2019. Enabling Transparent Memory-Compression for +Commodity Memory Systems. In HPCA. +[108] Georgios Zervas, Hui Yuan, Arsalan Saljoghei, Qianqiao Chen, and Vaibhawa Mishra. 2018. Optically Disaggregated +Data Centers with Minimal Remote Memory Latency: Technologies, Architectures, and Resource Allocation. JOCN +(2018). +[109] Qizhen Zhang, Yifan Cai, Sebastian G. Angel, Vincent Liu, Ang Chen, and B. T. Loo. 2020. Rethinking Data Management +Systems for Disaggregated Data Centers. In CIDR. +[110] Yang Zhou, Hassan M. G. Wassel, Sihang Liu, Jiaqi Gao, James Mickens, Minlan Yu, Chris Kennelly, Paul Turner, +David E. Culler, Henry M. Levy, and Amin Vahdat. 2022. Carbink: Fault-Tolerant Far Memory. In OSDI. +[111] J. Ziv and A. Lempel. 1977. A Universal Algorithm for Sequential Data Compression. IEEE Transactions on Information +Theory (1977). +[112] Pengfei Zuo, Jiazhao Sun, Liu Yang, Shuangwu Zhang, and Yu Hua. 2021. One-sided RDMA-Conscious Extendible +Hashing for Disaggregated Memory. In ATC. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:32 +Christina Giannoula, et al. +APPENDIX +A +Extended Results +A.1 +Network Bandwidth Utilization +Figure 19 compares the bandwidth utilization across the network of a compute component and a +memory component achieved by various data movement schemes. +pr +nw +bf +bc +sp +hp +dr +rs +GM +0 +20 +40 +60 +80 +Network Bandwidth + Utilization (%) +stch-lat=100 bw-fact=1/2 +pr +nw +bf +bc +sp +hp +dr +rs +GM +0 +20 +40 +60 +80 +Network Bandwidth + Utilization (%) +stch-lat=400 bw-fact=1/4 +Fig. 19. Bandwidth utilization (%) across the network of a compute component and a memory component +achieved by various data movement schemes. +We make three key observations. First, LC typically reduces the network bandwidth utilization +over Remote (by 2.49× on average across all workloads and network configurations), because fewer +bytes are transferred through the network, since remote pages are migrated in a compressed format. +Note that LC improves the total execution time over Remote, and thus in a few workloads, e.g., pr, +the network bandwidth utilization might be higher within a smaller execution time. Second, PQ +decreases the network bandwidth utilization over Remote in workloads with poor spatial locality +within pages (e.g., nw), since the selection granularity unit effectively schedules more cache line +movements and fewer page migrations. Instead, PQ might slightly increase the network bandwidth +utilization over Remote in workloads with medium spatial locality within pages (e.g., bf, bc), since +the selection granularity unit enables both cache line and page migrations to leverage both the ability +to prioritize critical cache line requests and the benefits of data locality within pages. In workloads +with high spatial locality within pages (e.g., dr, rs), PQ favors more page migrations and fewer cache +line movements, thus achieving similar network bandwidth utilization to Remote. Third, DaeMon +greatly decreases the network bandwidth utilization over Remote by 2.32× on average across all +workloads and network configurations (not graphed). DaeMon effectively transfers remote pages +in a compressed format and on-the-fly selects the granularity of data migrations to significantly +reduce the bandwidth consumption across the network of fully disaggregated systems. +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +Remote +LC +PQ +DaeMonDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems +21:33 +A.2 +Sensitivity Study to Switch Latency +Figure 20 compares DaeMon’s performance over Remote’s performance averaged across all work- +loads, when varying the switch latency of the network. When the fixed switch latency becomes +very high dominating the total data movement costs, DaeMon has lower benefits over Remote, since +DaeMon does not hide the propagation and switching delays in network components (e.g., fixed +processing costs of the packet inside network switches). However, even with a very high switch +latency in the order of microsecond, i.e., 1𝜇s (=1000ns), DaeMon outperforms Remote by 1.49× on +average across all workloads. +100ns +200ns +300ns +400ns +500ns +600ns +800ns +1000ns +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Speedup +GeoMean +Remote +DaeMon +Fig. 20. Performance benefits of DaeMon over Remote, when varying the switch latency of the network. +A.3 +Sensitivity Study to Network Bandwidth +To evaluate bandwidth-limited scenarios, Figure 21 compares DaeMon’s performance normalized to +Remote’s performance in mutithreaded workloads running on 8 OoO cores of a compute component, +when varying the bandwidth factor of the network, e.g., up to having a very low bandwidth factor +of 1/16 (i.e., network bandwidth is 16× slower than the DRAM bus bandwidth) between a compute +component and memory component. We find that on average DaeMon’s benefits increase over +the widely-adopted approach of moving data at page granularity, i.e., Remote, since DaeMon +even more significantly alleviates bandwidth bottlenecks and data movement overheads under +bandwidth-constrained scenarios. +kc +tr +pr +nw +bf +bc +ts +sp +sl +GM +0 +1 +2 +3 +4 +5 +Speedup +8.9 +12.6 +11.5 +13.7 +10.5 +stch-lat=100 +1/2 +1/4 +1/8 +1/16 +Fig. 21. Performance benefits of DaeMon normalized to Remote using multithreaded workloads, when varying +the bandwidth factor of the network between a compute component and memory component. +A.4 +Performance Benefits With Multiple Memory Components +Figure 22 evaluates the performance of DaeMon normalized to Remote’s performance, when +increasing the number of memory components in the system having the same network configuration +for each memory component, i.e., 100ns switch latency and a bandwidth factor of 1/4. We evaluated +distributing memory pages with either a round-robin way or randomly across multiple remote +memory components, and draw the same key observations for both distributions. Similarly to +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + +21:34 +Christina Giannoula, et al. +Figure 17, we observe that when pages are distributed across multiple memory components and +the system provides larger aggregate network and memory bandwidth, data access costs decrease. +For example, when increasing the number of memory components from 2 to 4, the remote data +access latency decreases by 1.39× on average across all workloads. However, even when data access +costs affect less the total execution time of applications, DaeMon still further mitigates data access +overheads: DaeMon outperforms the widely-adopted Remote approach by 2.09× and 1.88× on +average across all workloads, when using 2 and 4 memory components, respectively. +kc +tr +pr +nw +bf +bc +ts +sp +sl +hp +pf +dr +rs +GM +0 +1 +2 +3 +4 +5 +Speedup +9.2 +23.6 +15.1 +14.9 +stch-lat=100 bw-fact=1/4 +1 MC +2 MCs +4 MCs +Fig. 22. Performance benefits of DaeMon normalized to Remote, when increasing the number of memory +components having 100ns switch latency and a bandwidth factor of 1/4 for each memory component. +Received October 2022; revised December 2022; accepted January 2023 +Proc. ACM Meas. Anal. Comput. Syst., Vol. 6, No. 1, Article 21. Publication date: March 2022. + diff --git a/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/load_file.txt b/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1da14966ee4d2fd8eb7f5cbfbbf1ed70778f22b4 --- /dev/null +++ b/d9AyT4oBgHgl3EQfjfgx/content/tmp_files/load_file.txt @@ -0,0 +1,2198 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf,len=2197 +page_content='21 DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems CHRISTINA GIANNOULA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Canada and National Technical University of Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Greece KAILONG HUANG∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Canada JONATHAN TANG∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Canada NECTARIOS KOZIRIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' National Technical University of Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Greece GEORGIOS GOUMAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' National Technical University of Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Greece ZESHAN CHISHTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Intel Corporation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' USA NANDITA VIJAYKUMAR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Canada Resource disaggregation offers a cost effective solution to resource scaling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' utilization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and failure-handling in data centers by physically separating hardware devices in a server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Servers are architected as pools of processor, memory, and storage devices, organized as independent failure-isolated components interconnected by a high-bandwidth network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A critical challenge, however, is the high performance penalty of accessing data from a remote memory module over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Addressing this challenge is difficult as disaggregated systems have high runtime variability in network latencies/bandwidth, and page migration can significantly delay critical path cache line accesses in other pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This paper conducts a characterization analysis on different data movement strategies in fully disaggregated systems, evaluates their performance overheads in a variety of workloads, and introduces DaeMon, the first software-transparent mechanism to significantly alleviate data movement overheads in fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, to enable scalability to multiple hardware components in the system, we enhance each compute and memory unit with specialized engines that transparently handle data migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, to achieve high performance and provide robustness across various network, architecture and application characteristics, we implement a synergistic approach of bandwidth partitioning, link compression, decoupled data movement of multiple granularities, and adaptive granularity selection in data movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate DaeMon in a wide variety of workloads at different network and architecture configurations using a state-of-the-art accurate simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon improves system performance and data access costs by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='39× and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='06×, respectively, over the widely-adopted approach of moving data at page granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CCS Concepts: • General and reference → Performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Experimentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' • Com- puter systems organization → Architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' • Hardware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Additional Key Words and Phrases: data movement, data access, memory access, hardware support, hard- ware mechanism, high performance, memory systems, memory disaggregation, resource disaggregation, disaggregated systems, workload characterization, benchmarking, performance characterization ∗Equal contribution to this research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Authors’ addresses: Christina Giannoula, christina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='giann@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='com, University of Toronto, Canada, National Technical University of Athens, Greece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kailong Huang, University of Toronto, Canada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Jonathan Tang, University of Toronto, Canada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nectarios Koziris, National Technical University of Athens, Greece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Georgios Goumas, National Technical University of Athens, Greece;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Zeshan Chishti, Intel Corporation, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nandita Vijaykumar, University of Toronto, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1145/3508041 Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='00414v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='AR] 1 Jan 2023 21:2 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Reference Format: Christina Giannoula, Kailong Huang, Jonathan Tang, Nectarios Koziris, Georgios Goumas, Zeshan Chishti, and Nandita Vijaykumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, 1, Article 21 (March 2022), 34 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1145/ 3508041 1 Introduction With recent advances in network technologies [33, 41, 63, 79, 80] that enable high bandwidth networks, resource disaggregation [33, 87] has emerged as a promising technology for data cen- ters [16, 33, 37, 54, 87, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Resource disaggregation proposes the physical separation of hardware devices (CPU, accelerator, memory, and disk) in a server as independent and failure-isolated com- ponents connected over a high-bandwidth network such as RDMA [63] and Gen-Z [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Compared to monolithic servers that tightly integrate these components (Figure 1a), disaggregated systems can greatly improve resource utilization, as memory/storage components can be shared across applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' resource scaling, as hardware components can be flexibly added, removed, or upgraded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and failure handling, as the entire server does not need to be replaced in the event of a fault in a device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, resource disaggregation can significantly decrease data center costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Disaggregated systems comprise multiple compute, memory and storage components, intercon- nected over a high-bandwidth network (Figure 1b), each independently managed by a specialized kernel module (monitor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Typically, each compute component includes a small amount (a few GBs) of main memory (henceforth referred to as local memory) to improve memory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, almost all the memory in the data center is separated as network-attached disaggregated memory components to maximize resource sharing and independence in failure handling (different from typical hybrid memory architectuers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, the majority of the application working sets is accessed from the disaggregated memory components (henceforth referred to as remote memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Each memory component includes its own controller and can be flexibly shared by many compute components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, disaggregated systems can provide high memory capacity for applications with large working sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', bioinformatics, graph processing and neural networks) at lower cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fine-grain microsecond-latency networking technologies [36, 41, 63, 79, 80] that interconnect all hardware components have made fully disaggregated systems feasible, being only 2-8× slower than DRAM bus bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, since a large fraction of the application’s data (typically ∼80%) [33, 54, 87] is located and accessed from remote memory, the higher latencies of remotely accessing data over the network can cause large performance penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Alleviating data access overheads is challenging in disaggregated systems for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, disaggregated systems are not monolithic and comprise independently managed entities: each component has its own hardware controller, its resource allocation is transparent from other components and a specialized kernel monitor uses its own functionality/implementation to manage the component it runs on (only communicates with other monitors via network messaging if there is a need to access remote resources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This characteristic necessitates a distributed and disaggregated solution that can scale to a large number of independent components in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, there is high variability in data access latencies as they depend on the location of the remote memory component and the contention with other compute components that share the same memory components and network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Data placements can also vary during runtime or between multiple executions, since data is dynamically allocated in one or more remote memory components and hardware updates can flexibly change the architecture of the memory component and the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, data is typically migrated at page granularity [5, 6, 10, 35, 54, 57, 87, 103, 109] as it enables: (i) transparency to avoid modifications to existing OS/applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) low metadata overheads for address translation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (iii) leveraging spatial locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:3 network across components (a) (b) Local Memory CPU Compute Component processor monitor VA -> PA Table Remote Memory Controller memory monitor Memory Component VA -> PA Table DRAM Cache (fast) CPU Monolithic Server Monolithic OS Main Memory (slow) VA -> PA Table Remote Memory Controller memory monitor Memory Component VA -> PA Table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (a) A monolithic system versus (b) a disaggregated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, we observe in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 that moving memory pages in disaggregated systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', moving data at a large granularity over the network, can significantly increase bandwidth consumption and slow down accesses to cache lines in other concurrently accessed pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Recent works on hybrid memory systems [3, 4, 23, 26–29, 42, 45, 47, 52, 58–61, 64, 82, 83, 103], for example, those that integrate die-stacked DRAM [44] caches aim to address the high page movement costs between main memory and the DRAM cache [23, 42, 60, 61, 83] with mechanisms to move data at smaller granularities [23, 43, 60, 61, 76, 96, 97], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', cache line, or by using page placement/hot page selection mechanisms [3, 4, 26–29, 45, 47, 52, 58, 59, 64, 82, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, these prior works are tailored for a monolithic tightly-integrated architecture (Figure 1a), and are not suitable for disaggregated systems (See § 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' These works assume centralized data management/allocation (unlike in disaggregated systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For instance, software runtimes [3, 4, 26–29, 45, 47, 52, 58, 59, 64, 82, 103] running on CPUs in hybrid systems leverage TLBs/page tables to track page hotness and move pages across different memory devices (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Instead, in fully disaggregated systems all hardware memory functionalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', TLBs, page tables) of remote pages are moved to the memory components themselves [37, 87] (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus they cannot be used to track page hotness at the CPU side to implement intelligent page placement/movement in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Similarly, hardware-based approaches [43, 55, 76, 96] add centralized hardware units in the CPU to track metadata for pages in second-tier memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This however would incur high hardware costs in disaggregated systems that enable large amounts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', TBs) of remote memory [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Requiring each compute component to control/track a large number of pages in remote memory components would impose significant hardware costs and scalability challenges, and thus might annihilate the benefits of resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Moreover, disaggregated systems incur significant variations in access latencies and bandwidth based on the current network architecture and concurrent jobs sharing the memory components/network, which are not addressed by prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This necessitates a solution primarily designed for robustness to this variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In this work, we analyze different data movement strategies in fully disaggregated systems, and introduce DaeMon, an efficient software-transparent mechanism to alleviate data movement overheads in disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon provides (i) high performance on dynamic workload demands, (ii) robustness to variations in architectures, network characteristics and application behavior, and (iii) independence and scalability to multiple compute components and memory components that are managed transparently to each other and are flexibly added/removed in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon consists of two key ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, we offload data migrations to dedicated hardware engines, named DaeMon compute and memory engine, that are added at each compute component and memory component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This key idea enables independence and scalability to a large Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:4 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' number of compute components and memory components of disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Compared to a centralized design, DaeMon’s distributed management of data movement enables simultaneous processing of data movement across multiple components and decreases the processing costs and queuing delays to serve data requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, we leverage the synergy of three key techniques to provide robustness to the high variability in network latencies/bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1) We use a bandwidth partitioning approach to enable the decoupled movement of data at two granularities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', page and cache line, and prioritize cache line granularity data moves over page moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This design enables low access latencies to remote memory for the cache line requests on the critical path, while the associated pages can be still be moved independently at slower rates to retain the benefits of spatial locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2) We design an adaptive approach to decide on-the-fly if a request should be served by a cache line, page or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Via selective granularity data movement, we provide robustness to variations in network, architectures and application characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 3) We leverage hardware link compression when migrating pages to reduce network bandwidth consumption and alleviate queuing delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The synergy of the afforementioned key techniques provides a robust solution for disaggregated systems: decoupled multiple granularity data movement effectively prioritizes cache line requests on the critical path, and migrates pages at a slower rate leveraging compression to reduce bandwidth consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The adaptive granularity selection mechanism effectively adapts to the characteristics of the application data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', by favoring moving more pages if application data is highly compressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The decoupled cache line granularity movement also enables the use of more sophisticated and effective compression algorithms (with relatively high compression latency) for page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate DaeMon using a range of capacity intensive workloads with different memory access patterns from machine learning, high-performance-computing, graph processing, and bioinfor- matics domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Over the widely-adopted approach of moving data at page granularity, DaeMon decreases memory access latencies by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='06× on average, and improves system performance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='39× on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We demonstrate that DaeMon provides (i) robustness and significant performance benefits on various network/architecture configurations and application behavior (Figures 8 and 13), (ii) scalability to multiple hardware components and networks, (Figure 17), and (iii) adaptivity to dynamic workload demands, even when multiple heterogeneous jobs are concurrently executed in the disaggregated system (Figure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This paper makes the following contributions: We heavily modify a state-of-the-art simulator to develop and evaluate the overheads of different data movement strategies in fully disaggregated systems, analyze the challenges of providing efficient data movement in such systems, and develop DaeMon, an adaptive distributed data movement mechanism for fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We enable decoupled data movement at two granularities, and migrate the requested critical data quickly at cache line granularity and the corresponding pages opportunistically without stalling critical cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We dynamically control the data movement granularity to effectively adapt to the current system load and application behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We employ a high-latency compression scheme to further reduce bandwidth consumption during page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate DaeMon using a wide range of capacity intensive workloads, various architecture/net- work configurations, and in multi-workload executions of concurrent heterogeneous jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We demonstrate that DaeMon significantly outperforms the state-of-the-art data movement strategy, and constitutes a robust and scalable approach for data movement in fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:5 2 Background and Motivation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 Baseline Disaggregated System Figure 2 shows the baseline organization of the disaggregated system, which includes several compute components and memory components as network-attached components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To improve performance, each compute component tightly includes a few GBs of main memory, referred to as local memory, which can typically host 20-25% of the application’s memory footprint [33, 84, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Each memory component includes its own controller and connects multiple DIMM modules, referred to as remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Cores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='On-Chip ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Caches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Coherent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Interconnect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='FPGA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Control Logic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Translation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Local Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='VA -> PA Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Translation Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Remote Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='VA -> PA Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Remote ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Compute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='processor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Remote ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Remote ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Compute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='processor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='monitor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' High-level organization of a disaggregated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We assume distributed OS modules that coordinate and communicate with each other via network messaging when needed, similar to [37, 54, 87]: processor and memory kernel monitors run at compute components and memory components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The memory allocation/management of remote memory is performed at the memory component itself [37, 87], transparently to compute components, enabling the different components to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The on-chip caches and the local memory of compute components are typically indexed by virtual addresses [87], and remote data is requested from memory components using virtual addresses [16, 37, 54, 87] (unlike in hybrid memory systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The data management is typically performed at page granularity [16, 54, 87] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 4KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The local memory of the compute component can be treated as a cache with tags [87] or a local virtual to physical translation mapping [54] can be used (either approach works with DaeMon, however we assume the second approach in our evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The physical memory addresses of the local memory can be found by accessing and traversing metadata (tags or local page tables) kept in a dedicated (pre-reserved) DRAM memory space (kernel metadata is directly indexed via physical addresses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When an local memory miss happens, either (i) the processor kernel module of the compute component triggers a page fault and fetches the requested page from remote memory components [87], or (ii) dedicated software runtimes co-designed with hardware primitives [16] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', supported in FPGA-based controllers as shown in Figure 2a) handle remote data requests on demand completely eliminating expensive page faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Either approach works with DaeMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We assume that the controllers of memory components implement hardware-based address translation (Figure 2b) to access pages in remote memory as proposed in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:6 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Jobs running at different compute components can share read-only pages located at multiple memory components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Similarly to prior state-of-the-art works [16, 37, 87], we assume that the system does not support writable shared pages across compute components, since they are rare across datacenter jobs [37, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 Data Movement Overheads in Fully Disaggregated Systems Prior state-of-the-art works [14, 35, 37, 46, 54, 74, 75, 99, 109, 112] typically enable data management at page granularity for three compelling reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, the memory allocation and management is transparent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', requires little to no modification to OS or application code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, the coarse granularity enables low metadata overheads for address translation in local memory and remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Managing local memory as a cache at cache line granularity would incur prohibitively high metadata overheads [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, page movements enable exploiting spatial locality in common memory access patterns [43, 56, 102], and increase the number of accesses served from the lower cost local memory instead of remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 3 compares the performance of different data movement strategies in disaggregated systems across various workloads (See Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate one memory component and one compute component having local memory to fit ∼20% of the application’s working set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We use 100ns/400ns latency [33, 54] to model the propagation and switching delays on the network (referred to as switch latency), and configure the network bandwidth between the compute component and the memory component (referred to as bandwidth factor) to be 1/4× the DRAM bus bandwidth [33, 87] of the local memory or remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We compare six configurations: (i) Local: all accesses are served from the local memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) cache-line: accessing data from remote memory at cache line granularity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and directly writing data to the Last Level Cache (LLC) of the compute component (local memory is not used),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (iii) Remote: accessing data from remote memory at page granularity (moving pages to local memory) accounting for all network-related overheads,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (iv) page-free: remote accesses incur the latency of one cache line granularity remote access and the corresponding page is transferred to local memory at zero cost (spatial locality is leveraged),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (v) cache-line+page: requesting data from remote memory at both cache line (moved to LLC) and page granularity (moved to local memory) and servicing data requests using the latency of the packet that arrives earlier to compute component (accounting for all network-related overheads),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (vi) DaeMon: accessing data from remote memory using DaeMon (accounting for all network-related overheads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We make four observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, Remote, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', the typically-used approach of moving data at page granularity, incurs significant performance slowdowns, on average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='86×, over the monolithic Local configuration due to transferring large amounts of data over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In addition to the large network bandwidth consumption, migrating pages can slow down critical path accesses to data in other concurrently accessed pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, page-free achieves almost the same performance as the Local scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A small penalty is incurred as the first access to a page in remote memory incurs cache line granularity latency access cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, since the whole page is migrated to local memory for free, performance significantly improves thanks to spatial locality benefits of migrating pages in addition to the requested cache line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, migrating pages to local memory is critical to achieving high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, cache-line outperforms Remote in some latency- bound workloads with poor spatial locality, however its performance benefits depend on network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For example, in tr, cache-line outperforms Remote by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='42× when the switch latency is 100ns, while it incurs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='82× performance slowdown over Remote with 400ns switch latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fourth, the cache-line+page scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', naively moving data at both granularities, is still inefficient (only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='11× better than Remote), since critical cache lines are still queued behind large pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:7 kc tr pr nw bf bc ts sp sl hp pf dr rs GM 0 2 4 Slowdown stch-lat=100 bw-fact=1/4 10 11 39 39 cache-line Remote page-free cache-line+page DaeMon kc tr pr nw bf bc ts sp sl hp pf dr rs GM 0 2 4 6 8 Slowdown stch-lat=400 bw-fact=1/4 10 11 39 39 12 10 cache-line Remote page-free cache-line+page DaeMon Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Data movement overheads in disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Overall, we draw two conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (i) Page migrations incur high performance penalties and can significantly slow down the critical path cache line requests to other concurrently accessed pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, if the overheads of migrating pages can be mitigated, moving data at page granu- larity offers a critical opportunity to alleviate remote access costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) There is no one-size-fits-all granularity in data movements to always perform best across all network configurations and appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Depending on the spatial locality and the network load, some applications benefit from cache line-only accesses that avoids unnecessary congestion of pages in the network, while some applications significantly benefit from page movements that leverage spatial locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To this end, we design DaeMon to significantly reduce data movement costs across various application, network and architecture characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 3 demonstrates that DaeMon significantly outperforms the Remote and cache-line+page schemes by on average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='38× and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='14×, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 3 DaeMon: Our Approach DaeMon is an adaptive and scalable data movement mechanism for fully disaggregated systems that supports low-overhead page migration, enables software transparency, and provides robustness to variations in memory component placements, network architectures and application behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon comprises two key ideas: (1) Disaggregated Hardware Support for Data Movement Acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We enhance each compute component and memory component with specialized engines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', DaeMon compute and memory engine (Figure 4), respectively, to manage data movements across the network of disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon engines enable independence and high scalability to a large number of compute components/memory components that are flexibly added/removed in disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Moreover, distributed management of data migrations at multiple DaeMon engines in- creases the execution parallelism and decreases the processing costs and queuing delays to serve data requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (2) Synergy of Three Key Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon incorporates three synergistic key techniques shown in Figure 4: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:8 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote Memory Controller Compute Component Memory Component Local Memory CPU LLC Page Queue Sub-block Queue (De) Compr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Unit Cachelines Compressed Pages Network DaeMon Compute Engine DaeMon Memory Engine Page Queue Sub-block Queue Selection Granularity Unit (De) Compr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Unit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' High-level overview of DaeMon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (I) Decoupled Multiple Granularity Data Movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, we integrate two separate hardware queues to manage and serve data requests from remote memory at two granularities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', cache line (via the sub-block queue) and page (via the page queue) granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Cache line requests are directly moved to Last Level Cache (LLC) of the compute component to avoid additional metadata overheads and eliminate memory latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Page requests are moved to local memory of the compute component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, we prioritize moving cache lines over moving pages via a bandwidth partitioning approach: a queue controller serves cache line and page requests with a predefined fixed ratio to ensure that at any given time a certain fraction of the bandwidth resources is always allocated to serve cache line requests quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon implements both network and remote memory bus bandwidth partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This technique provides two benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, retaining page migrations in DaeMon (i) enables software-transparency, (ii) allows maintaining metadata for DRAM at page granularity (thus incurring low metadata overheads), and (iii) exploits the performance benefits of data (spatial) locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, cache line data movements that are on the critical path are quickly served, and have fewer slowdowns from expensive page movements that may have been previously triggered, since DaeMon effectively prioritizes cache line movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (II) Selection Granularity Data Movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To handle network, architecture and application variability in disaggregated systems, we design an dynamic approach to decide whether a request should be served by a cache line, page, or both, depending on application and network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' At DaeMon’s engine of each compute component, we include two separate hardware buffers to track pending data migrations for both cache line and page granularity, and a selection granularity unit to control the granularity of upcoming data requests based on the utilization of the above buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The utilization of these buffers allows us to capture dynamic information regarding the current traffic in the system and the application behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', locality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Our proposed selection granularity data movement enables robustness against fluctuations in network, architecture and application characteristics (we explain how this is implemented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (III) Link Compression on Page Movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We leverage the decoupled page movement to use a high-latency link compression scheme (with high compression ratio), when moving pages across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We integrate hardware compression units at both the compute components and memory components to highly compress pages moved over the network: the page is compressed before it is being transferred over the network, and decompressed when it arrives at the destination (before it is written in memory modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Link compression on page movements reduces the network bandwidth consumption and alleviates network bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Overall, DaeMon cooperatively integrates all three key techniques, the synergy of which provides a versatile solution: (1) Prioritizing requested cache lines helps DaeMon to tolerate high (de)compression latencies in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:9 page migrations over the network, while also leveraging benefits of page migrations (low metadata overheads, spatial locality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (2) Moving compressed pages consumes less network bandwidth, helping DaeMon to reserve part of the bandwidth to effectively prioritize critical path cache line accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (3) Selection granularity movement helps DaeMon to adapt to the application data compressibility: if the pages are highly compressible, the number of pending page migrations is relatively low, thus DaeMon favors moving data more at page granularity instead of cache line granularity (and vice-versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4 DaeMon: Detailed Design We design DaeMon to be a disaggregated solution: a DaeMon compute engine is added at each compute component of the system to handle data requests to remote memory, and a DaeMon memory engine is integrated at the controller of each memory component of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 5 shows our proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Network Compute Component Memory Component CPU Cores LLC Local Memory Coherent Interconnect Remote Memory FPGA DaeMon Compute Engine Control Logic Controller DaeMon Memory Engine Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proposed architecture for compute component and memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The baseline architecture of each compute component includes a chiplet-based CPU+FPGA architecture (this CPU+FPGA integrated design has also been proposed to prior state-of-the-art work [16, 40]), which is expected to have small cost [16] compared to the overall cost savings enabled by disaggregated systems, while it is also socket compatible to current systems [25, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The FPGA has three communication paths: i) a coherent path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', CPU-FPGA coherent links, to access the CPU on-chip cache hierarchy, ii) an interface (channel-based connection) to access the local memory, and iii) an external connection to the network controller to move data to/from remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We propose extending the FPGA by adding a new lightweight hardware component to handle data requests, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', the DaeMon compute engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Each memory component includes its own controller [37, 54, 87], that has two communication paths: a channel-based connection to DIMM modules of remote memory, and an external connection to the network, which is used to move data from/to compute components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We propose extending the controller of each memory component by adding a new hardware component to handle data movements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', the DaeMon memory engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In our study, we assume that the local memory is an inclusive cache for the remote memory, which contains all application data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The local memory implements an approximate LRU replacement policy, similar to prior state-of-the-art work [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 Enabling Decoupled Multiple Granularity Data Movement Figure 6 shows the detailed design of the DaeMon compute engine and DaeMon memory engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon engine includes two queues to handle requests at each granularity: cache line granularity via the sub-block queue 1 , and page granularity via the page queue 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' It also includes a queue Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:10 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' controller 7 to serve requests from both queues, and a packet buffer 6 to temporarily keep arrived packets, while they are being processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='LLC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='LLC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Sub-block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Packet Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Compression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Decompression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='DaeMon Compute Engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Buffer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Inflight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Sub-block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Buffer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Inflight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Page Buffer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Selection Granularity Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Page ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Dirty Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Dirty Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Buffer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Sub-block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='DaeMon Memory Engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Page ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Queue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Packet Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Decompression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Compression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Unit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Buffer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Detailed design of DaeMon engines for the compute (left) and memory (right) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Approximate Bandwidth Partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To prioritize cache line data movements while also ensuring that page movements are not aggressively stalled, we design an approximate bandwidth partitioning approach between the cache line and page movements, and configure the queue controller to serve cache line and page requests with a predefined fixed ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Assuming that cache line and page requests transfer 64B and 4KB of data, respectively, and having a bandwidth partitioning ratio of 25% (Figure 11 presents a sensitivity study on this ratio), 25% of the bandwidth is reserved for cache lines as follows: for each page request issued through the network, which results in transferring 4KB data, the queue controller needs to serve 4096/64 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25/(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25) ≈ 21 cache line requests, each transferring 64B of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To ensure this approximate partitioning is always maintained, we retain this alternate serving of page and cache line requests even if either queue is empty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', requests may not issued in all cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon implements an approximate bandwidth partitioning both in the network across components of the system and when accessing data from remote memory modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 Selecting the Data Movement Granularity DaeMon compute engine additionally includes two separate hardware buffers to track data requests which are scheduled to be moved or in the process of being migrated (henceforth referred to as inflight): (i) the inflight sub-block buffer for the cache line granularity requests 3 , and (ii) the inflight page buffer for the page granularity requests 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Both buffers are used to track pending data migrations and avoid requesting the same data multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon compute engine includes a selection granularity unit 5 which throttles data requests to avoid requesting the same data multiple times, and decides at which granularity the request should be served (cache line, page, or both granularities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Scheduling Page Granularity Data Movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When DaeMon compute engine receives a data request, the selection granularity units checks (i) the utilization of the inflight page buffer, and (ii) if the corresponding page has already been scheduled to be moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If the page has already been requested or the inflight page buffer is full, the selection granularity unit does not request the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus at any given time, the number of pages scheduled to be moved is automatically limited by the selection granularity unit, also limiting storage/area overheads to track the pending page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If the inflight page buffer is not full, the selection granularity units schedules the page migration by adding a new entry in the page queue and the inflight page buffer, marking the page as scheduled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the queue controller issues the movement, the corresponding entry is released in the page Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:11 queue, and the page entry in the inflight page buffer is marked as moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the requested page arrives, the corresponding entry is released (invalid state) in the inflight page buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The page is written to local memory and all pending requests are serviced via local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Any entries in the inflight sub-block buffer with requests to cache lines in the same page are removed and thus, any data packets that arrive in the future with cache lines from the same page are simply ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In DaeMon, we retain existing data management and address translation mechanisms at page granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Local page table updates at the compute component are only performed in page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Scheduling Cache Line Granularity Data Movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To decide whether a cache line gran- ularity movement should be made, the selection granularity unit checks (i) the utilization of the inflight sub-block buffer and (ii) if the corresponding page was already scheduled to be moved (by a previous request).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' There are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, if the corresponding page is not scheduled to be moved according to the inflight page buffer, the selection granularity unit always schedules a cache line granularity data movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, if the corresponding page is already scheduled to be moved, the selection granularity unit sends the cache line only if: (i) the sub-block buffer has lower utilization than the page buffer and (ii) the page is not already in the process of migration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', the page is in the page queue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Otherwise, it drops the request as the page has already been requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This avoids unnecessarily sending cache lines when the corresponding page is likely to arrive faster and when the sub-block queue is likely to be slow due to oversaturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If a cache line is scheduled, a new entry is added both in the sub-block queue and the inflight sub-block buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the queue controller issues the movement, the corresponding entry is released in the sub-block queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the requested cache line arrives at the compute component, the corresponding entry is released in the sub-block buffer, and the data is directly written to LLC through the FPGA-based coherent interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The above mechanism enables an adaptive approach for the data movement granularity based on the dynamic network/architecture and application characteristics: (1) If there is high locality within pages, there are fewer pages requested, and the sub-block buffer fills up faster than the page buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, DaeMon favors issuing pages and throttles cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If there is low locality within pages, the page buffer fills up faster than the sub-block buffer, since cache line requests are served with a higher rate than page requests (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 21:1 cache lines versus pages requests for 25% bandwidth ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, DaeMon favors issuing cache line movements and throttles page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (2) If both the page and sub-block buffers are fully utilized, DaeMon detects bandwidth constrained scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In bandwidth constrained scenarios, DaeMon favors issuing more cache line movements to alleviate bandwidth bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the bottleneck is mitigated, (inflight buffers are not fully utilized), DaeMon schedules more page movements to obtain locality benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (3) Additionally, when using link compression to transfer pages, DaeMon is able to adapt to the compressibility of the application data: if the pages are highly compressible, the inflight page buffer empties at a faster rate and thus DaeMon favors sending more page migrations (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 Handling Dirty Data Dirty data (cache lines/pages) is always directly written to remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Data (cache line or page granularity) can be in one of the three states: (i) local: when data is cached in on-chip caches (for cache lines) or local memory (for pages), (ii) remote: when data is only in remote memory, and (iii) inflight: when data is being migrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' With DaeMon, data can be present simultaneously in two states: for example, local as a cache line (in the cache hierarchy of the compute component) and inflight as a page or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This poses coherence issues if the processor writes to data in the above state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:12 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' There are two scenarios: (i) if a page arrives to compute component before a prioritized cache line, any modifications to the page may be overwritten by the stale cache line that arrives later, and (ii) if a dirty cache line is evicted from the LLC while the corresponding page is in transit, the modifications would be lost when the page arrives to compute component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' As explained, in the (i) scenario, when a page arrives, the corresponding entries in the inflight sub-block buffer with requests to cache lines in the same page are removed and thus, any data packets that arrive in the future with cache lines from the same page are simply ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In the (ii) scenario, for every dirty cache line that gets evicted by the LLC and also misses in the local memory, its corresponding page can be either inflight or in remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To ensure correctness, DaeMon compute engine first checks if there is an inflight page request in the inflight page buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If there is no inflight page request (according to the inflight page buffer), the evicted dirty cache line is directly migrated to remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In the other case, we need to retain the dirty cache line until the page arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We include a dirty unit 8 in the DaeMon compute engine with a dirty data buffer that temporarily stores these dirty cache lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the corresponding inflight page arrives, the DaeMon compute engine flushes the dirty cache line(s) from the dirty buffer to local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Prior works [2, 16] observe that typically a few cache lines (1-8 cache lines) or all cache lines of a page are accessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, when the evicted dirty cache lines of the same page increase beyond a predefined threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 8 cache lines), the DaeMon compute engine flushes all dirty cache lines to remote memory, and marks the corresponding entry for that page in the inflight page buffer as throttled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the inflight page arrives, the DaeMon compute engine ignores it, since its entry is in the throttled state, and sends a new request for that page to receive the up-to-date data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This enables lower area/storage overheads for the dirty data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 Link Compression in Page Migrations Approaches for data compression are typically of two types: (i) latency-optimized compression schemes [8, 21, 73, 104, 105], which optimize/minimize the (de)compression latencies, and (ii) ratio-optimized compression schemes [1, 49, 93, 111], which provide higher compression ratios while incurring relatively high (de)compression latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We select a ratio-optimized compression scheme in DaeMon based on two observations (§6): (i) in disaggregated systems, queueing delays and network latencies can be significant, thus compression benefits outweigh the high (de)compression latencies, and (ii) DaeMon prioritizes cache lines that are on the critical path, thus we can tolerate relatively high (de)compression latencies for page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon engines include (de)compression units 9 10 that compress pages transferred through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We implement a hardware design similar to IBM MXT [1, 93], using the LZ77 compres- sion algorithm [111], and operating at 1KB granularity at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (De)Compression units include 4 engines, each of which operates on 256B of data and uses a 256B shared dictionary, incurring in total a 64-cycle latency according to [1, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 DaeMon’s Hardware Structures We estimate the overheads of DaeMon’s hardware structures for each compute component assuming a 64-core CPU, using CACTI [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The sizes of the DaeMon sub-block and page queues and the sub-block and page buffers have been selected based on the maximum possible number of pending data migrations at a time, which is determined by the number of the available LLC MSHRs (Miss Status Holding Registers) in a typical CPU system, and is independent of the workloads’ patterns and the mix of workloads that are running at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For the hardware structures at each memory component, we scale the sizes of the DaeMon sub-block and page queues, assuming that each memory component can concurrently serve up to 4 compute components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Table 1 presents Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:13 the hardware overheads of DaeMon compute engine (C) and DaeMon memory engine (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 7 shows an inflight sub-block buffer entry, an inflight page buffer entry, and a dirty data buffer entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hardware Entries Size Access Area Energy Structure (KB) Cost (ns) Cost (mm2) Cost (nJ) Sub-block Queue (C) 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='038 Sub-block Queue (M) 512 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='039 Page Queue (C) 256 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='038 Page Queue (M) 1024 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='041 Inflight Sub-block Buffer (C) 128 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='046 Inflight Page Buffer (C) 256 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='096 Dirty Data Buffer (C) 256 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='046 Packet Buffer (C) 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='044 Packet Buffer (M) 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='047 2 × Dictionary Table (C,M) 1024 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='020 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon’s hardware overheads for C: compute engine and M: memory engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Address State Dirty Cache Line Offset Cache Line Offset Address State Data Address 00: scheduled 01: moved 10: throttled 11: invalid 0: scheduled 1: invalid 0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='11010 32 bits 2 bits 64 bits 64 bits 32 bits 1 bit 512 bits 32 bits b) Inflight Page Buffer Entry a) Inflight Sub-block Buffer Entry c) Dirty Data Buffer Entry 0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='11010 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' An inflight sub-block buffer entry, an inflight page buffer entry, and a dirty data buffer entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Sub-block Queue (SRAM), 128 entries: The sub-block queue size is limited by the available LLC MSHRs of the compute component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Page Queue (SRAM) - 256 entries: The page queue has 256 entries, since DaeMon serves requests from the page queue at a smaller rate than the sub-block queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Inflight Sub-block Buffer (CAM) - 128 entries: Similar to the sub-block queue, this buffer has 128 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We design this hardware structure to be indexed using the corresponding page address to achieve smaller area costs, since at a given time there may be multiple inflight cache line requests to the same page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Each entry (Figure 7a) includes the page address, the state (scheduled or invalid), and a 64-bit queue that is used to indicate the offsets within the page of the inflight cache requests by (re)setting the corresponding bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Inflight Page Buffer (CAM) - 256 entries: An inflight page buffer entry (Figure 7b) includes the page address, the state that can be scheduled, moved, throttled (when the page needs to be re-requested) or invalid, and a 64-bit queue to indicate the offsets of the dirty cache lines of the inflight page that are temporarily kept in the dirty data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Dirty Data Buffer (SRAM) - 256 entries: A dirty data buffer entry (Figure 7c) includes the evicted cache line and its address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Packet Buffer (SRAM) - 8KB: We use an 8KB buffer to temporarily store arrived data packets until they are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Dictionary Tables for (De)Compression (CAM) - 2KB: DaeMon proposes 4 engines at each (de)compression unit, each of them has 256B CAM [1, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In total, we estimate each dictionary table as 1KB CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:14 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Overall, DaeMon’s hardware overheads are due to the cache memories corresponding to the sub-block and page queues, the sub-block and page buffers, and the dictionary tables used for data compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The total sizes of the DaeMon cache memories are ∼34KB and 40KB for the DaeMon compute and memory engine, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, DaeMon’s hardware overheads are similar to that of the small L1 cache memory of a modern state-of-the-art processor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Intel Xeon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We conclude that our proposed hardware structures incur very modest hardware and financial costs to be integrated into the compute components and memory components of disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 Handling Failures DaeMon handles compute component, memory component and network failures using fault- tolerance approaches of prior works [16, 54, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' If the compute component fails (CPU or DaeMon compute engine), the application needs to be restarted potentially on a different compute com- ponent of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Network failures are handled using timeouts: DaeMon engines can trigger timeouts when pending page or cache line requests have not arrived after a long time, or when ACK messages have not been received for migrations of dirty data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The exploration of the timeout period value is left for is future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Finally, memory component failures are handled via data replication, similarly to prior work [87]: DaeMon can send the evicted dirty data to more than one memory component, and wait to receive ACK messages from all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 DaeMon Extensions Prefetching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon can flexibly support hardware/software-based prefetchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Existing CPU prefetchers might generate data requests, which DaeMon can normally serve by migrating the prefetched data at a cache line granularity, page granularity or both granularities, via on our proposed selection granularity scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Page prefetchers [62] might generate page-granularity data requests, which DaeMon can serve by migrating the prefetched data at page granularity or throttling the page request based on our selection granularity scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Large Pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon can be easily extended to support large granularity pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 2MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To effectively prioritize cache line requests over page requests, DaeMon’s predefined ratio for the approximate bandwidth partitioning needs to be properly configured based on the size of the large page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To enable multiple page sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', both 4KB and 2MB), we could enhance DaeMon to split large pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 2MB) to consecutive page requests of smaller sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 4KB) issued in the page queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 5 Methodology Simulation Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We use Sniper [17, 18], a state-of-the-art accurate simulator, and we heavily modified it to model a disaggregated system with one compute component and multiple memory components interconnected across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We present detailed evaluation results using one memory component and provide a characterization study of multiple memory components with various network configurations in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For the network across components, we use both (i) a fixed latency of 100ns/400ns [33, 54] to model propagation and switching delays inside network (referred to as switch latency), and (ii) a variable latency of modeling the current bandwidth utilization at each simulation interval (100K ns) when configuring the network bandwidth to be 2-8× less than DRAM bandwidth [33, 87] (referred to as bandwidth factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For the compute component, we configure a state-of-the-art CPU server with on-chip cache memories of typical sizes and x86 OoO cores of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6GHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The local memory size is configured to fit ∼20% of each application’s working set, and we evaluate LRU replacement policy [87] in local memory, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The aforementioned configuration is consistent with prior state-of-the-art works in disaggregated systems [33, 54, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For both the local memory and remote memory, we evaluate Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:15 a DDR4 memory model with 17GB/s bus bandwidth, and we simulate hardware-based address translation for memory pages, having overhead as one DRAM access cost per lookup, as explained in prior state-of-the-art work [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate access overheads in DaeMon queues/buffers using CACTI [66] (See Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Table 2 lists the parameters of our simulated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CPU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 GHz, 4-way OoO x86 cores, 224-entry ROB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' L1 Instr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Cache 32 KB, 4-way associativity, LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' L1 Data Cache 32 KB, 8-way associativity, 4-cycle access latency, LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' L2 Cache 256 KB, 8-way associativity, 8-cycle access latency, LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LLC 4MB, 16-way associativity, 30-cycle access latency, LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Local Memory 2400MHz, 15ns process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' latency, 17GB/s bus bandwidth [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Network 2-8× less than bus bandwidth, 100-400ns switching latency [33, 87];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote Memory 2400MHz, 15ns process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' latency, 17GB/s bus bandwidth [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Configuration of simulated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate various workloads with different memory access patterns from various application domains including graph processing, machine learning, bioinformatics, linear algebra, data analytics, and HPC domains, shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The dynamic working sets at any given point at runtime range from 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2MB to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='32GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In a fully disaggregated system, the application working set (irrespective of the size) is primarily housed in remote memory to provide the benefits of improved elasticity, heterogeneity, and failure isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, we configure the local memory size to fit ∼20% of each application’s working set (similar to prior state-of-the-art work [33, 54, 87]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' All data is initially located at remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We simulate most workloads to full execution and for slower long running workloads, we simulate 1B instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Workload ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Input Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='K-Core Decomposition (kc) [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Graph Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1M vertices x 10M edges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Triangle Counting (tr) [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Graph Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1M vertices x 10M edges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Page Rank (pr) [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Graph Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1M vertices x 10M edges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Needle Wunsch (nw) [20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Bioinformatics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4096 base pairs per sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Breath First Search (bf) [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Graph Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1M vertices x 10M edges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Betweenness Centrality (bc) [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Graph Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1M vertices x 10M edges ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Timeseries (ts) [106] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Data Analytics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='262144 elements in sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='Sparse Matrix Vector Multipl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (sp) [51] Linear Algebra pkustk14 matrix Sparse Lengths Sum (sl) [67] Machine Learning Kaggle Criteo 10GB Dataset High Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Conjugate Gradient (hp) [39] HPC 104 x 104 x 104 Particle Filter (pf) [20] HPC 4096 x 4096, 30000 particles Darknet19 (dr) [81] Machine Learning dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='jpg (768 x 576 pixels) Resnet50 (rs) [81] Machine Learning dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='jpg (768 x 576 pixels) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Summary of workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6 Evaluation We evaluate six schemes: (i) Remote: the typically-used approach [16, 54, 87] of moving data to/from remote memory at page granularity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) LC: DaeMon’s link compression for page movement without enabling cache line granularity data movement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', moving data at page granularity with LZ-based link compression enabled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (iii) BP: enabling only DaeMon’s decoupled multiple granularity data movement with 25% bandwidth partitioning ratio for cache line movements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', moving data always at both granularities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (iv) PQ: enabling DaeMon’s both decoupled multiple granularity and selection granularity data movement with 25% bandwidth partitioning ratio for cache line movements (without enabling data compression in page migrations);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (v) DaeMon: DaeMon’s complete design Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:16 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' enabling all its three techniques (25% bandwidth partitioning ratio);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (vi) Local: the monolithic approach where all the data fits in local memory of the compute component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 8 compares all schemes with different network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Our evaluated workloads exhibit three patterns and we make the following observations.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LC BP PQ DaeMon LocalDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:17 First, kc, tr, pr, and nw exhibit relatively poor spatial locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In such workloads, BP effectively prioritizes critical cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, PQ provides significant benefits thanks to dynamically selecting the data movement granularity: the page buffer saturates faster than the sub-block buffer given the poor locality and the higher servicing rate of the cache line requests in the queue controller, thus the selection granularity unit enables the movement of more cache lines and fewer pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This results to reduced access latencies as critical path cache line requests are no longer stalled behind many page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, bf, bc, and ts exhibit medium spatial locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In such workloads, both LC and PQ decrease data access costs using different approaches: LC enables exploiting more spatial locality by moving more pages, while PQ accelerates accesses to the critical path cache line requests, both of which benefit these workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, the remaining workloads exhibit high spatial locality within pages, thus page migration is critical to leverage data locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In these workloads, BP incurs high performance slowdowns, since it is oblivious to application behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Instead, PQ effectively enables more page movements and throttles cache line movements by tracking pending data requests, thus achieving similar system performance to Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LC performs better for sp, sl, hp, and pf, since these workloads have higher data compressibility than dr and rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fourth, when network bandwidth is more constrained, LC provides even higher performance over Remote, while PQ is unaffected by bandwidth as the bandwidth partitioning approach prioritizes cache line movements even with low available bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fifth, PQ is slightly affected by the switch latency (Please also see Figure 20 in Appendix § A): PQ outperforms Remote by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='60× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='51× for 100ns and 400ns switch latency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The slightly lower benefits are due to PQ’s inability to hide network switch latencies in critical cache line movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Instead, LC is unaffected by switch latency, as page movement incurs much higher overheads (due to very high network processing and queueing delays) over the smaller switch latency, which link compression is able to alleviate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Finally, DaeMon provides high performance benefits for all three classes of workloads with different locality characteristics thanks to synergistically integrating both LC and PQ: (i) PQ helps hide the (de)compression latencies in LC and migrate fewer pages in order to prioritize critical path cache line movements, and (ii) LC releases network bandwidth resources and helps recover the lost spatial locality in pages by moving more pages with the available network bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' dr and rs show only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='05× speedup over Remote as neither LC nor PQ is able to provide speedups due to the poor application data compressibility and high spatial locality within pages (which favors moving pages rather than cache lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon’s adaptive approach also provides high performance benefits across all network configurations: (i) when the switch latencies are high, cache lines movements are slowed down and the sub-block queue fills up faster, thus DaeMon favors moving more pages, which is more effective at high network switch latencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (ii) the approximate bandwidth partitioning approach effectively prioritizes cache line over page movements even when network bandwidth is constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, DaeMon significantly outperforms the state-of-the-art Remote scheme by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='85×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='36×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='97× for 1/2, 1/4, and 1/8 bandwidth factor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Overall, we conclude that DaeMon’s cooperative techniques provide a robust approach to alleviate data movement overheads across various network characteristics and application behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Memory Access Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 9 compares the average data access costs (latencies) achieved by various schemes normalized to Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Due to space limitations, in the remaining plots, we present a representative subset of our evaluated workloads, but we report geometric mean values across all evaluated workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Please also see Figure 19 in Appendix § A, which compares the network bandwidth utilization achieved by the various data movement schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:18 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' pr nw bf bc sp hp dr rs GM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Access Costs stch-lat=100 bw-fact=1/2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 pr nw bf bc sp hp dr rs GM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Access Costs stch-lat=400 bw-fact=1/4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Data access costs achieved by various schemes normalized to Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We make three observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, LC improves data access costs over Remote by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='12× across all network configurations (not graphed), because it reduces the network processing costs and queueing delays by sending fewer bytes through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' PQ improves access costs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='06× over Remote across all configurations) by prioritizing critical path cache line movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, PQ significantly reduces data access costs in workloads with poor page locality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', pr, nw), since critical path cache line movements are not stalled by migrating pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, in applications with high data locality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', dr, rs), although PQ reduces data access costs by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='43× over Remote, it improves performance by only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='05×, because the selection granularity unit favors sending pages for workloads with high locality and a few requests are served at cache line granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, DaeMon significantly reduces data access costs by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='06× over Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon employs link compression to migrate more pages with lower network overhead over PQ, thus exploiting more data locality, while also leveraging the ability to prioritize critical cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In pr, DaeMon can achieve lower access latency than Local, since serving requests from both local memory and remote memory increases the effective aggregate memory bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hit Ratio in Local Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 10 presents the hit ratio in local memory, and is thus a measure of the page movement benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To prioritize cache lines, PQ throttles some page migrations, thus reducing the local memory hit rate as a tradeoff for reduced access latencies to critical path cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, DaeMon enables moving more pages over PQ thanks to link compression, while still retaining the cache line prioritization benefits of PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The numbers shown over each bar for DaeMon present the additional pages that were moved in DaeMon as a percentage over PQ, thanks to the reduced bandwidth consumption provided by link compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A zero value indicates that neither PQ nor DaeMon has throttled any page movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We draw three findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, Remote has on average 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7% hit ratio in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thanks to high spatial locality, all workloads benefit from page migration, leading to high hit rates: even workloads with relatively poor spatial locality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', nw) have 90% hit ratio in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, PQ decreases the hit ratio in local memory by up to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4% over Remote, because PQ throttles page movements in some workloads to prioritize cache line requests, thus increasing the number of accesses to remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, DaeMon recovers most of the lost local memory hits, achieving on average only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4% worse hit ratio over Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Leveraging link compression in DaeMon reduces Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote LC PQ DaeMon LocalDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:19 pr nw bf bc sp hp dr rs GM 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Hit Ratio (%) stch-lat=100 bw-fact=1/2 46 45 100 100 0 0 0 0 72 pr nw bf bc sp hp dr rs GM 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Hit Ratio (%) stch-lat=400 bw-fact=1/4 25 70 91 82 0 100 100 100 70 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hit ratio in local memory achieved by various schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' network bandwidth consumption and significantly increases the number of pages that can be migrated over PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Across all configurations (not graphed) DaeMon migrates 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9% of the pages throttled by PQ via leveraging link compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We conclude that DaeMon enables both leveraging the benefits of data locality within pages and the prioritization of critical path cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Bandwidth Partitioning Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 11 presents a sensitivity study on the bandwidth parti- tioning ratio between the cache line and page movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' pr nw bf bc sp hp dr rs GM 1 2 3 4 5 6 7 Speedup stch-lat=100 bw-fact=1/2 10 11 12 11 14 16 16 10% PQ 25% PQ 50% PQ 80% PQ 10% DaeMon 25% DaeMon 50% DaeMon 80% DaeMon pr nw bf bc sp hp dr rs GM 1 2 3 4 5 Speedup stch-lat=400 bw-fact=1/2 10 10 11 9 9 11 13 10% PQ 25% PQ 50% PQ 80% PQ 10% DaeMon 25% DaeMon 50% DaeMon 80% DaeMon Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of PQ and DaeMon normalized to Remote varying the bandwidth partitioning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We draw three findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, a higher bandwidth partitioning ratio (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 50%) than DaeMon’s default 25% ratio, incurs slowdowns in workloads of medium and high spatial locality, and only Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote LC PQ DaeMon21:20 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' improves performance in workloads with very low locality within pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', pr, nw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' This is because high bandwidth partitioning ratios favor cache line movements and throttle a higher number of page movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, since cache line data movements are affected more by the switch latency compared to page movements, the performance benefits of higher bandwidth partitioning ratios reduce at higher switch latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For example, in pr, the 50% bandwidth partitioning ratio outperforms 25% ratio by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='19× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='08× using DaeMon at 100ns and 400ns switch latency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Finally, across all different bandwidth factors (not graphed), DaeMon’s default 25% ratio outperforms the 50% ratio by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='02× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='04× for 100ns and 400ns switch latency, respectively, and the 80% ratio by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='07× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='33× for 100ns and 400ns switch latency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We conclude that DaeMon’s default 25% ratio on average performs best across all various network and application characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Compression Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 12 compares the performance of LC normalized to Remote with three compression schemes: (i) fpcbdi: a latency-optimized hybrid scheme of BDI [73] and FPC [8] with 4-cycle (de)compression latency per cache line [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) fve: the latency-optimized FVE [91] scheme using a 256B dictionary table and having 6-cycle (de)compression latency per cache line [91];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (iii) LZ: DaeMon’s compression ratio-optimized LZ-based scheme [49, 93] (See details on § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' pr nw bf bc sp hp dr rs GM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 Speedup stch-lat=100 bw-fact=1/2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 fpcbdi fve LZ pr nw bf bc sp hp dr rs GM 0 1 2 3 4 Speedup stch-lat=100 bw-fact=1/8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 fpcbdi fve LZ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of LC varying the compression scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We observe that LZ always outperforms Remote, despite the high (de)compression latencies, because the network overheads are significantly higher, indicating that link compression is a highly effective solution for disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' dr and rs show little performance improvement with LZ, because the application data is less compressible (their compression ratio is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='42× versus 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='47× on average across all evaluated workloads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Moreover, LZ outperforms fpcbdi and fve across all network configurations (not graphed) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='54× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='44× on average, respectively, since it achieves higher compression ratios (on average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='92× and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='73× higher compression ratio than fpcbdi and fve respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The benefits of LZ over fpcbdi and fve are even higher in the more bandwidth limited configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', with 1/8 bandwidth factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, we conclude that the high network overheads in disaggregated systems favor compression algorithms that provide higher compression ratios, since the benefits of the reduced bandwidth consumption outweigh the higher (de)compression latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:21 Network Disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figures 13 and 14 compare the IPC and the hit ratio in local memory respectively, of LC, PQ and DaeMon, when the network traffic varies during runtime: we simulate contention from other compute components that share the same network, by artificially injecting packets inside the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluate pr and nw, as they incur the highest data movement costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Executed Instructions 1e8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 IPC nw stch-lat=100 bw-fact=1/2 LC PQ DaeMon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Executed Instructions 1e9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 IPC pr stch-lat=100 bw-fact=1/2 LC PQ DaeMon Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of LC, PQ, DaeMon, when creating artificial disturbance in the network during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Executed Instructions 1e8 86 88 90 92 94 96 98 100 Hit Ratio (%) nw net-lat=100 bw-fact=1/2 LC PQ DaeMon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0 Executed Instructions 1e9 86 88 90 92 94 96 98 Hit Ratio (%) pr net-lat=100 bw-fact=1/2 LC PQ DaeMon Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hit ratio in local memory of LC, PQ and DaeMon, when creating artificial disturbance in the network during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:22 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon outperforms both LC and PQ by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='85× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='19×, respectively, even when network traffic varies during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon effectively adapts to varying application behavior and network conditions at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For example, in nw, in the first 50M instructions, DaeMon benefits more from LC as the application has high bandwidth consumption and higher locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In the next 100M instructions, the workload exhibits less data locality within pages, and DaeMon benefits more from PQ, which provides significant performance benefits over LC by effectively prioritizing critical path cache line requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In the last part of execution, DaeMon again leverages the benefits of LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, we conclude that DaeMon provides a versatile approach to dynamic and variable runtime application and network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Multithreaded Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 15 shows DaeMon’s performance benefits for multithreaded workloads on 8 OoO cores, thus evaluating more bandwidth-limited executions compared to that of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Please also see Figure 21 in Appendix § A which evaluates even more bandwidth-limited executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Across all workloads and network configurations (not graphed), DaeMon outperforms the typically-used Remote scheme by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='73× on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When network bandwidth is very limited, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 1/16 bandwidth factor (Figure 21), DaeMon’s benefits are even higher, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='95× over Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' kc tr pr nw bf bc ts sp sl GM 0 2 4 6 8 10 Speedup stch-lat=100 bw-fact=1/2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 LC PQ DaeMon Local kc tr pr nw bf bc ts sp sl GM 0 2 4 6 8 10 Speedup stch-lat=400 bw-fact=1/2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 LC PQ DaeMon Local kc tr pr nw bf bc ts sp sl GM 02468 10 12 14 Speedup stch-lat=400 bw-fact=1/4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 LC PQ DaeMon Local Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Speedup achieved by various schemes in multithreaded workloads normalized to Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' FIFO Replacement Policy in Local Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 16 compares DaeMon and Local normalized to Remote, when using First-In-First-Out (FIFO) replacement policy in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Across all workloads and network configurations (not graphed), DaeMon outperforms the widely-adopted Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:23 Remote scheme by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='63×, when using a FIFO replacement policy in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon is orthogonal to the replacement policy used in local memory, and can be used synergistically with any arbitrary replacement policy in local memory to even further reduce data access costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Overall, DaeMon can significantly mitigate the data movement overheads in fully disaggregated systems independently on the number of data migrations happens during runtime: even when a small number of data migrations happens during runtime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', thanks to sophisticated approaches such as intelligent replacement policies in local memory, hot page placement/selection techniques, page prefetchers), DaeMon can even further alleviate the data movement costs by dynamically selecting the granularity of data movements, prioritizing the critical cache line requests, and opportunistically moving compressed pages at slower rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, we conclude that DaeMon can work synergistically with sophisticated replacement policies in local memory, page prefetchers and intelligent page placement/movement techniques to even further improve system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' pr nw bf bc sp hp dr rs GM 0 1 2 3 4 5 Speedup stch-lat=100 bw-fact=1/2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 Remote DaeMon Local pr nw bf bc sp hp dr rs GM 0 1 2 3 4 5 Speedup stch-lat=100 bw-fact=1/4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 Remote DaeMon Local Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of Local and DaeMon over Remote, when using FIFO replacement policy in local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Multiple Memory Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 17 compares Remote and DaeMon normalized to Local, when varying the number of memory components and having a different network configuration for each memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Please also see Figure 22 in Appendix § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluated distributing memory pages with either a round-robin way or randomly across remote memory components, and draw the same key observation for both distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When adding more memory components using the same network configuration with that of when having one memory component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', having 100ns switch latency and 1/4 bandwidth factor for each memory component), performance of both Remote and DaeMon improves over Local: memory pages are distributed across multiple memory components and the system provides larger aggregate network and memory bandwidth, thus data migrations incur smaller overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Finally, DaeMon significantly outperforms Remote by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25× across all workload-architecture combinations, and constitutes a scalable solution for large-scale disaggregated systems with multiple hardware components and various architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:24 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 MC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 MC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 MC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 0 1 2 3 4 5 6 7 8 Slowdown 39 21 40 42 21 50 42 78 nw ts sp dr Remote DaeMon #memory components stch-lat bw-fact MC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 1 100 1/4 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 2 100-100 1/4-1/4 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 2 400-400 1/4-1/8 MC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 2 100-100 1/8-1/8 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 4 100-100-100-100 1/4-1/4-1/4-1/4 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 4 100-400-100-400 1/4-1/8-1/4-1/8 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 4 400-400-400-400 1/8-1/8-1/8-1/8 MC4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 4 100-100-100-100 1/8-1/16-1/8-1/16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of Remote and DaeMon over Local when using multiple memory components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Multiple Concurrent Workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 18 shows DaeMon’s performance benefits when concur- rently running multiple workloads on a compute component with 4 OoO cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The performance of each core is normalized to that of the same core using Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The local memory hosts ∼15% and ∼9% of each application’s working set, when running 2 and 4 workloads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon outperforms Remote by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='96× across all multiple-workload experiments, thus being highly efficient and performant when multiple heterogeneous jobs concurrently run in the disaggregated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' bf-nw bf-ts pr-nw nw-sp tr-nw bc-ts nw-ts ts-dr ts-sp bf-dr-ts-nw bf-dr-ts-sp pr-dr-sp-nw pr-nw-ts-sp bc-dr-ts-sp tr-dr-ts-sp bc-nw-ts-sp tr-nw-ts-sp dr-ts-sp-nw 0 1 2 3 4 5 6 7 Speedup stch-lat=100 bw-fact=1/2 Core 1 Core 2 Core 3 Core 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance of DaeMon over Remote when running multiple concurrent workloads in a 4-CPU compute component and a memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 Key Takeaways and Recommendations This section summarizes our key takeaways and recommendations extracted from our evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Key Takeaway #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' There is no one-size-fits-all granularity in data movements: the best-performing granularity at each time depends on the network/system load and the application data access patterns, which can significantly vary across applications and within application during runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 8 demonstrates that some applications significantly benefit from the prioritization of critical cache line data movements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', pr, nw), and some applications only benefit from page migrations that leverage data locality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', dr, rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 12 shows that some applications have highly compressible data, and thus greatly benefit from compressed page granularity data movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Finally, Figure 13 proves that the application behavior and network traffic can highly vary during runtime, and thus the best-performing data movement granularity needs to adapt to the application characteristics and network/system conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, we recommend that system and hardware designers of disaggregated systems implement system-level solutions and hardware mechanisms that dynami- cally change and adapt their configurations and selection methods to the availability of the system resources and the runtime behavior of the heterogeneous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Key Takeaway #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Typical datacenter applications exhibit high data locality within memory pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 4KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 10 shows that Remote achieves high data locality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', always has at least 90% hit ratio in local memory, across a wide variety of datacenter workloads with diverse access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Therefore, migrating data at a large granularity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', page granularity, is very effective and critical to achieving high system performance in fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To this end, we suggest that hardware and system designers of disaggregated systems retain coarse-grained data migration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', page granularity data migration), since it both enables high performance and maintains low metadata overheads for address translation in local memory and remote memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Key Takeaway #3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Aggressively prioritizing the cache line granularity data movements that are on the critical path might hurt performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 11 shows that a high bandwidth partitioning ratio, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 50% or 80% bandwidth partitioning ratio, which significantly prioritizes the cache line granularity data movements over the page granularity data movements, incurs significant performance slowdowns in workloads with medium and high spatial locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' As a result, we suggest that hardware and system designers of data movement solutions tailored for disaggregated systems always ensure that page migrations are not aggressively stalled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Key Takeaway #4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Distributed and disaggregated data movements solutions are highly effective and efficient in fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' disaggregated systems are distributed architectures and comprise multiple hardware devices, each of them is independently and transparently managed from other hardware components in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Our evaluations in Figures 17 and 22 show that distributed and disaggregated solutions for data movement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', DaeMon) better leverage the available aggregate network and memory bandwidth in the system, and enable high scalability to large-scale disaggregated systems with multiple hardware components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To this end, we recommend that hardware architects design distributed hardware mechanisms for fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 7 Related Work To our knowledge, this is first work to (i) analyze and alleviate the data movement problem in fully disaggregated systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (ii) enable prioritized and decoupled movement of data at multiple granularities simultaneously to reduce access latencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (iii) propose a dynamic selection granularity mechanism with approximate bandwidth partitioning to effectively leverage both cache line and page movement depending on application and network characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' and (iv) implement a synergistic solution of link compression, bandwidth partitioning, and adaptive granularity selection in data movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We discuss prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:26 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Disaggregated Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Several prior works [5, 6, 10, 14–16, 33–35, 37, 38, 46, 54, 57, 70, 74, 75, 78, 87, 99, 108–110, 112] propose OS modules, system-level solutions, programming frameworks, software management systems, architectures and emulators for disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' These works do not tackle the data movement problem in disaggregated systems, and thus DaeMon is orthogonal to these proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MIND [54] proposes memory sharing among compute components by implementing coherence and address translation in network switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kona [16] is a software runtime to track cache line granularity accesses to remote memory, and eliminate page faults by decoupling the application memory access tracking from the virtual memory page size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, Kona and MIND do not mitigate data movement overheads in disaggregated systems, as data is always moved at page granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, DaeMon is largely orthogonal to these works and could be used to further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Clio [37] proposes a disaggregated system that virtualizes and manages remote memory at the hardware level (independently to compute components), and eliminates expensive page faults in memory components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Clio accesses remote data at a byte granularity via dedicated API, however not being transparent to programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' As explained in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2, moving data always at a small granularity can cause significant performance penalties in many applications, and does not provide robustness against fluctuations in network characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Instead, DaeMon is software-transparent, robust and significantly alleviates data movement costs via decoupled and selective data movement at multiple granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [56] propose a disaggregated architecture and characterize moving data only at cache line or page granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The authors show that the page-based configuration outperforms the cache line configuration at most common patterns (as observed in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2), however it does not address the high performance penalties of page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Maruf and Chowdhury [62] propose a page prefetching scheme for disaggregated systems, which however can only help applications with high locality within pages, and does not capture the significant variability in data access costs of fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon is orthogonal to page prefetchers and can work synergistically with them to even further improve performance, as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We leave the experimentation of their synergy for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hybrid Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Numerous works for hybrid memory systems propose data placement schemes [3, 19, 22, 28, 29, 32, 45, 47, 59, 82, 100], or selection methods [4, 26, 27, 43, 52, 53, 58, 64, 76, 89, 96, 103] to identify hot memory pages that are migrated to die-stacked DRAM, that is organized as a cache of a larger main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Compared to these approaches, first, intelligent page placement/movement is orthogonal to DaeMon, and cannot by itself address the high overheads caused by remote page migrations across the network, that can be significantly slower than that within the server and more latency/bandwidth-constrained in the context of fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, these prior works assume a monolithic centralized system where TLBs/page tables can be leveraged to track page hotness of remote pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', [4, 26, 27, 52, 58, 64, 103]) or that memory allocation/placement is handled by the server itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', [3, 28, 29, 32, 45, 47, 55, 59, 82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, in disaggregated systems, address translation and memory management are distributed across memory components and cannot be used to track pages at the CPU server side, while compute components and memory components are managed by independent kernel monitors that have no visibility/control of other components or data management/placement across components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Similarly, hardware-based approaches [43, 55, 76, 96] for hybrid systems add centralized hardware units at the server side to store page tracking metadata for the second-tier main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For example, Chop [43] adds 4MB of metadata to track 16GB of second-tier memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' These schemes would incur significant area overheads (in the order of GBs) to track large amounts of remote memory (in the order of TBs) enabled by disaggregated systems [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Requiring each compute component to track a large number of pages enabled by multiple remote memory components would cause scalability issues and significantly limit the benefits of resource disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thus, designing an effective Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:27 scalable hot page selection scheme for fully disaggregated systems is an open challenge, and DaeMon could work in conjunction with such schemes to further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, all these prior works do not handle variability in data access costs of disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' disaggregated systems necessitate an adaptive mechanism given the significant variations in access latencies and bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fourth, applying/adapting the design of prior schemes tailored for tightly-integrated hybrid systems in disaggregated systems might incur significantly higher overheads and require important modifications than that described in the original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A few recent works design hardware schemes for commodity servers to enable moving data only at cache line granularity [23, 60, 61, 97] or a larger sub-block granularity (a few cache lines) [42, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Ekman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [30] evaluate a critical-block first approach, where each 8KB page is split in blocks of 2KB data, and the requested (critical) 2KB block of data is transferred first, and written in DRAM cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' As we show in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2, moving data at a single granularity (page or cache line) can incur high performance costs and does not provide robustness towards significant variations in network bandwidth and latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hardware Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Prior works propose compression schemes [1, 8, 12, 21, 24, 31, 48, 50, 68, 69, 71–73, 77, 86, 91, 92, 101, 104, 105, 107] for cache memory, main memory and memory bus links in CPUs/GPUs [65, 85, 90, 98], and selection methods to dynamically enable/disable compression [7, 9, 95], or find the best-performing compression scheme [11, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' These works integrate ratio-optimized or latency-optimized compression schemes depending on the particular context and system’s characteristics they target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Our work enables link compression in page movements synergistically with decoupled multiple granularity data movement, which allows us to tolerate the high compression latencies of ratio-optimized compression schemes such as LZ [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 8 Conclusion DaeMon is the first adaptive data movement solution for fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon supports low-cost page migration, scales elastically to multiple hardware components, enables software transparency, and provides robustness across various architecture/network characteristics and the application behavior by effectively monitoring pending cache line and page movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Our evaluations using a state-of-the-art accurate simulator show that DaeMon significantly improves system performance and data access costs for a wide range of applications under various architecture and network configurations, and when multiple jobs are simultaneously running in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We conclude that DaeMon is an efficient, scalable and robust solution to alleviate data movement overheads in disaggregated systems, and hope that this work encourages further studies of the data movement problem in disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Acknowledgments We thank the anonymous reviewers from SIGMETRICS 2023, and our shepherd, Abhishek Chandra, for their comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We also thank Konstantinos Kanellopoulos and Ivan Fernandez for their help on technical aspects of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The final version of our paper is also available on arXiv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:28 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Abali, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Franke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Poff, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Saccone, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Schulz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Herger, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Memory Expansion Technology (MXT): Software Support and Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' IBM Journal of Research and Development (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [2] Atul Adya, Robert Grandl, Daniel Myers, and Henry Qin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Fast Key-Value Stores: An Idea Whose Time Has Come and Gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HotOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [3] Neha Agarwal, David Nellans, Mark Stephenson, Mike O’Connor, and Stephen W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Keckler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Page Placement Strategies for GPUs within Heterogeneous Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [4] Neha Agarwal and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wenisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thermostat: Application-Transparent Page Management for Two-Tiered Main Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [5] Marcos K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Aguilera, Nadav Amit, Irina Calciu, Xavier Deguillard, Jayneel Gandhi, Stanko Novaković, Arun Ra- manathan, Pratap Subrahmanyam, Lalith Suresh, Kiran Tati, Rajesh Venkatasubramanian, and Michael Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote Regions: A Simple Abstraction for Remote Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [6] Marcos K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Aguilera, Nadav Amit, Irina Calciu, Xavier Deguillard, Jayneel Gandhi, Pratap Subrahmanyam, Lalith Suresh, Kiran Tati, Rajesh Venkatasubramanian, and Michael Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote Memory in the Age of Fast Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SoCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Alameldeen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Adaptive Cache Compression for High-Performance Processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [8] Alaa Alameldeen and David Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Frequent Pattern Compression: A Significance-Based Compression Scheme for L2 Caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [9] Alaa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Alameldeen and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Interactions Between Compression and Prefetching in Chip Multipro- cessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [10] Sebastian Angel, Mihir Nanavati, and Siddhartha Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Disaggregation and the Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HotCloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [11] Angelos Arelakis, Fredrik Dahlgren, and Per Stenstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' HyComp: A Hybrid Cache Compression Method for Selection of Data-Type-Specific Compression Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [12] Angelos Arelakis and Per Stenstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' SC2: A Statistical Compression Cache Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [13] JEDEC Solid State Technology Assn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' JESD79-4B: DDR4 SDRAM Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [14] Laurent Bindschaedler, Ashvin Goel, and Willy Zwaenepoel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hailstorm: Disaggregated Compute and Storage for Distributed LSM-Based Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [15] Dhantu Buragohain, Abhishek Ghogare, Trishal Patel, Mythili Vutukuru, and Purushottam Kulkarni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DiME: A Performance Emulator for Disaggregated Memory Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In APSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [16] Irina Calciu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Talha Imran, Ivan Puddu, Sanidhya Kashyap, Hasan Al Maruf, Onur Mutlu, and Aasheesh Kolli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Rethinking Software Runtimes for Disaggregated Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [17] Trevor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Carlson, Wim Heirman, and Lieven Eeckhout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Sniper: Exploring the Level of Abstraction for Scalable and Accurate Parallel Multi-Core Simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [18] Trevor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Carlson, Wim Heirman, Stijn Eyerman, Ibrahim Hur, and Lieven Eeckhout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' An Evaluation of High-Level Mechanistic Core Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' TACO (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [19] Chia-Hao Chang, Adithya Kumar, and Anand Sivasubramaniam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' To Move or Not to Move?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Page Migration for Irregular Applications in over-Subscribed GPU Memory Systems with DynaMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SYSTOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [20] Shuai Che, Michael Boyer, Jiayuan Meng, David Tarjan, Jeremy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Sheaffer, Sang-Ha Lee, and Kevin Skadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Rodinia: A Benchmark Suite for Heterogeneous Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In IISWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [21] Xi Chen, Lei Yang, Robert P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Dick, Li Shang, and Haris Lekatsas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' C-Pack: A High-Performance Microprocessor Cache Compression Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' VLSI (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [22] Chiachen Chou, Aamer Jaleel, and Moinuddin Qureshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' BATMAN: Techniques for Maximizing System Bandwidth of Memory Systems with Stacked-DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MEMSYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [23] Chia Chen Chou, Aamer Jaleel, and Moinuddin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Qureshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CAMEO: A Two-Level Memory Organization with Capacity of Main Memory and Flexibility of Hardware-Managed Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [24] Esha Choukse, Mattan Erez, and Alaa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Alameldeen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Compresso: Pragmatic Main Memory Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [25] David Cock, Abishek Ramdas, Daniel Schwyn, Michael Giardino, Adam Turowski, Zhenhao He, Nora Hossle, Dario Korolija, Melissa Licciardello, Kristina Martsenko, Reto Achermann, Gustavo Alonso, and Timothy Roscoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Enzian: An Open, General, CPU/FPGA Platform for Systems Software Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [26] Xiangyu Dong, Yuan Xie, Naveen Muralimanohar, and Norman P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Jouppi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Simple but Effective Heterogeneous Main Memory with On-Chip Memory Controller Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [27] Thaleia Dimitra Doudali, Sergey Blagodurov, Abhinav Vishnu, Sudhanva Gurumurthi, and Ada Gavrilovska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kleio: A Hybrid Memory Page Scheduler with Machine Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [28] Thaleia Dimitra Doudali, Daniel Zahka, and Ada Gavrilovska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Cori: Dancing to the Right Beat of Periodic Data Movements over Hybrid Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In IPDPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:29 [29] Subramanya R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Dulloor, Amitabha Roy, Zheguang Zhao, Narayanan Sundaram, Nadathur Satish, Rajesh Sankaran, Jeff Jackson, and Karsten Schwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Data Tiering in Heterogeneous Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In EuroSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [30] Magnus Ekman and Per Stenstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Cost-Effective Main Memory Organization for Future Servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In IPDPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Ekman and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Stenstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Robust Main-Memory Compression Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Feeley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Morgan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Pighin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Karlin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Levy, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Thekkath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Implementing Global Memory Management in a Workstation Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [33] Peter X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gao, Akshay Narayan, Sagar Karandikar, Joao Carreira, Sangjin Han, Rachit Agarwal, Sylvia Ratnasamy, and Scott Shenker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Network Requirements for Resource Disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In OSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [34] Donghyun Gouk, Sangwon Lee, Miryeong Kwon, and Myoungsoo Jung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Direct Access, High-Performance Memory Disaggregation with DirectCXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [35] Juncheng Gu, Youngmoon Lee, Yiwen Zhang, Mosharaf Chowdhury, and Kang G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Efficient Memory Disaggregation with Infiniswap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In NSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [36] Chuanxiong Guo, Haitao Wu, Zhong Deng, Gaurav Soni, Jianxi Ye, Jitu Padhye, and Marina Lipshteyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' RDMA over Commodity Ethernet at Scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SIGCOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [37] Zhiyuan Guo, Yizhou Shan, Xuhao Luo, Yutong Huang, and Yiying Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Clio: A Hardware-Software Co-Designed Disaggregated Memory System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [38] Sangjin Han, Norbert Egi, Aurojit Panda, Sylvia Ratnasamy, Guangyu Shi, and Scott Shenker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Network Support for Resource Disaggregation in Next-Generation Datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HotNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [39] HPCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' High Performance Conjugate Gradient Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='com/hpcg-benchmark/hpcg [40] Ranggi Hwang, Taehun Kim, Youngeun Kwon, and Minsoo Rhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Centaur: A Chiplet-Based, Hybrid Sparse-Dense Accelerator for Personalized Recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [41] Intel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Intel Omni-Path Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='intel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='com/content/www/us/en/high-performance-computing- fabrics/omni-path-driving-exascale-computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='html [42] Djordje Jevdjic, Stavros Volos, and Babak Falsafi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Die-Stacked DRAM Caches for Servers: Hit Ratio, Latency, or Bandwidth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Have It All with Footprint Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [43] Xiaowei Jiang, Niti Madan, Li Zhao, Mike Upton, Ravishankar Iyer, Srihari Makineni, Donald Newell, Yan Solihin, and Rajeev Balasubramonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CHOP: Adaptive Filter-Based DRAM Caching for CMP Server Platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [44] Hongshin Jun, Jinhee Cho, Kangseol Lee, Ho-Young Son, Kwiwook Kim, Hanho Jin, and Keith Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' HBM DRAM Technology and Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In IMW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [45] Sudarsun Kannan, Ada Gavrilovska, Vishal Gupta, and Karsten Schwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' HeteroOS: OS Design for Heterogeneous Memory Management in Datacenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [46] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Katrinis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syrivelis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Pnevmatikatos, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Zervas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Theodoropoulos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Koutsopoulos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hasharoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Raho, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Pinto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Espina, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Lopez-Buedo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nemirovsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Roca, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Klos, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Berends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Rack-Scale Disaggregated Cloud Data Centers: The dReDBox Project Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In DATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [47] Jonghyeon Kim, Wonkyo Choe, and Jeongseob Ahn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Exploring the Design Space of Page Management for Multi-Tiered Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [48] Jungrae Kim, Michael Sullivan, Esha Choukse, and Mattan Erez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Bit-Plane Compression: Transforming Data for Better Compression in Many-Core Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [49] Seikwon Kim, Seonyoung Lee, Taehoon Kim, and Jaehyuk Huh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Transparent Dual Memory Compression Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In PACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kjelso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gooch, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Jones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Design and Performance of a Main Memory Hardware Data Compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In EUROMICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [51] Fredrik Kjolstad, Stephen Chou, David Lugato, Shoaib Kamil, and Saman Amarasinghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Taco: A Tool to Generate Tensor Algebra Kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [52] Jagadish B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kotra, Haibo Zhang, Alaa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Alameldeen, Chris Wilkerson, and Mahmut T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kandemir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CHAMELEON: A Dynamically Reconfigurable Heterogeneous Memory System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [53] Andres Lagar-Cavilla, Junwhan Ahn, Suleiman Souhlal, Neha Agarwal, Radoslaw Burny, Shakeel Butt, Jichuan Chang, Ashwin Chaugule, Nan Deng, Junaid Shahid, Greg Thelen, Kamil Adam Yurtsever, Yu Zhao, and Parthasarathy Ranganathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Software-Defined Far Memory in Warehouse-Scale Computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [54] Seung-seob Lee, Yanpeng Yu, Yupeng Tang, Anurag Khandelwal, Lin Zhong, and Abhishek Bhattacharjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MIND: In-Network Memory Management for Disaggregated Data Centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [55] Yang Li, Saugata Ghose, Jongmoo Choi, Jin Sun, Hui Wang, and Onur Mutlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Utility-Based Hybrid Memory Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In CLUSTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [56] Kevin Lim, Jichuan Chang, Trevor Mudge, Parthasarathy Ranganathan, Steven K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Reinhardt, and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wenisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Disaggregated Memory for Expansion and Sharing in Blade Servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [57] Kevin Lim, Yoshio Turner, Jose Renato Santos, Alvin AuYoung, Jichuan Chang, Parthasarathy Ranganathan, and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wenisch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' System-Level Implications of Disaggregated Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:30 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [58] Haikun Liu, Yujie Chen, Xiaofei Liao, Hai Jin, Bingsheng He, Long Zheng, and Rentong Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hardware/Software Cooperative Caching for Hybrid DRAM/NVM Memory Architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [59] Lei Liu, Shengjie Yang, Lu Peng, and Xinyu Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hierarchical Hybrid Memory Management in OS for Tiered Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' TPDS (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [60] Gabriel Loh and Mark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Supporting Very Large DRAM Caches with Compound-Access Scheduling and MissMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' IEEE Micro (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [61] Gabriel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Loh and Mark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Efficiently Enabling Conventional Block Sizes for Very Large Die-Stacked DRAM Caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [62] Hasan Al Maruf and Mosharaf Chowdhury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Effectively Prefetching Remote Memory with Leap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [63] Mellanox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Mellanox Innova Adapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='com/en-us/networking/products/data-processing- unit/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='mtag=programmable_adapter_cards [64] Mitesh R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Meswani, Sergey Blagodurov, David Roberts, John Slice, Mike Ignatowski, and Gabriel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Loh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Heterogeneous Memory Architectures: A HW/SW Approach for Mixing Die-Stacked and Off-Package Memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [65] Sparsh Mittal and Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Vetter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Survey Of Architectural Approaches for Data Compression in Cache and Main Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' TPDS (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [66] Naveen Muralimanohar, Rajeev Balasubramonian, and Norm Jouppi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Optimizing NUCA Organizations and Wiring Alternatives for Large Caches with CACTI 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [67] Maxim Naumov, Dheevatsa Mudigere, Hao-Jun Michael Shi, Jianyu Huang, Narayanan Sundaraman, Jongsoo Park, Xiaodong Wang, Udit Gupta, Carole-Jean Wu, Alisson G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Azzolini, Dmytro Dzhulgakov, Andrey Mallevich, Ilia Cherniavskii, Yinghai Lu, Raghuraman Krishnamoorthi, Ansha Yu, Volodymyr Kondratenko, Stephanie Pereira, Xianjie Chen, Wenlin Chen, Vijay Rao, Bill Jia, Liang Xiong, and Misha Smelyanskiy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Deep Learning Recommendation Model for Personalization and Recommendation Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [68] Tri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nguyen, Adi Fuchs, and David Wentzlaff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CABLE: A CAche-Based Link Encoder for Bandwidth-Starved Manycores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [69] Tri M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nguyen and David Wentzlaff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MORC: A Manycore-Oriented Compressed Cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [70] Vlad Nitu, Boris Teabe, Alain Tchana, Canturk Isci, and Daniel Hagimont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Welcome to Zombieland: Practical and Energy-Efficient Memory Disaggregation in a Datacenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In EuroSys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [71] Sungbo Park, Ingab Kang, Yaebin Moon, Jung Ho Ahn, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Edward Suh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' BCD Deduplication: Effective Memory Compression Using Partial Cache-Line Deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [72] Gennady Pekhimenko, Vivek Seshadri, Yoongu Kim, Hongyi Xin, Onur Mutlu, Phillip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gibbons, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kozuch, and Todd C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Mowry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Linearly Compressed Pages: A Low-Complexity, Low-Latency Main Memory Compression Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [73] Gennady Pekhimenko, Vivek Seshadri, Onur Mutlu, Phillip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gibbons, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kozuch, and Todd C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Mowry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Base-Delta-Immediate Compression: Practical Data Compression for on-Chip Caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In PACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [74] Ivy Peng, Roger Pearce, and Maya Gokhale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' On the Memory Underutilization: Exploring Disaggregated Memory on HPC Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SBAC-PAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [75] Christian Pinto, Dimitris Syrivelis, Michele Gazzetti, Panos Koutsovasilis, Andrea Reale, Kostas Katrinis, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Peter Hofstee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ThymesisFlow: A Software-Defined, HW/SW co-Designed Interconnect Stack for Rack-Scale Memory Disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [76] Andreas Prodromou, Mitesh Meswani, Nuwan Jayasena, Gabriel Loh, and Dean M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Tullsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MemPod: A Clustered Architecture for Efficient and Scalable Migration in Flat Address Space Multi-level Memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [77] Cheng Qian, Libo Huang, Qi Yu, Zhiying Wang, and Bruce Childers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' CMH: Compression Management for Improving Capacity in the Hybrid Memory Cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [78] Pramod Subba Rao and George Porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Is Memory Disaggregation Feasible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Case Study with Spark SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ANCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [79] RDMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' RDMA Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='rdmaconsortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='org/ [80] RDMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gen-Z Core Specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' https://genzconsortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='org/ [81] Joseph Redmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2013–2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Darknet: Open Source Neural Networks in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' http://pjreddie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='com/darknet/ [82] Zhenyuan Ruan, Malte Schwarzkopf, Marcos K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Aguilera, and Adam Belay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' AIFM: High-Performance, Application-Integrated Far Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In OSDI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [83] Jee Ho Ryoo, Mitesh R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Meswani, Andreas Prodromou, and Lizy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' John.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' SILC-FM: Subblocked InterLeaved Cache-Like Flat Memory Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [84] Amedeo Sapio, Ibrahim Abdelaziz, Abdulla Aldilaijan, Marco Canini, and Panos Kalnis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In-Network Computation is a Dumb Idea Whose Time Has Come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HotNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [85] Vijay Sathish, Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Schulte, and Nam Sung Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Lossless and Lossy Memory I/O Link Compression for Improving Performance of GPGPU Workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In PACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:31 [86] Ali Shafiee, Meysam Taassori, Rajeev Balasubramonian, and Al Davis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' MemZip: Exploring Unconventional Benefits from Memory Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [87] Yizhou Shan, Yutong Huang, Yilun Chen, and Yiying Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LegoOS: A Disseminated, Distributed OS for Hardware Resource Disaggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In OSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [88] Julian Shun and Guy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Blelloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Ligra: A Lightweight Graph Processing Framework for Shared Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In PpopP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [89] Gagandeep Singh, Rakesh Nadig, Jisung Park, Rahul Bera, Nastaran Hajinazar, David Novo, Juan Gómez-Luna, Sander Stuijk, Henk Corporaal, and Onur Mutlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Sibyl: Adaptive and Extensible Data Placement in Hybrid Storage Systems Using Online Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [90] Martin Thuresson, Lawrence Spracklen, and Per Stenstrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Memory-Link Compression Schemes: A Value Locality Perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [91] Martin Thuresson and Per Stenström.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Accommodation of the Bandwidth of Large Cache Blocks Using Cache/Memory Link Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ICPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [92] Yingying Tian, Samira M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Khan, Daniel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Jiménez, and Gabriel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Loh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Last-Level Cache Deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [93] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Tremaine, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wazlowski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Har, Kwok-Ken Mak, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Arramreddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Pinnacle: IBM MXT in a Memory Controller Chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' IEEE Micro (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [94] Shin-Yeh Tsai and Yiying Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LITE Kernel RDMA Support for Datacenter Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [95] Irina Chihaia Tuduce and Thomas Gross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Adaptive Main Memory Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [96] Evangelos Vasilakis, Vassilis Papaefstathiou, Pedro Trancoso, and Ioannis Sourdis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' LLC-Guided Data Migration in Hybrid Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In IPDPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [97] Vasilakis, Evangelos and Papaefstathiou, Vassilis and Trancoso, Pedro and Sourdis, Ioannis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Hybrid2: Combining Caching and Migration in Hybrid Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [98] Nandita Vijaykumar, Gennady Pekhimenko, Adwait Jog, Abhishek Bhowmick, Rachata Ausavarungnirun, Chita Das, Mahmut Kandemir, Todd C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Mowry, and Onur Mutlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Case for Core-Assisted Bottleneck Acceleration in GPUs: Enabling Flexible Data Compression with Assist Warps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ISCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [99] Chenxi Wang, Haoran Ma, Shi Liu, Yuanqi Li, Zhenyuan Ruan, Khanh Nguyen, Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Bond, Ravi Netravali, Miryung Kim, and Guoqing Harry Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Semeru: A Memory-Disaggregated Managed Runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In OSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [100] Johannes Weiner, Niket Agarwal, Dan Schatzberg, Leon Yang, Hao Wang, Blaise Sanouillet, Bikash Sharma, Tejun Heo, Mayank Jain, Chunqiang Tang, and Dimitrios Skarlatos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' TMO: Transparent Memory Offloading in Datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [101] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wilson, Scott F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Kaplan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Smaragdakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' The Case for Compressed Caching in Virtual Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [102] Dong Hyuk Woo, Nak Hee Seong, Dean L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Lewis, and Hsien-Hsin Sean Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' An Optimized 3D-stacked Memory Architecture by Exploiting Excessive, High-Density TSV Bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' HPCA (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [103] Zi Yan, Daniel Lustig, David Nellans, and Abhishek Bhattacharjee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Nimble Page Management for Tiered Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ASPLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [104] Jun Yang, Rajiv Gupta, and Chuanjun Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Frequent Value Encoding for Low Power Data Buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' TODAES (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [105] Jun Yang, Youtao Zhang, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Frequent Value Compression in Data Caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In MICRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [106] Chin-Chia Michael Yeh, Yan Zhu, Liudmila Ulanova, Nurjahan Begum, Yifei Ding, Hoang Anh Dau, Diego Furtado Silva, Abdullah Mueen, and Eamonn Keogh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Matrix Profile I: All Pairs Similarity Joins for Time Series: A Unifying View that Includes Motifs, Discords and Shapelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ICDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [107] Vinson Young, Sanjay Kariyappa, and Moinuddin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Qureshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Enabling Transparent Memory-Compression for Commodity Memory Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In HPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [108] Georgios Zervas, Hui Yuan, Arsalan Saljoghei, Qianqiao Chen, and Vaibhawa Mishra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Optically Disaggregated Data Centers with Minimal Remote Memory Latency: Technologies, Architectures, and Resource Allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' JOCN (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [109] Qizhen Zhang, Yifan Cai, Sebastian G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Angel, Vincent Liu, Ang Chen, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Loo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Rethinking Data Management Systems for Disaggregated Data Centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In CIDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [110] Yang Zhou, Hassan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Wassel, Sihang Liu, Jiaqi Gao, James Mickens, Minlan Yu, Chris Kennelly, Paul Turner, David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Culler, Henry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Levy, and Amin Vahdat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Carbink: Fault-Tolerant Far Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In OSDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [111] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Ziv and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Lempel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A Universal Algorithm for Sequential Data Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' IEEE Transactions on Information Theory (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' [112] Pengfei Zuo, Jiazhao Sun, Liu Yang, Shuangwu Zhang, and Yu Hua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' One-sided RDMA-Conscious Extendible Hashing for Disaggregated Memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In ATC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:32 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' APPENDIX A Extended Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 Network Bandwidth Utilization Figure 19 compares the bandwidth utilization across the network of a compute component and a memory component achieved by various data movement schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' pr nw bf bc sp hp dr rs GM 0 20 40 60 80 Network Bandwidth Utilization (%) stch-lat=100 bw-fact=1/2 pr nw bf bc sp hp dr rs GM 0 20 40 60 80 Network Bandwidth Utilization (%) stch-lat=400 bw-fact=1/4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Bandwidth utilization (%) across the network of a compute component and a memory component achieved by various data movement schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We make three key observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' First, LC typically reduces the network bandwidth utilization over Remote (by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='49× on average across all workloads and network configurations), because fewer bytes are transferred through the network, since remote pages are migrated in a compressed format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Note that LC improves the total execution time over Remote, and thus in a few workloads, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', pr, the network bandwidth utilization might be higher within a smaller execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Second, PQ decreases the network bandwidth utilization over Remote in workloads with poor spatial locality within pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', nw), since the selection granularity unit effectively schedules more cache line movements and fewer page migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Instead, PQ might slightly increase the network bandwidth utilization over Remote in workloads with medium spatial locality within pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', bf, bc), since the selection granularity unit enables both cache line and page migrations to leverage both the ability to prioritize critical cache line requests and the benefits of data locality within pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' In workloads with high spatial locality within pages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', dr, rs), PQ favors more page migrations and fewer cache line movements, thus achieving similar network bandwidth utilization to Remote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Third, DaeMon greatly decreases the network bandwidth utilization over Remote by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='32× on average across all workloads and network configurations (not graphed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' DaeMon effectively transfers remote pages in a compressed format and on-the-fly selects the granularity of data migrations to significantly reduce the bandwidth consumption across the network of fully disaggregated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Remote LC PQ DaeMonDaeMon: Architectural Support for Efficient Data Movement in Disaggregated Systems 21:33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 Sensitivity Study to Switch Latency Figure 20 compares DaeMon’s performance over Remote’s performance averaged across all work- loads, when varying the switch latency of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' When the fixed switch latency becomes very high dominating the total data movement costs, DaeMon has lower benefits over Remote, since DaeMon does not hide the propagation and switching delays in network components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', fixed processing costs of the packet inside network switches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, even with a very high switch latency in the order of microsecond, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 1𝜇s (=1000ns), DaeMon outperforms Remote by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='49× on average across all workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 100ns 200ns 300ns 400ns 500ns 600ns 800ns 1000ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='00 Speedup GeoMean Remote DaeMon Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance benefits of DaeMon over Remote, when varying the switch latency of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='3 Sensitivity Study to Network Bandwidth To evaluate bandwidth-limited scenarios, Figure 21 compares DaeMon’s performance normalized to Remote’s performance in mutithreaded workloads running on 8 OoO cores of a compute component, when varying the bandwidth factor of the network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', up to having a very low bandwidth factor of 1/16 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', network bandwidth is 16× slower than the DRAM bus bandwidth) between a compute component and memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We find that on average DaeMon’s benefits increase over the widely-adopted approach of moving data at page granularity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Remote, since DaeMon even more significantly alleviates bandwidth bottlenecks and data movement overheads under bandwidth-constrained scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' kc tr pr nw bf bc ts sp sl GM 0 1 2 3 4 5 Speedup 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='5 stch-lat=100 1/2 1/4 1/8 1/16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance benefits of DaeMon normalized to Remote using multithreaded workloads, when varying the bandwidth factor of the network between a compute component and memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='4 Performance Benefits With Multiple Memory Components Figure 22 evaluates the performance of DaeMon normalized to Remote’s performance, when increasing the number of memory components in the system having the same network configuration for each memory component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', 100ns switch latency and a bandwidth factor of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' We evaluated distributing memory pages with either a round-robin way or randomly across multiple remote memory components, and draw the same key observations for both distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Similarly to Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 21:34 Christina Giannoula, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Figure 17, we observe that when pages are distributed across multiple memory components and the system provides larger aggregate network and memory bandwidth, data access costs decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' For example, when increasing the number of memory components from 2 to 4, the remote data access latency decreases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='39× on average across all workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' However, even when data access costs affect less the total execution time of applications, DaeMon still further mitigates data access overheads: DaeMon outperforms the widely-adopted Remote approach by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='09× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='88× on average across all workloads, when using 2 and 4 memory components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' kc tr pr nw bf bc ts sp sl hp pf dr rs GM 0 1 2 3 4 5 Speedup 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content='9 stch-lat=100 bw-fact=1/4 1 MC 2 MCs 4 MCs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Performance benefits of DaeMon normalized to Remote, when increasing the number of memory components having 100ns switch latency and a bandwidth factor of 1/4 for each memory component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Received October 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' revised December 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' accepted January 2023 Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' ACM Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 6, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' 1, Article 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} +page_content=' Publication date: March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfjfgx/content/2301.00414v1.pdf'} diff --git a/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/2301.04318v1.pdf.txt b/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/2301.04318v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f5f79d6b8d18b51e5cd7d62b612ecfe716b9e81 --- /dev/null +++ b/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/2301.04318v1.pdf.txt @@ -0,0 +1,1832 @@ +Beyond Graph Convolutional Network: An Interpretable Regularizer-centered +Optimization Framework +Shiping Wang1,2, Zhihao Wu1,2, Yuhong Chen1,2, Yong Chen3* +1 College of Computer and Data Science, Fuzhou University +2 Fujian Provincial Key Laboratory of Network Computing and Intelligent Information Processing, Fuzhou University +3 School of Computer Science, Beijing University of Posts and Telecommunications +shipingwangphd@163.com, zhihaowu1999@gmail.com, yhchen2320@163.com, alphawolf.chen@gmail.com. +Abstract +Graph convolutional networks (GCNs) have been attracting +widespread attentions due to their encouraging performance +and powerful generalizations. However, few work provide a +general view to interpret various GCNs and guide GCNs’ de- +signs. In this paper, by revisiting the original GCN, we in- +duce an interpretable regularizer-centerd optimization frame- +work, in which by building appropriate regularizers we can +interpret most GCNs, such as APPNP, JKNet, DAGNN, and +GNN-LF/HF. Further, under the proposed framework, we de- +vise a dual-regularizer graph convolutional network (dubbed +tsGCN) to capture topological and semantic structures from +graph data. Since the derived learning rule for tsGCN con- +tains an inverse of a large matrix and thus is time-consuming, +we leverage the Woodbury matrix identity and low-rank ap- +proximation tricks to successfully decrease the high computa- +tional complexity of computing infinite-order graph convolu- +tions. Extensive experiments on eight public datasets demon- +strate that tsGCN achieves superior performance against quite +a few state-of-the-art competitors w.r.t. classification tasks. +Introduction +Owing to the powerful ability to aggregate neighborhood in- +formation, Graph Convolutional Network (GCN) has been +successfully applied to diverse domains, such as computer +vision [1, 2, 3], recommender systems [4, 5], privacy pre- +serving [6], and traffic forecasting [7, 8]. Rooted in a series +of theoretical foundations, GCN extends convolution opera- +tions to the non-Euclidean spaces and effectively propagates +label signals, and therefore its variants have been extensively +employed for a variety of graph-related tasks, including clas- +sification [9, 10], clustering [11, 12] and link prediction +[13, 14]. In a nutshell, GCN generates the graph embedding +with the well-established graph convolutional layers gath- +ering semantics from neighbors according to the network +topology, which are revealed to be the most critical com- +ponent. +Although GCN has behaved well in many machine learn- +ing tasks, lots of studies have pointed out its certain draw- +backs and made efforts for further improvements. Bo et al. +[15] indicated that the propagation mechanism could be con- +sidered as a special form of low-pass filter, and presented a +*Corresponding author +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +GCN with an adaptive frequency. Zhang et al. [16] argued +that most GCN-based methods ignored the global informa- +tion and proposed SHNE, which leveraged the structure and +feature similarity to capture latent semantics. Wang et al. +[17] revealed that the original GCN aggregated informa- +tion from node neighbors inadequately, and then developed +a multi-channel GCN by utilizing feature-based semantic +graph. In spite of the performance boosts of these GCN +variants, they didn’t establish a generalized framework, i.e., +these approaches understood and enhanced GCN from cer- +tain and non-generalizable perspectives, thereby they are ex- +ceedingly difficult to be further developed, and with limited +interpretability. +Consequently, it is expected to construct a unified frame- +work for various GCNs with better interpretability; however, +it is a pity that this kind of work is still in shortage. Zhao et +al. [18] linked GCN and Graph-regularized PCA (GPCA), +and then proposed a multi-layer network by stacking the +GPCA layers. Zhu et al. [19] attempted to interpret exist- +ing GCN-based methods with a unified optimization frame- +work, under which they devised an adjustable graph filter +for a new GCN variant. Yang et al. [20] designed a family of +graph convolutional layers inspired by the updating rules of +two typical iterative algorithms. Although these efforts have +contributed to better understanding of GCNs, they only ex- +plained GCNs in partial aspects, promoting the expectation +of a more comprehensive analysis of GCNs. +To tackle the aforementioned issues, this paper induces an +interpretable regularizer-centered optimization framework, +which provides a novel perspective to digest various GCNs, +i.e., this framework captures the common essential proper- +ties of existing state-of-the-art GCN variants and could de- +fines them just by devising different regularizers. Moreover, +in light of the analyses on current representative GCNs, we +find that most of the existing approaches only consider cap- +turing the topological regularization, while the feature-based +semantic structure is underutilized, and hence this motivates +us to design a dual-regularizer graph convolutional network +(called tsGCN) within the regularizer-centered optimization +framework for the fullest explorations of both structures +and semantics from graph data. Due to the high computa- +tional complexity of performing infinite-order graph con- +volutions, the unified framework provides a straightforward +way employing truncated polynomials to approximate the +arXiv:2301.04318v1 [cs.LG] 11 Jan 2023 + +graph Laplacian, similar to the truncated Chebyshev poly- +nomials by vanilla GCN, restricting the message passing of +a single graph convolution to the first-order neighborhood. +The main contributions of this paper can be summarized +as the following three aspects: +• Propose a regularizer-centered constrained optimization +framework, which interprets various existing GCNs with +specific regularizers. +• Establish a new dual-regularizer graph convolutional net- +work (tsGCN), which exploits topological and semantic +structures of the given data; and develop an efficient algo- +rithm to reduce the computational complexity of solving +infinite-order graph convolutions. +• Conduct a series of experiments to show that tsGCN per- +forms much better than many SOTA GCNs, and also con- +sumes much less time than the newly GNN-HF/LF. +Related Work +Graph Convolutional Networks +The original GCN was first introduced by Kipf et al. [21], +who generalized the convolution operations from the Eu- +clidean domain to the non-Euclidean domain. SGC [22] as- +sumed that the nonlinear transform of GCN was not that +significant, and then devised a simplified GCN by remov- +ing the nonlinear activation functions and collapsing the +weight matrices. PPNP [23] employed the relationship be- +tween PageRank and GCN for the improvement on the prop- +agation mechanism of GCN, and an iterative version called +APPNP was further proposed to reduce the high compu- +tational complexity. Attempting to adaptively learn the in- +fluence radii for each node and task, JKNet [24] combined +various aggregations at the last layer and was able to learn +representations of different orders for graph substructures. +GNN-LF and GNN-HF [19] considered the low-pass and the +high-pass filter as the convolution kernels to improve GCN’s +expressive power, respectively. AdaGCN [25] leveraged Ad- +aboost strategy for the enhancement of GCN, allowing in- +formation to be shared between layers. To sum up, a main +characteristic of these methods is exploring GCN from the +perspectives of redesigning information aggregation strate- +gies or modifying graph convolutions, and few work try to +construct a unified framework to interpret various GCNs and +reveal the underlying common principles. +Further Insights on GCNs +Quite a few studies have been devoted to explore the mech- +anisms of GCN for deeper insights. Li et al. [26] indicated +that the convolutional operation of GCN was a special form +of Laplacian smoothing, attributed to which GCN suffered +from the so-called over-smoothing problem. Specifically, the +performance of GCN will decrease as the number of layers +increases, which has been validated by many other studies. +However, Liu et al. [27] held a different opinion that the en- +tanglement of two steps in GCN damages the performance +of the deep GCN, where the two steps were explained as +propagation and transformation. Based on this view, they de- +coupled the two operations and further presented a deeper +GCN. Zhu et al. [19] also decomposed the convolution op- +eration of GCN into two separate stages, called aggregation +and transformation, and focused on the aggregation process, +formulating an optimization objective to interpret it. Yang et +al. [28] explored network topology refinement, leveraging a +topology optimization process for the explanation. Oono et +al. [29] analyzed the forward propagation of GCN and in- +terpreted it with a specific dynamical system, allowing GCN +to be related to the underlying topological structures. Over- +all, these studies have contributed to the interpretability of +GCNs, and also let researchers better understand GCNs. In +this paper, we build a unified optimization framework from a +novel view of graph regularizers to interpret and understand +GCNs. +Mathematical Notations +For the convenience of formal descriptions, derivations, and +analyses, necessary notations are narrated as below. A graph +is denoted as G = (V, E, A), where V marks the vertex set +with |V| = N (N is the total number of nodes in graph G), +E marks the edge set, and A = [Aij]N×N marks an affinity +matrix of which Aij measures the similarity between the i- +th and the j-th node. In addition, D = [Dij]N×N represents +the degree matrix of G with Dii = �N +j=1 Aij, and then the +normalized symmetrical graph Laplacian of G is computed +as �L = I − �A with �A = D− 1 +2 AD− 1 +2 . +Revisiting Graph Convolutional Network +For a graph G = (V, E, A), the svd of its graph Laplacian is +L = UΛU⊤, where U ∈ RN×N is comprised of orthonor- +mal eigenvectors and Λ = diag(λ1, · · · , λN) is a diagonal +matrix with λi denoting the i-th eigenvalue and λi ≥ λj +(i = 1, · · · , N). Essentially, this decomposition induces a +Fourier transform on the graph domain, where eigenvectors +correspond to Fourier components and eigenvalues represent +frequencies of the graph. For an input signal x ∈ RN defined +on the graph G, the corresponding graph Fourier transform +of x is �x = U⊤x, and its inverse transform is derived as +x = U�x. Consequently, the graph convolution between the +signal x and the filter g ∈ RN is +g ∗ x = U(�g ⊙ �x) = U((U⊤g) ⊙ (U⊤x)), +(1) +where ⊙ is the Hadamard product between two vectors. Par- +ticularly, denoting gΘ = diag(Θ) := U⊤g parameterized +by Θ ∈ RN, the graph convolution between x and g can be +rewritten as +g ∗ x = U(�g ⊙ �x) = UgΘU⊤x, +(2) +where Θ is regarded as the filter coefficients to be optimized. +Especially, Θ is assumed to be the polynomials of the spec- +trums of the graph Laplacian [30], i.e., +Θ = Θ(Λ) = +K +� +i=1 +ΘiΛi, +(3) +where K is the order of Chebyshev polynomials. By fixing +K = 2, the graph convolutional network (GCN) [21] takes +an effective form +g ∗ x = θ(I + L)x, +(4) + +Methods +Propagation Rules +Regularizer L(H(l); G) +Projective Set +GCN +H(l) = σ +� +�AH(l−1)Θ(l)� +Tr +� +H(l)⊤�LH(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +SGC +H(l) = σ +� +�AH(l−1)Θ(l)� +Tr +� +H(l)⊤�LH(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +APPNP +H(l) = σ +� +(1 − α) �AH(l−1) + αH(0)� +Tr +� +1 +1−αH(l)⊤ �A−1(H(l) − 2αH(0)) +� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +JKNet +H(l) = σ +��K +k=1 αk �AkH(l−1)Θ(l)� +Tr +� +H(l)⊤ �A−1(I + β�L)H(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +DAGNN +H(l) = σ +��K +k=0 αk �AkH(l−1)� +Tr +� +H(l)⊤(I + β�L)H(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +GNN-HF +H(l) = σ +� +(I + α�L)−1(I + β�L)H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + β�L)−1(I + α�L)H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex. +GNN-LF +H(l) = σ +� +(I + α �A)−1(I + β �A)H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + β �A)−1(I + α �A)H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +tsGCN +H(l) = σ +� +(I + α�LG + β�LX )−1H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + α�LG + β�LX )H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +Table 1: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework. +where Θ = [θ] is a parameter to be optimized. When ex- +tending single channel signal x and filter θ to multi-channel +H(l) ∈ RN×dl and Θ(l) ∈ Rdl×fl, the GCN is converted to +H(l) = σ( �AH(l−1)Θ(l)), +(5) +where �A is a normalized version of I + �A, σ(·) is an acti- +vation function, and H(l) ∈ RN×dl is the output of the l-th +layer with H(0) = X being the input feature matrix. +An Interpretable Regularizer-centered +Optimization Framework for GCNs +Given the input H(l−1) of the (l)-th layer, GCN can compute +the output H(l) by minimizing +L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1 +2Tr(H(l)⊤�LH(l)) +(6) +s.t. H(l) ≥ 0, +where +1 +2Tr(H(l)⊤�LH(l)) = +1 +4 +�N +j=1 +�N +i=1 Aij|| h(l) +i +√Dii − +h(l) +j +√ +Djj ||2 +2 with H(l) = [h(l) +1 ; · · · ; h(l) +N ]; it is a normalized reg- +ularizer to preserve the pairwise similarity of any two nodes +in the given graph. Besides, the −Tr(H(l)⊤H(l−1)Θ(l)) is +actually a fitting loss term bewteen H(l) and H(l−1)Θ(l), +i.e., ||H(l)−H(l−1)Θ(l)||2 +F with H(l−1) and Θ(l) fixed when +optimizing H(l). Note that the square term ||H(l)||2 +F is a L2- +regularized smoother, which can be ignored or absorbed in +the second graph regularizer Tr(H(l)⊤�LH(l)). +Taking derivative of L with respect to H(l) and setting it +to zero, we obtain H(l+) as +H(l+) = (I − �A)−1H(l−1)Θ(l); +(7) +and then there yields +H(l) = σ +� +H(l+)� +, +(8) +when the nonnegative constraints H(l) ≥ 0 are further con- +sidered. Notice that σ(·) is the ReLU(·) activation function. +Here, if the matix inverse (I− ˜A)−1 = �∞ +i=0 ˜Ai is approxi- +mated by the first-order expansion, i.e., (I− ˜A)−1 ≈ I+ ˜A, +then Eq. (8) will lead to the updating rule (5) of GCN. +Usually, the activation functions in GCN are ReLU(·) and +Softmax(·), which could be converted to different projec- +tion optimizations. Concretely, the ReLU(·) activation func- +tion is equivalent to project a point x onto the non-negative +plane S+ = {s ∈ Rd|s ≥ 0}, i.e., +ReLU(x) = arg min +y∈S+ +−x⊤y + 1 +2||y||2 +2. +(9) +By the way, we denote S = {s ∈ Rd}, which corresponds +to an identity activation function. In terms of the Softmax(·) +activation function, it can be regarded as projecting x onto +the set Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}, i.e., +Softmax(x) = arg min +y∈Ssimplex +−x⊤y + y⊤ log(y), +(10) +where y⊤ log(y) = �d +i=1 yi log(yi) is the negative entropy +of y [31]. In fact, with respect to other activation functions, +they can also be equivalent to project a point onto some fea- +sible set with some metric. +Up to present, we have actually utilized a constrained op- +timization problem to interpret GCN, including information +propagations (i.e., Eq. (7)) and the nonlinear activation func- +tions (i.e., ReLU(·) and Softmax(·)). +The above analyses can not only explain the vanilla GCN, +but also stimulate a regularizer-centered optimization frame- +work that can further unify various GCNs. By extending the +optimization (6), a more general framework is written as +L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1 +2L(H(l); G) +(11) +s.t. H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex. +Under this framework, different regularizers could derive +different GCNs, for example, + +Algorithm 1: Topological and Semantic Regularized GCN +Require: Graph data G = (V, E, A), labels y, number of +layers L, and hyperparameters {α, β, r}. +Ensure: Predicted label set {y∗ +i }N +i=n+1. +1: Initialize model parameters {H(l), Θ(l)}L +l=1; +2: Compute the joint graph Laplacian α�LG + β�LX and its +low-rank factorization WV⊤; +3: Substitute the matrix inverse (I + α�LG + β�LX )−1 with +I − W(I + V⊤W)−1V⊤; +4: while not convergent do +5: +Calculate hidden layers {H(l)}L +l=1 by Eq. (14); +6: +Update weights: Θ(l+1) ← Θ(l) − η +∂L +∂Θ(l) ; +7: end while +8: return The predicted labels: y∗ +i = arg maxj H(L) +ij . +• If L(H(l); G) = Tr +� +H(l)⊤(I + µ�L)−1(I + λ�L)H(l)� +with λ = β + +1 +α − 1, µ = β, and �L = I − �A, +then +it +induces +the +updating +rule +H(l) += +σ +� +(I + α �A)−1(I + β �A)H(l−1)Θ(l)� +, +which +cor- +responds to GNN-HF [19]. +• If L(H(l); G) = Tr +� +H(l)⊤(I + µ �A)−1(I + λ �A)H(l)� +with λ += +−αβ+2α−1 +αβ−α+1 +and µ += +1 +β − 1, then +it +gives +rise +to +the +updating +rule +H(l) += +σ +� +(I + α �A)−1(I + β �A)H(l−1)Θ(l)� +, +which +cor- +responds to GNN-LF [19]. +For more cases, their results are summarized in Table 4, and +the derivation details can refer to those of the original GCN +(from Eq. (7) to Eq. (10)) and the supplementary. +Remarks. The work [19] is most similar to our work with +the same research idea: they both want to propose a unified +framework to interpret the current GCNs and guide the de- +sign of new GCN variants; however, they are realized in dif- +ferent ways. To be specific, (1) Zhu et al. [19] develop an +optimization framework to explain different GCNs’ prop- +agation processes; whereas we propose a constrained op- +timization framework not only to interpret various GCNs’ +propagation processes, but also explain the nonlinear activa- +tion layers; (2) [19] unifies various GCNs via devising vari- +ous fitting items which are essentially constructed by limited +graph filters; while our work derives different GCNs through +designing different regularizers. To sum up, our work inter- +prets the whole (not partial) GCNs with regularizer-centered +constrained optimizations. +tsGCN: Topological and Semantic Regularized +Graph Convolutional Network +One finding from most existing GCNs is that they often ig- +nored feature-based semantic structures, which can weaken +the representation learning abilities of graph networks, then +Table 2: Dataset statistics. +Datasets +#Nodes +#Edges +#Classes +#Features +#Train/Val/Test +Cora +2,708 +5,429 +7 +1,433 +140/500/1,000 +Citeseer +3,327 +4,732 +6 +3,703 +120/500/1,000 +Pubmed +19,717 +44,338 +3 +500 +60/500/1,000 +ACM +3,025 +13,128 +3 +1,870 +60/500/1,000 +BlogCatalog +5,196 +171,743 +6 +8,189 +120/500/1,000 +CoraFull +19,793 +65,311 +70 +8,710 +1,400/500/1,000 +Flickr +7,575 +239,738 +9 +12,047 +180/500/1,000 +UAI +3,067 +28,311 +19 +4,973 +367/500/,1000 +we focus on two regularizers, i.e., +L1(H(l); G) = 1 +2Tr +� +{H(l)}⊤(1 +2I + α�LG)H(l) +� +, +(12) +L2(H(l); X) = 1 +2Tr +� +{H(l)}⊤(1 +2I + β�LX )H(l) +� +, +(13) +where �LG is a graph Laplacian from the given adjacency ma- +trix (e.g., �LG = �L), and �LX is a graph Laplacian calculated +from the pairwise similarity of any two graph nodes. Hence, +we devise a dual-regularizer, i.e., L(H(l)) = L1(H(l); G) + +L2(H(l); X), and if it is under the optimization framework +(19), then there yields the following updating rule +H(l) = σ +� +(I + α�LG + β�LX )−1H(l−1)Θ(l)� +. +(14) +Since this method seeks to preserve both the topological and +semantic structures for more accurate presentations, we call +it tsGCN (i.e., Topological and Semantic regularized GCN). +Notably, the computational complexity of (I + α�LG + +β�LX )−1 is O(N 3), which tends to be unaffordable in prac- +tical applications. To this end, a low-rank approximation is +operated, i.e., α�LG +β�LX ≈ WV⊤, where W, V ∈ RN×r +with r ≪ N. This leads to the Woodbury matrix identity: +(I + WV⊤)−1 = I − W(I + V⊤W)−1V⊤, +(15) +of which the computational complexity costs O(N 2). +Given that the optimal M∗ of the following problem +min +M∈RN×N: rank(M)=r ||M − (α�LG + β�LX )||2 +F +(16) +is attained at the r-truncated singular value decomposition +of α�LG + β�LX , i.e., M∗ = UΣU⊤, where Σ ∈ Rr×r is a +diagonal matrix containing the r largest singular values. An +optimal {W∗, V∗} to α�LG + β�LX ≈ WV⊤ can be given +by an analytic form of W∗ = V∗ = UΣ +1 +2 . +To obtain the optimum {W∗, V∗}, the iterative algorithm +[32] with O(N 2) is leveraged as +Z(t+1) ← (α�LG + β�LX )U(t), +(17) +{U(t+1), R(t+1)} ← QR(Z(t+1)), +(18) +where QR(·) denotes the QR-decomposition. Note that this +algorithm can converge to the r largest eigenvalues R(t+1) +and its corresponding eigenvectors Z(t+1) when the iterative +number t is large enough. Finally, there will be W∗ = V∗ = +U(t+1)[R(t+1)] +1 +2 . +Gathering all analyses mentioned above, the procedure for +tsGCN is summarized in Algorithm 1. + +Table 3: Accuracy and F1-score (mean% and standard deviation%) of all methods, where the best results are in red and the +second-best are in blue. Note that GraphSAGE fails to work on the ACM dataset, and thus its results are marked with “—”. +Metrics +Methods / Datasets +Cora +Citeseer +Pubmed +ACM +BlogCatalog +CoraFull +Flickr +UAI +Chebyshev +76.2 (0.7) +69.3 (0.4) +74.0 (0.8) +82.8 (1.4) +68.3 (1.6) +57.2 (1.1) +38.5 (1.6) +49.7 (0.4) +GraphSAGE +76.7 (0.6) +64.4 (0.9) +75.5 (0.2) +— +57.8 (0.7) +59.9 (0.7) +32.7 (1.0) +41.7 (1.4) +GAT +79.1 (0.8) +68.3 (0.5) +78.4 (0.3) +84.6 (0.5) +67.1 (1.7) +62.4 (0.4) +40.4 (0.9) +49.7 (3.0) +GCN +80.6 (1.4) +69.1 (1.5) +77.6 (1.3) +88.8 (0.5) +84.2 (0.6) +62.8 (0.4) +51.0 (1.2) +58.5 (1.1) +SGC +79.3 (1.0) +66.4 (1.7) +76.8 (2.0) +80.8 (2.7) +81.3 (0.2) +62.9 (2.2) +51.0 (0.1) +56.5 (3.5) +APPNP +78.0 (0.1) +65.8 (0.2) +78.0 (0.0) +88.2 (0.0) +87.7 (0.3) +63.1 (0.5) +57.5 (0.2) +62.3 (1.2) +JKNet +83.1 (0.1) +72.3 (0.1) +80.1 (0.2) +82.3 (0.6) +75.7 (0.1) +62.6 (0.0) +54.0 (0.3) +45.6 (0.5) +DAGNN +81.9 (0.7) +70.0 (1.1) +80.6 (0.7) +87.4 (0.9) +84.6 (1.9) +65.6 (0.3) +54.6 (5.9) +46.7 (12.4) +GNN-LF +81.1 (0.5) +72.3 (0.9) +80.0 (0.4) +90.8 (0.5) +86.7 (0.6) +63.5 (0.9) +56.6 (0.6) +36.6 (19.8) +GNN-HF +80.7 (0.2) +68.8 (1.3) +77.7 (0.2) +91.2 (0.5) +84.5 (0.4) +63.0 (0.7) +60.7 (0.4) +54.8 (1.4) +tsGCN (inv) +80.3 (0.3) +73.3 (0.4) +78.4 (0.3) +85.1 (1.6) +87.8 (6.3) +67.0 (0.9) +53.3 (12.6) +64.2 (1.8) +ACC +tsGCN +82.0 (0.3) +73.1 (0.4) +82.4 (0.1) +92.8 (0.3) +92.3 (0.5) +67.9 (0.9) +79.1 (3.0) +67.9 (0.6) +Chebyshev +76.3 (0.7) +65.4 (0.8) +73.9 (0.7) +82.5 (1.4) +64.3 (1.6) +40.0 (0.5) +38.4 (1.5) +39.1 (0.2) +GraphSAGE +76.7 (0.5) +60.7 (0.5) +74.7 (0.2) +— +54.7 (0.6) +51.9 (0.6) +31.0 (1.1) +35.3 (1.0) +GAT +77.1 (0.7) +64.6 (0.5) +78.2 (0.2) +84.8 (0.5) +66.3 (1.9) +46.4 (0.4) +38.1 (1.1) +40.8 (1.3) +GCN +79.4 (1.4) +65.2 (2.4) +77.2 (1.4) +88.9 (0.5) +82.4 (0.5) +52.8 (0.8) +50.0 (1.7) +45.0 (1.1) +SGC +77.7 (0.9) +61.5 (1.7) +76.5 (2.3) +81.1 (2.6) +80.7 (0.3) +53.2 (2.1) +44.2 (0.2) +46.7 (1.7) +APPNP +77.6 (0.1) +63.2 (0.2) +77.7 (0.0) +88.3 (0.0) +85.7 (0.3) +48.2 (0.7) +56.9 (0.2) +48.6 (1.6) +JKNet +82.3 (0.3) +67.8 (0.1) +79.3 (0.3) +82.2 (0.6) +75.0 (0.1) +51.3 (0.1) +51.1 (0.5) +31.7 (1.5) +DAGNN +80.0 (0.7) +65.7 (0.7) +80.7 (0.7) +87.5 (0.9) +83.8 (2.4) +53.0 (0.9) +55.5 (6.7) +39.3 (11.2) +GNN-LF +79.1 (0.7) +66.7 (0.4) +80.2 (0.5) +90.9 (0.5) +85.9 (0.6) +50.5 (1.9) +54.3 (1.0) +29.7 (15.1) +GNN-HF +78.6 (0.3) +64.3 (1.7) +78.1 (0.2) +91.3 (0.5) +83.8 (0.4) +49.0 (1.1) +58.6 (0.6) +44.9 (0.8) +tsGCN (inv) +78.5 (0.3) +69.6 (0.4) +78.7 (0.3) +85.1 (1.5) +85.2 (7.1) +57.2 (1.1) +52.9 (15.8) +48.5 (0.8) +F1 +tsGCN +80.5 (0.5) +69.0 (0.3) +82.4 (0.1) +92.8 (0.4) +90.1 (0.6) +58.7 (0.7) +79.3 (2.9) +50.1 (0.1) +Experiment +This section will show tsGCN’s effectiveness and efficiency +via comprehensive experiments. +Datasets +Cora, Citeseer and Pubmed are citation networks, and Cora- +Full is a larger version of Cora; ACM is a paper network, and +BlogCatalog and Flickr are social networks; UAI has been +utilized for community detection. The detailed statistics of +the above eight public datasets are concluded in Table 2. +Compared Methods +Two types of methods are employed here for comparisons. +Chebyshev [33], GraphSAGE [34] and GAT [35] are clas- +sical graph neural networks. GCN, SGC [22], APPNP [23], +JKNet [24], DAGNN [27], GNN-LF and GNN-HF [19] are +selected as state-of-the-art GCN variants. +Parameter Setups +For all experiments, we randomly split samples into a small +set of 20 labeled samples per class for training, a set of 500 +samples for validating and a set of 1, 000 samples for testing. +In terms of the ten baseline methods, all their configurations +are set as the default in their original papers. With respect to +tsGCN, following the vanilla GCN, the learning rate, weight +decay and the size of hidden units are set to 1 × 10−2, 5 × +10−4 and 32, respectively. The hyperparameters α and β are +selected in {0.1, 0.2, . . . , 1.0} for different datasets, and r is +chosen in {⌊ d +211 ⌋, ⌊ d +210 ⌋, . . . , ⌊ d +23 ⌋}, where d is the feature +dimension of the original data. +Semi-supervised Classification +Performance Comparisons. The semi-supervised classifi- +cation task is conducted on selected datasets, whose results +are recorded in Table 3. Specifically, we compare our tsGCN +with the ten baseline methods in terms of both accuracy and +F1-score, marking the best and second-best results on each +dataset. Note that tsGCN (inv) denotes tsGCN without the +low-rank approximation, which directly calculates the ma- +trix inverse in Eq. (14). From Table 3, we have the following +observations: +• tsGCN achieves the best performances on most datasets, +and is only slightly inferior to the JKNet method on the +smallest Cora dataset. +• tsGCN yields higher scores than JKNet and APPNP, es- +pecially on Pubmed, CoraFull, BlogCatalog, and Flickr, +where the first two are relatively large datasets and the +latter two have dense edges. tsGCN even outperforms the +second-best approach GNN-HF by about 20% on Flickr. +It is worth mentioning that tsGCN utilizes high-order +information by the infinite-order graph convolution, and +JKNet and APPNP also develop different techniques for the +same goal. Hence, the advantage of tsGCN over APPNP and +JKNet implies that the infinite-order graph convolution im- +plemented by the low-rank approximation not only requires +less computational complexity, but also effectively captures +high-order neighborhood information and filters significant +noises. +Runtime Comparisons. This section collects the train- +ing time (i.e., runtime) of all methods on two selected large +datasets, i.e., Pubmed and CoraFull, as exhibited in Fig. 1(a): +the first three columns correspond to classical GNNs, while + +(a) Runtime +(b) Classification Accuracy +Figure 1: (a) All methods’ runtime on two large datasets. (b) The classification accuracy of tsGCN w.r.t. (α, β) on all datasets. +(a) Cora +(b) Citeseer +(c) Pubmed +(d) ACM +(e) BlogCatalog +(f) CoraFull +(g) Flickr +(h) UAI +Figure 2: The classification accuracy of tsGCN w.r.t. hyperparameters α and β on all datasets. +the rest are GCNs. From Fig. 1(a), we find that tsGCN takes +much less runtime than Chebyshev, GAT, and GraphSAGE; +however, it performs moderately well among the state-of- +the-art GCN variants. Specifically, tsGCN is (1) inferior to +SGC, JKNet, and DAGNN; (2) well-matched with the orig- +inal GCN; (3) but advantageous over the recently proposed +GNN-LF and GNN-HF. +Parameter Sensitivity Analysis +Fig. 1(b) curves the accuracy of tsGCN w.r.t. various ranks +by fixing other parameters α and β. Considering that differ- +ent datasets hold different distributions, their optimal ranks +to the optimization (16) are also different. For example, in +regard to the curves on BlogCatalog and ACM, their accu- +racy first go up to a high value and then keep steady, which +indicates that when rank r = ⌊d/512⌋, the low-rank approx- +imation is effective and efficient enough. When it comes to +the curve on Pubmed, the trend of its performance mono- +tonically decreases as rank r becomes bigger, which implies +that a very low-rank (i.e., r = ⌊d/2048⌋) approximation is +sufficient enough to preserve abundant information for good +results. However, with respect to the other curves such as on +Flickr and Cora, the y-axis’ scores generally rise to a peak +first and then fall continuously as the rank r increases. For +these cases, the optimal ranks differ at their peaks. +Fig. 2 plots the accuracy of tsGCN w.r.t. (α, β) by fixing +the optimal ranks. On Cora, Citeseer, Pubmed, BlogCatalog, +and CoraFull, tsGCN performs well with large α and small +β; while, on ACM, Flickr, and UAI, tsGCN generates high +accuracy when these two parameters are both large. +For detailed settings of these hyperparameters, please ref- +erence the codes and datasets to be released on Github. +Ablation Study +The results of GCN, tsGCN-s, tsGCN-t, tsGCN (inv), and ts- +GCN are plotted in Fig. 3 (notice that tsGCN-s and tsGCN-t +are with semantic and topological regularizer, respectively), +telling us: +• The performance is unsatisfactory when the two regular- +izers are adopted alone, while tsGCN can always effec- +tively fuse the two to better capture underlying structures. +• tsGCN (inv) is even worse than single-regularizer model +on some datasets, indicating that the infinite-order graph + +Accuracy (%) +Accuracy (%) +F1-score (%) +F1-score (%) +Figure 3: Accuracy and F1-score of tsGCN and its variants on all datasets. +(a) Chebyshev +(b) GraphSAGE +(c) GAT +(d) GCN +(e) SGC +(f) APPNP +(g) JKNet +(h) DAGNN +(i) GNN-LF +(j) GNN-HF +(k) tsGCN (inv) +(l) tsGCN +Figure 4: Different methods’ t-SNE visualizations on BlogCatalog, where each color corresponds to one class. +convolutions implemented by the matrix inverse can pull- +in instability to the model. +• Compared to GCN, tsGCN (inv) performs comparable or +even worse, whereas tsGCN shows substantial improve- +ments on all datasets, which indicates that the low-rank +approximation enhances the robustness of infinite-order +graph convolutions. +Data Visualization +Fig. 4 exhibits the graph representations learned by different +methods on BlogCatalog. As can be seen clearly, the results +of the three classical graph neural networks, i.e., Chebyshev, +GraphSAGE and GAT, are unsatisfactory; while for the other +competitors, there are: +• Both tsGCN (inv) and tsGCN are better than other GCNs, +which indicates that the dual-regularizer can extract more +accurate inter-relationships from the topological and se- +mantic structures. +• Comparing the embeddings learned by tsGCN with those +of tsGCN (inv), classes in the former sub-figure are more +clearly recognized and the within-clusters are more com- +pact, which testifies the effectiveness of the low-rank ap- +proximation. +In a nutshell, the embeddings of the proposed model show +the best inter-class separation and intra-class aggregation. +Conclusion +By revisiting GCN, this paper puts forward an interpretable +regularizer-centered optimization framework, in which the +connections between existing GCNs and diverse regularizers +are revealed. It’s worth mentioning that this framework pro- +vides a new perspective to interpret existing work and guide +new GCNs just by designing new graph regularizers. Im- +pressed by the significant effectiveness of the feature based +semantic graph, we further combine it with nodes’ topolog- +ical structures, and develop a novel dual-regularizer graph +convolutional network, called tsGCN. Since the analytical +updating rule of tsGCN contains a time-consuming matrix +inverse, we devise an efficient algorithm with low-rank ap- +proximation tricks. Experiments on node classification tasks +demonstrate that tsGCN performs much better than quite a +few state-of-the-art competitors, and also exhibit that tsGCN +runs much faster than the very recently proposed GCN vari- +ants, e.g., GNN-HF and GNN-LF. +Acknowledgments +This work is in part supported by the National Natu- +ral Science Foundation of China (Grant Nos. U21A20472 +and 62276065), the Natural Science Foundation of Fujian +Province (Grant No. 2020J01130193). +Supplementary +In this supplementary, we mainly present specific details to +link various GCNs with various graph regularizers under the +regularizer-centered optimization framework. Besides, more +experimental settings and results are provided to further en- +rich the main paper. +The Framework Review +An interpretable regularizer-centered constrained optimiaza- +tion framework is induced as +arg min +H(l) L = −Tr(H(l)⊤H(l−1)Θ(l)) +� +�� +� +fitting ++ 1 +2L(H(l); G) +� +�� +� +regularization +(19) + +s.t. H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex, +with the aim to unify various GCNs in an interpretable way, +and also to guide the design of new GCN variants. Note that +the first term in optimization (19) is equivalent to the fitting +loss between the forward propagation H(l−1)Θ(l) and the +output H(l), while the second term is the priors-based graph +regularizer. Besides, S, S+ and Ssimplex are separately de- +fined to be +S = {s ∈ Rd}, +(20) +S+ = {s ∈ Rd|s ≥ 0}, +(21) +and +Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}. +(22) +The above three sets correspond to the Identity(·), Relu(·), +and Softmax(·) activation functions frequently used in graph +convolutional networks, respectively. +It’s claimed that by designing different regularizers, this +framework can give birth to different GCN methods. In the +following, we will give specific details about how they could +be derived from optimization (19). +Link Various GCNs with Various Regularizers +Theorem 1. The updating rule of the vanilla GCN [21] +H(l) = σ +� +�AH(l−1)Θ(l)� +, l ∈ [L], +(23) +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l) +(24) +s.t. S(l) ∈ {S or S+ or Ssimplex}, +where +J (l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +H⊤�LH +� +. +(25) +Proof. Taking derivative of J (l) w.r.t. H, we obtain +∂J (l) +∂H += −H(l−1)Θ(l) + �LH; +(26) +if it ( ∂J (l) +∂H ) is further set to zero, then there yields +H∗ = (I − �A)−1H(l−1)Θ(l). +(27) +By projecting H∗ onto S(l), we could arrive at +H(l) = σ(H∗). +(28) +Notably, (I − �A)−1 = �∞ +i=0 �Ai; and when its first-order +approximation is leveraged, i.e., (I − �A)−1 ≈ I + ˜A = �A, +Eq. (28) gives birth to the updating rule (23). +The above analyses reveal that when the regularizer is de- +signed to 1 +2Tr +� +H⊤�LH +� +, the framework (19) could gener- +ate the vanilla GCN [21]. +Theorem 2. Given H(0) = f MLP +Θ +(X) and α ∈ [0, 1), the +updating rule of APPNP [23] +H(l) = σ +� +(1 − α) �AH(l−1) + αH(0)� +, l ∈ [L], +(29) +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l), +(30) +s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, +where +J (l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +1 +1 − αH⊤ �A−1(H − 2αH(0)) +� +. +(31) +Proof. Taking the derivative of J (l) w.r.t. H and setting it +to zero, we come to +∂J (l) +∂H += −H(l−1) + +1 +1 − α +�A−1(H − αH(0)) = 0, (32) +which leads to +H∗ = (1 − α) �AH(l−1) + αH(0). +(33) +By projecting H∗ onto S(l), we could achieve +H(l) = σ +� +(1 − α) �AH(l−1) + αH(0)� +, l ∈ [L], +(34) +which completes the proof. +The above analyses reveal that when the regularizer is de- +vised to 1 +2Tr +� +1 +1−αH⊤ �A−1(H − 2αH(0)) +� +, the framework +(19) could give birth to APPNP [23]. +Theorem 3. The updating rule of JKNet [24] +H(l) = σ +� K +� +k=1 +αk �AkH(l−1)Θ(l) +� +, l ∈ [L], +(35) +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l) +(36) +s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, +where +J (l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +H⊤ �A−1(I + β�L)H +� +. +(37) +Proof. Taking the derivative of J (l) w.r.t. H and setting it +to zero, we have +∂J (l) +∂H += −H(l−1)Θ(l) + �A−1(I + β�L)H = 0, +(38) +which leads to +H∗ = +1 +β + 1 +� +I − +β +β + 1 +�A +�−1 +�AH(l−1)Θ(l). +(39) + +Methods +Propagation Rules +Regularizer L(H(l); G) +Projective Set +GCN +H(l) = σ +� +�AH(l−1)Θ(l)� +Tr +� +H(l)⊤�LH(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +SGC +H(l) = σ +� +�AH(l−1)Θ(l)� +Tr +� +H(l)⊤�LH(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +APPNP +H(l) = σ +� +(1 − α) �AH(l−1) + αH(0)� +Tr +� +1 +1−αH(l)⊤ �A−1(H(l) − 2αH(0)) +� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +JKNet +H(l) = σ +��K +k=1 αk �AkH(l−1)Θ(l)� +Tr +� +H(l)⊤ �A−1(I + β�L)H(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +DAGNN +H(L) = σ +��K +k=0 αk �AkH(0)� +Tr +� +H(l)⊤(I + β�L)H(l)� +� +S(l) = S, l ∈ [L−1], +S(L) = Ssimplex +GNN-HF +H(l) = σ +� +(I + α�L)−1(I + β�L)H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + β�L)−1(I + α�L)H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex. +GNN-LF +H(l) = σ +� +(I + α �A)−1(I + β �A)H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + β �A)−1(I + α �A)H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +tsGCN +H(l) = σ +� +(I + α�LG + β�LX )−1H(l−1)Θ(l)� +Tr +� +H(l)⊤(I + α�LG + β�LX )H(l)� +� +S(l) = S+, l ∈ [L−1], +S(L) = Ssimplex +Table 4: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework. +It is noted that the spectral radius of +β +β+1 �A is smaller than +one, indicating +� +I − +β +β + 1 +�A +�−1 += +∞ +� +k=0 +� +β +β + 1 +�A +�k +. +(40) +If its (K−1)-order approximation is employed, then there +goes +� +I − +β +β + 1 +�A +�−1 +≈ +K−1 +� +k=0 +βk +(β + 1)k �Ak, +(41) +which suggests that H∗ can be approximated by +H∗ = +K +� +k=1 +βk−1 +(β + 1)k �AkH(l−1)Θ(l). +(42) +If denote αk = +βk−1 +(β+1)k (k ∈ [K]), then {αk}∞ +k=1 is a set +of parameters with �∞ +k=1 αk = +1 +β+1 +1 +1− +β +β+1 = 1. +By projecting H∗ onto S(l), we can realize +H(l) = σ +� K +� +k=1 +αk �AkH(l−1)Θ(l) +� +, l ∈ [L], +(43) +which completes the proof. +The above analyses reveal that when the regularizer is +devised to 1 +2Tr +� +H⊤ �A−1(I + β�L)H +� +, the framework (19) +can produce JKNet [24]. +Theorem 4. Given H(0) = f MLP +Θ +(X) and a trainable pro- +jection vector α ∈ RK+1, the updating rule of DAGNN [27] +H(l) = σ +� K +� +k=0 +αk �AkH(0) +� +, +(44) +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l) +(45) +s.t. H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, +where +J (l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +H⊤(I + β�L)H +� +. +(46) +Proof. Taking the derivative of J (l) w.r.t. H and setting it +to zero, we can harvest +∂J (l) +∂H += −H(0) + (I + β�L)H = 0. +(47) +Similar to the proof of Theorem 3, the K-order approxi- +mation of +� +I + β�L +�−1 += +1 +β+1 +� +I − +β +β+1 �A +�−1 +is utilized, +and then we obtain +H∗ = +K +� +k=0 +αk �AkH(0). +(48) +By projecting H∗ onto S(l), we can arrive at +H(l) = σ +� K +� +k=0 +αk �AkH(0) +� +, l ∈ [L], +(49) +which completes the proof. +The above analyses reveal that when the regularizer is +devised to 1 +2Tr +� +H⊤(I + β�L)H +� +, the framework (19) can +produce DAGNN [27]. +Theorem 5. The updating rule of GNN-HF [19] +H(l) = σ((β + 1 +α)I ++ (1 − β − 1 +α) �A−1(I + β�L)H(l−1)Θ(l)), +(50) + +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l) +(51) +s.t. H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex, +where +J (l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +H⊤(I + µ�L)−1(I + λ�L)H +� +(52) +with λ = β + 1 +α − 1 and µ = β. +Proof. Taking the derivative of J (l) w.r.t. H and setting it +to zero, we own +∂J (l) +∂H += −H(l−1)Θ(l)+(I+µ�L)−1(I+λ�L)H = 0. (53) +which yields +H∗ = (I + λ�L)−1(I + µ�L)H(l−1)Θ(l). +(54) +Substituting λ = β + 1 +α − 1 and µ = β into Eq. (54), we +obtain +(I + λ�L)−1 = +� +(1 + λ)I − λ �A +�−1 += +� +(β + 1 +α)I + (1 − β − 1 +α) �A +�−1 +. +(55) +By projecting H∗ onto S(l), we can ahieve +H(l) = σ (H∗) , l ∈ [L], +(56) +which completes the proof. +The above analyses reveal that when the regularizer is de- +vised to 1 +2Tr +� +H⊤(I + µ�L)−1(I + λ�L)H +� +, the framework +(19) can produce GNN-HF [19]. +Theorem 6. The updating rule of GNN-LF [19] +H(l) = σ((βI + (1 − β) �A ++ ( 1 +α − 1)�L)−1(βI + (1 − β) �A)H(l−1)), +(57) +is equivalent to solving the following optimization +H(l) = arg min +H∈S(l) J (l) +(58) +s.t. H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex, +where +J(l) = −Tr +� +H⊤H(l−1)Θ(l)� ++ 1 +2Tr +� +H⊤(I + µ �A)−1(I + λ �A)H +� +(59) +with λ = −αβ+2α−1 +αβ−α+1 +and µ = 1 +β − 1. +Proof. Taking the derivative of J (l) w.r.t. H and setting it +to zero, we can get +∂J (l) +∂H += −H(l−1)Θ(l)+(I+µ �A)−1(I+λ �A)H = 0, (60) +which leads to +H∗ = (I + λ �A)−1(I + µ �A)H(l−1)Θ(l). +(61) +Absorbing the scale +αβ +αβ−α+1 into the to-be-learnt variable +Θ(l), and substituting λ = −αβ+2α−1 +αβ−α+1 +and µ = 1 +β − 1 into +Eq. (61), we can harvest +αβ +αβ − α + 1(I + λ �A)−1(I + µ �A) += +αβ +αβ − α + 1(I + −αβ + 2α − 1 +αβ − α + 1 +�A)−1(I + ( 1 +β − 1) �A) += +� +(1 − 1 +β + 1 +αβ )I + (−1 + 2 +β − 1 +αβ ) �A +�−1 +(I + ( 1 +β − 1) �A) += +� +(β − 1 + 1 +α)I + (−β + 2 − 1 +α) �A +�−1 +(βI + (1 − β) �A). +(62) +By projecting H∗ onto S(l), we can attain +H(l) = σ (H∗) , l ∈ [L], +(63) +For notation consistency, we denote λ and µ as α and β +in Table 4, completing the proof. +The above analyses reveal that when the regularizer is set +to 1 +2Tr +� +H⊤(I + µ �A)−1(I + λ �A)H +� +, the framework (19) +can generate GNN-LF [19]. +More Experimental Settings and Results +In this part, we provide more experimental settings and re- +sults for tsGCN, including hyperparameter settings, t-SNE +visualizations of various methods, and the parameter sensi- +tivity analysis of tsGCN w.r.t. F1-score. +Hyperparameter Settings. The detailed values of several +hyperpaerameters are recorded in Table 5, which can be used +to reproduce the reported experimental results. And the code +is also provided as a supplementary file. +More visualizations. We draw the t-SNE of embeddings +generated by all methods on all datasets from Fig. 5 to +Fig. 11, from which we have the following observations: +• The results are generally matched with the quantitative +performance, i.e., tsGCN achieves better results than the +others on most datasets. +• Embeddings generated by tsGCN achieve better inter- +class separation alongside intra-class clustering than +those generated by tsGCN (inv), even when their quanti- +tative performance is comparable. +Parameter Sensitivity. It can be seen that the F1-scores of +tsGCN w.r.t. (α, β) hold the similar trends with the classifi- +cation accuracies of tsGCN. + +Table 5: Specific (α, β, r) and other parameters of tsGCN on all datasets. +Datasets/Parameters +α +β +r +Learning rate +Weight decay +Hidden units +Cora +1.0 +0.2 +⌊d/16⌋ +1 × 10−2 +5 × 10−4 +32 +Citeseer +1.0 +0.4 +⌊d/16⌋ +1 × 10−2 +5 × 10−4 +32 +Pubmed +1.0 +0.3 +⌊d/2048⌋ +1 × 10−2 +5 × 10−4 +32 +ACM +1.0 +0.9 +⌊d/64⌋ +1 × 10−2 +5 × 10−4 +32 +BlogCatalog +1.0 +0.5 +⌊d/64⌋ +1 × 10−2 +5 × 10−4 +32 +CoraFull +1.0 +0.1 +⌊d/8⌋ +1 × 10−2 +5 × 10−4 +32 +Flickr +1.0 +1.0 +⌊d/64⌋ +1 × 10−2 +5 × 10−4 +32 +UAI +1.0 +1.0 +⌊d/16⌋ +1 × 10−2 +5 × 10−4 +32 +Figure 5: Different methods’ t-SNE visualizations on Cora, where each color corresponds to one class. +Figure 6: Different methods’ t-SNE visualizations on Citeseer, where each color corresponds to one class. + +Figure 7: Different methods’ t-SNE visualizations on Pubmed, where each color corresponds to one class. +Figure 8: Different methods’ t-SNE visualizations on ACM, where each color corresponds to one class. Note that GraphSAGE +fails to run on ACM. +Figure 9: Different methods’ t-SNE visualizations on CoraFull, where each color corresponds to one class. + +Figure 10: Different methods’ t-SNE visualizations on Flickr, where each color corresponds to one class. +Figure 11: Different methods’ t-SNE visualizations on UAI, where each color corresponds to one class. +(a) Cora +(e) BlogCatalog +(b) Citeseer +(f) CoraFull +(c) Pubmed +(g) Flickr +(d) ACM +(h) UAI +Figure 12: The classification F1-scores of tsGCN w.r.t. different hyperparameters α and β on all datasets. + +References +[1] Z. Chen, X. Wei, P. Wang, Y. Guo, Multi-label im- +age recognition with graph convolutional networks, in: +CVPR, 2019, pp. 5177–5186. +[2] W. Nie, Y. Zhao, A. Liu, Z. Gao, Y. Su, Multi-graph +convolutional network for unsupervised 3d shape re- +trieval, in: MM, 2020, pp. 3395–3403. +[3] Y. Wang, M. Cao, Z. Fan, S. Peng, Learning to detect +3d facial landmarks via heatmap regression with graph +convolutional network, in: AAAI, 2022, pp. 2595– +2603. +[4] F. Xu, J. Lian, Z. Han, Y. Li, Y. Xu, X. Xie, Relation- +aware graph convolutional networks for agent-initiated +social e-commerce recommendation, in: CIKM, 2019, +pp. 529–538. +[5] Y. Chen, L. Huang, C. Wang, J. Lai, Hybrid-order +gated graph neural network for session-based recom- +mendation, IEEE Transactions on Industrial Informat- +ics 18 (3) (2022) 1458–1467. +[6] H. Hu, L. Cheng, J. P. Vap, M. Borowczak, Learn- +ing privacy-preserving graph convolutional network +with partially observed sensitive attributes, in: WWW, +2022, pp. 3552–3561. +[7] B. Yu, H. Yin, Z. Zhu, Spatio-temporal graph convolu- +tional networks: A deep learning framework for traffic +forecasting, in: IJCAI, 2018, pp. 3634–3640. +[8] W. Chen, L. Chen, Y. Xie, W. Cao, Y. Gao, X. Feng, +Multi-range attentive bicomponent graph convolu- +tional network for traffic forecasting, in: AAAI, 2020, +pp. 3529–3536. +[9] Y. Zhang, S. Pal, M. Coates, D. ¨Ustebay, Bayesian +graph +convolutional +neural +networks +for +semi- +supervised +classification, +in: +AAAI, +2019, +pp. +5829–5836. +[10] M. Yang, Y. Shen, R. Li, H. Qi, Q. Zhang, B. Yin, A +new perspective on the effects of spectrum in graph +neural networks, in: ICML, 2022, pp. 25261–25279. +[11] S. Fan, X. Wang, C. Shi, E. Lu, K. Lin, B. Wang, +One2multi graph autoencoder for multi-view graph +clustering, in: WWW, 2020, pp. 3070–3076. +[12] H. Zhu, P. Koniusz, Simple spectral graph convolution, +in: ICLR, 2021, pp. 1–11. +[13] H. Chen, H. Yin, X. Sun, T. Chen, B. Gabrys, K. Mu- +sial, Multi-level graph convolutional networks for +cross-platform anchor link prediction, in: KDD, 2020, +pp. 1503–1511. +[14] N. Halliwell, Evaluating explanations of relational +graph convolutional network link predictions on +knowledge graphs, in: AAAI, 2022, pp. 12880–12881. +[15] D. Bo, X. Wang, C. Shi, H. Shen, Beyond low- +frequency information in graph convolutional net- +works, in: AAAI, 2021, pp. 3950–3957. +[16] Z. Zhang, C. Chen, Y. Chang, W. Hu, X. Xing, Y. Zhou, +Z. Zheng, Shne: Semantics and homophily preserv- +ing network embedding, IEEE Transactions on Neural +Networks and Learning Systems (2021) 1–12. +[17] X. Wang, M. Zhu, D. Bo, P. Cui, C. Shi, J. Pei, Am- +gcn: Adaptive multi-channel graph convolutional net- +works, in: KDD, 2020, pp. 1243–1253. +[18] L. Zhao, L. Akoglu, Connecting graph convolutional +networks and graph-regularized pca, arXiv preprint +arXiv:2006.12294 (2020). +[19] M. Zhu, X. Wang, C. Shi, H. Ji, P. Cui, Interpreting and +unifying graph neural networks with an optimization +framework, in: WWW, 2021, pp. 1215–1226. +[20] Y. Yang, T. Liu, Y. Wang, J. Zhou, Q. Gan, Z. Wei, +Z. Zhang, Z. Huang, D. Wipf, Graph neural networks +inspired by classical iterative algorithms, in: ICML, +2021, pp. 11773–11783. +[21] T. N. Kipf, M. Welling, Semi-supervised classification +with graph convolutional networks, in: ICLR, 2017, +pp. 1–13. +[22] F. Wu, A. H. S. Jr., T. Zhang, C. Fifty, T. Yu, +K. Q. Weinberger, Simplifying graph convolutional +networks, in: ICML, 2019, pp. 6861–6871. +[23] J. Klicpera, A. Bojchevski, S. G¨unnemann, Predict +then propagate: Graph neural networks meet person- +alized pagerank, in: ICLR, 2019, pp. 1–15. +[24] K. +Xu, +C. +Li, +Y. +Tian, +T. +Sonobe, +K. +ichi +Kawarabayashi, S. Jegelka, Representation learning on +graphs with jumping knowledge networks, in: ICML, +2018, pp. 5449–5458. +[25] K. Sun, Z. Zhu, Z. Lin, Adagcn: Adaboosting graph +convolutional networks into deep models, in: ICLR, +2021, pp. 1–15. +[26] Q. Li, Z. Han, X. Wu, Deeper insights into graph con- +volutional networks for semi-supervised learning, in: +AAAI, 2018, pp. 3538–3545. +[27] M. Liu, H. Gao, S. Ji, Towards deeper graph neural +networks, in: KDD, 2020, pp. 338–348. +[28] L. Yang, Z. Kang, X. Cao, D. Jin, B. Yang, Y. Guo, +Topology optimization based graph convolutional net- +work, in: AAAI, 2019, pp. 4054–4061. +[29] K. Oono, T. Suzuki, Graph neural networks exponen- +tially lose expressive power for node classification, in: +ICLR, 2020, pp. 1–8. +[30] D. K. Hammond, P. Vandergheynst, R. Gribonval, +Wavelets on graphs via spectral graph theory, Applied +and Computational Harmonic Analysis 30 (2011) 129– +150. +[31] B. Amos, Differentiable optimization-based modeling +for machine learning, Ph.D. thesis, Carnegie Mellon +University (2019). +[32] J. Sun, Z. Xu, Neural diffusion distance for image seg- +mentation, in: NeurIPS, 2019, pp. 1441–1451. +[33] M. Defferrard, X. Bresson, P. Vandergheynst, Convo- +lutional neural networks on graphs with fast localized +spectral filtering, in: NeurIPS, 2016, pp. 1–9. +[34] W. L. Hamilton, Z. Ying, J. Leskovec, Inductive repre- +sentation learning on large graphs, in: NeurIPS, 2017, +pp. 1024–1034. + +[35] P. Velickovic, G. Cucurull, A. Casanova, A. Romero, +P. Li`o, Y. Bengio, Graph attention networks, in: ICLR, +2018, pp. 1–12. + diff --git a/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/load_file.txt b/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8705d467b315e25526ad9fe121354af84e985a7 --- /dev/null +++ b/dtE3T4oBgHgl3EQfGwnT/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf,len=1050 +page_content='Beyond Graph Convolutional Network: An Interpretable Regularizer-centered Optimization Framework Shiping Wang1,2, Zhihao Wu1,2, Yuhong Chen1,2, Yong Chen3* 1 College of Computer and Data Science, Fuzhou University 2 Fujian Provincial Key Laboratory of Network Computing and Intelligent Information Processing, Fuzhou University 3 School of Computer Science, Beijing University of Posts and Telecommunications shipingwangphd@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='com, zhihaowu1999@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='com, yhchen2320@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='com, alphawolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='chen@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Abstract Graph convolutional networks (GCNs) have been attracting widespread attentions due to their encouraging performance and powerful generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' However, few work provide a general view to interpret various GCNs and guide GCNs’ de- signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In this paper, by revisiting the original GCN, we in- duce an interpretable regularizer-centerd optimization frame- work, in which by building appropriate regularizers we can interpret most GCNs, such as APPNP, JKNet, DAGNN, and GNN-LF/HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Further, under the proposed framework, we de- vise a dual-regularizer graph convolutional network (dubbed tsGCN) to capture topological and semantic structures from graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Since the derived learning rule for tsGCN con- tains an inverse of a large matrix and thus is time-consuming, we leverage the Woodbury matrix identity and low-rank ap- proximation tricks to successfully decrease the high computa- tional complexity of computing infinite-order graph convolu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Extensive experiments on eight public datasets demon- strate that tsGCN achieves superior performance against quite a few state-of-the-art competitors w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Introduction Owing to the powerful ability to aggregate neighborhood in- formation, Graph Convolutional Network (GCN) has been successfully applied to diverse domains, such as computer vision [1, 2, 3], recommender systems [4, 5], privacy pre- serving [6], and traffic forecasting [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Rooted in a series of theoretical foundations, GCN extends convolution opera- tions to the non-Euclidean spaces and effectively propagates label signals, and therefore its variants have been extensively employed for a variety of graph-related tasks, including clas- sification [9, 10], clustering [11, 12] and link prediction [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In a nutshell, GCN generates the graph embedding with the well-established graph convolutional layers gath- ering semantics from neighbors according to the network topology, which are revealed to be the most critical com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Although GCN has behaved well in many machine learn- ing tasks, lots of studies have pointed out its certain draw- backs and made efforts for further improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [15] indicated that the propagation mechanism could be con- sidered as a special form of low-pass filter, and presented a Corresponding author Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' GCN with an adaptive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [16] argued that most GCN-based methods ignored the global informa- tion and proposed SHNE, which leveraged the structure and feature similarity to capture latent semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [17] revealed that the original GCN aggregated informa- tion from node neighbors inadequately, and then developed a multi-channel GCN by utilizing feature-based semantic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In spite of the performance boosts of these GCN variants, they didn’t establish a generalized framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', these approaches understood and enhanced GCN from cer- tain and non-generalizable perspectives, thereby they are ex- ceedingly difficult to be further developed, and with limited interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Consequently, it is expected to construct a unified frame- work for various GCNs with better interpretability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' however, it is a pity that this kind of work is still in shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [18] linked GCN and Graph-regularized PCA (GPCA), and then proposed a multi-layer network by stacking the GPCA layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [19] attempted to interpret exist- ing GCN-based methods with a unified optimization frame- work, under which they devised an adjustable graph filter for a new GCN variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [20] designed a family of graph convolutional layers inspired by the updating rules of two typical iterative algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Although these efforts have contributed to better understanding of GCNs, they only ex- plained GCNs in partial aspects, promoting the expectation of a more comprehensive analysis of GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To tackle the aforementioned issues, this paper induces an interpretable regularizer-centered optimization framework, which provides a novel perspective to digest various GCNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', this framework captures the common essential proper- ties of existing state-of-the-art GCN variants and could de- fines them just by devising different regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Moreover, in light of the analyses on current representative GCNs, we find that most of the existing approaches only consider cap- turing the topological regularization, while the feature-based semantic structure is underutilized, and hence this motivates us to design a dual-regularizer graph convolutional network (called tsGCN) within the regularizer-centered optimization framework for the fullest explorations of both structures and semantics from graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Due to the high computa- tional complexity of performing infinite-order graph con- volutions, the unified framework provides a straightforward way employing truncated polynomials to approximate the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='04318v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='LG] 11 Jan 2023 graph Laplacian, similar to the truncated Chebyshev poly- nomials by vanilla GCN, restricting the message passing of a single graph convolution to the first-order neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The main contributions of this paper can be summarized as the following three aspects: Propose a regularizer-centered constrained optimization framework, which interprets various existing GCNs with specific regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Establish a new dual-regularizer graph convolutional net- work (tsGCN), which exploits topological and semantic structures of the given data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' and develop an efficient algo- rithm to reduce the computational complexity of solving infinite-order graph convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Conduct a series of experiments to show that tsGCN per- forms much better than many SOTA GCNs, and also con- sumes much less time than the newly GNN-HF/LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Related Work Graph Convolutional Networks The original GCN was first introduced by Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [21], who generalized the convolution operations from the Eu- clidean domain to the non-Euclidean domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' SGC [22] as- sumed that the nonlinear transform of GCN was not that significant, and then devised a simplified GCN by remov- ing the nonlinear activation functions and collapsing the weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' PPNP [23] employed the relationship be- tween PageRank and GCN for the improvement on the prop- agation mechanism of GCN, and an iterative version called APPNP was further proposed to reduce the high compu- tational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Attempting to adaptively learn the in- fluence radii for each node and task, JKNet [24] combined various aggregations at the last layer and was able to learn representations of different orders for graph substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' GNN-LF and GNN-HF [19] considered the low-pass and the high-pass filter as the convolution kernels to improve GCN’s expressive power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' AdaGCN [25] leveraged Ad- aboost strategy for the enhancement of GCN, allowing in- formation to be shared between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To sum up, a main characteristic of these methods is exploring GCN from the perspectives of redesigning information aggregation strate- gies or modifying graph convolutions, and few work try to construct a unified framework to interpret various GCNs and reveal the underlying common principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Further Insights on GCNs Quite a few studies have been devoted to explore the mech- anisms of GCN for deeper insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [26] indicated that the convolutional operation of GCN was a special form of Laplacian smoothing, attributed to which GCN suffered from the so-called over-smoothing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Specifically, the performance of GCN will decrease as the number of layers increases, which has been validated by many other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' However, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [27] held a different opinion that the en- tanglement of two steps in GCN damages the performance of the deep GCN, where the two steps were explained as propagation and transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Based on this view, they de- coupled the two operations and further presented a deeper GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [19] also decomposed the convolution op- eration of GCN into two separate stages, called aggregation and transformation, and focused on the aggregation process, formulating an optimization objective to interpret it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [28] explored network topology refinement, leveraging a topology optimization process for the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Oono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [29] analyzed the forward propagation of GCN and in- terpreted it with a specific dynamical system, allowing GCN to be related to the underlying topological structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Over- all, these studies have contributed to the interpretability of GCNs, and also let researchers better understand GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In this paper, we build a unified optimization framework from a novel view of graph regularizers to interpret and understand GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Mathematical Notations For the convenience of formal descriptions, derivations, and analyses, necessary notations are narrated as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' A graph is denoted as G = (V, E, A), where V marks the vertex set with |V| = N (N is the total number of nodes in graph G), E marks the edge set, and A = [Aij]N×N marks an affinity matrix of which Aij measures the similarity between the i- th and the j-th node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In addition, D = [Dij]N×N represents the degree matrix of G with Dii = �N j=1 Aij, and then the normalized symmetrical graph Laplacian of G is computed as �L = I − �A with �A = D− 1 2 AD− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Revisiting Graph Convolutional Network For a graph G = (V, E, A), the svd of its graph Laplacian is L = UΛU⊤, where U ∈ RN×N is comprised of orthonor- mal eigenvectors and Λ = diag(λ1, · · · , λN) is a diagonal matrix with λi denoting the i-th eigenvalue and λi ≥ λj (i = 1, · · · , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Essentially, this decomposition induces a Fourier transform on the graph domain, where eigenvectors correspond to Fourier components and eigenvalues represent frequencies of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' For an input signal x ∈ RN defined on the graph G, the corresponding graph Fourier transform of x is �x = U⊤x, and its inverse transform is derived as x = U�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Consequently, the graph convolution between the signal x and the filter g ∈ RN is g ∗ x = U(�g ⊙ �x) = U((U⊤g) ⊙ (U⊤x)), (1) where ⊙ is the Hadamard product between two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Par- ticularly, denoting gΘ = diag(Θ) := U⊤g parameterized by Θ ∈ RN, the graph convolution between x and g can be rewritten as g ∗ x = U(�g ⊙ �x) = UgΘU⊤x, (2) where Θ is regarded as the filter coefficients to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Especially, Θ is assumed to be the polynomials of the spec- trums of the graph Laplacian [30], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Θ = Θ(Λ) = K � i=1 ΘiΛi, (3) where K is the order of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' By fixing K = 2, the graph convolutional network (GCN) [21] takes an effective form g ∗ x = θ(I + L)x, (4) Methods Propagation Rules Regularizer L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) Projective Set GCN H(l) = σ � �AH(l−1)Θ(l)� Tr � H(l)⊤�LH(l)� � S(l) = S+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex SGC H(l) = σ � �AH(l−1)Θ(l)� Tr � H(l)⊤�LH(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex APPNP H(l) = σ � (1 − α) �AH(l−1) + αH(0)� Tr � 1 1−αH(l)⊤ �A−1(H(l) − 2αH(0)) � � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex JKNet H(l) = σ ��K k=1 αk �AkH(l−1)Θ(l)� Tr � H(l)⊤ �A−1(I + β�L)H(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex DAGNN H(l) = σ ��K k=0 αk �AkH(l−1)� Tr � H(l)⊤(I + β�L)H(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex GNN-HF H(l) = σ � (I + α�L)−1(I + β�L)H(l−1)Θ(l)� Tr � H(l)⊤(I + β�L)−1(I + α�L)H(l)� � S(l) = S+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' GNN-LF H(l) = σ � (I + α �A)−1(I + β �A)H(l−1)Θ(l)� Tr � H(l)⊤(I + β �A)−1(I + α �A)H(l)� � S(l) = S+, l ∈ [L−1], S(L) = Ssimplex tsGCN H(l) = σ � (I + α�LG + β�LX )−1H(l−1)Θ(l)� Tr � H(l)⊤(I + α�LG + β�LX )H(l)� � S(l) = S+, l ∈ [L−1], S(L) = Ssimplex Table 1: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' where Θ = [θ] is a parameter to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' When ex- tending single channel signal x and filter θ to multi-channel H(l) ∈ RN×dl and Θ(l) ∈ Rdl×fl, the GCN is converted to H(l) = σ( �AH(l−1)Θ(l)), (5) where �A is a normalized version of I + �A, σ(·) is an acti- vation function, and H(l) ∈ RN×dl is the output of the l-th layer with H(0) = X being the input feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' An Interpretable Regularizer-centered Optimization Framework for GCNs Given the input H(l−1) of the (l)-th layer, GCN can compute the output H(l) by minimizing L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1 2Tr(H(l)⊤�LH(l)) (6) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) ≥ 0, where 1 2Tr(H(l)⊤�LH(l)) = 1 4 �N j=1 �N i=1 Aij|| h(l) i √Dii − h(l) j √ Djj ||2 2 with H(l) = [h(l) 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' h(l) N ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' it is a normalized reg- ularizer to preserve the pairwise similarity of any two nodes in the given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Besides, the −Tr(H(l)⊤H(l−1)Θ(l)) is actually a fitting loss term bewteen H(l) and H(l−1)Θ(l), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', ||H(l)−H(l−1)Θ(l)||2 F with H(l−1) and Θ(l) fixed when optimizing H(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that the square term ||H(l)||2 F is a L2- regularized smoother, which can be ignored or absorbed in the second graph regularizer Tr(H(l)⊤�LH(l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking derivative of L with respect to H(l) and setting it to zero, we obtain H(l+) as H(l+) = (I − �A)−1H(l−1)Θ(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (7) and then there yields H(l) = σ � H(l+)� , (8) when the nonnegative constraints H(l) ≥ 0 are further con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Notice that σ(·) is the ReLU(·) activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Here, if the matix inverse (I− ˜A)−1 = �∞ i=0 ˜Ai is approxi- mated by the first-order expansion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', (I− ˜A)−1 ≈ I+ ˜A, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (8) will lead to the updating rule (5) of GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Usually, the activation functions in GCN are ReLU(·) and Softmax(·), which could be converted to different projec- tion optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Concretely, the ReLU(·) activation func- tion is equivalent to project a point x onto the non-negative plane S+ = {s ∈ Rd|s ≥ 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', ReLU(x) = arg min y∈S+ −x⊤y + 1 2||y||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (9) By the way, we denote S = {s ∈ Rd}, which corresponds to an identity activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In terms of the Softmax(·) activation function, it can be regarded as projecting x onto the set Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Softmax(x) = arg min y∈Ssimplex −x⊤y + y⊤ log(y), (10) where y⊤ log(y) = �d i=1 yi log(yi) is the negative entropy of y [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In fact, with respect to other activation functions, they can also be equivalent to project a point onto some fea- sible set with some metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Up to present, we have actually utilized a constrained op- timization problem to interpret GCN, including information propagations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (7)) and the nonlinear activation func- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', ReLU(·) and Softmax(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses can not only explain the vanilla GCN, but also stimulate a regularizer-centered optimization frame- work that can further unify various GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' By extending the optimization (6), a more general framework is written as L = −Tr(H(l)⊤H(l−1)Θ(l)) + 1 2L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) (11) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Under this framework, different regularizers could derive different GCNs, for example, Algorithm 1: Topological and Semantic Regularized GCN Require: Graph data G = (V, E, A), labels y, number of layers L, and hyperparameters {α, β, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Ensure: Predicted label set {y∗ i }N i=n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1: Initialize model parameters {H(l), Θ(l)}L l=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 2: Compute the joint graph Laplacian α�LG + β�LX and its low-rank factorization WV⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3: Substitute the matrix inverse (I + α�LG + β�LX )−1 with I − W(I + V⊤W)−1V⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 4: while not convergent do 5: Calculate hidden layers {H(l)}L l=1 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 6: Update weights: Θ(l+1) ← Θ(l) − η ∂L ∂Θ(l) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 7: end while 8: return The predicted labels: y∗ i = arg maxj H(L) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' If L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) = Tr � H(l)⊤(I + µ�L)−1(I + λ�L)H(l)� with λ = β + 1 α − 1, µ = β, and �L = I − �A, then it induces the updating rule H(l) = σ � (I + α �A)−1(I + β �A)H(l−1)Θ(l)� , which cor- responds to GNN-HF [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' If L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) = Tr � H(l)⊤(I + µ �A)−1(I + λ �A)H(l)� with λ = −αβ+2α−1 αβ−α+1 and µ = 1 β − 1, then it gives rise to the updating rule H(l) = σ � (I + α �A)−1(I + β �A)H(l−1)Θ(l)� , which cor- responds to GNN-LF [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' For more cases, their results are summarized in Table 4, and the derivation details can refer to those of the original GCN (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (7) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (10)) and the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The work [19] is most similar to our work with the same research idea: they both want to propose a unified framework to interpret the current GCNs and guide the de- sign of new GCN variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' however, they are realized in dif- ferent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To be specific, (1) Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [19] develop an optimization framework to explain different GCNs’ prop- agation processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' whereas we propose a constrained op- timization framework not only to interpret various GCNs’ propagation processes, but also explain the nonlinear activa- tion layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (2) [19] unifies various GCNs via devising vari- ous fitting items which are essentially constructed by limited graph filters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' while our work derives different GCNs through designing different regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To sum up, our work inter- prets the whole (not partial) GCNs with regularizer-centered constrained optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' tsGCN: Topological and Semantic Regularized Graph Convolutional Network One finding from most existing GCNs is that they often ig- nored feature-based semantic structures, which can weaken the representation learning abilities of graph networks, then Table 2: Dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Datasets #Nodes #Edges #Classes #Features #Train/Val/Test Cora 2,708 5,429 7 1,433 140/500/1,000 Citeseer 3,327 4,732 6 3,703 120/500/1,000 Pubmed 19,717 44,338 3 500 60/500/1,000 ACM 3,025 13,128 3 1,870 60/500/1,000 BlogCatalog 5,196 171,743 6 8,189 120/500/1,000 CoraFull 19,793 65,311 70 8,710 1,400/500/1,000 Flickr 7,575 239,738 9 12,047 180/500/1,000 UAI 3,067 28,311 19 4,973 367/500/,1000 we focus on two regularizers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', L1(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) = 1 2Tr � {H(l)}⊤(1 2I + α�LG)H(l) � , (12) L2(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' X) = 1 2Tr � {H(l)}⊤(1 2I + β�LX )H(l) � , (13) where �LG is a graph Laplacian from the given adjacency ma- trix (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', �LG = �L), and �LX is a graph Laplacian calculated from the pairwise similarity of any two graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hence, we devise a dual-regularizer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', L(H(l)) = L1(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) + L2(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' X), and if it is under the optimization framework (19), then there yields the following updating rule H(l) = σ � (I + α�LG + β�LX )−1H(l−1)Θ(l)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (14) Since this method seeks to preserve both the topological and semantic structures for more accurate presentations, we call it tsGCN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Topological and Semantic regularized GCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Notably, the computational complexity of (I + α�LG + β�LX )−1 is O(N 3), which tends to be unaffordable in prac- tical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To this end, a low-rank approximation is operated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', α�LG +β�LX ≈ WV⊤, where W, V ∈ RN×r with r ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' This leads to the Woodbury matrix identity: (I + WV⊤)−1 = I − W(I + V⊤W)−1V⊤, (15) of which the computational complexity costs O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Given that the optimal M∗ of the following problem min M∈RN×N: rank(M)=r ||M − (α�LG + β�LX )||2 F (16) is attained at the r-truncated singular value decomposition of α�LG + β�LX , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', M∗ = UΣU⊤, where Σ ∈ Rr×r is a diagonal matrix containing the r largest singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' An optimal {W∗, V∗} to α�LG + β�LX ≈ WV⊤ can be given by an analytic form of W∗ = V∗ = UΣ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' To obtain the optimum {W∗, V∗}, the iterative algorithm [32] with O(N 2) is leveraged as Z(t+1) ← (α�LG + β�LX )U(t), (17) {U(t+1), R(t+1)} ← QR(Z(t+1)), (18) where QR(·) denotes the QR-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that this algorithm can converge to the r largest eigenvalues R(t+1) and its corresponding eigenvectors Z(t+1) when the iterative number t is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Finally, there will be W∗ = V∗ = U(t+1)[R(t+1)] 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gathering all analyses mentioned above, the procedure for tsGCN is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Table 3: Accuracy and F1-score (mean% and standard deviation%) of all methods, where the best results are in red and the second-best are in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that GraphSAGE fails to work on the ACM dataset, and thus its results are marked with “—”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Metrics Methods / Datasets Cora Citeseer Pubmed ACM BlogCatalog CoraFull Flickr UAI Chebyshev 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) GraphSAGE 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) — 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) GAT 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0) GCN 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) SGC 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 (0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) GNN-LF 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) GNN-HF 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8) tsGCN (inv) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8) F1 tsGCN 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='6) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='7) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1) Experiment This section will show tsGCN’s effectiveness and efficiency via comprehensive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Datasets Cora, Citeseer and Pubmed are citation networks, and Cora- Full is a larger version of Cora;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' ACM is a paper network, and BlogCatalog and Flickr are social networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' UAI has been utilized for community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The detailed statistics of the above eight public datasets are concluded in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Compared Methods Two types of methods are employed here for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chebyshev [33], GraphSAGE [34] and GAT [35] are clas- sical graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' GCN, SGC [22], APPNP [23], JKNet [24], DAGNN [27], GNN-LF and GNN-HF [19] are selected as state-of-the-art GCN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Parameter Setups For all experiments, we randomly split samples into a small set of 20 labeled samples per class for training, a set of 500 samples for validating and a set of 1, 000 samples for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In terms of the ten baseline methods, all their configurations are set as the default in their original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' With respect to tsGCN, following the vanilla GCN, the learning rate, weight decay and the size of hidden units are set to 1 × 10−2, 5 × 10−4 and 32, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The hyperparameters α and β are selected in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0} for different datasets, and r is chosen in {⌊ d 211 ⌋, ⌊ d 210 ⌋, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' , ⌊ d 23 ⌋}, where d is the feature dimension of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Semi-supervised Classification Performance Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The semi-supervised classifi- cation task is conducted on selected datasets, whose results are recorded in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Specifically, we compare our tsGCN with the ten baseline methods in terms of both accuracy and F1-score, marking the best and second-best results on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that tsGCN (inv) denotes tsGCN without the low-rank approximation, which directly calculates the ma- trix inverse in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' From Table 3, we have the following observations: tsGCN achieves the best performances on most datasets, and is only slightly inferior to the JKNet method on the smallest Cora dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' tsGCN yields higher scores than JKNet and APPNP, es- pecially on Pubmed, CoraFull, BlogCatalog, and Flickr, where the first two are relatively large datasets and the latter two have dense edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' tsGCN even outperforms the second-best approach GNN-HF by about 20% on Flickr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' It is worth mentioning that tsGCN utilizes high-order information by the infinite-order graph convolution, and JKNet and APPNP also develop different techniques for the same goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hence, the advantage of tsGCN over APPNP and JKNet implies that the infinite-order graph convolution im- plemented by the low-rank approximation not only requires less computational complexity, but also effectively captures high-order neighborhood information and filters significant noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Runtime Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' This section collects the train- ing time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', runtime) of all methods on two selected large datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Pubmed and CoraFull, as exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1(a): the first three columns correspond to classical GNNs, while (a) Runtime (b) Classification Accuracy Figure 1: (a) All methods’ runtime on two large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (b) The classification accuracy of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (α, β) on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (a) Cora (b) Citeseer (c) Pubmed (d) ACM (e) BlogCatalog (f) CoraFull (g) Flickr (h) UAI Figure 2: The classification accuracy of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' hyperparameters α and β on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' the rest are GCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1(a), we find that tsGCN takes much less runtime than Chebyshev, GAT, and GraphSAGE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' however, it performs moderately well among the state-of- the-art GCN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Specifically, tsGCN is (1) inferior to SGC, JKNet, and DAGNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (2) well-matched with the orig- inal GCN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (3) but advantageous over the recently proposed GNN-LF and GNN-HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Parameter Sensitivity Analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1(b) curves the accuracy of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' various ranks by fixing other parameters α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Considering that differ- ent datasets hold different distributions, their optimal ranks to the optimization (16) are also different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' For example, in regard to the curves on BlogCatalog and ACM, their accu- racy first go up to a high value and then keep steady, which indicates that when rank r = ⌊d/512⌋, the low-rank approx- imation is effective and efficient enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' When it comes to the curve on Pubmed, the trend of its performance mono- tonically decreases as rank r becomes bigger, which implies that a very low-rank (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', r = ⌊d/2048⌋) approximation is sufficient enough to preserve abundant information for good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' However, with respect to the other curves such as on Flickr and Cora, the y-axis’ scores generally rise to a peak first and then fall continuously as the rank r increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' For these cases, the optimal ranks differ at their peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 2 plots the accuracy of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (α, β) by fixing the optimal ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' On Cora, Citeseer, Pubmed, BlogCatalog, and CoraFull, tsGCN performs well with large α and small β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' while, on ACM, Flickr, and UAI, tsGCN generates high accuracy when these two parameters are both large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' For detailed settings of these hyperparameters, please ref- erence the codes and datasets to be released on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Ablation Study The results of GCN, tsGCN-s, tsGCN-t, tsGCN (inv), and ts- GCN are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3 (notice that tsGCN-s and tsGCN-t are with semantic and topological regularizer, respectively), telling us: The performance is unsatisfactory when the two regular- izers are adopted alone, while tsGCN can always effec- tively fuse the two to better capture underlying structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' tsGCN (inv) is even worse than single-regularizer model on some datasets, indicating that the infinite-order graph Accuracy (%) Accuracy (%) F1-score (%) F1-score (%) Figure 3: Accuracy and F1-score of tsGCN and its variants on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (a) Chebyshev (b) GraphSAGE (c) GAT (d) GCN (e) SGC (f) APPNP (g) JKNet (h) DAGNN (i) GNN-LF (j) GNN-HF (k) tsGCN (inv) (l) tsGCN Figure 4: Different methods’ t-SNE visualizations on BlogCatalog, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' convolutions implemented by the matrix inverse can pull- in instability to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Compared to GCN, tsGCN (inv) performs comparable or even worse, whereas tsGCN shows substantial improve- ments on all datasets, which indicates that the low-rank approximation enhances the robustness of infinite-order graph convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Data Visualization Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 4 exhibits the graph representations learned by different methods on BlogCatalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' As can be seen clearly, the results of the three classical graph neural networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', Chebyshev, GraphSAGE and GAT, are unsatisfactory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' while for the other competitors, there are: Both tsGCN (inv) and tsGCN are better than other GCNs, which indicates that the dual-regularizer can extract more accurate inter-relationships from the topological and se- mantic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Comparing the embeddings learned by tsGCN with those of tsGCN (inv), classes in the former sub-figure are more clearly recognized and the within-clusters are more com- pact, which testifies the effectiveness of the low-rank ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In a nutshell, the embeddings of the proposed model show the best inter-class separation and intra-class aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Conclusion By revisiting GCN, this paper puts forward an interpretable regularizer-centered optimization framework, in which the connections between existing GCNs and diverse regularizers are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' It’s worth mentioning that this framework pro- vides a new perspective to interpret existing work and guide new GCNs just by designing new graph regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Im- pressed by the significant effectiveness of the feature based semantic graph, we further combine it with nodes’ topolog- ical structures, and develop a novel dual-regularizer graph convolutional network, called tsGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Since the analytical updating rule of tsGCN contains a time-consuming matrix inverse, we devise an efficient algorithm with low-rank ap- proximation tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Experiments on node classification tasks demonstrate that tsGCN performs much better than quite a few state-of-the-art competitors, and also exhibit that tsGCN runs much faster than the very recently proposed GCN vari- ants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', GNN-HF and GNN-LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Acknowledgments This work is in part supported by the National Natu- ral Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' U21A20472 and 62276065), the Natural Science Foundation of Fujian Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 2020J01130193).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Supplementary In this supplementary, we mainly present specific details to link various GCNs with various graph regularizers under the regularizer-centered optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Besides, more experimental settings and results are provided to further en- rich the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The Framework Review An interpretable regularizer-centered constrained optimiaza- tion framework is induced as arg min H(l) L = −Tr(H(l)⊤H(l−1)Θ(l)) � �� � fitting + 1 2L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) � �� � regularization (19) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) ∈ {S+ or S}, l ∈ [L − 1], H(L) ∈ Ssimplex, with the aim to unify various GCNs in an interpretable way, and also to guide the design of new GCN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that the first term in optimization (19) is equivalent to the fitting loss between the forward propagation H(l−1)Θ(l) and the output H(l), while the second term is the priors-based graph regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Besides, S, S+ and Ssimplex are separately de- fined to be S = {s ∈ Rd}, (20) S+ = {s ∈ Rd|s ≥ 0}, (21) and Ssimplex = {s ∈ Rd|1⊤s = 1, s ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (22) The above three sets correspond to the Identity(·), Relu(·), and Softmax(·) activation functions frequently used in graph convolutional networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' It’s claimed that by designing different regularizers, this framework can give birth to different GCN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' In the following, we will give specific details about how they could be derived from optimization (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Link Various GCNs with Various Regularizers Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The updating rule of the vanilla GCN [21] H(l) = σ � �AH(l−1)Θ(l)� , l ∈ [L], (23) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l) (24) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(l) ∈ {S or S+ or Ssimplex}, where J (l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � H⊤�LH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H, we obtain ∂J (l) ∂H = −H(l−1)Θ(l) + �LH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (26) if it ( ∂J (l) ∂H ) is further set to zero, then there yields H∗ = (I − �A)−1H(l−1)Θ(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (27) By projecting H∗ onto S(l), we could arrive at H(l) = σ(H∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (28) Notably, (I − �A)−1 = �∞ i=0 �Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' and when its first-order approximation is leveraged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', (I − �A)−1 ≈ I + ˜A = �A, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (28) gives birth to the updating rule (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is de- signed to 1 2Tr � H⊤�LH � , the framework (19) could gener- ate the vanilla GCN [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Given H(0) = f MLP Θ (X) and α ∈ [0, 1), the updating rule of APPNP [23] H(l) = σ � (1 − α) �AH(l−1) + αH(0)� , l ∈ [L], (29) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l), (30) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, where J (l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � 1 1 − αH⊤ �A−1(H − 2αH(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (31) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking the derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H and setting it to zero, we come to ∂J (l) ∂H = −H(l−1) + 1 1 − α �A−1(H − αH(0)) = 0, (32) which leads to H∗ = (1 − α) �AH(l−1) + αH(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (33) By projecting H∗ onto S(l), we could achieve H(l) = σ � (1 − α) �AH(l−1) + αH(0)� , l ∈ [L], (34) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is de- vised to 1 2Tr � 1 1−αH⊤ �A−1(H − 2αH(0)) � , the framework (19) could give birth to APPNP [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The updating rule of JKNet [24] H(l) = σ � K � k=1 αk �AkH(l−1)Θ(l) � , l ∈ [L], (35) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l) (36) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, where J (l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � H⊤ �A−1(I + β�L)H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking the derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H and setting it to zero, we have ∂J (l) ∂H = −H(l−1)Θ(l) + �A−1(I + β�L)H = 0, (38) which leads to H∗ = 1 β + 1 � I − β β + 1 �A �−1 �AH(l−1)Θ(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (39) Methods Propagation Rules Regularizer L(H(l);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G) Projective Set GCN H(l) = σ � �AH(l−1)Θ(l)� Tr � H(l)⊤�LH(l)� � S(l) = S+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex SGC H(l) = σ � �AH(l−1)Θ(l)� Tr � H(l)⊤�LH(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex APPNP H(l) = σ � (1 − α) �AH(l−1) + αH(0)� Tr � 1 1−αH(l)⊤ �A−1(H(l) − 2αH(0)) � � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex JKNet H(l) = σ ��K k=1 αk �AkH(l−1)Θ(l)� Tr � H(l)⊤ �A−1(I + β�L)H(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex DAGNN H(L) = σ ��K k=0 αk �AkH(0)� Tr � H(l)⊤(I + β�L)H(l)� � S(l) = S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex GNN-HF H(l) = σ � (I + α�L)−1(I + β�L)H(l−1)Θ(l)� Tr � H(l)⊤(I + β�L)−1(I + α�L)H(l)� � S(l) = S+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' l ∈ [L−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S(L) = Ssimplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' GNN-LF H(l) = σ � (I + α �A)−1(I + β �A)H(l−1)Θ(l)� Tr � H(l)⊤(I + β �A)−1(I + α �A)H(l)� � S(l) = S+, l ∈ [L−1], S(L) = Ssimplex tsGCN H(l) = σ � (I + α�LG + β�LX )−1H(l−1)Θ(l)� Tr � H(l)⊤(I + α�LG + β�LX )H(l)� � S(l) = S+, l ∈ [L−1], S(L) = Ssimplex Table 4: Different regularizers can derive different GCN variants under the regularizer-centered optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' It is noted that the spectral radius of β β+1 �A is smaller than one, indicating � I − β β + 1 �A �−1 = ∞ � k=0 � β β + 1 �A �k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (40) If its (K−1)-order approximation is employed, then there goes � I − β β + 1 �A �−1 ≈ K−1 � k=0 βk (β + 1)k �Ak, (41) which suggests that H∗ can be approximated by H∗ = K � k=1 βk−1 (β + 1)k �AkH(l−1)Θ(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (42) If denote αk = βk−1 (β+1)k (k ∈ [K]), then {αk}∞ k=1 is a set of parameters with �∞ k=1 αk = 1 β+1 1 1− β β+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' By projecting H∗ onto S(l), we can realize H(l) = σ � K � k=1 αk �AkH(l−1)Θ(l) � , l ∈ [L], (43) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is devised to 1 2Tr � H⊤ �A−1(I + β�L)H � , the framework (19) can produce JKNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Given H(0) = f MLP Θ (X) and a trainable pro- jection vector α ∈ RK+1, the updating rule of DAGNN [27] H(l) = σ � K � k=0 αk �AkH(0) � , (44) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l) (45) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) = S, l ∈ [L − 1], H(L) = Ssimplex, where J (l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � H⊤(I + β�L)H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (46) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking the derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H and setting it to zero, we can harvest ∂J (l) ∂H = −H(0) + (I + β�L)H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (47) Similar to the proof of Theorem 3, the K-order approxi- mation of � I + β�L �−1 = 1 β+1 � I − β β+1 �A �−1 is utilized, and then we obtain H∗ = K � k=0 αk �AkH(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (48) By projecting H∗ onto S(l), we can arrive at H(l) = σ � K � k=0 αk �AkH(0) � , l ∈ [L], (49) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is devised to 1 2Tr � H⊤(I + β�L)H � , the framework (19) can produce DAGNN [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The updating rule of GNN-HF [19] H(l) = σ((β + 1 α)I + (1 − β − 1 α) �A−1(I + β�L)H(l−1)Θ(l)), (50) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l) (51) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex, where J (l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � H⊤(I + µ�L)−1(I + λ�L)H � (52) with λ = β + 1 α − 1 and µ = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking the derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H and setting it to zero, we own ∂J (l) ∂H = −H(l−1)Θ(l)+(I+µ�L)−1(I+λ�L)H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (53) which yields H∗ = (I + λ�L)−1(I + µ�L)H(l−1)Θ(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (54) Substituting λ = β + 1 α − 1 and µ = β into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (54), we obtain (I + λ�L)−1 = � (1 + λ)I − λ �A �−1 = � (β + 1 α)I + (1 − β − 1 α) �A �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (55) By projecting H∗ onto S(l), we can ahieve H(l) = σ (H∗) , l ∈ [L], (56) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is de- vised to 1 2Tr � H⊤(I + µ�L)−1(I + λ�L)H � , the framework (19) can produce GNN-HF [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The updating rule of GNN-LF [19] H(l) = σ((βI + (1 − β) �A + ( 1 α − 1)�L)−1(βI + (1 − β) �A)H(l−1)), (57) is equivalent to solving the following optimization H(l) = arg min H∈S(l) J (l) (58) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H(l) = S+, l ∈ [L − 1], H(L) = Ssimplex, where J(l) = −Tr � H⊤H(l−1)Θ(l)� + 1 2Tr � H⊤(I + µ �A)−1(I + λ �A)H � (59) with λ = −αβ+2α−1 αβ−α+1 and µ = 1 β − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Taking the derivative of J (l) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H and setting it to zero, we can get ∂J (l) ∂H = −H(l−1)Θ(l)+(I+µ �A)−1(I+λ �A)H = 0, (60) which leads to H∗ = (I + λ �A)−1(I + µ �A)H(l−1)Θ(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (61) Absorbing the scale αβ αβ−α+1 into the to-be-learnt variable Θ(l), and substituting λ = −αβ+2α−1 αβ−α+1 and µ = 1 β − 1 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (61), we can harvest αβ αβ − α + 1(I + λ �A)−1(I + µ �A) = αβ αβ − α + 1(I + −αβ + 2α − 1 αβ − α + 1 �A)−1(I + ( 1 β − 1) �A) = � (1 − 1 β + 1 αβ )I + (−1 + 2 β − 1 αβ ) �A �−1 (I + ( 1 β − 1) �A) = � (β − 1 + 1 α)I + (−β + 2 − 1 α) �A �−1 (βI + (1 − β) �A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (62) By projecting H∗ onto S(l), we can attain H(l) = σ (H∗) , l ∈ [L], (63) For notation consistency, we denote λ and µ as α and β in Table 4, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The above analyses reveal that when the regularizer is set to 1 2Tr � H⊤(I + µ �A)−1(I + λ �A)H � , the framework (19) can generate GNN-LF [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' More Experimental Settings and Results In this part, we provide more experimental settings and re- sults for tsGCN, including hyperparameter settings, t-SNE visualizations of various methods, and the parameter sensi- tivity analysis of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hyperparameter Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' The detailed values of several hyperpaerameters are recorded in Table 5, which can be used to reproduce the reported experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' And the code is also provided as a supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' More visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' We draw the t-SNE of embeddings generated by all methods on all datasets from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 5 to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 11, from which we have the following observations: The results are generally matched with the quantitative performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', tsGCN achieves better results than the others on most datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Embeddings generated by tsGCN achieve better inter- class separation alongside intra-class clustering than those generated by tsGCN (inv), even when their quanti- tative performance is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Parameter Sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' It can be seen that the F1-scores of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (α, β) hold the similar trends with the classifi- cation accuracies of tsGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Table 5: Specific (α, β, r) and other parameters of tsGCN on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Datasets/Parameters α β r Learning rate Weight decay Hidden units Cora 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='2 ⌊d/16⌋ 1 × 10−2 5 × 10−4 32 Citeseer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='4 ⌊d/16⌋ 1 × 10−2 5 × 10−4 32 Pubmed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='3 ⌊d/2048⌋ 1 × 10−2 5 × 10−4 32 ACM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='9 ⌊d/64⌋ 1 × 10−2 5 × 10−4 32 BlogCatalog 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='5 ⌊d/64⌋ 1 × 10−2 5 × 10−4 32 CoraFull 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='1 ⌊d/8⌋ 1 × 10−2 5 × 10−4 32 Flickr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 ⌊d/64⌋ 1 × 10−2 5 × 10−4 32 UAI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='0 ⌊d/16⌋ 1 × 10−2 5 × 10−4 32 Figure 5: Different methods’ t-SNE visualizations on Cora, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 6: Different methods’ t-SNE visualizations on Citeseer, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 7: Different methods’ t-SNE visualizations on Pubmed, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 8: Different methods’ t-SNE visualizations on ACM, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Note that GraphSAGE fails to run on ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 9: Different methods’ t-SNE visualizations on CoraFull, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 10: Different methods’ t-SNE visualizations on Flickr, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Figure 11: Different methods’ t-SNE visualizations on UAI, where each color corresponds to one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' (a) Cora (e) BlogCatalog (b) Citeseer (f) CoraFull (c) Pubmed (g) Flickr (d) ACM (h) UAI Figure 12: The classification F1-scores of tsGCN w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' different hyperparameters α and β on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' References [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Guo, Multi-label im- age recognition with graph convolutional networks, in: CVPR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 5177–5186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Nie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Su, Multi-graph convolutional network for unsupervised 3d shape re- trieval, in: MM, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3395–3403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Peng, Learning to detect 3d facial landmarks via heatmap regression with graph convolutional network, in: AAAI, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 2595– 2603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Lian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Han, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xie, Relation- aware graph convolutional networks for agent-initiated social e-commerce recommendation, in: CIKM, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 529–538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Lai, Hybrid-order gated graph neural network for session-based recom- mendation, IEEE Transactions on Industrial Informat- ics 18 (3) (2022) 1458–1467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Vap, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Borowczak, Learn- ing privacy-preserving graph convolutional network with partially observed sensitive attributes, in: WWW, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3552–3561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu, Spatio-temporal graph convolu- tional networks: A deep learning framework for traffic forecasting, in: IJCAI, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3634–3640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Feng, Multi-range attentive bicomponent graph convolu- tional network for traffic forecasting, in: AAAI, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3529–3536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Pal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Coates, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' ¨Ustebay, Bayesian graph convolutional neural networks for semi- supervised classification, in: AAAI, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 5829–5836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Qi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yin, A new perspective on the effects of spectrum in graph neural networks, in: ICML, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 25261–25279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Fan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Lu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Lin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, One2multi graph autoencoder for multi-view graph clustering, in: WWW, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3070–3076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Koniusz, Simple spectral graph convolution, in: ICLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gabrys, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Mu- sial, Multi-level graph convolutional networks for cross-platform anchor link prediction, in: KDD, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1503–1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Halliwell, Evaluating explanations of relational graph convolutional network link predictions on knowledge graphs, in: AAAI, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 12880–12881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shen, Beyond low- frequency information in graph convolutional net- works, in: AAAI, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3950–3957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [16] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xing, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zheng, Shne: Semantics and homophily preserv- ing network embedding, IEEE Transactions on Neural Networks and Learning Systems (2021) 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [17] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Pei, Am- gcn: Adaptive multi-channel graph convolutional net- works, in: KDD, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1243–1253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Akoglu, Connecting graph convolutional networks and graph-regularized pca, arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='12294 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Ji, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cui, Interpreting and unifying graph neural networks with an optimization framework, in: WWW, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1215–1226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wipf, Graph neural networks inspired by classical iterative algorithms, in: ICML, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 11773–11783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Kipf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Welling, Semi-supervised classification with graph convolutional networks, in: ICLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Fifty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Weinberger, Simplifying graph convolutional networks, in: ICML, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 6861–6871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Klicpera, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bojchevski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' G¨unnemann, Predict then propagate: Graph neural networks meet person- alized pagerank, in: ICLR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Tian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Sonobe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' ichi Kawarabayashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Jegelka, Representation learning on graphs with jumping knowledge networks, in: ICML, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 5449–5458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Lin, Adagcn: Adaboosting graph convolutional networks into deep models, in: ICLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [26] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Han, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Wu, Deeper insights into graph con- volutional networks for semi-supervised learning, in: AAAI, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 3538–3545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Ji, Towards deeper graph neural networks, in: KDD, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 338–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Kang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Jin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Guo, Topology optimization based graph convolutional net- work, in: AAAI, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 4054–4061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Oono, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Suzuki, Graph neural networks exponen- tially lose expressive power for node classification, in: ICLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hammond, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Vandergheynst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Gribonval, Wavelets on graphs via spectral graph theory, Applied and Computational Harmonic Analysis 30 (2011) 129– 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [31] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Amos, Differentiable optimization-based modeling for machine learning, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' thesis, Carnegie Mellon University (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Xu, Neural diffusion distance for image seg- mentation, in: NeurIPS, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1441–1451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Defferrard, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bresson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Vandergheynst, Convo- lutional neural networks on graphs with fast localized spectral filtering, in: NeurIPS, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Hamilton, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Ying, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Leskovec, Inductive repre- sentation learning on large graphs, in: NeurIPS, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1024–1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' [35] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Velickovic, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Cucurull, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Casanova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Romero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Li`o, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' Bengio, Graph attention networks, in: ICLR, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} +page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfGwnT/content/2301.04318v1.pdf'} diff --git a/gNFKT4oBgHgl3EQftS6b/content/tmp_files/2301.11886v1.pdf.txt b/gNFKT4oBgHgl3EQftS6b/content/tmp_files/2301.11886v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd57f0623a92ff08410e2edebe6f7f1fafdd5206 --- /dev/null +++ b/gNFKT4oBgHgl3EQftS6b/content/tmp_files/2301.11886v1.pdf.txt @@ -0,0 +1,1600 @@ +Machine Learning Over Heuristic: +a Learned Cache Eviction Framework with Minimal Overhead +Dongsheng Yang1, Daniel S. Berger2, Kai Li1, and Wyatt Lloyd2 +1Princeton University +2Microsoft Research +Abstract +Recent work shows the effectiveness of Machine Learning +(ML) to reduce cache miss ratios by making better eviction de- +cisions than heuristics. However, state-of-the-art ML caches +require many predictions to make an eviction decision, mak- +ing them impractical for high-throughput caching systems. +This paper introduces Machine learning At the Tail (MAT), +a framework to build efficient ML-based caching systems by +integrating an ML module with a traditional cache system +based on a heuristic algorithm. MAT treats the heuristic al- +gorithm as a “filter” to receive high-quality samples to train +an ML model and likely candidate objects for evictions. We +evaluate MAT on 8 production workloads, spanning storage, +in-memory caching, and CDNs. The simulation experiments +show MAT reduces the number of costly ML predictions-per- +eviction from 63 to 2, while achieving comparable miss ratios +to the state-of-the-art ML cache system. We compare a MAT +prototype system with an LRU-based caching system in the +same setting and show that achieve similar request rates. +1 +Introduction +Software caching systems are ubiquitous in modern comput- +ing infrastructure. Examples of large-scale use cases include +include content delivery networks (CDNs), in-memory caches, +and storage systems. CDNs protect expensive and scarce In- +ternet backbone bandwidth and are expected to serve 72% +of Internet traffic by 2022 [16]. In-memory caches protect +computationally expensive services are extensively used in +the data centers of Facebook [32] and Twitter [42]. Storage +caches reduce the data movement of large objects in the net- +work and an essential part of cloud services [22]. +Caching systems seek to minimize their miss ratio, i.e., +the fraction of requests not served by the cache. The lower +the miss ratios, the lower the load on backend servers and +Internet traffic (for CDNs). To decide which objects to keep +in the cache, current caching systems [3, 6, 12, 32] rely on +heuristic algorithms, such as Least Recently Used (LRU), and +First In First out (FIFO), and Least Frequently Used (LFU). +Recent work [8, 13, 35, 38, 40] shows that machine learning +based eviction algorithms (ML-based caching systems) signif- +icantly outperform these heuristics by using a history of past +access patterns to predict future access patterns. These accu- +rate predictions reduce miss ratios by up to 25% compared to +heuristic caches [35]. +Bringing ML-based caching systems from research to pro- +duction faces a key challenge due to their computational over- +head and hardware cost. In particular, ML-based caching sys- +tems are not yet applicable in systems with high throughput +demands [10, 23] or when CPU resources are scarce due to +being coloated with other applications [12]. +The overhead of ML-based caches is significantly higher +than heuristic caching systems for two reasons. First, ML- +based caching systems need to update the model online fre- +quently to retrain with more recent access patterns. For ex- +ample, a state-of-the-art ML-based caching system for CDNs, +LRB [35], uses all cache requests to generate training entries, +which leads to a large training data volume and a slow training +process. +Second, ML-based caches require running many predic- +tions to find an object to evict. For example, LRB samples 64 +eviction candidates randomly within the cache to run predic- +tions. Running 64 predictions per eviction can be slow and +expensive especially in bursty production systems that can +face pressure to evict hundreds of thousands of evictions in a +second due to burst arrivals. +While the overheads of ML-based caches are known, it is +less known which of their decisions are actually required to +improve miss ratios compared to heuristics. An answer to +this question can guide applying costly ML predictions only +where they are needed. In fact, when comparing the eviction +decisions of heuristics to an offline optimal algorithm, we find +that they evict most of the objects that the optimal algorithm +evicts, but they sometimes evict objects they should keep. This +leads to our main insight that heuristic algorithms can serve +as good filters. The ML algorithm will only run predictions +on the objects evicted by the heuristic algorithm, instead of +1 +arXiv:2301.11886v1 [cs.OS] 27 Jan 2023 + +all objects in the cache. It can dramatically reduce the number +of predictions without affecting miss ratios. +We propose an efficient ML-based caching framework, Ma- +chine learning At the Tail (MAT), which builds on the insight +that we can effectively pair a heuristic with an ML predic- +tor. We define the tail as evictions of the heuristic algorithm, +e.g., the least recently used items in LRU. MAT feeds the tail +objects into a novel ML predictor. This ML predictor then de- +cides which objects to keep in the cache and which objects to +truly evict. This allows MAT to identify good objects to evict +using only a few predictions because the heuristic’s tail is a +small subset of objects in the cache. Similarly, it allows MAT +to focus its training on this small subset of objects. In turn, +this means MAT does far less computation per eviction than +prior ML-based caching system. But, because the heuristic’s +tail contains nearly all the objects that should be evicted, MAT +can achieve the same miss ratio as state-of-the-art ML-based +caching systems. +An additional challenge in many caches is handling scenar- +ios where computation power becomes scarce during certain +time intervals such as request load spiking or an increase in +higher-priority work that is collocated with the cache [12]. +MAT’s design robustly handles these scarce computation sce- +narios by falling back to the heuristic algorithm when the ML +model cannot keep up with the request load. +To compare MAT with a variety of algorithms including +LRB, the state-of-the-art ML cache, We have also imple- +mented it in a cache simulator and run experiments with 8 +production workloads from CDNs, in-memory caches, and +storage caches. Our results show that while achieving compa- +rable miss ratios, MAT dramatically reduces the overhead for +ML: it reduces the average number of predictions per evic- +tion by 31 times (from 63 to 2) and the average prediction +overhead per eviction including metadata feature building +overhead by 21 times (from 300us to 9.3us). +We have implemented MAT in Cachelib [12], which is an +open-source cache system developed by Facebook. We com- +pare its performance with the Cachelib instance with LRU al- +gorithm. Our end-to-end evaluation shows that MAT achieves +similar request rates to the LRU-based caching system. +Section 2 elaborates on the motivation of our work. Sec- +tion 3 details the observation that leads to our design. In +Section 4 we present the design of our framework MAT. Sec- +tion 5 describes how MAT is implemented in the simulator +and in the Cachelib prototype. Section 6 presents an evalu- +ation of MAT. We cover related work in Section 7 and we +conclude in Section 8. +2 +Background and Motivation +This section covers background on offline caching algorithm, +heuristic caching algorithms, and ML caching algorithms. +We will use examples from each group to study the decision +quality of heuristic algorithms in Section 3. +2.1 +Optimal Offline Algorithm +In 1966, Belady proposed the MIN algorithm that evicts a data +object whose next access occurs furthest in the future [11]. +This algorithm provably optimal for caching equal-sized data +objects [31] such as cacheline or video chunks. Since knowl- +edge about future accesses is not typically available in an +online setting, we call Belady’s MIN algorithm an offline +optimal or oracle algorithm. +2.2 +Heuristic Caching Algorithms +Figure 1: Heuristic cache algorithms that maintain the +rank of objects in a priority queue. +The most common class of caching algorithms used in +production systems are based on heuristics. A heuristic is +designed to decide which object to evict from a cache to +admit an object that is currently not cached. Most heuristic +algorithms form an explicit or implicit ranking of objects in +the cache. If the cache request is a miss, it inserts the requested +object into the ranking and evicts the object with the lowest +rank. If the cache request is a hit, it re-ranks the requested +object in the cache. Figure 1 shows the typical structure of a +heuristic caching algorithm. +A well-known example is Least-Recently-Used (LRU), +which maintains a queue to implicitly rank objects by their +most recent access times. LRU inserts the most recently ac- +cessed object at the head of the queue and evicts the one at +the tail of the queue. +The main advantage of heuristic caching algorithms is their +simplicity and efficiency. For example, LRU can be imple- +mented with a doubly-linked list as its priority queue, and a +hash table to speedup the lookup operation. This implemen- +tation of LRU is the default algorithm in many production +systems such as Cachelib [12]. +The main drawback of heuristic caching algorithms is that +they work well for certain workloads or access patterns while +working poorly for others [35]. +2.3 +ML-Based Caching Algorithms +Machine learning is changing how caches are designed. An +ML-based cache trains a model with past access patterns and +then uses the model to predict which objects in the cache +2 + +Object index +Lookup +Reinsert +Requests +Head +Tail +Insert +Evict +Priority queueshould be evicted. Recent studies [13, 35, 41, 44] show that +ML-based approaches can adapt to different workloads dy- +namically and can reduce wide area network traffic by around +20% compared to the state-of-the-art heuristic algorithms. +Two Key Challenges. +There are two key challenges with +ML-based caching systems. The first is the overhead for train- +ing ML models. Adapting to recent access patterns requires +training and updating the model frequently. This overhead +can be significant in space and time as hardware accelerators +are usually not equipped on caching servers. +The second is the overhead for making eviction decisions. +To mimic the optimal offline oracle in a straightforward way, +the ML-based caching system needs to predict the next access +times of all objects in the cache and evicts the one with the +furthest time in the future. The prediction overhead would +be prohibitively high for large caches. Therefore, ML-based +caching systems for software caching have to decide which +subset of objects to run predictions on. +Figure 2: The state-of-the-art ML-based caching system, +Learning Relaxed Belady (LRB). +ML-Based Caching by Sampling. +LRB [35] is the state- +of-the-art ML-based caching system and it is based on sam- +pling for both training and for eviction selection. Figure 2 +shows how it trains the ML model and uses the ML model to +make evictions. +LRB overcomes the first challenge by using a Gradient +Boosted Decision Tree (GBDT), a relatively simple ML ap- +proach. It trains and updates the GBDT model online with +a relatively small training dataset (about 128K objects) ran- +domly sampled from a window of recent request history. The +training overhead is small enough to update the model every +few seconds. +It overcomes the second challenge by relaxing the eviction +criteria of the Belady’s MIN algorithm. Instead of finding +the object in the cache whose next access time occurs the +furthest in the future, it picks any object whose predicted next +access time is far enough. With this approximation, LRB ap- +proach runs predictions on only k randomly sampled objects +in the cache, where k is set to 64. When k = 64, with a high +probability, LRB can find at least one object whose predicted +next access time is close to the largest next access time in the +cache. +LRB achieved better miss ratios than 14 state-of-the-art +heuristic caching algorithms over various cache sizes with 6 +production CDN workloads. Since LRB is designed for CDN +workloads whose objects are quite large and requests rates +are low, it can afford some computational overhead. However, +we find that LRB requires more than two orders of magni- +tude more CPU resources than heuristic algorithms. We seek +to reduce this overhead to enable us to deploy ML-based +algorithms in high-throughput environments, application- +embedded environments, or low-power environments such +as the Internet edge. +Insertion-time ML caching algorithms. +Another cate- +gory of ML caching algorithms runs a prediction on each +object as its request arrives [13]. It maintains a data structure +that remembers the ranking of the predicted scores of all ob- +jects in the cache and chooses the lowest ranked object for an +eviction. +The prediction overhead of this method is significant for +two main reasons. First, the number of requests can be larger +than the number of evictions by an order of magnitude, de- +pending on cache miss ratios. Second, after updating the ML +model with new training data, the previous predictions are not +consistent with the new model. It needs to rerun predictions +of all objects in the cache to make the ranking up-to-date and +consistent [44]. If the frequency of retraining the model is +high, the total cost for predictions can be extremely high. +ML-based caching systems make better eviction decisions +and thus they can save cost on storage media and network +bandwidth. However, ML-based caching systems require a lot +more computation than heuristic algorithms. In high through- +put caching systems, the available computation power is not +enough for the ML-based caching systems to keep up with the +line speed. In other cases, it is also highly preferred to reduce +the computational overhead of ML-based caching systems +to save up power budget for other applications such as edge +computing. Thus, our goal is to have as good eviction deci- +sions as the state-of-the-art ML-based caching systems, while +have orders of magnitude lower computational overhead. +3 +Heuristic Algorithms as Filters +The key idea in this paper is to use a heuristic caching al- +gorithm as a filter in front of an ML-based caching system +to reduce the predictions per eviction and the samples for +training an ML model. The question is how good heuristic +algorithms are as such filters. +3 + +Label +Requests +Training +Candidates +LRB Metadata +Sample +Randomly +Object index +Model +Sample +Randomly- +Eviction +Candidates +Choose one to evictLRU +FIFO +LFUDA +LRUK +Belady’s MIN +TTAT +TTAT +TTAT +TTAT +TTAT +CDN1 +55% +90% +65% +90% +47% +87% +363% +97% +0% +100% +CDN2 +47% +95% +57% +95% +36% +96% +487% +92% +0% +100% +CDN3 +42% +94% +67% +91% +23% +95% +41% +96% +0% +100% +Wikipedia +129% +94% +196% +87% +86% +95% +89% +93% +0% +100% +Table 1: Fractions of evicted objects whose TTA < T and TTA > T by 5 caching algorithms (LRU, FIFO, LFUDA, LRUK +and Belady’s MIN) with 4 workloads (CDN1, CDN2, CDN3 and Wikipedia). All fractions are normalized to the total +number of objects evicted by Belady’s MIN. +To answer this question, we compare the distribution of +evicted objects by a heuristic algorithm to Belady’s MIN +(optimal offline) algorithm. A good filter should pass over +most good eviction candidates that Belady’s MIN evicts, even +at the cost of passing over some bad candidates. We will first +look at LRU algorithm as a filter and then look at several other +heuristic algorithms. +Compare LRU to Belady’s MIN +Figure 3: Time-To-next-Access (TTA) distribution of the +evicted objects by LRU and Belady’s MIN algorithms. +The result is collected from the Wikipedia trace with +256GB cache size. +Figure 3 compares the distributions of evicted objects by +LRU and Belady’s MIN for Wikipedia workload with a cache +size of 256 GB. The figure groups evicted objects according to +the log of their Time-To-next-Accesses (TTAs) from the time +when they are evicted by a given algorithm. The time is the +logical time, which means it is a counter that increments on +each request by 1. The figure plots the percentage of evictions +in each group normalized by the total number of evictions of +the Belady’s MIN algorithm. +The threshold T in the figure separates good eviction de- +cisions from bad ones. All evicted objects by Belady’s MIN +have their TTAs > T (on the right hand side of the threshold), +none have TTAs < T (on the left hand side of threshold). +We have two observations about the distributions of LRU. +First, the total number of evicted objects whose TTAs > T by +LRU is close to that by Belady’s MIN. This means that LRU +evicts most of objects that Belady’s MIN does. In other words, +LRU rarely keeps objects it should evict. This indicates that in +most cases, LRU does not filter out most of the good eviction +candidates. +Second, the number of objects evicted by LRU on the left +hand side of the threshold is similar to that on the right hand +side. In other words, although LRU frequently evicts objects +it should keep, one of every two evictions is a good decision +on average for this workload. This means that using LRU as a +filter, we can reduce the number of predictions from 64 to 2! +Compare Other Heuristics to Belady’s MIN. +Table 1 shows the distributions of good and bad eviction +decisions with four workloads by LRU, FIFO, LFUDA, LRUK +and Belady’s MIN algorithms. As Belady’s MIN is an optimal +offline algorithm, 100% of its evicted objects have TTA > T. +The main observation is that all heuristic algorithms in the +table can serve as filters well. They can evict 87-97% of the +objects (TTA > T) Belady’s MIN evicts. +Among these heuristic algorithms as filters, LRU and +LFUDA are better overall. LRU evicts 90%, 95%, 94%, and +94% of the objects that Belady’s MIN evicts with CD1, CDN2, +DDN3 and Wiki workloads respectively. LFUDA evicts 87%, +96%, 95%, and 95% of those that Belady’s MIN evicts respec- +tively. LFUDA has the smallest fractions (23-86%) of evicted +objects whose TTA < T. +LRUK achieves the highest coverage (92-97%) of the ob- +jects that Belady’s MIN evicts, but it has large fractions of bad +eviction decisions with CDN1 and CDN2 worklaods (363% +and 487% respectively). +FIFO can also be a good filter but it is slightly worse for +Wiki workload, evicting 87% of the objects that Belady’s +MIN evicts. +Filtering Training Samples +Using a heuristic caching algorithm as a filter can deliver bet- +ter training samples than random sampling. Our key insight is +that since the ML-based caching algorithm takes objects from +the tail of the heuristic algorithm as eviction candidates, the +ML model needs to learn only from such historical candidates +to make better predictions. +4 + +Threshold T +Fraction of evicted objects +50% +Should not evict +Should evict +40% +LRU +■Belady +30% +20% +10% +0% +¥1112　131415　16　17　18　1920 ≥21 +≤10 +log(Time-To-next-Access)Figure 4: The architecture and the data flow of the MAT system. The ML module makes predictions on objects removed +from the tail of the heuristic cache and decides to insert the objects back or evict the objects. +4 +Machine Learning at the Tail +This section describes our approach called Machine Learning +at the Tail (MAT). Section 4.1 describes the two components +of MAT, which are the heuristic cache system and the ML +module, and their interfaces. Section 4.2 describes and dis- +cusses how MAT uses a dynamic threshold to decide which +object should be evicted. Section 4.3 introduces an imple- +mentation of the machine learning method of MAT. Finally, +Section 4.3 describes how training data are generated in MAT. +4.1 +Architecture and Algorithm +Figure 4 shows the architecture of MAT, consisting of two +main modules: a heuristic cache and an ML module. +Heuristic cache. +It is a traditional cache using a priority- +queue based heuristic algorithm. It needs to provide two calls +to interface with the ML module: +• RemoveFromTail(): removes an object from the the tail +of the priority queue(s) of the heuristic algorithm and +return the object to the ML model. +• Insert(x, rank): insert object x back to a priority queue +of the heuristic algorithm. The ML model can inform +the heuristic algorithm to insert to a specific position of +the queue by providing the rank. +In the case that the heuristic algorithm uses multiple queues, +such as Segmented LRU, 2Q, and TinyLFU, the two proce- +dures need to pick one of the queues. +The MAT framework is general since most heuristic al- +gorithms use priority queue(s). We have implemented and +experimented MAT with LRU, 2Q [25] and TinyLFU [19] +algorithms. The two calls are simple to implement. +ML Module. +In the MAT design, it consists of a training +pipeline and a prediction (or inference) pipeline. The predic- +tion pipeline implements Evict() which returns an object for +eviction. +The main data structure in the training pipeline is a training +dataset, which is the recent historical candidates from the +candidate queue. The training thread uses the dataset to train +a model and update the current model with the newly trained +model. +Two kinds of threads are used in the prediction pipeline: +ML threads and eviction thread, connected by two queues: +candidate queue and eviction queue. +An ML thread removes a batch of candidate objects from +the candidate queue, predicts the time-to-next-access (TTA) +of each object in a way similar to that of LRB [35]. If the +TTA is greater than threshold T, the object will be put on the +eviction queue. +The eviction thread is responsible for removing ob- +jects from the tail of the heuristic algorithm (by calling +RemoveFromTail()) and putting them in the candidate queue, +and insert them back into the priority queue(s) (by calling +Insert()). +When the cache system needs to evict an object from the +cache, it will call Evict() which will remove an object from +the eviction queue and return it as the object for eviction. If +the eviction queue is empty, Evict() will remove an object +at the tail of priority queue(s) of the heuristic algorithm. In +either case, the cache system will evict the returned object +from the cache and also removes it from its related priority +queue. +The main advantage to allow the cache system to go ahead +when the eviction queue is empty is that the cache system can +run at a speed (or throughput) similar to the heuristic cache +system with a ML module. In this case, eviction decisions +can fall back to the heuristic algorithm. The cache system +continues functioning well when the ML thread is slow or +even fails. +5 + +Requests +Training +Training +data +thread +Cache +Heuristic cache +Candidate queue +algorithm gueue +RemoveFromTail() +Eviction +ML +Model +thread +thread +Insert(), Evict() +Eviction queue +Heuristic cache system +ML module4.2 +Eviction Decision +Algorithm 1: MAT Eviction +Input: The expected number of predictions per +eviction k; The TTA threshold T; +r := 1; +while r ≤ L do +obj := Heuristic.RemoveFromTail(); +TTA := MLModel.Predict(obj); +if TTA ≥ T then +break; +else +r += 1; +Heuristic.Insert(obj, TTA); +end +end +if r > k then +T *= (1-δ); +end +if r < k then +T *= (1+δ); +end +return obj; +As shown in Algorithm 1, the ML thread evaluates eviction +candidates one at a time by predicting its TTA and compares +it with a threshold T. If the TTA of object x is ≥ T, object x +will be put on the eviction queue. +If the number of iterations reaches a limit L, it means none +of the L candidates satisfies TTA ≥ T. In this case, we will +choose the one with the largest TTA for eviction, putting it +on the eviction queue. Our system uses L = 10 to bound the +maximal cost for an eviction decision. +The threshold T is dynamically adjusted to achieve a target +average number of predictions per eviction, denoted as k. If it +takes fewer than k iterations to find an object whose TTA ≥ T, +we will increase the threshold slightly T = (1+δ)T. If it takes +more than k iterations, we will decrease the threshold slightly +T = (1−δ)T. +The rationale to adjust the threshold T dynamically is to +tolerate variations of the workloads as request distributions +change over time. We find that there is no optimal constant +value of T for an entire workload. Our system uses δ = 1e−4 +as the default. +What happen to the objects inserted back to the priority +queue? These are the objects that the heuristic algorithm +would like to evict, but the ML model disagrees. When an +object is inserted back into the priority queue, it may take a +while for it to reach the tail of the queue again. By then, its +metadata (e.g., the time since last access) has changed. The +next time it becomes a candidate, the ML model might choose +to evict it. +40 +60 +80 +100 +Number of Requests (Million) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +TTA Threshold T +1e8 +Wikipedia +(a) Optimal threshold +2 +4 +8 +16 +32 +64 +# predictions per eviction (k) +0 +10 +20 +30 +Miss ratio reduction of + Threshold over Top-1 (%) +CDN1 +CDN2 +CDN3 +Wikipedia +(b) Threshold vs. Top-1 +Figure 5: Why using dynamic threshold to select object +to evict. (a) The best TTA threshold is continuously changing. +(b) The dynamic threshold method reduces up to 30% misses +of the Top-1 method. +Why does MAT uses threshold T to make eviction deci- +sions? An alternative is to choose the object with the Top-1 +TTA among k eviction candidates from the heuristic algo- +rithm. As shown in Figure 5, there are two advantages of us- +ing threshold T over the Top-1 method. First, in the case that +the heuristic algorithm continuously sends good candidates +with TTA ≥ T, each will be selected by our threshold method, +whereas the Top-1 method will miss k −1 good candidates. +Second, in the situation that the heuristic algorithm contin- +uously sends bad candidates with TTA (< T), our threshold +method choose the one with the largest TTA among L can- +didates, whereas the Top-1 method chooses the the best one +among k candidates. Since k ≪ L, the Top-1 method may +choose many bad candidates. +4.3 +Machine Learning Methods +MAT is a general framework that can run with any supervised +ML module. Here, we will introduce one ML method as an +example. While this is our implementation of MAT, many +other implementations can also work well. +Machine learning models. +The cache request stream can +be viewed as time-series data. To predict the TTA of an object, +we are interested in the past access pattern of the object and +also in the context of other objects. The access pattern of this +object can be viewed as low dimension tabular data, while the +context is sequence data. +We have experimented with several simple and efficient +ML approaches including Linear Regression (LR), Gradi- +ent Boosted Decision Trees (GBDT), Multi-layer Perceptron +(MLP), and Recursive Neural Network (RNN). By default, +MAT uses the GDBT model as it has a better trade-off be- +tween accuracy and computational overhead. +Metadata of input objects +The ML module in the MAT- +based caching system needs to learn from the past access +patterns to perform predictions in order to make the eviction +6 + +decisions to minimize cache miss ratios. We call such data +the metadata for ML and they vary depending on the choice +of ML models. +To study MAT, we use metadata similar to those in +LRB [35]. The metadata keeps track of three clusters of fea- +tures for each object: +• Deltai is the interval between the ith and the i+1th most +recent accesses (e.g., delta1 is the interval between the +most recent and second most recent accesses). +• Exponential Decayed Counter (EDC) is a counter that is +incremented on each access and is halved after a certain +time. EDCi is halved after each 2i requests in the cache. +• Each object also has static features that are not related +to access, such as object size, object class, etc. +By default, we maintain 32 deltas, 10 EDCs and 2 static fea- +tures and the size of the metadata for each object is at most +192 Bytes. Objects with fewer past accesses will take less +space. +4.4 +Training Dataset +In the MAT framework, a training dataset is a set of recent +historical eviction candidates, as opposed to all objects. +The training data generation involves a tagging phase +and a labelling phase. When an object is selected as +an eviction candidate by the heuristic algorithm (calling +RemoveFromTail()), it is tagged as eligible for training. +When an object is requested, if it is tagged, the tag is re- +set and MAT calculates the true label of TTA with regard to +its last access. Then the object features and the label are in- +serted into the training batch. Once the training batch reaches +a predefined batch size (e.g., 1 million ), it is used to retrain a +ML model online and then replaces the current model. +This method uses the heuristic algorithm to filter out ob- +jects that are not heuristically determined eviction candidates. +The intuition is that they are not relevant to predictions, so +excluding them will not affect the learning of the ML model. +In fact, it reduces the noise in the training data and can im- +prove the accuracy of the ML model. Depending on the miss +ratio, the tagged objects can usually be 10% to 50% of all the +objects, so the amount of the training data can be reduced by +50% to 90%. +4.5 +Time-To-next-Access Prediction +The main goal of using the ML model is to predict the Time- +To-next-Access (TTA) of a given object. The inference opera- +tion with the ML model outputs the predicted distance to the +next access, which is defined as the difference between the +timestamps of the last access and the next access of an object. +TTA is calculated as this predicted distance minus the time +passed since the last access. +In the case that the predicted distance to the next access is +shorter than the time passed since the last access, we use the +time passed since the last access minus the predicted distance +to the next access as an estimation of TTA. +The intuition is that when the time passed since the last +access is only slightly larger than the predicted distance to +the next access, we still have confidence in the ML prediction +and we estimate the TTA to be small. +However, if the object still does not come and the time +past since last access becomes much larger than the predicted +distance to the next access, we want to stop keeping this object +in the cache. +5 +Implementations +5.1 +Optimizations +Batched predictions. +The basic MAT makes one eviction +decision at a time. To take advantage of modern processors, +MAT can exploit the data parallelism for eviction decisions. +This approach runs parallel predictions on B objects before +the tail of the heuristic algorithm. The predicted TTAs are +recorded in the metadata. When the ML model receives an +eviction candidate from the heuristic algorithm, it can directly +use the recorded TTA for making an eviction decision. If there +is no recorded TTA, it will initiate a new parallel prediction +task. +The batch size B has influence on the parallelism and the +miss ratio. If B is too small, the parallelism is not fully realized. +If B is too large, the prediction results are stall and the miss +ratio will be hurt. In our design, we use B = 64. +5.2 +Prototype +We have implemented a MAT prototype in Cachelib [12], +which is an open-source C++ caching library. Our implemen- +tation adds about 1,000 lines of code, with about 900 for MAT +itself and about 100 lines for integrating it into Cachelib. We +use LightGBM [27] to implement Gradient Boosted Decision +Trees. The LightGBM model has 32 trees and each tree has +no more than 32 leaves. The bagging frequency is 5 and the +bagging fraction is 0.8. The learning rate is 0.1. +5.3 +Simulators +To compare MAT to LRB, we also integrated MAT-LRU into +LRB’s simulation framework [2], The simulator measures +the miss ratios of caching algorithms by replaying all cache +requests in the traces. It only maintains metadata of objects +and does not allocate physical space for the objects. +The advantages are that it can simulate cache sizes much +larger than the memory on the simulation machine and the sys- +tem is always bottlenecked on the caching algorithm so that +we can measure the running time of the caching algorithms. +7 + +Trace +Type +Object Size +Number of Requests +Requested Bytes +Default +Cache Size +Mean +Max +Total +Unique +Total +Unique +CDN1 +CDN +2 MB +2 MB +300 M +31 M +585 TB +60 TB +4 TB +CDN2 +CDN +2 MB +2 MB +220 M +19 M +430 TB +38 TB +4 TB +CDN3 +CDN +451 KB +1 GB +200 M +22 M +72 TB +9.5 TB +4 TB +Wikipedia [7] +CDN +116 KB +1.3 GB +200 M +15 M +7.9 TB +1.7 TB +256 GB +Memcachier [4] +In-memory +4.6 KB +1 MB +500 M +9 M +1 TB +40 GB +1 GB +InMem +In-memory +337 B +400 KB +500 M +62 M +159 GB +19 GB +8 GB +IBM merged [1] +Storage +3.1 M +4 MB +500 M +30 M +1832 TB +89 TB +16 TB +Microsoft [5] +Storage +445 KB +6 MB +200 M +48 M +5.1 TB +2 TB +512 GB +Table 2: Overview of the traces used for evaluation. +We mainly use the simulator to perform an apple-to-apple +comparison between MAT and LRB. LRB is only available in +the simulator because it is non-trivial to implement a bug-free +LRB in the Cachelib prototype. The results in Section 6.2 and +6.3 are collected from the simulator. +6 +Evaluation +Our evaluation answers the following questions: +• How many predictions does MAT need for each eviction +decision to achieve comparable miss ratios to SOTA? +• What is the software overhead of MAT? +• What performance can MAT prototype system achieve? +• Is MAT sensitive to heuristic algorithm choices? +• How well can MAT tolerate slow ML predictions? +In the following, we will first describe our experimental setup, +implementations, and experimental results to answer these +questions. +6.1 +Experimental Setup +Two hardware settings are used in our experiments. All simu- +lation experiments are run on servers in Cloudlab [18], each +with two 2 GHz Intel E5-2683v3 CPUs (14 physical cores) +and 256 GB of RAM. +Prototype experiments are run on two servers in Microsoft +Azure cloud, each with a 2.4 GHz AMD EPYC 7763 CPU (48 +cores) and 378 GB of RAM. The two servers are connected +to 40 Gbps local area network. +Workloads. +We use 4 CDN traces and 4 other workloads +in our experiments. Table 2 shows the characteristics of these +workloads. In addition, +• CDN1, CDN2 are collected from different caching servers +(located in different regions) of same anonymous service +provider. +• CDN3 is collected from the caching server of another video +service provider. +• Wikipedia trace is from wikipedia servers. +• Memcachier is from a in-memory application cache. +• InMem is an anonymous trace collected from an in-memory +key value store of social media company. +• IBM merged is a combined workload of 99 traces from IBM +object store, which is a cloud storage service. We merge the +traces based on request timestamps. +• Microsoft is a storage trace from Microsoft. +Warmup. +The first 50 million requests of each trace are +used as a warm-up period. The byte miss ratios and through- +put are measured after the warmup period. +We measure 3 aspects of algorithms. +• Byte Miss Ratio: This metric is the total size of the objects +that are not present in the cache when requested divided by +the total size of the objects of all requests. This metric is an +indicator of the network traffic volume, which is the main +optimization goal for CDN caching. +• Request Processing Rate: This metric measures the number +of requests processed by the caching system per second. It +is greatly influenced by the object size in the workload. +• Software Overhead: Software overhead refers to the extra +computational overhead introduced by the ML algorithms. +We measure the normalized running time of each part of +the machine learning pipeline, including trainings, predic- +tions, and building features. The running time is normalized +by the number of evictions. In addition, we measure the +number of predictions and the number of training entries +incurred by the ML based caching algorithm for each evic- +tion. +8 + +256 +512 +1024 +2048 +4096 +Cache Size (MB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Byte Miss Ratio +MIN* +LRB(64) +LRU +MAT(2) +MAT(3) +MAT(4) +1 +2 +4 +8 +16 +Cache Size (TB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Byte Miss Ratio +(a) CDN1 +1 +2 +4 +8 +16 +Cache Size (TB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Byte Miss Ratio +(b) CDN2 +1 +2 +4 +8 +16 +Cache Size (TB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Byte Miss Ratio +(c) CDN3 +64 +128 +256 +512 +1024 +Cache Size (GB) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Byte Miss Ratio +(d) Wikipedia +256 +512 +1024 +2048 +4096 +Cache Size (MB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Byte Miss Ratio +(e) Memcachier +2 +4 +8 +16 +32 +Cache Size (GB) +0.00 +0.05 +0.10 +0.15 +Byte Miss Ratio +(f) InMem +4 +8 +16 +32 +64 +Cache Size (TB) +0.0 +0.2 +0.4 +0.6 +Byte Miss Ratio +(g) IBM merged +128 +256 +512 +1024 +2048 +Cache Size (GB) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Byte Miss Ratio +(h) Microsoft +Figure 6: Miss ratio comparisons in the simulator. The algorithm names show the number of predictions per eviction. MAT +has better or similar miss ratios compared to LRB while MAT has an order of magnitude fewer number of predictions. +6.2 +Predictions per Eviction +To answer the first question, we would like to experimen- +tally evaluate how many predictions MAT needs to make +an eviction decision, while achieving similar miss ratios to +the LRB [35] with various workloads. We conduct our ex- +periments using the simulator with 8 workloads as shown in +Figure 6. Our experiments compare 3 MAT cases (2, 3, and 4 +predictions per eviction) with LRB which runs 64 predictions +per eviction (default). +The results show that MAT reduces the number of predic- +tions by 31 times compared to LRB without degrading the +miss ratios. MAT with 2, 3, and 4 predictions per eviction +have similar miss ratios to LRB with 64 predictions. +The differences among the 4 cases are small. Figure 6a- +6d are the results on the 4 CDN traces and the miss ratios of +MAT(2), MAT(3), and MAT(4) are almost identical. MAT(2)’s +miss ratios are slightly better than LRB(64)’s miss ratios on +CDN1 and Wikipedia, while LRB(64) is slightly better on +CDN1 and CDN2. Figure 6e, 6f are for in-memory traces. +Compared to LRB(64), MAT(2) has 1% and 5% average rela- +tive reductions in the miss ratios on Memcachier and InMem, +respectively. Figure 6g, 6h show the results on storage traces. +MAT(2) has in average 1% relatively lower miss ratio than +LRB(64) on IBM merged. LRB(64) has in average 2% rela- +tively lower miss ratio than MAT(2) on Microsoft workload. +All ML algorithms achieve significant improvements over +the LRU approach. For instance, on CDN1 with a 2 TB cache +MAT has a 18% miss ratio compared to LRU’s 21%, which +would reduce wide-area traffic by (21%-18%)/21%=14%. +When the cache sizes are large, the differences among them +diminish. +However, there is still a significant gap between these algo- +rithms and the the optimal offline algorithm. As MAT frame- +work can reduce the number of predictions to 2, it allows +the community to explore more sophisticated ML models to +reduce this gap. +6.3 +Software Overhead +To see how much overhead MAT can reduce compared to +LRB with similar ML modules, we run both in the same +simulation environment with 256 GB cache size. +Table 3 shows the average prediction overhead per eviction. +The average prediction overhead including the feature build- +ing time per eviction is reduced by reduction is 32X (from +300us to 9.3us). +LRB +MAT +Reduction +Number of predictions +63.2 +2.0 +32 times +Prediction time (us) +240 +6.4 +38 times +Feature building time (us) +60 +2.9 +21 times +Total time (us) +300 +9.3 +32 times +Table 3: Average overhead per eviction (256 GB cache +size, Wikipedia workload). +9 + +128 +256 +512 +1024 +2048 +Cache Size (GB) +0.2 +0.3 +0.4 +Object Miss Ratio +LRU +MAT-LRU +2Q +MAT-2Q +Tiny +MAT-Tiny +128 +256 +512 +1024 +2048 +Cache Size (GB) +0.2 +0.3 +0.4 +Byte Miss Ratio +(a) CDN1 +128 +256 +512 +1024 +2048 +Cache Size (GB) +0.20 +0.25 +0.30 +0.35 +Byte Miss Ratio +(b) CDN2 +128 +256 +512 +1024 +2048 +Cache Size (GB) +0.2 +0.3 +0.4 +Byte Miss Ratio +(c) CDN3 +8 +16 +32 +64 +128 +256 +Cache Size (GB) +0.4 +0.5 +0.6 +0.7 +0.8 +Byte Miss Ratio +(d) Wikipedia +Figure 7: The byte miss ratios of MAT with LRU, 2Q, and TinyLFU as the base algorithms. MAT in average reduce the +byte miss ratio of LRU, 2Q, and TinyLFU by relatively 12%, 12%, and 10% respectively. +MAT reduces the average number of predictions by 31X +(from 63.2 to 2) while reducing the average feature building +time by 21X (from 60us to 2.9us). Feature building refers to +converting the object metadata to the feature matrix which +serves as the input of the ML model, for both training and +predictions. +Table 4 shows the average overheads per eviction of LRB +and MAT. The average training time reduction of MAT is +9.5X (from 160us to 16.9us). +LRU +MAT +Reduction +Number of training samples +8.6 +0.93 +9 times +Training time (us) +100 +14 +7 times +Table 4: Average training overhead per eviction ( 256 GB +cache size, Wikipedia workload). +The average number of training samples per eviction is +reduced by 9.2X (from 8.6 to 0.93). The average training time +overhead is reduced by 7X (from 100us to 14us). +In summary, the total overhead of MAT’s ML module is +23.3 us per eviction, 17X reduction over LRB. +6.4 +Prototype Performance +To evaluate the performance of MAT prototype, we compare +its performance with Cachelib with LRU algorithm. We are +interested in the request processing rate of MAT prototype +compared to that of LRU-based caching system. Since both +MAT and LRU are implemented in Cachelib. We conduct +experiments in the same setting. +To run such experiments, we extended the Cachebench +module in the Cachelib library to support running experi- +ments over a network. We implement a Cachebench client +instance, a Cachebench server instance, and an Nginx server +instance. The Cachebench client instance reads requests from +a trace file and sends them through HTTP to the Nginx server +using CURL. The Nginx server accepts requests and forwards +them to the Cachebench server instance using Fastcgi. The +Cachebench server runs either prototype which executes the +requests and returns the results using the Fastcgi API to the +Nginx server. The Nginx server then forwards the requests +back to the Cachebench client. +Trace +Cache Size +MAT +LRU +Wikipedia +32 GB +23787 req/s +24465 req/s +128 GB +25117 req/s +25577 req/s +512 GB +30258 req/s +31181 req/s +Memcachier +Infinite +52883 req/s +51380 req/s +Table 5: Request rates of MAT and LRU prototypes with +Wikipedia and Memcachier workloads. +The main result is that MAT prototype achieves similar +request rates as LRU prototype. Table 5 shows the request +rates of MAT and LRU with Wikipedia and Memcacheir work- +loads. +For Wikipedia workload whose average object size is +116KB, MAT achieves 23,878, 25,117, and 30,258 re- +quests/sec with cache sizes of 32GB, 128GB, and 512GB +respectively. It is 2.8%, 1.8%, and 3.0% slower than LRU. +The these results include the warmup period. Without the +warmup, we expect MAT will achieve higher request rates +than above since its miss ratio is substantially lower than +LRU. +The reason for using Memcachier workload is to test maxi- +mal request rates as its average object size is relatively small +(4.6KB). MAT achieves 52,883 requests/sec whereas LRU +achieves 51,380 requests/sec. In this case, MAT is 2.9% faster. +6.5 +Heuristic Algorithm Choices +To understand the effects of different heuristic algorithm +choices, we compare three heuristic algorithms (LRU, 2Q +and TinyLFU) in Cachelib with their corresponding MAT im- +plementations (MAT-LRU, MAT-2Q and MAT-TinyLFU) as +10 + +shown in Figure 7. We conduct experiments with 4 workloads: +CDN1, CDN2, CDN3 and Wikipedia. +The experiments show two results. First, MAT is not sensi- +tive to heuristic algorithm choices. MAT-LRU, MAT-2Q and +MAT-TinyLFU achieve similar miss ratios with all 4 work- +loads. MAT-TinyLFU achieves slightly worse miss ratios than +MAT-LRU and MAT-2Q with CDN2 workload. +Second, MAT can correct the eviction mistakes by its +heuristic algorithm. Dramatic examples are TinyLFU with +CDN1 and CDN2. In both cases, TinyLFU has much higher +miss ratios than LRU and 2Q. However, MAT-TinyLFU can +achieve miss ratios similar to those of MAT-LRU and MAT- +2Q. The ML model in MAT can effectively make good evic- +tion decisions, adapting to different request patterns. +6.6 +Tolerance to Slow ML Predictions +256 +1024 +Cache Size (MB) +0.00 +0.05 +0.10 +0.15 +0.20 +Reduction in Miss Ratio (%) +MAT(0us) +MAT(1us) +MAT(10us) +MAT(100us) +MAT(1ms) +LRU +Figure 8: MAT with slow Predictions. When the ML model +is slower, the miss ratio degenerates gracefully. +MAT has the ability to fall back to its heuristic algorithm +when its ML threads cannot keep up with the rate of evictions. +To answer the question how well MAT can tolerate slow ML +predictions, we conduct experiments on CDN1 by stalling +each prediction for 1 us, 10 us, 100 us, and 1 ms. Note that a +prediction itself takes 3 us on average. +Figure 8 shows that the miss ratio reduction of MAT us- +ing LRU as the baseline. MAT can always deliver better or +equal miss ratio to LRU and MAT degrades gracefully when +the ML model stalls. When the stall is 10 us, which means +the ML model runs at 25% of its full speed, the miss ratio +reduction is about 40% of the full speed ML model. This +makes MAT particularly practical for deployment because it +can run elastically with any available amount of computation +resources. +7 +Related Work +This section discusses related work on learning-based cache +algorithms and heuristic cache algorithms. We also review +systems targeting other aspects of CDN cache system. +Learning-based cache replacement +Existing works on ap- +plying machine learning cache algorithms can be categorized +into supervised learning and reinforcement learning. Most +supervised learning approaches use regression. LRB [35] is +the most similar approach to MAT. As we have seen in Sec- +tion 6, LRB processing time overhead is 17× higher than +MAT. LFO [13] uses boosting tress to predict and imitate the +admission policy of OPT. It requires complex offline training, +and thus is not practical to adapt quickly to workload changes. +LFO also performed poorly in prior experiments [35]. Similar +to regression on a single value, [44] learns the next access +distribution, and uses it to compute a utility. However, to get +the distribution training data, it is also limited to offline train- +ing. Different from regression, Parrot [29] takes a ranking +approach. Instead of predicting the time to next access, it di- +rectly learns to rank objects. This approach is more end-to-end +but the computation overhead is much higher than regression. +Another line of works [17, 28, 38] apply reinforcement +learning to cache algorithms. Instead of predicting the time +to next access, the eviction policy is directly modeled and op- +timized. Unfortunately, feedback loops in production systems +would be in the tens of millions of steps, which exceeds the +capability of current reinforcement learning frameworks. Con- +sequently, the performance of reinforcement-learning-based +approaches is worse than supervised learning. +MAT is the first learning-based algorithm that can provide +both a low miss ratio and high throughput. This is because its +design provides efficient online training and inference. +Heuristic-based cache replacement +Over the past five +decades, many cache algorithms based on heuristics have +been proposed. Prominent examples include LRU, FIFO, +SLRU [26], 2Q [25], LIRS [24], ARC [33], LeCaR [37], and +LHD [10]. MAT is agnostic to most base heuristic algorithms +and can use any priority-queue based heuristic as its base +algorithm. +Learning-augmented +cache +replacement +Several +learning-augmented replacement algorithms have been pro- +posed to combine heuristic with learning-based algorithms. +To the best of our knowledge, previous works [9, 30, 34, 39] +focus on theoretical analysis on the competitive ratio of +learning-augmented cache, or only evaluate the cache miss +ratio regardless of the overhead [15]. MAT is the first +learning-augmented cache that achieves a low byte miss +ratio and high throughput on production cache software and +workloads. +Cache admission policies +Multiple systems focus on learn- +ing smart cache admission policies while relying on exist- +ing cache replacement heuristics such as TinyLFU [19, 20], +AdaptSize [14], Flashield [21], and CacheLib [12]. While +there is no public implementation of Flashield, our evaluation +11 + +of TinyLFU, AdaptSize, and CacheLib shows that admission +policies are not sufficient to achieve state-of-the-art miss ra- +tios. LRB outperforms both systems’ miss ratio, and MAT +achieves similar miss ratios to LRB while significantly im- +proving throughput and reducing overhead. +High-throughput +in-memory +caching +systems +MemC3 [23] and LHD [10] show how to significantly +increase the throughput of in-memory caching systems based +on replacement heuristics. The throughput challenges faced +by a ML-based caching systems are different from the setting +in MemC3. Some of MemC3’s techniques, such as faster +hashing and fast approximations for a heuristic like LRU, +are complementary to MAT. In fact, MemC3’s replacement +heuristic can be used to create candidates for MAT’s eviction +filter. LHD’s primary design based on sampling is similar to +LRB. MAT effectively overcomes the challenges of sampling +which requires too many evaluations and calls to a prediction +model. SegCache [43] is designed for small objects with +TTLs. It groups objects with similar TTLs together to reduce +memory fragmentation and thus improves the hit ratios. +CDN cache systems +Many works optimize CDN cache sys- +tems from other aspects. RIPQ [36] co-locates small writes to +reduce SSD write amplification. AViC [8] designs the eviction +algorithm based on the video chunk sequential accessed at a +constant speed and leverages the properties of video delivery +to optimize the hit ratio. The design of MAT is flexible for +general cache and can be applied together with these systems. +8 +Conclusion +MAT is proposed as a general framework for building an +efficient ML-based caching system by adding an ML module +to an existing cache system based on a heuristic algorithm. +The key idea behind MAT is to treat a heuristic algorithm +as a filter for the ML module. Most heuristic algorithms can +serve as good filters, as we demonstrate that they evict most +objects an optimal algorithm evicts while evicting some ob- +jects they should keep. The role of ML module is to correct +these mistakes. +We show several simulation and prototyping results. First, +it reduces the average number of predictions per eviction +to 2, a 31 times reduction compared to the state-of-the-art +ML-based caching system while achieving comparable miss +ratios. Second, MAT is not sensitive to the choice of heuristic +algorithms. Third, MAT can fall back to the heuristic algo- +rithm which allows it to run efficiently, tolerating slow ML +inferences or lack of computing power. +The ML module used in our implementations is Gradient +Boosted Decision Tree (GBDT) due to its simplicity and effi- +ciency. Other ML methods that are less efficient may further +reduce miss ratios. Since MAT framework dramatically re- +duces the ML overhead to merely 2 predictions per eviction, +it enables us to explore some sophisticated ML approaches. +References +[1] Ibm object storage. +http://iotta.snia.org/traces/ +key-value/36305. +[2] Lrb github repository. +https://github.com/sunnyszy/ +lrb. +[3] memcached - a distributed memory object caching system. +http://memcached.org/, . +[4] Memcachier. https://www.memcachier.com/, . +[5] Microsoft. +http://iotta.snia.org/traces/block-io/ +388. +[6] redis. https://redis.io/. +[7] Wikipedia +trace. +http://lrb.cs.princeton.edu/ +wiki2019.tr.tar.gz. +[8] Z. Akhtar, Y. Li, R. Govindan, E. Halepovic, S. Hao, Y. Liu, and +S. Sen. Avic: a cache for adaptive bitrate video. In Proceedings +of the 15th International Conference on Emerging Networking +Experiments And Technologies, pages 305–317, 2019. +[9] A. Antoniadis, C. Coester, M. Elias, A. Polak, and B. Simon. +Online metric algorithms with untrusted predictions. In In- +ternational Conference on Machine Learning, pages 345–355. +PMLR, 2020. +[10] N. Beckmann, H. Chen, and A. Cidon. LHD: Improving cache +hit rate by maximizing hit density. In USENIX NSDI, pages +389–403, 2018. +[11] L. A. Belady. A study of replacement algorithms for a virtual- +storage computer. IBM Systems journal, 5(2):78–101, 1966. +[12] B. Berg, D. S. Berger, S. McAllister, I. Grosof, S. Gunasekar, +J. Lu, M. Uhlar, J. Carrig, N. Beckmann, M. Harchol-Balter, +et al. The cachelib caching engine: Design and experiences +at scale. In 14th {USENIX} Symposium on Operating Sys- +tems Design and Implementation ({OSDI} 20), pages 753–768, +2020. +[13] D. S. Berger. Towards lightweight and robust machine learning +for cdn caching. In ACM HotNets, pages 134–140, 2018. +[14] D. S. Berger, R. Sitaraman, and M. Harchol-Balter. Adapt- +size: Orchestrating the hot object memory cache in a content +delivery network. In USENIX NSDI, pages 483–498, 2017. +[15] J. Chł˛edowski, A. Polak, B. Szabucki, and K. T. ˙Zołna. Robust +learning-augmented caching: An experimental study. In Inter- +national Conference on Machine Learning, pages 1920–1930. +PMLR, 2021. +[16] CISCO. +Cisco visual networking index: Forecast and +trends +2022, February +2019. +Available +at https: +//www.cisco.com/c/en/us/solutions/collateral/ +service-provider/visual-networking-index-vni/ +white-paper-c11-741490.pdf, accessed 09/18/19. +[17] R. Costa and J. Pazos. Mlcache: A multi-armed bandit policy +for an operating system page cache. Technical report, Technical +Report. University of British Columbia, 2017. +[18] D. Duplyakin, R. Ricci, A. Maricq, G. Wong, J. Duerig, E. Eide, +12 + +L. Stoller, M. Hibler, D. Johnson, K. Webb, A. Akella, K. Wang, +G. Ricart, L. Landweber, C. Elliott, M. Zink, E. Cecchet, S. Kar, +and P. Mishra. The design and operation of CloudLab. In Pro- +ceedings of the USENIX Annual Technical Conference (ATC), +pages 1–14, July 2019. URL https://www.flux.utah.edu/ +paper/duplyakin-atc19. +[19] G. Einziger and R. Friedman. Tinylfu: A highly efficient cache +admission policy. In IEEE Euromicro PDP, pages 146–153, +2014. +[20] G. Einziger, O. Eytan, R. Friedman, and B. Manes. Adaptive +software cache management. In ACM Middleware, pages 94– +106, 2018. +[21] A. Eisenman, A. Cidon, E. Pergament, O. Haimovich, +R. Stutsman, M. Alizadeh, and S. Katti. Flashield: a hybrid +key-value cache that controls flash write amplification. In +USENIX NSDI, pages 65–78, 2019. +[22] O. Eytan, D. Harnik, E. Ofer, R. Friedman, and R. Kat. It’s +time to revisit {LRU} vs.{FIFO}. In 12th USENIX Workshop +on Hot Topics in Storage and File Systems (HotStorage 20), +2020. +[23] B. Fan, D. G. Andersen, and M. Kaminsky. MemC3: Compact +and concurrent memcache with dumber caching and smarter +hashing. In USENIX NSDI, pages 371–384, 2013. +[24] S. Jiang and X. Zhang. LIRS: an efficient low inter-reference +recency set replacement policy to improve buffer cache perfor- +mance. ACM SIGMETRICS, 30(1):31–42, 2002. +[25] T. Johnson and D. Shasha. 2Q: A low overhead high perfor- +mance buffer management replacement algorithm. In VLDB, +pages 439–450, 1994. +[26] R. Karedla, J. S. Love, and B. G. Wherry. Caching strategies +to improve disk system performance. IEEE Computer, 27(3): +38–46, 1994. +[27] G. Ke, Q. Meng, T. Finley, T. Wang, W. Chen, W. Ma, Q. Ye, +and T.-Y. Liu. Lightgbm: A highly efficient gradient boosting +decision tree. In Advances in Neural Information Processing +Systems, pages 3146–3154, 2017. +[28] V. Kirilin, A. Sundarrajan, S. Gorinsky, and R. K. Sitaraman. +Rl-cache: Learning-based cache admission for content delivery. +IEEE Journal on Selected Areas in Communications, 38(10): +2372–2385, 2020. +[29] E. Liu, M. Hashemi, K. Swersky, P. Ranganathan, and J. Ahn. +An imitation learning approach for cache replacement. In +International Conference on Machine Learning, pages 6237– +6247. PMLR, 2020. +[30] T. Lykouris and S. Vassilvtiskii. Competitive caching with ma- +chine learned advice. In International Conference on Machine +Learning, pages 3296–3305. PMLR, 2018. +[31] R. L. Mattson, J. Gecsei, D. R. Slutz, and I. L. Traiger. Eval- +uation techniques for storage hierarchies. In IBM Systems +journal, volume 9, pages 78–117, 1970. +[32] S. McAllister, B. Berg, J. Tutuncu-Macias, J. Yang, S. Gu- +nasekar, J. Lu, D. S. Berger, N. Beckmann, and G. R. Ganger. +Kangaroo: Caching billions of tiny objects on flash. In Pro- +ceedings of the ACM SIGOPS 28th Symposium on Operating +Systems Principles, pages 243–262, 2021. +[33] N. Megiddo and D. S. Modha. ARC: A self-tuning, low over- +head replacement cache. In USENIX FAST, volume 3, pages +115–130, 2003. +[34] D. Rohatgi. Near-optimal bounds for online caching with +machine learned advice. In Proceedings of the Fourteenth +Annual ACM-SIAM Symposium on Discrete Algorithms, pages +1834–1845. SIAM, 2020. +[35] Z. Song, D. S. Berger, K. Li, A. Shaikh, W. Lloyd, S. Ghorbani, +C. Kim, A. Akella, A. Krishnamurthy, E. Witchel, et al. Learn- +ing relaxed belady for content distribution network caching. +In 17th {USENIX} Symposium on Networked Systems Design +and Implementation ({NSDI} 20), pages 529–544, 2020. +[36] L. Tang, Q. Huang, W. Lloyd, S. Kumar, and K. Li. RIPQ: +advanced photo caching on flash for facebook. In USENIX +FAST, pages 373–386, 2015. +[37] G. Vietri, L. V. Rodriguez, W. A. Martinez, S. Lyons, J. Liu, +R. Rangaswami, M. Zhao, and G. Narasimhan. Driving cache +replacement with ML-based LeCaR. In USENIX HotStorage, +2018. +[38] H. Wang, H. He, M. Alizadeh, and H. Mao. Learning caching +policies with subsampling. In NeurIPS Machine Learning for +Systems Workshop, 2019. +[39] A. Wei. Better and simpler learning-augmented online caching. +arXiv preprint arXiv:2005.13716, 2020. +[40] G. Yan and J. Li. Rl-bélády: A unified learning framework for +content caching. In Proceedings of the 28th ACM International +Conference on Multimedia, pages 1009–1017, 2020. +[41] G. Yan, J. Li, and D. Towsley. Learning from optimal caching +for content delivery. In Proceedings of the 17th International +Conference on emerging Networking EXperiments and Tech- +nologies, pages 344–358, 2021. +[42] J. Yang, Y. Yue, and K. Rashmi. A large scale analysis of +hundreds of in-memory cache clusters at twitter. +In 14th +{USENIX} Symposium on Operating Systems Design and Im- +plementation ({OSDI} 20), pages 191–208, 2020. +[43] J. Yang, Y. Yue, and R. Vinayak. Segcache: a memory-efficient +and scalable in-memory key-value cache for small objects. In +18th USENIX Symposium on Networked Systems Design and +Implementation (NSDI 21), pages 503–518, 2021. +[44] G. Zhou and M. Maas. Learning on distributed traces for data +center storage systems. Proceedings of Machine Learning and +Systems, 3, 2021. +13 + diff --git a/gNFKT4oBgHgl3EQftS6b/content/tmp_files/load_file.txt b/gNFKT4oBgHgl3EQftS6b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce9ced6c5ecb022c314bf612ff0919787cb98b47 --- /dev/null +++ b/gNFKT4oBgHgl3EQftS6b/content/tmp_files/load_file.txt @@ -0,0 +1,997 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf,len=996 +page_content='Machine Learning Over Heuristic: a Learned Cache Eviction Framework with Minimal Overhead Dongsheng Yang1, Daniel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Berger2, Kai Li1, and Wyatt Lloyd2 1Princeton University 2Microsoft Research Abstract Recent work shows the effectiveness of Machine Learning (ML) to reduce cache miss ratios by making better eviction de- cisions than heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' However, state-of-the-art ML caches require many predictions to make an eviction decision, mak- ing them impractical for high-throughput caching systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This paper introduces Machine learning At the Tail (MAT), a framework to build efficient ML-based caching systems by integrating an ML module with a traditional cache system based on a heuristic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' MAT treats the heuristic al- gorithm as a “filter” to receive high-quality samples to train an ML model and likely candidate objects for evictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We evaluate MAT on 8 production workloads, spanning storage, in-memory caching, and CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The simulation experiments show MAT reduces the number of costly ML predictions-per- eviction from 63 to 2, while achieving comparable miss ratios to the state-of-the-art ML cache system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We compare a MAT prototype system with an LRU-based caching system in the same setting and show that achieve similar request rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 1 Introduction Software caching systems are ubiquitous in modern comput- ing infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Examples of large-scale use cases include include content delivery networks (CDNs), in-memory caches, and storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' CDNs protect expensive and scarce In- ternet backbone bandwidth and are expected to serve 72% of Internet traffic by 2022 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In-memory caches protect computationally expensive services are extensively used in the data centers of Facebook [32] and Twitter [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Storage caches reduce the data movement of large objects in the net- work and an essential part of cloud services [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Caching systems seek to minimize their miss ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=', the fraction of requests not served by the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The lower the miss ratios, the lower the load on backend servers and Internet traffic (for CDNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' To decide which objects to keep in the cache, current caching systems [3, 6, 12, 32] rely on heuristic algorithms, such as Least Recently Used (LRU), and First In First out (FIFO), and Least Frequently Used (LFU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Recent work [8, 13, 35, 38, 40] shows that machine learning based eviction algorithms (ML-based caching systems) signif- icantly outperform these heuristics by using a history of past access patterns to predict future access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' These accu- rate predictions reduce miss ratios by up to 25% compared to heuristic caches [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Bringing ML-based caching systems from research to pro- duction faces a key challenge due to their computational over- head and hardware cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In particular, ML-based caching sys- tems are not yet applicable in systems with high throughput demands [10, 23] or when CPU resources are scarce due to being coloated with other applications [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The overhead of ML-based caches is significantly higher than heuristic caching systems for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' First, ML- based caching systems need to update the model online fre- quently to retrain with more recent access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' For ex- ample, a state-of-the-art ML-based caching system for CDNs, LRB [35], uses all cache requests to generate training entries, which leads to a large training data volume and a slow training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Second, ML-based caches require running many predic- tions to find an object to evict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' For example, LRB samples 64 eviction candidates randomly within the cache to run predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Running 64 predictions per eviction can be slow and expensive especially in bursty production systems that can face pressure to evict hundreds of thousands of evictions in a second due to burst arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' While the overheads of ML-based caches are known, it is less known which of their decisions are actually required to improve miss ratios compared to heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' An answer to this question can guide applying costly ML predictions only where they are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In fact, when comparing the eviction decisions of heuristics to an offline optimal algorithm, we find that they evict most of the objects that the optimal algorithm evicts, but they sometimes evict objects they should keep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This leads to our main insight that heuristic algorithms can serve as good filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The ML algorithm will only run predictions on the objects evicted by the heuristic algorithm, instead of 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='11886v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='OS] 27 Jan 2023 all objects in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' It can dramatically reduce the number of predictions without affecting miss ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We propose an efficient ML-based caching framework, Ma- chine learning At the Tail (MAT), which builds on the insight that we can effectively pair a heuristic with an ML predic- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We define the tail as evictions of the heuristic algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=', the least recently used items in LRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' MAT feeds the tail objects into a novel ML predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This ML predictor then de- cides which objects to keep in the cache and which objects to truly evict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This allows MAT to identify good objects to evict using only a few predictions because the heuristic’s tail is a small subset of objects in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Similarly, it allows MAT to focus its training on this small subset of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In turn, this means MAT does far less computation per eviction than prior ML-based caching system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' But, because the heuristic’s tail contains nearly all the objects that should be evicted, MAT can achieve the same miss ratio as state-of-the-art ML-based caching systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' An additional challenge in many caches is handling scenar- ios where computation power becomes scarce during certain time intervals such as request load spiking or an increase in higher-priority work that is collocated with the cache [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' MAT’s design robustly handles these scarce computation sce- narios by falling back to the heuristic algorithm when the ML model cannot keep up with the request load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' To compare MAT with a variety of algorithms including LRB, the state-of-the-art ML cache, We have also imple- mented it in a cache simulator and run experiments with 8 production workloads from CDNs, in-memory caches, and storage caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Our results show that while achieving compa- rable miss ratios, MAT dramatically reduces the overhead for ML: it reduces the average number of predictions per evic- tion by 31 times (from 63 to 2) and the average prediction overhead per eviction including metadata feature building overhead by 21 times (from 300us to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='3us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We have implemented MAT in Cachelib [12], which is an open-source cache system developed by Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We com- pare its performance with the Cachelib instance with LRU al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Our end-to-end evaluation shows that MAT achieves similar request rates to the LRU-based caching system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Section 2 elaborates on the motivation of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Sec- tion 3 details the observation that leads to our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In Section 4 we present the design of our framework MAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Sec- tion 5 describes how MAT is implemented in the simulator and in the Cachelib prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Section 6 presents an evalu- ation of MAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We cover related work in Section 7 and we conclude in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 2 Background and Motivation This section covers background on offline caching algorithm, heuristic caching algorithms, and ML caching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We will use examples from each group to study the decision quality of heuristic algorithms in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='1 Optimal Offline Algorithm In 1966, Belady proposed the MIN algorithm that evicts a data object whose next access occurs furthest in the future [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This algorithm provably optimal for caching equal-sized data objects [31] such as cacheline or video chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Since knowl- edge about future accesses is not typically available in an online setting, we call Belady’s MIN algorithm an offline optimal or oracle algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='2 Heuristic Caching Algorithms Figure 1: Heuristic cache algorithms that maintain the rank of objects in a priority queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The most common class of caching algorithms used in production systems are based on heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' A heuristic is designed to decide which object to evict from a cache to admit an object that is currently not cached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Most heuristic algorithms form an explicit or implicit ranking of objects in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' If the cache request is a miss, it inserts the requested object into the ranking and evicts the object with the lowest rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' If the cache request is a hit, it re-ranks the requested object in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Figure 1 shows the typical structure of a heuristic caching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' A well-known example is Least-Recently-Used (LRU), which maintains a queue to implicitly rank objects by their most recent access times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' LRU inserts the most recently ac- cessed object at the head of the queue and evicts the one at the tail of the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The main advantage of heuristic caching algorithms is their simplicity and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' For example, LRU can be imple- mented with a doubly-linked list as its priority queue, and a hash table to speedup the lookup operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This implemen- tation of LRU is the default algorithm in many production systems such as Cachelib [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The main drawback of heuristic caching algorithms is that they work well for certain workloads or access patterns while working poorly for others [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='3 ML-Based Caching Algorithms Machine learning is changing how caches are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' An ML-based cache trains a model with past access patterns and then uses the model to predict which objects in the cache 2 Object index Lookup Reinsert Requests Head Tail Insert Evict Priority queueshould be evicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Recent studies [13, 35, 41, 44] show that ML-based approaches can adapt to different workloads dy- namically and can reduce wide area network traffic by around 20% compared to the state-of-the-art heuristic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Two Key Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' There are two key challenges with ML-based caching systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The first is the overhead for train- ing ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Adapting to recent access patterns requires training and updating the model frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' This overhead can be significant in space and time as hardware accelerators are usually not equipped on caching servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The second is the overhead for making eviction decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' To mimic the optimal offline oracle in a straightforward way, the ML-based caching system needs to predict the next access times of all objects in the cache and evicts the one with the furthest time in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The prediction overhead would be prohibitively high for large caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Therefore, ML-based caching systems for software caching have to decide which subset of objects to run predictions on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Figure 2: The state-of-the-art ML-based caching system, Learning Relaxed Belady (LRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' ML-Based Caching by Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' LRB [35] is the state- of-the-art ML-based caching system and it is based on sam- pling for both training and for eviction selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Figure 2 shows how it trains the ML model and uses the ML model to make evictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' LRB overcomes the first challenge by using a Gradient Boosted Decision Tree (GBDT), a relatively simple ML ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' It trains and updates the GBDT model online with a relatively small training dataset (about 128K objects) ran- domly sampled from a window of recent request history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The training overhead is small enough to update the model every few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' It overcomes the second challenge by relaxing the eviction criteria of the Belady’s MIN algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Instead of finding the object in the cache whose next access time occurs the furthest in the future, it picks any object whose predicted next access time is far enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' With this approximation, LRB ap- proach runs predictions on only k randomly sampled objects in the cache, where k is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' When k = 64, with a high probability, LRB can find at least one object whose predicted next access time is close to the largest next access time in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' LRB achieved better miss ratios than 14 state-of-the-art heuristic caching algorithms over various cache sizes with 6 production CDN workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Since LRB is designed for CDN workloads whose objects are quite large and requests rates are low, it can afford some computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' However, we find that LRB requires more than two orders of magni- tude more CPU resources than heuristic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' We seek to reduce this overhead to enable us to deploy ML-based algorithms in high-throughput environments, application- embedded environments, or low-power environments such as the Internet edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Insertion-time ML caching algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Another cate- gory of ML caching algorithms runs a prediction on each object as its request arrives [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' It maintains a data structure that remembers the ranking of the predicted scores of all ob- jects in the cache and chooses the lowest ranked object for an eviction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The prediction overhead of this method is significant for two main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' First, the number of requests can be larger than the number of evictions by an order of magnitude, de- pending on cache miss ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Second, after updating the ML model with new training data, the previous predictions are not consistent with the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' It needs to rerun predictions of all objects in the cache to make the ranking up-to-date and consistent [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' If the frequency of retraining the model is high, the total cost for predictions can be extremely high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' ML-based caching systems make better eviction decisions and thus they can save cost on storage media and network bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' However, ML-based caching systems require a lot more computation than heuristic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In high through- put caching systems, the available computation power is not enough for the ML-based caching systems to keep up with the line speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' In other cases, it is also highly preferred to reduce the computational overhead of ML-based caching systems to save up power budget for other applications such as edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' Thus, our goal is to have as good eviction deci- sions as the state-of-the-art ML-based caching systems, while have orders of magnitude lower computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' 3 Heuristic Algorithms as Filters The key idea in this paper is to use a heuristic caching al- gorithm as a filter in front of an ML-based caching system to reduce the predictions per eviction and the samples for training an ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' The question is how good heuristic algorithms are as such filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Requests ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='LRB Metadata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Randomly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Object index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Randomly- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Eviction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Candidates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Choose one to evictLRU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='FIFO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='LFUDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='LRUK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='Belady’s MIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFKT4oBgHgl3EQftS6b/content/2301.11886v1.pdf'} +page_content='TTA 0 to create an observable +space of dimension p. This yields a trajectory in the form y(t) = S(x(t)) ∈ Rp, +where we define the sampling map +S : Rn → Rp, +x �→ +� +������ +µ(x) +µ(F τ(x)) +µ(F 2τ(x)) +... +µ(F (p−1)τ(x)) +� +������ +. +(9) +An important question is how invariant sets of system (1) in Rn are re- +produced in the observable space Rp. In particular, when the full state space +trajectory x(t) resides on a d-dimensional invariant manifold M, will y(t) also +do so? Takens’s embedding theorem gives an affirmative answer. It states that +if µ is generic and no small integer multiple of τ coincides with the period of +any possible periodic orbit of (1) lying in M, then for +p ≥ 2d + 1, +(10) +6 + +the manifold M will have a diffeomorphic copy +˜ +M in Rp via the mapping +(9) [47]. Whereas Takens’s theorem was formulated only for scalar observable +functions, this result has since been extended to multi-dimensional µ as long as +the total observable space dimension exceeds 2d [65]. +Both the nonlinear geometry and dynamics of M and the observable function +influence the geometry of ˜ +M. It is therefore difficult to predict its geometry for +a general flow map. Around the fixed point q = S(0) ∈ Rp, however, the O(1) +expansion of ˜ +M, i.e., its tangent space Tq ˜ +M, can be directly determined, as we +will show next. Note that since the flow map is the identity at the origin, q lies +on the diagonal in the observable space, with each of its identical components +given by µ(0). +We start by rewriting (1) in modal coordinates: +˙z = f(z) = Λz + E−1g(Ez), +(11) +where E = [e1, . . . , en] contains the eigenvectors of A and Λ = diag(λ1, . . . , λn) +the corresponding eigenvalues, which we assume to be distinct. We define modal +coordinates z ∈ Cn by letting z = E−1x. Whereas the observable function is +defined as a function of x, it is notationally convenient to define it as a function +of z, as µ(x) = µ(Ez). +Let M be a d-dimensional invariant manifold of (1) intersecting the origin +0 ∈ Rn, where it is tangent to a set of d eigenvectors e1, e2, . . . , ed of A with +corresponding eigenvalues λ1, . . . , λd. We define the Vandermonde matrix V ∈ +Cp×d of the d eigenvalues governing the linearized dynamics on M as Vjk = +eλkjτ, i.e., +V = +� +������ +1 +1 +. . . +1 +eλ1τ +eλ2τ +. . . +eλdτ +e2λ1τ +e2λ2τ +. . . +e2λdτ +... +... +... +... +e(p−1)λ1τ +e(p−1)λ2τ +. . . +e(p−1)λdτ +� +������ +. +(12) +Theorem 1. Under the assumptions of a generic observable function µ : Rn → +R and distinct eigenvalues λ1 ̸= . . . ̸= λd, the tangent space of the observable +manifold +˜ +M at the fixed point can be written +Tq ˜ +M = range V . +(13) +Proof. See Appendix A.1. +This result is illustrated in Fig. 1. Note that the observable function must +have full rank, as spelled out in the following remark. +Remark 1. For (13) to hold, we must have +∂µ +∂zk |0 ̸= 0 ∀k ∈ {1, . . . , d}, which +defines the genericity of µ. If the gradient of the observable function is orthog- +onal to any of the eigenvectors e1, . . . , ed, the sampling map S will not be an +embedding of M. +7 + +Manifold in full phase space +Manifold in delay embedding space +� +���� +1 +eλ1τ +e2λ1τ +... +� +���� +� +���� +1 +eλ2τ +e2λ2τ +... +� +���� +Sampling map +S : Rn → Rp +Tangent space +embedding DS(T0M) +x1 +x2 +x3,...,n +y1 +y2 +y3,...,p +M +T0M +˜ +M +Tq ˜ +M +Figure 1: Delay embedding of the tangent space T0M of an invariant manifold +M. The full state space manifold M (left) has a diffeomorphic copy +˜ +M in the +observable space (right) by Takens’s theorem. The shape of the reconstructed +manifold +˜ +M depends on the flow map, but its tangent space, Tq ˜ +M, is directly +given by the eigenvalues at the fixed point, independent of the geometry of M +and the observable function µ. +This should be kept in mind particularly when dealing with symmetries of +engineering structures, as we will show in our examples below. +Theorem 2. The columns of V are eigenvectors of the linearized delay-embedded +system at the fixed point. Indeed, the dynamics in the observable space can be +written +˙y = V ΛV †(y − q) + o(|y − q|) +(14) +Proof. See Appendix A.2. +Corollary 1. In the observable space Rp, the timelag τ and the eigenvalues +λk fully determine the tangent space and the linear part of the dynamics. In +particular, the linear dynamics are independent of both the full eigenvectors and +the observable function. +In the following, we will demonstrate how this structure can be exploited for +parametrizing spectral submanifolds from data, when the corresponding eigen- +values are approximately known. +Finally, when the observable function is multi-dimensional, the tangent space +is influenced by the relative dependency of each component µℓ of the observable +function on each modal coordinate zk. +Theorem 3. For a multidimensional observable µ : Rn → Rq with components +8 + +µ1, . . . , µq, the tangent space Tq ˜ +M ⊂ Rpq can be expressed as +Tq ˜ +M = range +� +���� +V diag +� +∂µ1 +∂z +��� +0 +� +... +V diag +� +∂µq +∂z +��� +0 +� +� +���� . +(15) +Proof. See Appendix A.3. +When the observable function is a set of displacements, the linearized multi- +dimensional observable function +∂µ +∂z +��� +0 corresponds to the mode shapes of the +system in terms of those displacements. +Therefore, if the mode shapes and +eigenvalues of the observed system are known, we can directly compute the +tangent space of ˜ +M. In the special case of a scalar observable, the tangent space +is independent of the observable function and we do not need any information +about the mode shapes. +3.2 +Delay-embedded spectral submanifold reconstruction +These theoretical results can be exploited as a constraint to aid SSM identi- +fication from data. In the case of a scalar signal and with the eigenvalues of +interest λ1, . . . , λd approximately known, we select the matrix representation of +the tangent space T appearing in (3) as the Vandermonde matrix (12), i.e., +T := V . +(16) +We have seen that the gradient of a multi-dimensional observable func- +tion µ enters the expression for the tangent space (15). +While an expres- +sion for this gradient is typically not available in experiments, mode shapes +ˆE = [ˆe1, . . . , ˆed] ∈ Rq×d are often known from theory or obtained experimen- +tally. Here, each mode shape ˆek ∈ Rq describes how the eigenvector ek ∈ Rn is +observed. Specifically, they are related by +ˆek = ck +∂µ +∂x +���� +0 +ek, +(17) +where ck ∈ C is a nonzero constant that only rescales the eigenvectors. We +select T as the columnwise Kronecker product of the Vandermonde matrix and +the observable mode shapes, i.e., +T := +� +�� +V diag (ˆe1) +... +V diag (ˆed) +� +�� . +(18) +A sketch of the geometry is shown in Fig. 2. In the case of unknown mode +shapes, it may be possible to first project low-amplitude data onto the delay- +embedded eigenvectors and then extract the observable mode shapes via SVD, +although we do not explore this idea further in this work. +9 + +q +µ1(x(t + τ)) +µ1(x(t)) +µ2(x(t)) +V +ˆE +T +˜ +M ⊂ Rpq +Figure 2: The tangent space Tq ˜ +M of the delay-embedded manifold +˜ +M for a +q-dimensional observable function µ is the range of the matrix T , defined as the +columnwise Kronecker product of the Vandermonde matrix V and the mode +shapes ˆE in terms of the observable. +Prescribing T and projecting the data onto its columns yields modal reduced +coordinates. This diagonalization of the system simplifies the learning of the +geometry and the reduced dynamics of the SSM via the algorithm outlined in +Sect. 2.2. +Choosing proper delay-embedding parameters to reconstruct nonlinear sys- +tems can be a challenge. For the linear part of the system, however, our results +suggest picking the timelag τ and embedding dimensionality p so as to obtain +numerically favorable reduced coordinates along the SSM. We ideally want the +columns of the Vandermonde matrix (12) to be orthogonal in order to maxi- +mize the signal-to-noise ratio in each of the observed modes. To this end, we +formulate a minimization problem, +(κ⋆, p⋆) = argmin +κ,p∈N+ +��V (κ∆t, p)⊤V (κ∆t, p) − I +�� +F , +(19) +in which the columns of V are normalized and ∥·∥F denotes the Frobenius norm. +Since the timelag is an integer multiple of the sampling timestep, τ = κ∆t, (19) +defines an optimization over a set of discrete variables which can be solved +simply by brute force. +Bearing in mind the nonlinear part of the system, however, an optimal choice +of delay parameters is not as straightforward. Increasing the timelag and em- +bedding dimension tends to curve the SSM, requiring higher orders of approx- +10 + +imation and in extreme cases folding the manifold, so that it can no longer be +parametrized as a graph. Taking into account the nonlinear part of the SSM +reconstruction, therefore, we typically want the total delay embedding to be as +small as possible. While solving (19) gives some guidance, a suitable choice of τ +and p will also depend on the nonlinearity of the system in the data range and +the amount of signal noise. +4 +Applications +We now apply our method to three datasets: two from simulations and one from +experiments. The eigenvalues in these examples are known either from theory +or simulations. We infer the delay-embedded tangent space accordingly before +parametrizing the SSM. The examples include an oscillator chain, a clamped- +clamped beam and tank sloshing. +4.1 +Two-degree-of-freedom oscillator with nonlinear springs +As our first example, we consider an oscillator chain of two masses, both at- +tached with linear springs to each other and to the ground. In addition, the +spring connecting the left mass to the ground has a quadratic softening non- +linearity and the spring connecting the masses is of cubic hardening type. The +masses and linear spring stiffnesses are set to 1, the softening parameter is −2 +and the hardening parameter is 1. Each of the springs also has a linear damp- +ing coefficient of 0.03. Fig. 3a shows the configuration. The sampling time is +∆t = 0.1 s. +m +x1(t) +k, κ +c +m +x2(t) +k, γ +c +k +c +(a) +-0.5 +0.5 +0 +-0.5 +x4 +0.5 +0 +0 +x1 +x3 +0.5 -0.5 +M +Data +Re E1 +Im E1 +T0M +(b) +0.5 +-0.5 +0.5 +0 +0 +x1(t) +x1(t + 30"t) +0.5 +0 +x1(t + 15"t) +-0.5 +-0.5 +~ +M +Data +Re V1 +Im V1 +T0 ~ +M +(c) +Figure 3: (a) Setup for the two-degree-of-freedom oscillator example with two +nonlinear springs. (b) The slow 2D SSM (gray) in the full state space, along +with its tangent space (red). (c) The delay-embedded SSM in the observable +space. +We compute an initial condition on the slow 2D SSM for the single training +trajectory using SSMTool. Our observable function is the first mass displace- +ment, µ(x) = x1. The trajectory in the full phase space and the SSM are shown +in Fig. 3b. The first two eigenvectors span the tangent space of the SSM. +11 + +Next, we delay embed the trajectory with a timelag τ = 15∆t and embedding +dimension p = 5, and seek the 2D SSM in this observable space using fastSSM. +We obtain reduced coordinates by projection of the trajectory data onto the +columns of the Vandermonde matrix V as predicted by our theory. +Fig. 3c +shows the SSM in the first three coordinates of the observable space. Indeed, +the tangent space of this observable space is identical to the column space of +V . +0.5 +-0.5 +0.5 +0 +0 +x2(t) +x2(t + 30"t) +0.5 +0 +x2(t + 15"t) +-0.5 +-0.5 +~ +M +Data +Re V1 +Im V1 +T0 ~ +M +(a) +0.5 +-0.5 +0 +0 +x3(t) + x4(t) +x3(t + 30"t) + x4(t + 30"t) +0.5 +0.5 +x3(t + 15"t) + x4(t + 15"t) +0 +-0.5 +-0.5 +~ +M +Data +Re V1 +Im V1 +T0 ~ +M +(b) +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +!x1(t + 30"t) + x2(t + 30"t) +-0.2 +0.1 +-0.1 +!x1(t) + x2(t) +0 +!x1(t + 15"t) + x2(t + 15"t) +0 +-0.1 +0.1 +-0.2 +Data +(c) +Figure 4: (a,b) Changing the observable function leads to different SSM geome- +tries, but the tangent space remains the same as in Fig. 3c. (c) A nongeneric +observable function however, observing only the second mode, does not embed +the manifold. +Corollary 1 predicts that this tangent space will be independent of the ob- +servable function, provided that it is generic. To illustrate this, we plot the +delay-embedded SSM for various observable functions, µ(x) = x1 (Fig. 3c), +µ(x) = x2 (Fig. 4a), and µ(x) = ˙x1 + ˙x2 (Fig. 4b). These different observable +functions clearly produce different SSM geometries, but the eigenvectors and +tangent spaces of the manifolds all agree. +One exception is when we observe the distance between the masses, µ(x) = +x2 − x1 (Fig. 4c). In this case, the delay-embedded trajectory no longer lies on +an invariant manifold, as is evident by the nonsmooth cusp in the data at the +origin. The reason is that this observable is non-generic precisely in the sense +of our theory; it coincides with the mode shape of the second, fast mode of the +full system. This means that the observable function acts orthogonally to the +slow SSM at the fixed point and thus the delay mapping is not an embedding, +by Remark 1. +Next, we pick µ(x) = x2 and use fastSSM to approximate the cubic order +reduced dynamics on the SSM from the data. +Computing the normal form +yields +� ˙ρ1 +˙θ1 +� += +� −0.0014 ρ13 − 0.0148 ρ1 +1.0025 − 0.0919 ρ12 +� +. +(20) +The trajectory projected onto the columns of V is shown in Fig. 5a. Integrating +the obtained normal form and mapping back to the observable space yields a +good reconstruction of the training data. +Finally, following Theorem 3, we demonstrate how to determine the tangent +12 + +-0.1 +0 +0.1 +Re(V yy) +-0.1 +0 +0.1 +Im(V yy) +(a) +0 +500 +1000 +1500 +2000 +time +-0.2 +0 +0.2 +0.4 +x2(t) +Simulation +Prediction +(b) +0.4 +0.2 +-0.2 +0 +x1(t) +0 +0.5 +x2(t + 15"t) +0.2 +-0.2 +0.4 +x1(t + 15"t) +0 +-0.4 +-0.5 +~ +M +Data +Re V1 +Im V1 +T0 ~ +M +(c) +Figure 5: (a) Projection of the data onto the delay-embedded tangent space +predicted by our theory. (b) fastSSM predicts a model that successfully recon- +structs the decay of the trajectory. (c) A view of the SSM in a delay-embedded +space from a multi-dimensional observable, with the tangent space predicted by +our theory. +space of the SSM at the fixed point when the observable is a vector. When +we choose µ(x) = [x1, x2]⊤, unlike for a scalar observable function, the tangent +space orientation is influenced not only by the eigenvalues, but also by the shape +of the first mode. This first mode shape corresponds to the masses moving in +unison, i.e. +ˆe1 = ˆe2 = +� 1 +1 +� +. +(21) +Then, by (18), we obtain vectors spanning the tangent space as the columns of +the matrix +T = +� V diag(ˆe1) +V diag(ˆe2) +� += +� V +V +� +, +(22) +where V is the Vandermonde matrix (12). A view of the SSM and its tangent +space in this 10-dimensional observable space is shown in Fig. 5c. +The relation of this mode shape to the observable function is given by (17). +In particular, we can compute the derivative of the observable function with +respect to the modal coordinates as +� +∂µ1 +∂z1 (0) +∂µ1 +∂z2 (0) +∂µ2 +∂z1 (0) +∂µ2 +∂z2 (0) +� += +� c1 +c2 +c1 +c2 +� +, +(23) +where c1, c2 ∈ C are nonzero constants depending on the scaling of the eigen- +vectors. +For simplicity, in (21) we chose c1 = c2 = 1, such that T is the +Vandermonde matrix vertically stacked twice. +4.2 +6D SSM in a nonlinear finite-element model of a beam +We train an SSM-reduced model with data from numerical simulations of a +finite-element (FE) representation of a clamped-clamped von K´arm´an nonlinear +beam [66]. This example was previously studied in Refs. [31, 37], which identified +13 + +the slowest 2D SSM in the delay-embedded observable space, predicted the +forced response and analyzed the radius of convergence of the analytical normal +form. +Here, thanks to our results on the tangent spaces of delay-embedded +SSMs, we can extend the analysis to the six-dimensional SSM emanating from +the three slowest modes of the linear part of the system. +Each node in the FE model has three degrees of freedom: axial deformation +u, transverse deflection w, and rotation w′. The von K´arm´an axial strain is +given by +ϵ11 = u′(x) + 1 +2 (w′(x))2 − zw′′(x). +(24) +The axial stress is given by +σ = Eϵ11 + c˙ϵ11, +(25) +where E = 70 GPa denotes the Young’s modulus and c = 1.0 × 106 Pa · s the +material rate of viscous damping. Based on a convergence analysis, we set the +number of elements to 12, resulting in a 33-degree of freedom mechanical system, +i.e., a 66-dimensional phase space. We set the beam length to 1000 mm, width +50 mm, and thickness 20 mm. The sampling time is ∆t = 0.0955 s. +w(t) +w(t) +w(t) +Figure 6: von K´arm´an beam: schematic first, second, and third mode shapes. +The scalar observable must be generic in the sense that it must have contri- +butions from all modes of interest. For instance, the midpoint displacement +is not excited by the second mode. Instead, we choose the shown transverse +displacement at 1/4 of the beam length. +By Remark 1, the observable function µ must have significant contributions +from all modes zk that we wish to model. For example, the midpoint displace- +ment chosen as observable function in Refs. [31, 37] was sufficient to model the +2D SSM, but cannot be employed for higher-dimensional SSMs. This is be- +cause the antisymmetric shape of the second mode has zero displacement at the +14 + +midpoint (see Fig. Figure 6), i.e. +∂µ +∂z3 +(0) = ∂µ +∂z4 +(0) = 0. +(26) +Instead, we choose the transverse displacement of the beam at one fourth of the +total length, µ = w(l/4), as this degree of freedom has nonzero contributions +from all three mode shapes. +For our data-driven modeling objectives, we need training data containing +the first three modes. To generate initial conditions for such trajectories, we +use linear combinations of the mode shapes of the system computed from its +linear part. Since the SSM is normally attracting, these trajectories will quickly +approach it and we can use them to train our reduced-order model. With this +method, we produce three trajectories close to the 6D SSM with different ini- +tial conditions, of which we use two as training data and one as test data. For +validation purposes, we also pick the individual mode shapes as initial condi- +tions and use as test data. The individual modal contributions in these initial +conditions were chosen as follows: +Initial +Mode +Type +condition +1 +2 +3 +1 +1 +0 +0 +Test +2 +0 +1 +0 +Test +3 +0 +0 +1 +Test +4 +0.8 +-0.8 +0.8 +Train +5 +-0.1 +0.8 +0.8 +Train +6 +-0.6 +-0.2 +-0.8 +Test +We choose the delay embedding parameters guided by the observations in +Sect. 3.2. +Setting κ = 1 such that the timelag τ = ∆t and the embedding +dimension to p = 50 gives a local optimum of the function (19) with the com- +puted eigenvalues, while still keeping the maximal delay κp moderate to prevent +folding of the embedding. +Fig. 7a shows the delay embedding of the single-mode trajectories 1-3, cor- +responding to the first three modes, in three of the 50 delay coordinates. These +trajectories visualize the orientations of the corresponding eigenspaces in the +observable space. Indeed, minimization of (19) corresponds to making these +planes orthogonal, simplifying their identification. +Fig. 7b similarly displays the delay embedding of the first training trajectory +along with a visualization of the columns of the Vandermonde matrix as vectors. +Our delay theory predicts that projection of the data onto these vectors yields +modal coordinates, as shown in Fig. 7c. This space will serve as the reduced +coordinates of the SSM. +After projection onto these eigenvectors, we approximate the geometry of the +6D SSM with a 3rd order polynomial. For the reduced dynamics in fastSSM, we +also use a 3rd order approximation. We compute the normal form of this reduced +dynamics up to 7th order and obtain our model for the reduced dynamics. The +15 + +w(t + 16"t) +w(t) +w(t + 8"t) +Mode 1 +Mode 2 +Mode 3 (#2) +(a) +w(t + 16"t) +w(t) +w(t + 8"t) +Traj. 4 +Re V1 +Re V3 +Re V5 +(b) +(V yy)2 +(V yy)3 +(V yy)1 +Projection of traj. 4 onto V +(c) +Figure 7: (a) The trajectories with single modal contributions visualize the +modal subspaces in the delay-embedded space. The third mode data has been +scaled by a factor 2 to increase visibility. (b) The same delay-embedded view +of the first training trajectory, along with the delay-embedded eigenvectors. (c) +After projection of this trajectory onto the eigenvectors, the modal structure +becomes clear. +terms up to third order in polar form are found by fastSSM to be of the form +� +� +� +� +� +� +� +� +˙ρ1 +ρ1 ˙θ1 +˙ρ2 +ρ2 ˙θ2 +˙ρ3 +ρ3 ˙θ3 +� +� +� +� +� +� +� +� += +� +� +� +� +� +� +� +� +0.3058 ρ13 + 2.088ρ1 ρ22 − 3.091ρ1 +102.0 ρ13 + 82.70 ρ22ρ1 + 657.2ρ1 +−2.705 ρ12ρ2 + 1.723 ρ23 − 23.72ρ2 +95.64 ρ12ρ2 + 115.6 ρ23 + 1812ρ2 +−8.968ρ3 ρ12 − 13.27ρ3 ρ22 − 88.47ρ3 +115.9 ρ12ρ3 + 85.04 ρ22ρ3 + 3558ρ3 +� +� +� +� +� +� +� +� ++ O(5). +(27) +We transform the initial conditions from the observable space to the normal +form and integrate our model to predict signal decay. This produces a normal- +ized mean trajectory error (as defined in [31]) of 2.2 % on the test data. Some +of the predictions are shown in Fig. 8. +0 +0.05 +0.1 +0.15 +0.2 +time [s] +-5 +0 +5 +u [m] +#10-3 +Training data +Reconstruction +(a) +0 +0.05 +0.1 +0.15 +0.2 +time [s] +-5 +0 +5 +u [m] +#10-3 +Test data +Reconstruction +(b) +0 +0 +1 +0.5 +;3 +;2 +;1 +0.5 +0.5 +1 +0 +1 +Traj. 1 +Traj. 2 +Traj. 3 +Traj. 4 +Traj. 5 +Traj. 6 +(c) +Figure 8: (a,b) Predictions from fastSSM for the decaying trajectories 5 and 6 +(c) Phase portrait of the trajectories after transformation to the normal form. +We also visualize our reduced-order model by plotting the instantaneous +frequency and damping as predicted by the normal form (27) for varying ampli- +tudes of mode 1 and 2. For instance, our model predicts hardening of the first +16 + +mode with respect to both the first and the second modal amplitudes (Fig. 9a), +a decrease in the instantaneous damping of mode 1 with respect to mode 2 +(Fig. 9b), and independence of the third instantaneous frequency with respect +to itself (Fig. 9c). The predictions for each of the trajectories are included for +reference. +650 +1 +700 +1 +_31 +750 +;2 +0.5 +;1 +800 +0.5 +0 +0 +(a) +-3 +1 +-2 +1 +_;1=;1 +;2 +0.5 +-1 +;1 +0.5 +0 +0 +(b) +3550 +1 +3600 +1 +_33 +;3 +0.5 +3650 +;1 +0.5 +0 +0 +(c) +Figure 9: Visualization of the normal form (27) with the trajectories for (a) +instantaneous frequency and (b) damping of mode 1, as well as (c) frequency of +mode 3. +4.3 +Multimodal sloshing of water in a tank +4 +B. B¨auerlein and K. Avila +Figure 2. Sketch of the experiment. A motor (a) drives an eccentric disk which converts the +rotary motion of the motor via a pushing rod (b) into a quasi-harmonic horizontal oscillation of +the platform. A positioning sensor (c) directly records the motion of the platform on which +the tank (d), two high speed cameras (e) and an USB-camera (f) are mounted. For the +PIV measurements a light sheet (g) is provided by a laser passing through a cylinder lens +(implemented in the stationary laser guiding arm). +dynamics. We find that neither the exact surface shape, nor the frequency spectrum +are useful to determine the nonlinear resonance maxima. The key indicator is the +phase-lag between driving and response. We systematically investigate the role of initial +conditions, characterise the sloshing amplitude with the motion of the liquid’s centre +of mass and directly measure the damping coefficient. The results obtained with our +approach are compared to common approaches used in the literature. The paper is +structured as follows. In the next section, we describe the experimental methods and in +§3 the quantitative characterisation of the sloshing phenomena. In §4 and §5, the Duffing +and multimodal model of sloshing are respectively described and briefly compared to +our measured data. Detailed measurements of large-amplitude sloshing are presented +in §6 with focus on the nonlinear dynamics of the system, including multiplicity and +competition of several flow states. The experimental response curves obtained for several +amplitudes are presented and compared to the Duffing and multimodal model in §7. An +assessment of the strengths and weakness of these models in capturing the experimentally +measured response is given in §8 before the conclusion in §9. +2. Methods +Our experiments were performed in a rectangular container subjected to harmonic +horizontal excitation. As illustrated in figure 1, the flow is quasi-two-dimensional. Slosh- +ing waves reaching from a quasi-planar surface, up to run-up at the tank walls and +wave-breaking were investigated. A distinct feature of the sloshing waves in an oscillated +(or pitched) tank is their asymmetric shape leading to an oscillation of the liquid’s +centre of mass (shown as a red dot in figure 1). Many fundamental studies consider +sloshing in wavemaker tanks (Taylor 1953; Fultz 1962; Chester 1968a). A key difference +between oscillated and wavemaker tanks is that in the latter the primary resonant mode +is symmetric and the liquid’s centre of mass is steady in the lateral direction. +2.1. Experimental setup +A sketch of our experimental setup is shown in figure 2. The tank (width w = 500 mm, +depth l = 50 mm) is mounted on a platform and filled with water at room temperature +(a) +0 +100 +200 +300 +400 +500 +x [mm] +Mode 1 +Mode 2 +Mode 3 +Mode 4 +(b) +Figure 10: (a) Experimental setup for tank sloshing (adapted from [67]) (b) The +first four sloshing mode shapes. +For our final example, we apply our results to sloshing experiments. Sloshing +models have a wide range of industrial applications, including fluid container +interaction with ship motion [68], road transportation of fluids [69], damping +devices in towers [70], and fuel tank design in spacecraft [71, 72]. +A tank +partially filled with water exhibits several nonlinear phenomena under horizontal +harmonic excitation [73]. On the one hand, intensified fluid motion can alter +the instantaneous damping and frequency of the first sloshing mode [74]. On +the other hand, increasing the amplitude further activates several nonlinearly +coupled modes of the system and gives rise to a range of different wave motions +[75, 76]. +17 + +Our training data comes from experiments described in Ref. [67] with a rect- +angular tank of width w = 500 mm and thickness 50 mm, partially filled with +water up to a height of h = 400 mm. The tank was attached to a horizon- +tally moving platform harmonically excited by a motor at different frequencies. +Then, once the system had reached a steady state, the motor was turned off. +Depending on the forcing frequency, this periodic response exhibited planar, +wave-breaking, or three-periodic motion. The three-periodic forced state was +characterized by an increase in the response amplitude every third forcing cycle, +while the wave-breaking response was defined as overturning of the water close +to the walls [67]. A camera detected the surface profile h with the sampling +time ∆t = 0.01 s. Figure 10a displays the experimental setup. +While previous work successfully captured the dynamics of the main sloshing +mode using a 2D SSM for the center of mass signal [31] and the full surface +profile [37], here, we model the decay from a multimodal state by identifying +a 6D SSM, corresponding to the nonlinear extension of the three dominant +oscillatory modes. We train on three decaying measurements: Trajectory 1 and +2 start at a three-periodic state, and Trajectory 3 starts at a wave-breaking +state. +The observable vector µ is the surface profile measured at 1 771 points along +the tank width. Since this function is multi-dimensional, in order to apply our +theory on delay-embedded tangent spaces, we need an estimate of the eigen- +values and linear mode shapes in our observable. The eigenfrequencies can be +computed from potential theory [74] as +ωk = +� +gπ +w k tanh +� +πk h +w +� +, +(28) +which scales approximately with the square root of the mode number k for +our configuration. +The first five eigenfrequencies are [7.80, 11.1, 13.6, 15.7, +17.6] rad/s, with an approximate 1:2 resonance between frequencies 1 and 4. +The mode shapes by the same theory are +ˆek = cos(kx/w), +x ∈ [0, w], +(29) +shown in Fig. 10b. For the tangent space, in principle, we also need the lin- +ear damping of each mode. In practice, this real part of the eigenvalues has +very little influence on V for limited delay embedding and we pick the values +[−0.05, −0.07, −0.08, −0.09, −0.1] based on previous fits of the first mode and +the assumption of increasing damping with the mode number. +Based on (19), we delay-embed the data with timelag τ = 5∆t and dimension +p = 47. +A projection of the delay-embedded data onto the eigenvectors T +predicted by our theory appears to yield modal coordinates, as indicated in +Fig. 11a. +Consequently, the norm of these projections can be used as a heuristic mea- +sure of the modal content in the signal. This procedure should be used with +caution, since it does not take manifold curvature into account, but it can be +18 + +-200 +-2000 +0 +2000 +(T yy)3 +(T yy)1 +0 +(T yy)2 +0 +200 +2000 +-2000 +Projection of traj. 2 onto T +(a) +0 +10 +20 +30 +40 +time [s] +0 +1000 +2000 +3000 +Projected amplitude +Mode 1 +Mode 2 +Mode 3 +Mode 4 +Mode 5 +(b) +0 +5 +10 +15 +time [s] +0 +100 +200 +300 +Projected amplitude +Mode 1 +Mode 2 +Mode 3 +Mode 4 +Mode 5 +(c) +Figure 11: (a) Projecting one of the trajectories onto the tangent space vec- +tors unveils the modal structure. +(b) By projecting the trajectory onto the +eigenvectors and taking the absolute value, we can estimate the relative modal +contributions in the signal (c) A zoomed-in view of (b) indicates that modes 1, +2, and 4 dominate. +employed to provide an initial guess for the SSM dimension. In Fig. 11b, we +plot the absolute value of the projection onto each modal subspace of T over +time for Trajectory 2. This plot shows that the first mode dominates, while the +zoomed-in view (Fig. 11c) indicates that the second and fourth modes appear to +be the most prevalent of the higher modes throughout the decay. The third and +fifth mode are present at first but quickly die out. All amplitudes are decaying +except for the fourth mode, which instead initially grows. Based on this analy- +sis, we will identify a 6D SSM emanating from the spectral subspace of modes +1, 2, and 4. This choice also takes SSM theory into account, by which the 1:2 +resonance requires that the modal subspace of the fourth mode is included in +the spectral subspace of the SSM. We choose to start our training data after +1.2 s, as the third and fifth modal amplitudes are small thereafter and we expect +the trajectory to lie sufficiently close to the SSM. +With an SSM parametrization order m = 4, reduced dynamics order r = 3, +and normal form order h = 3, we compute the SSM geomety and dynamics and +integrate our reduced-order model to predict the decay from the various flow +states. This yields a normalized mean trajectory error (NMTE) [31] of 2.6 %. +fastSSM successfully detects and accounts for the internal resonance by adding +phase-dependent terms to the computed normal form, which reads +˙ρ1 +ρ1 = −0.056 − 0.0069 sin(ψ − 0.26)ρ4 − 0.0015ρ2 +4 − 0.039ρ2 +2 + 0.023ρ2 +1 +˙θ1 = 7.78 + 0.0069 cos(ψ − 0.26)ρ4 + 0.040ρ2 +4 + 0.016ρ2 +2 − 0.82ρ2 +1 +˙ρ2 +ρ2 = −0.13 + 0.15ρ2 +4 − 0.89ρ2 +2 + 0.37ρ2 +1 +˙θ2 = 11.4 + 0.57ρ2 +4 − 0.0085ρ2 +2 − 2.2ρ2 +1 +˙ρ4 +ρ4 = −0.30 − 0.29ρ2 +4 + 0.67ρ2 +2 − 0.27 sin(ψ + 1.4)ρ2 +1 + 1.2ρ2 +1 +˙θ4 = 15.9 − 0.085ρ2 +4 + 1.2ρ2 +2 + 0.27 cos(ψ + 1.4)ρ2 +1 − 2.0ρ2 +1 +(30) +where ψ = θ4 − 2θ1 and the subscripts denote the corresponding mode number. +Looking at the linear part, we see that the eigenfrequencies are well captured. +19 + +Good agreement between the experimentally measured surface profile eleva- +tion at the tank’s leftmost point and the delay-embedded SSM-reduced predic- +tion is shown for the first period-three initial state in Fig. 12a and the wave- +breaking state in Fig. 12b. Further, our 6D reduced model can accurately predict +the full surface profile decay, with snapshots shown in Figure 13. +0 +5 +10 +15 +20 +time +-100 +0 +100 +200 +hx=0 [mm] +Original +Reconstruction +(a) +0 +10 +20 +30 +40 +time +-100 +-50 +0 +50 +100 +150 +hx=0 [mm] +Original +Reconstruction +(b) +0 +0.2 +0.4 +0.6 +0 +0.8 +;4 +;1 +0.5 +;2 +0.6 +0.4 +0.2 +0 +Traj. 1 +Traj. 2 +Traj. 3 +(c) +Figure 12: The prediction on the 6D SSM for the decay of (a) Trajectory 1 and +(b) Trajectory 3. (c) Phase portrait of the amplitudes of the normal form coor- +dinates on the SSM for each of the trajectories shows the modal contributions +and development for different intial flow states. +We project the training trajectories onto the SSM and transform them to +the normal form in polar coordinates. The development of the amplitudes in the +normal form are shown in Fig. 12c for each trajectory. This plot suggests that (i) +the wave-breaking motion (Traj. 3) does not seem to have any significant content +of the second mode, (ii) the amplitude of the fourth mode indeed increases after +motor detachment, (iii) there is a small oscillation in these signals not captured +by our model, which may be due to noise, insufficient separation of the modal +subspaces, a mode outside our model, or some other phenomenon. +0 +100 +200 +300 +400 +500 +x [mm] +-100 +0 +100 +200 +Elevation h [mm] +t = 1:2 s +Experiment +Simulation +0 +100 +200 +300 +400 +500 +x [mm] +t = 1:77 s +0 +100 +200 +300 +400 +500 +x [mm] +t = 14:49 s +Figure 13: The experimentally measured surface profile decay agrees with our +6D SSM model prediction for Trajectory 2. +We note that the combined higher modal content in the signal is small - +only about 10 % with respect to the first mode. This is because the data is +decaying from steady states induced by forcing near the first eigenfrequency. +Due to their symmetric shape, isolated forcing of the second and fourth modes +is not possible with horizontal harmonic excitation. Nevertheless, we are able +20 + +to capture these smaller oscillations on the SSM. The key technology allowing +this enhancement is the enforcement of the delay-embedded tangent space in +our SSM reconstruction, based on the theoretical eigenfrequencies and mode +shapes. +Due to the small activation of the higher modes, our model is expected to be +sensitive to noise. It is, however, stable with respect to changes in starting time, +manifold order, and delay parameters. A more robust model can be obtained by +decreasing the manifold dimension to 4, neglecting the relatively minor influence +of the fourth mode, resulting in an average NMTE of 3.9 %. Here, since our +objective was modal analysis of different flow states, we chose the more detailed +6D model. +5 +Conclusions +We have shown that for a scalar observation of an invariant manifold tangent +to a spectral subspace at a fixed point, the delay-embedded reconstruction of +the tangent space is dependent only on the corresponding eigenvalues of the full +system linearized at that point. In particular, we have proven that the columns +of a Vandermonde matrix, given by repeated multiplication of the exponential +of the eigenvalues times the timelag, are eigenvectors for the linearized system +in the observable space. Therefore, the Vandermonde matrix diagonalizes the +linear part of the delay-embedded dynamics. We have also shown that when sev- +eral quantities are measured and delay-embedded simultaneously, the tangent +space can be expressed given the Vandermonde matrix and the mode shapes +expressed in the observable function components. These results hold for any +invariant manifold tangent to a modal subspace with distinct eigenvalues, in- +cluding, e.g., classic stable manifolds. Here, our focus was the application of +this result to spectral submanifolds of hyperbolic fixed points. +In an attempt to exploit this uncovered structure, we have shown that +for data-driven SSM model reduction, when the eigenvalues are approximately +known, we can analytically predict the tangent space of the embedded SSM +a priori to achieve local modal decomposition and aid the reconstruction of the +nonlinear reduced dynamics. We have found that even for small activation of +higher modes, this trick helps modeling complex multimodal nonlinear dynam- +ics on an SSM, which in turn allows for analysis of modal energy interchange +and instantaneous frequencies. +Our theory assumes a generic observable function, which we describe in +more detail in our first and second example, and distinct eigenvalues. While +the second assumption is a generic one in a mathematical sense, it is not always +satisfied for engineering structures with symmetry. Vibrations in a square plate +is an example where our theory would fail, as it has repeated eigenvalues. Using +a vector-valued observable may help in differentiating between the modes in +such a case. Further, while technically covered by the theory, possible practical +difficulties related to the conditioning of the Vandermonde matrix include highly +different or very similar eigenvalues, or eigenvalues of different stability type. +21 + +In our third example with data from experiments, we also devised a new +heuristic scheme for using delay embedding to study modal contents in a signal. +With this method, that served as an initial guess, we projected the data onto +the respective prescribed modal subspaces, thereby implicitly assuming that the +SSM of each mode is nearly flat. An interesting development of this idea would +be its use as a filter, which could remove or keep certain frequencies in a signal. +Another idea would be to use the tangent space condition as a verification or +for iterative adjustment of the linear fit of the reduced dynamics. Finally, in +analogy with a Fourier analysis, it would be possible to estimate both instanta- +neous frequency and damping of a signal by singular value decomposition of the +trajectory in delay coordinates followed by analysis of the Vandermonde matrix +columns. This ties in with several other observations made in the literature; for +example, for a linear system, these columns agree with the recently proposed +notion of principal component trajectories [62]. Overall, we believe that our +findings shed more light on delay-embedding invariant manifolds and selecting +delay parameters in particular. For that reason, we expect these results to be +of use for a wide range of data-driven methods. +Acknowledgements +We are grateful to Kerstin Avila and Bastian B¨auerlein (U. Bremen) for sharing +their experimental surface profile data from Ref. [67] with us. +A +Appendix +A.1 +Proof of Theorem 1 +Let M be a d-dimensional invariant manifold of (1) containing the origin of Rn. +The tangent space of M at the origin can be written T0M = span {ek}k∈K, +where K ⊂ {1, . . . , n} is an index set labeling the d eigenvectors ek spanning +the spectral subspace from which the manifold is emanating. For example, for +a stable manifold, K = {k : Re λk < 0}. We also assume that the eigenvalues +in question are distinct, i ̸= k ⇔ λi ̸= λk, for i, k ∈ K. +To simplify the notation, we transform the full state space to modal coor- +dinates (11). We rewrite the observable function on the system x ∈ Rn as an +observable on the modal coordinate system z ∈ Cn as ˆµ(z) = µ(Ez). We de- +note by Φt = E ◦F t ◦E−1 the flow in Cn. Consider the sampling map in modal +coordinates +ˆS = +� +������ +ˆµ +ˆµ ◦ Φτ +ˆµ ◦ Φ2τ +... +ˆµ ◦ Φ(p−1)τ +� +������ +: Cn → Rp, +y = ˆS(z). +(31) +22 + +Under the conditions of Takens’s embedding theorem, the delay embedding +map Ψ = ˆS|M : M → +˜ +M is a smooth embedding with a smooth inverse +Ψ−1 : ˜ +M → M, and Ψ(0) = q. +In order to derive the tangent space Tq ˜ +M, we compute the derivative of the +embedding at 0: +DΨ(0) = +� +���� +Dˆµ(0) +Dˆµ(Φτ(0)) ◦ DΦτ(0) +... +Dˆµ(Φ(p−1)τ(0)) ◦ DΦ(p−1)τ(0) +� +���� = +� +���� +Dˆµ(0) +Dˆµ(0) ◦ eΛτ +... +Dˆµ(0) ◦ eΛ(p−1)τ +� +���� . +(32) +Now note that the jth component expressed in modal coordinates is +Dˆµ(0) ◦ eΛjτ(z) = +� +k∈K +∂ˆµ +∂zk +���� +0 +eλkjτzk, +j ∈ {1, . . . , p}. +(33) +We define the Vandermonde matrix V of the d eigenvalues {λk}k∈K governing +the linearized dynamics on M as Vjk = eλkjτ. We conclude that the tangent +space of the observable manifold at the fixed point in modal coordinates can be +written as +Tq ˜ +M = {DΨ(0)z, z ∈ Cn} = range +� +V diag +� ∂ˆµ +∂z +���� +0 +�� += range V , +(34) +where the diagonal matrix acts only as a rescaling of each component of z. +Therefore, one matrix representation of the tangent space of the manifold in +the observable space is V , which is independent both of the matrix E of full +system eigenvectors and the observable function µ. +Note that we must have +∂ ˆµ +∂zk |0 ̸= 0 +∀k ∈ K, which defines the genericity of +µ. In practice, this implies that the linearized observable function must contain +contributions from all modal coordinates that we wish to model. In addition, +note that for the embedding of the tangent space itself, i.e., the linear system, +p = d suffices. +A.2 +Proof of Theorem 2 +The flow on +˜ +M is +˜Φt = Ψ ◦ Φt ◦ Ψ−1. +(35) +We compute the ODE on +˜ +M, ˙y = ˜f(y), as +˜f = d +dt +˜Φt = DΨ ◦ f ◦ Ψ−1, +(36) +where f is given by (11). The derivative of ˜f at the fixed point is, therefore, +D ˜f(q) = DΨ(0) ◦ Λ ◦ DΨ−1(q) = V diag +� ∂ˆµ +∂z +���� +0 +� +Λ diag +� ∂ˆµ +∂z +���� +0 +�−1 +V † += V ΛV †, +(37) +23 + +where we used the commutative property of multiplication of diagonal matri- +ces, which eliminates the linearized observable function terms, and the fact +that DΨ−1(q) = diag +� +∂ ˆµ +∂z +��� +0 +�−1 +V † is well-defined. To see this, note that the +derivative of the delay embedding map composed with its inverse +DΨ(0) ◦ DΨ−1(q) = V diag +� ∂ˆµ +∂z +���� +0 +� +diag +� ∂ˆµ +∂z +���� +0 +�−1 +V † = V V †, +(38) +maps all points in T0M to themselves, since V has full rank under the assump- +tion of distinct eigenvalues {λk}k∈K. +Taylor-expanding the ODE on +˜ +M in the observable space yields +˙y = D ˜f(q)(y − q) + o(|y − q|) = V ΛV †(y − q) + o(|y − q|). +(39) +Therefore, under the assumptions of a generic observable function and distinct +eigenvalues, the tangent space Tq ˜ +M and the linearized dynamics D ˜f(q) in +the observable space Rp are both fully determined by the timelag τ and the +eigenvalues λk, k ∈ K. +In the special case that M = Rn, if p ≥ 2n + 1, the entire phase space can +be reconstructed. For a linear system, p = n suffices, and the delay embedding +reduces to a linear operator. +A.3 +Proof of Theorem 3 +For a vector-valued observable, µ : Rn → Rq, the delay embedding map reads +Ψ = +� +�� +Ψµ1 +... +Ψµq +� +�� : M → ˜ +M ⊂ Rpq, +DΨ(0) = +� +�� +DΨµ1(0) +... +DΨµq(0) +� +�� , +(40) +where Ψµℓ is the delay embedding map corresponding to the ℓth component of +the observable function µ. The derivatives are given by +DΨµℓ(0) = V diag +� ∂ˆµℓ +∂z +���� +0 +� +. +(41) +In this case, the tangent space is not independent of the observable function. +Instead, it is affected by the relative dependency of each component µℓ of the ob- +servable function on each modal coordinate zk. The tangent space can, however, +be expressed as +Tq ˜ +M = range +� +���� +V diag +� +∂ ˆµ1 +∂z +��� +0 +� +... +V diag +� +∂ ˆµq +∂z +��� +0 +� +� +���� . +(42) +24 + +References +[1] +J. L. Lumley. “The Structure of Inhomogeneous Turbulent Flows”. Atmo- +spheric Turbulence and Radio Wave Propagation (1967), pp. 166–177. +[2] +J. Awrejcewicz, V. A. Krys’ko, and A. F. Vakakis. “Order Reduction +by Proper Orthogonal Decomposition (POD) Analysis”. Nonlinear Dy- +namics of Continuous Elastic Systems. Springer, Berlin, Heidelberg, 2004, +pp. 279–320. +[3] +P. Schmid. “Dynamic mode decomposition of numerical and experimental +data”. J. Fluid Mech. 656 (2010), pp. 5–28. +[4] +J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor. Dynamic +Mode Decomposition. Philadelphia, PA: SIAM, 2016. +[5] +P. J. Schmid. “Dynamic Mode Decomposition and Its Variants”. Annual +Review of Fluid Mechanics 54.1 (2022), pp. 225–254. doi: 10 . 1146 / +annurev-fluid-030121-015835. +[6] +J. Page and R. Kerswell. “Koopman mode expansions between simple +invariant solutions”. J. Fluid Mech. 879 (2019), pp. 1–27. +[7] +S. L. Brunton, J. L. Proctor, and J. N. Kutz. “Discovering governing equa- +tions from data by sparse identification of nonlinear dynamical systems”. +Proceedings of the National Academy of Sciences 113.15 (2016), pp. 3932– +3937. +[8] +D. Bertsimas and W. Gurnee. Learning Sparse Nonlinear Dynamics via +Mixed-Integer Optimization. 2022. doi: 10.48550/ARXIV.2206.00176. +url: https://arxiv.org/abs/2206.00176. +[9] +J. N. Kutz and S. L. Brunton. “Parsimony as the ultimate regularizer +for physics-informed machine learning”. Nonlinear Dynamics 107.3 (Jan. +2022), pp. 1801–1817. doi: 10.1007/s11071-021-07118-3. url: https: +//doi.org/10.1007/s11071-021-07118-3. +[10] +S. Chen and S. A. Billings. “Neural networks for nonlinear dynamic system +modelling and identification”. International Journal of Control 56.2 (Aug. +1992), pp. 319–346. doi: 10 . 1080 / 00207179208934317. url: https : +//doi.org/10.1080/00207179208934317. +[11] +T. Daniel, F. Casenave, N. Akkari, and D. Ryckelynck. “Model order re- +duction assisted by deep neural networks (ROM-net)”. Adv. Model. and +Simul. in Eng. Sci. 7 (2020), p. 105786. +[12] +L. Salmela, N. Tsipinakis, A. Foi, C. Billet, J. M. Dudley, and G. Genty. +“Predicting ultrafast nonlinear dynamics in fibre optics with a recurrent +neural network”. Nature Machine Intelligence 3.4 (Feb. 2021), pp. 344– +354. doi: 10.1038/s42256-021-00297-z. url: https://doi.org/10. +1038/s42256-021-00297-z. +25 + +[13] +J.-C. Loiseau, S. L. Brunton, and B. R. Noack. “From the POD-Galerkin +method to sparse manifold models”. Model Order Reduction, Volume 3: +Applications. Ed. by P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, +W. Schilders, and L. M. Silveira. De Gruyter, Berlin, 2020, pp. 279–320. +[14] +Cabr´e, E. Fontich, and R. de la Llave. “The parameterization method for +invariant manifolds I: Manifolds associated to non-resonant subspaces”. +Indiana Univ. Math. J. 52.2 (2003), pp. 283–328. +[15] +A. Haro and R. de la Llave. “A parameterization method for the compu- +tation of invariant tori and their whiskers in quasi-periodic maps: rigorous +results”. J. Differential Eqs. 228.2 (2006), pp. 530–579. +[16] +G. Haller and S. Ponsioen. “Nonlinear normal modes and spectral sub- +manifolds: existence, uniqueness and use in model reduction”. Nonlinear +Dyn. 86.3 (2016), pp. 1493–1534. +[17] +S. Ponsioen, T. Pedergnana, and G. Haller. “Automated computation of +autonomous spectral submanifolds for nonlinear modal analysis”. J. Sound +and Vibration 420 (2018), pp. 269–295. +[18] +R. M. Rosenberg. “The Normal Modes of Nonlinear n-Degree-of-Freedom +Systems”. Journal of Applied Mechanics 29.1 (Mar. 1962), pp. 7–14. doi: +10.1115/1.3636501. url: https://doi.org/10.1115/1.3636501. +[19] +A. Vakakis. “Non-linear normal modes (NNMs) and their applications in +vibration theory: an overview”. Mechanical Systems and Signal Processing +11.1 (Jan. 1997), pp. 3–22. doi: 10.1006/mssp.1996.9999. url: https: +//doi.org/10.1006/mssp.1996.9999. +[20] +G. Kerschen, M. Peeters, J. Golinval, and A. Vakakis. “Nonlinear nor- +mal modes, Part I: A useful framework for the structural dynamicist”. +Mechanical Systems and Signal Processing 23.1 (Jan. 2009), pp. 170–194. +doi: 10.1016/j.ymssp.2008.04.002. url: https://doi.org/10.1016/ +j.ymssp.2008.04.002. +[21] +S. Shaw and C. Pierre. “Non-linear normal modes and invariant mani- +folds”. Journal of Sound and Vibration 150.1 (Oct. 1991), pp. 170–173. +doi: 10.1016/0022-460x(91)90412-d. url: https://doi.org/10. +1016/0022-460x(91)90412-d. +[22] +S. Shaw and C. Pierre. “Normal modes for non-linear vibratory systems”. +J. Sound and Vibration 164.1 (1993), pp. 85–124. +[23] +R. Szalai. “Invariant spectral foliations with applications to model order +reduction and synthesis”. Nonlinear Dyn. 101 (2020), pp. 2645–2669. +[24] +S. Jain, T. Thurner, M. Li, and G. Haller. SSMTool: Computation of +invariant manifolds and their reduced dynamics in high-dimensional me- +chanics problems. 2021. doi: 10 . 5281 / zenodo . 4614201. url: http : +//www.georgehaller.com. +[25] +S. Ponsioen, T. Pedergnana, and G. Haller. “Analytic prediction of iso- +lated forced response curves from spectral submanifolds”. Nonlinear Dyn. +98 (2019), pp. 2755–2773. +26 + +[26] +S. Ponsioen, S. Jain, and G. Haller. “Model reduction to spectral sub- +manifolds and forced-response calculation in high-dimensional mechanical +systems”. J. Sound and Vibration 488 (2020), p. 115640. +[27] +S. Jain and G. Haller. “How to compute invariant manifolds and their +reduced dynamics in high-dimensional finite element models?” Nonlinear +Dynamics 107.2 (Oct. 2021), pp. 1417–1450. doi: 10.1007/s11071-021- +06957-4. url: https://doi.org/10.1007/s11071-021-06957-4. +[28] +M. Li, S. Jain, and G. Haller. “Nonlinear analysis of forced mechanical +systems with internal resonance using spectral submanifolds – Part I: Pe- +riodic response and forced response curve”. Nonlinear Dynamics (2022). +[29] +M. Li and G. Haller. “Nonlinear analysis of forced mechanical systems +with internal resonance using spectral submanifolds – Part II: Bifurcation +and quasi-periodic response”. Nonlinear Dynamics (2022). doi: https: +//doi.org/10.1007/s11071-022-07476-6. +[30] +M. Li, S. Jain, and G. Haller. Model reduction for constrained mechanical +systems via spectral submanifolds. 2022. doi: 10.48550/ARXIV.2208. +03119. url: https://arxiv.org/abs/2208.03119. +[31] +M. Cenedese, J. Ax˚as, B. B¨auerlein, K. Avila, and G. Haller. “Data-driven +modeling and prediction of non-linearizable dynamics via spectral sub- +manifolds”. Nat. Commun. 13.1 (Feb. 2022). doi: 10.1038/s41467-022- +28518-y. url: https://doi.org/10.1038/s41467-022-28518-y. +[32] +M. Cenedese, J. Ax˚as, and G. Haller. SSMLearn. 2021. url: http://www. +georgehaller.com. +[33] +J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Sys- +tems and Bifircation of Vector Fields. Springer, New York, 1983. +[34] +M. Cenedese, J. Ax˚as, H. Yang, M. Eriten, and G. Haller. “Data-driven +nonlinear model reduction to spectral submanifolds in mechanical sys- +tems”. Philosophical Transactions of the Royal Society A: Mathematical, +Physical and Engineering Sciences 380.2229 (June 2022). doi: 10.1098/ +rsta.2021.0194. url: https://doi.org/10.1098/rsta.2021.0194. +[35] +B. Kasz´as, M. Cenedese, and G. Haller. “Dynamics-based machine learn- +ing of transitions in Couette flow”. Physical Review Fluids 7 (8 2022), +p. L082402. +[36] +J. I. Alora, M. Cenedese, E. Schmerling, G. Haller, and M. Pavone. Data- +Driven Spectral Submanifold Reduction for Nonlinear Optimal Control of +High-Dimensional Robots. 2022. doi: 10.48550/ARXIV.2209.05712. url: +https://arxiv.org/abs/2209.05712. +[37] +J. Ax˚as, M. Cenedese, and G. Haller. “Fast data-driven model reduction +for nonlinear dynamical systems”. Nonlinear Dyn. (2022). +[38] +M. O. Williams, I. G. Kevrekidis, and C. W. Rowley. “A Data–Driven +Approximation of the Koopman Operator: Extending Dynamic Mode De- +composition”. J Nonlinear Sci. 9 (2015), pp. 1307–1346. +27 + +[39] +D. Dylewsky, D. Barajas-Solano, T. Ma, A. M. Tartakovsky, and J. N. +Kutz. “Stochastically Forced Ensemble Dynamic Mode Decomposition +for Forecasting and Analysis of Near-Periodic Systems”. IEEE Access 10 +(2022), pp. 33440–33448. doi: 10.1109/ACCESS.2022.3161438. +[40] +S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz. +“Chaos as an intermittently forced linear system”. Nat. Commun. 8.1 +(May 2017). doi: 10.1038/s41467-017-00030-8. url: https://doi. +org/10.1038/s41467-017-00030-8. +[41] +J.-N. Juang and R. S. Pappa. “An eigensystem realization algorithm for +modal parameter identification and model reduction”. Journal of Guid- +ance, Control, and Dynamics 8.5 (Sept. 1985), pp. 620–627. doi: 10. +2514/3.20031. url: https://doi.org/10.2514/3.20031. +[42] +S. Pan and K. Duraisamy. “Data-Driven Discovery of Closure Models”. +SIAM Journal on Applied Dynamical Systems 17.4 (Jan. 2018), pp. 2381– +2413. doi: 10.1137/18m1177263. url: https://doi.org/10.1137/ +18m1177263. +[43] +A. Pikovsky. “Noise filtering in the discrete time dynamical systems”. Sov. +J. Commun. Technol. Electron 31.5 (1986), pp. 911–914. +[44] +G. Sugihara and R. M. May. “Nonlinear forecasting as a way of distin- +guishing chaos from measurement error in time series”. Nature 344.6268 +(Apr. 1990), pp. 734–741. doi: 10.1038/344734a0. url: https://doi. +org/10.1038/344734a0. +[45] +J. P. Crutchfield. “Prediction and stability in classical mechanics”. Senior +thesis in physics and mathematics, University of California, Santa Cruz. +1979. +[46] +N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. “Geometry +from a Time Series”. Phys. Rev. Lett. 45 (9 1980), pp. 712–716. doi: +10.1103/PhysRevLett.45.712. url: https://link.aps.org/doi/10. +1103/PhysRevLett.45.712. +[47] +F. Takens. “Detecting strange attractors in turbulence”. Dynamical Sys- +tems and Turbulence, Warwick 1980. Ed. by D. Rand and L. Young. +Springer Berlin Heidelberg, 1981, pp. 366–381. +[48] +T. Sauer, J. Yorke, and M. Casdagli. “Embedology”. J. Stat. Phys. 65 +(1991), pp. 579–616. +[49] +D. Broomhead and G. P. King. “Extracting qualitative dynamics from ex- +perimental data”. Physica D: Nonlinear Phenomena 20.2 (1986), pp. 217– +236. issn: 0167-2789. doi: https://doi.org/10.1016/0167-2789(86) +90031-X. url: https://www.sciencedirect.com/science/article/ +pii/016727898690031X. +28 + +[50] +M. Casdagli, S. Eubank, J. Farmer, and J. Gibson. “State space recon- +struction in the presence of noise”. Physica D: Nonlinear Phenomena 51.1 +(1991), pp. 52–98. issn: 0167-2789. doi: https://doi.org/10.1016/ +0167 - 2789(91 ) 90222 - U. url: https : / / www . sciencedirect . com / +science/article/pii/016727899190222U. +[51] +H. L. Yap and C. Rozell. “Stable Takens’ Embedding for Linear Dynamical +Systems”. Vol. 59. Dec. 2010, pp. 2948–2953. doi: 10.1109/TSP.2011. +2160629. +[52] +A. M. Fraser and H. L. Swinney. “Independent coordinates for strange +attractors from mutual information”. Phys. Rev. A 33 (2 Feb. 1986), +pp. 1134–1140. doi: 10.1103/PhysRevA.33.1134. url: https://link. +aps.org/doi/10.1103/PhysRevA.33.1134. +[53] +M. B. Kennel, R. Brown, and H. D. I. Abarbanel. “Determining embed- +ding dimension for phase-space reconstruction using a geometrical con- +struction”. Phys. Rev. A 45 (6 Mar. 1992), pp. 3403–3411. doi: 10.1103/ +PhysRevA . 45 . 3403. url: https : / / link . aps . org / doi / 10 . 1103 / +PhysRevA.45.3403. +[54] +H. D. I. Abarbanel and M. B. Kennel. “Local false nearest neighbors +and dynamical dimensions from observed chaotic data”. Phys. Rev. E 47 +(5 May 1993), pp. 3057–3068. doi: 10.1103/PhysRevE.47.3057. url: +https://link.aps.org/doi/10.1103/PhysRevE.47.3057. +[55] +K. H. Kraemer, M. Gelbrecht, I. Pavithran, R. I. Sujith, and N. Marwan. +“Optimal state space reconstruction via Monte Carlo decision tree search”. +Nonlinear Dynamics 108.2 (Mar. 2022), pp. 1525–1545. doi: 10.1007/ +s11071-022-07280-2. url: https://doi.org/10.1007/s11071-022- +07280-2. +[56] +E. Bozzo, R. Carniel, and D. Fasino. “Relationship between Singular Spec- +trum Analysis and Fourier analysis: Theory and application to the mon- +itoring of volcanic activity”. Computers and Mathematics with Applica- +tions 60.3 (Aug. 2010), pp. 812–820. doi: 10.1016/j.camwa.2010.05. +028. url: https://doi.org/10.1016/j.camwa.2010.05.028. +[57] +S. Pan and K. Duraisamy. “On the structure of time-delay embedding +in linear models of non-linear dynamical systems”. Chaos: An Interdis- +ciplinary Journal of Nonlinear Science 30.7 (July 2020), p. 073135. doi: +10.1063/5.0010886. url: https://doi.org/10.1063/5.0010886. +[58] +C. W. Rowley, I. Mezi´c, S. Bagheri, P. Schlachter, and D. Henningson. +“Spectral analysis of nonlinear flows”. J. Fluid Mech. 641 (2009), pp. 115– +127. +[59] +Z. Drmaˇc, I. Mezi´c, and R. Mohr. “Data Driven Koopman Spectral Anal- +ysis in Vandermonde–Cauchy Form via the DFT: Numerical Method and +Theoretical Insights”. SIAM Journal on Scientific Computing 41.5 (2019), +A3118–A3151. doi: 10.1137/18M1227688. url: https://doi.org/10. +1137/18M1227688. +29 + +[60] +M. Kamb, E. Kaiser, S. L. Brunton, and J. N. Kutz. Time-Delay Ob- +servables for Koopman: Theory and Applications. 2018. doi: 10.48550/ +ARXIV.1810.01479. url: https://arxiv.org/abs/1810.01479. +[61] +S. M. Hirsh, S. M. Ichinaga, S. L. Brunton, J. N. Kutz, and B. W. Brunton. +“Structured time-delay models for dynamical systems with connections to +Frenet–Serret frame”. Proceedings of the Royal Society A: Mathematical, +Physical and Engineering Sciences 477.2254 (Oct. 2021). doi: 10.1098/ +rspa.2021.0097. url: https://doi.org/10.1098/rspa.2021.0097. +[62] +D. Dylewsky, E. Kaiser, S. L. Brunton, and J. N. Kutz. “Principal com- +ponent trajectories for modeling spectrally continuous dynamics as forced +linear systems”. Phys. Rev. E 105 (1 Jan. 2022), p. 015312. doi: 10.1103/ +PhysRevE.105.015312. url: https://link.aps.org/doi/10.1103/ +PhysRevE.105.015312. +[63] +E. Bronstein, A. Wiegner, D. Shilo, and R. Talmon. “The spatiotemporal +coupling in delay-coordinates dynamic mode decomposition”. Chaos: An +Interdisciplinary Journal of Nonlinear Science 32.12 (Dec. 2022), p. 123127. +doi: 10.1063/5.0123101. url: https://doi.org/10.1063/5.0123101. +[64] +J. Ax˚as and G. Haller. fastSSM: Algorithm for fast computation of spec- +tral submanifolds from data. 2022. url: https://github.com/haller- +group/SSMLearn/tree/main/fastSSM. +[65] +E. R. Deyle and G. Sugihara. “Generalized Theorems for Nonlinear State +Space Reconstruction”. PLoS ONE 6.3 (2011). url: https://doi.org/ +10.1371/journal.pone.0018295. +[66] +S. Jain, P. Tiso, and G. Haller. “Exact nonlinear model reduction for a +von K´arm´an beam: slow-fast decomposition and spectral submanifolds”. +Journal of Sound and Vibration 423 (2018), pp. 195–211. +[67] +B. B¨auerlein and K. Avila. “Phase lag predicts nonlinear response maxima +in liquid-sloshing experiments”. J. Fluid Mech. 925 (2021), A22. doi: 10. +1017/jfm.2021.576. +[68] +S. Mitra, L. V. Hai, L. Jing, and B. C. Khoo. “A fully coupled ship motion +and sloshing analysis in various container geometries”. Journal of Marine +Science and Technology 17.2 (Feb. 2012), pp. 139–153. doi: 10.1007/ +s00773-012-0157-2. url: https://doi.org/10.1007/s00773-012- +0157-2. +[69] +F. Fleissner, A. Lehnart, and P. Eberhard. “Dynamic simulation of slosh- +ing fluid and granular cargo in transport vehicles”. Vehicle System Dy- +namics 48.1 (Jan. 2010), pp. 3–15. doi: 10.1080/00423110903042717. +url: https://doi.org/10.1080/00423110903042717. +[70] +J. Hickey, B. Broderick, B. Fitzgerald, and H. Moore. “Mitigation of +wind induced accelerations in tall modular buildings”. Structures 37 (Mar. +2022), pp. 576–587. doi: 10.1016/j.istruc.2022.01.037. url: https: +//doi.org/10.1016/j.istruc.2022.01.037. +30 + +[71] +F. Dodge. The New ”Dynamic Behavior of Liquids in Moving Contain- +ers”. Southwest Research Inst., 2000. url: https://books.google.ch/ +books?id=RltitwAACAAJ. +[72] +H. Abramson, ed. The dynamic behavior of liquids in moving contain- +ers: with applications to space vehicle technology / edited by H. Norman +Abramson. NASA SP-106. Washington, D.C: Scientific, Technical Infor- +mation Division, National Aeronautics, and Space Administration, 1966. +[73] +G. Taylor. “An experimental study of standing waves”. Proceedings of the +Royal Society of London. Series A. Mathematical and Physical Sciences +218.1132 (1953), pp. 44–59. +[74] +O. Faltinsen and A. Timokha. Sloshing. Cambridge University Press, 2009. +isbn: 9780521881111. url: https : / / books . google . ch / books ? id = +81qkPwAACAAJ. +[75] +G. S. Narimanov. “Movement of a tank partly filled by a fluid: the taking +into account of non-smallness of amplitude”. Prikl. Mat. Mekh. 21 (1957). +in Russian, pp. 513–524. +[76] +O. M. Faltinsen, O. F. Rognebakke, I. A. Lukovsky, and A. N. Timokha. +“Multidimensional modal analysis of nonlinear sloshing in a rectangu- +lar tank with finite water depth”. Journal of Fluid Mechanics 407 (Mar. +2000), pp. 201–234. doi: 10 . 1017 / s0022112099007569. url: https : +//doi.org/10.1017/s0022112099007569. +31 + diff --git a/htE3T4oBgHgl3EQfgQqz/content/tmp_files/load_file.txt b/htE3T4oBgHgl3EQfgQqz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b8a8d45001a878843495d171c53e0aa08bb7e7d5 --- /dev/null +++ b/htE3T4oBgHgl3EQfgQqz/content/tmp_files/load_file.txt @@ -0,0 +1,1748 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf,len=1747 +page_content='Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds Joar Ax˚as1, George Haller1∗ 1Institute for Mechanical Systems, ETH Z¨urich, Leonhardstrasse 21, 8092 Zurich, Switzerland 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2023 Abstract Delay embedding is a commonly employed technique in a wide range of data-driven model reduction methods for dynamical systems, including the Dynamic mode decomposition (DMD), the Hankel alternative view of the Koopman decomposition (HAVOK), nearest-neighbor predictions and the reduction to spectral submanifolds (SSMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In developing these applications, multiple authors have observed that delay embedding ap- pears to separate the data into modes, whose orientations depend only on the spectrum of the sampled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In this work, we make this ob- servation precise by proving that the eigenvectors of the delay-embedded linearized system at a fixed point are determined solely by the correspond- ing eigenvalues, even for multi-dimensional observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This implies that the tangent space of a delay-embedded invariant manifold can be pre- dicted a priori using an estimate of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We apply our results to three datasets to identify multimodal SSMs and analyse their nonlin- ear modal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While SSMs are the focus of our study, these results generalize to any delay-embedded invariant manifold tangent to a set of eigenvectors at a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, we expect this theory to be applicable to a number of data-driven model reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1 Background Much recent effort in nonlinear dynamics has focused on data-driven model reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Such algorithms return a simplified model of the system dynamics based on sampled trajectories from experiments or simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Com- monly pursued objectives for developing these methods include dimensionality reduction, sparsity, and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Prevalent methods include the proper orthogonal decomposition (POD) [1, 2] and the dynamic mode decomposition (DMD) [3, 4, 5], which fit models to data under various linearity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' E-mail: georgehaller@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='ch 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='04560v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='DS] 11 Jan 2023 Linear models cannot, however, capture characteristically nonlinear (or non- linearizable) phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Such phenomena include the coexistence of, and the transition between, isolated and compact stationary states, such as fixed points, limit cycles, and invariant tori [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To address this shortcoming, the Sparse iden- tification of nonlinear dynamics (SINDy) algorithm fits a sparse nonlinear model to training data using a library of nonlinear functions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' However, the choice of this library depends on the user [8] and the coordinate system used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Addition- ally, the size of the library scales up quickly with the problem dimensionality [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While neural networks can pattern-match nonlinear phenomena [10, 11, 12], the models they return are often difficult to interpret and generalize poorly outside the range of training data [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the last few years, spectral submanifolds (SSMs) have appeared as an alternative for model reduction in intrinsically nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' An SSM is the unique smoothest invariant manifold tangent to a nonresonant spectral sub- space emanating from a fixed point [14] or a periodic or quasiperiodic orbit [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, an attracting SSM is the ideal candidate for a low-dimensional model of a nonlinear system [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Concepts related to SSMs include nonlinear normal modes (NNMs) defined either as sets of periodic motions in conserva- tive systems [18, 19, 20] or invariant manifolds [21, 22] and invariant spectral foliations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, we will apply SSMs, as they are unique, exist under well- defined conditions in dissipative systems, can have arbitrary dimensions, and can include internally resonant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' After the computation of an SSM, we can project either equations or data onto it to reduce the system to a high-fidelity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Automated model re- duction to SSMs from equations [24] can successfully predict responses to small harmonic forcing [25, 26, 27] and bifurcations of those responses [28, 29], and has also been extended to constrained mechanical systems [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Recently, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [31] developed a data-driven method which identifies the SSM geometry and its re- duced dynamics to trajectories in an observable space [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This approach also transforms the SSM-reduced dynamics to a normal form, which describes the dynamics as sparsely as possible while maintaining essential nonlinearities [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' SSM-based model reduction has since been applied to both numerical and ex- perimental datasets in fluid and structural dynamics [34, 35] and control [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [37] showed how to improve the computational efficiency of data-driven SSM identification through a simplified formulation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Delay embedding is the method of reconstructing invariant sets by viewing a select number of measurements separated by a timelag as independent ob- servables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This method is routinely used to aid data-driven model identification in nonlinear dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Examples of model reduction methods based on delay embedding include the extended dynamic mode decomposition (DMD) [38, 39], the Hankel alternative view of the Koopman decomposition (HAVOK) [40], the eigensystem realization algorithm (ERA) [41], closure modeling [42], and nearest-neighbor prediction [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In addition, delay embedding has been extensively employed in SSM-based model reduction from data [31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For SSMs, a closer understanding of the delay embedding map improves fits to data and produces more accurate reduced-order models [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This has motivated 2 our present study on how invariant manifolds can be efficiently and accurately reconstructed in delay coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The main driver behind the introduction of delay embedding as a tool in dy- namical systems was the discovery that it could reconstruct strange attractors from scalar measurements of chaotic systems [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Floris Takens’s celebrated embedding theorem [47] and its later extension [48] show that, in principle, de- lay embedding recovers invariant sets from the full state space in a suitable observable space under generic assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In practice, however, the choice of the timelag and embedding dimension is critical to obtain robust models [49, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The many methods for choosing delay parameters for chaotic attractor reconstruction include minimization of the mutual information between subse- quent samples [52], minimization of false nearest neighbors [53, 54], and a Monte Carlo decision tree search formulation [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Recent work has also explored the geometric structure of delay embedded invariant sets in an effort to improve model order reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For periodic data, singular value decomposition (SVD) on the delay-embedded snapshot matrix has been shown to converge to a Fourier analysis [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The number of delays required to recover such periodic orbits equals the number of coefficients of the Fourier spectrum [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fitting a linear map between subsequent snapshots of such a delay-embedded periodic orbit produces a companion matrix, whose eigenvectors are given by the inverse Vandermonde matrix [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Furthermore, connections to convolutional coordinates [60] and the Frenet- Serret frame [61] have been made, and an interpretation of SVD modes in delay coordinates as principal component trajectories has been proposed [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For the special case of an observed signal composed of oscillating sinusoidal func- tions, the observable space contains invariant spaces determined by the signal frequencies [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Recently, it was shown that subsequent components of the DMD modes of delay-embedded linear systems are related by a multiplication of the corresponding eigenvalue [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In this work, we explore the local dynamics close to a fixed point of a nonlin- ear delay-embedded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We show that the linear part of the delay-embedded dynamics depends solely on the corresponding eigenvalues, and not on the ob- servable function and the full state space eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In particular, the eigen- vectors in the observable space are given by the columns of the Vandermonde matrix of the exponential of the eigenvalues multiplied by the timelag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Unlike available previous work, we do not attempt a linearization of the nonlinear dy- namics, nor do we restrict our attention to periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Instead, our results imply that the nonlinear delay-embedded system has an SSM whose tangent space coincides with the column space of this Vandermonde matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We exploit this structure to aid the data-driven identification of SSMs in three mechanical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We believe that these results enhance the understanding of delay embedding in reduced-order modeling and also reveal new opportunities for SSM-based model reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The structure of this paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' First, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2 briefly introduces SSM theory and summarizes a method for fast SSM-based data-driven modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3 outlines a new theory for delay-embedding tangent spaces of invariant 3 manifolds and discusses their application to SSM-based model reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4, we use these results to identify SSMs in examples of a 2-degree-of- freedom oscillator, simulations of multimodal vibrations in a von K´arm´an beam, and experiments of complex behavior in a sloshing tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5, we draw conclusions from these examples and discuss possible further extensions of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Finally, Appendix A contains the proofs of the results presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2 Model reduction to spectral submanifolds Here, we outline previous results on rigorous model order reduction to SSMs in smooth nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We also summarize fastSSM, the algorithm we use here to identify SSMs from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Spectral submanifold theory Consider a nonlinear, autonomous dynamical system of class Cl, l ∈ {N+, ∞, a}, where a denotes analyticity, in the form ˙x = Ax + g(x), x ∈ Rn, g ∼ O(|x|2), g : Rn → Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (1) Let us denote the flow map of the system by F t(x0) := x(t, x0), with x(t, x0) denoting the trajectory of (1) starting from x0 at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We assume that A ∈ Rn×n is diagonalizable and that the real parts of its eigenvalues are either all strictly negative or all strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We take d eigenvectors of A and denote their span by E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', a d-dimensional spectral subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In this step, we often choose the d slowest eigendirections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Provided that the d eigenvalues corresponding to E are non-resonant with the remaining n−d eigenvalues of A, the nonlinear system has a unique smoothest, invariant manifold M tangent to E at the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', T0M = E [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Following [16], we call M a spectral submanifold (SSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In case of a resonance between E and the rest of the spectrum of A, the d-dimensional SSM does not exist in general, and we must then include the resonant modal subspace in E to obtain a higher-dimensional SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' If all eigenvalues of A are stable, the slowest SSM attracts nearby trajectories, which makes it suitable for model order reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The open-source numerical package SSMTool computes SSMs from arbitrary finite-dimensional nonlinear systems [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' More recently, the SSMLearn pack- age was developed to find SSMs in data from nonlinear dynamical systems [31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, we will apply the simplified data-driven SSM algorithm fastSSM introduced by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 Fast data-driven model order reduction to spectral submanifolds The objective of dynamics-based machine learning is to reconstruct SSMs from data, and then use SSM-reduced models for predictions of the full system re- 4 sponse [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, we use fastSSM [64] to identify the SSM from snaphots of trajectories in an observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The procedure consists of two steps: man- ifold geometry detection and normal form computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The summary below follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [37], to which we refer for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Whereas that refer- ence differentiates between the algorithm for cubic polynomial approximations of two-dimensional SSMs and its extension to arbitrary order and dimension, here, we will simply refer to both algorithms as fastSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The SSM is parametrized in the graph style, that is, we construct M as a graph over the spectral subspace E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The data consists of snapshots y(ti) ∈ Rp in a p-dimensional observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For each trajectory we construct the snapshot matrix Y ∈ Rp×N from N snapshots as Y = � � | | | y(t1) y(t2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' y(tN) | | | � � (2) Let T ∈ Rp×d be a matrix whose columns approximately span the SSM tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In fastSSM, the standard procedure is to obtain T through SVD on the snapshot matrix Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' However, T can also be prescribed if the tangent space is known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Denoting by (·)† the Moore-Penrose pseudoinverse, we project each snapshot yi onto this subspace to obtain d-dimensional reduced coordinates ξ as ξ = T †y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (3) We write Ξ ∈ Cd×N for the projection of the snapshot matrix onto the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Next, we seek to approximate the embedding of M as the graph of a multi- variate polynomial of order m from the data: y(ξ) = Mξ1:m, M = [M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , Mm], Mi ∈ Rp×di, (4) where di is the number of d-variate monomials at order i and the superscript in (·)1:l denotes a vector of all monomials from order 1 up to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We obtain the manifold parametrization coefficients M ∈ Rp×d1:m by a polynomial regression, which yields the solution M = Y (Ξ1:m)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (5) The reduced dynamics are approximated by another O(r) polynomial re- gression, with a coefficient matrix G ∈ Cd×d1:r, in the form ˙ξ ≈ Gξ1:r, G = ˙Ξ(Ξ1:r)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (6) Finally, we compute the normal form [33] of the SSM-reduced dynamics up to order h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This amounts to a near-identity polynomial transformation with coefficients H ∈ Cd×d1:h from the new coordinates ζ ∈ Cd such that ξ = Hζ1:h = ζ + H2:hζ2:h, ˙ζ = Nζ1:h = Λζ + N2:hζ2:h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (7) 5 The normal form and the reduced dynamics are conjugate dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, we substitute (7) into (6) to obtain Dζ(Hζ1:h)Nζ1:h = G(Hζ1:h)1:r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (8) The matrices H and N are computed by solving (8) recursively at increasing orders with SSMTool [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This procedure requires that the training data lies sufficiently close to the SSM, which can be achieved by removing initial transients from the input signal, as identified by a spectral analysis on the training data [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Since the SSM built over the slowest d modes is unique and attracting, this method ensures relevant training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3 Delay-embedding the tangent spaces of invari- ant manifolds Here, we show how tangent spaces of invariant manifolds at a fixed point can be analytically recovered when the observable space arises from delay embedding of a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We also describe how the recovered tangent spaces facilitate the reconstruction of spectral submanifolds in such observable spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Theoretical results For the dynamical system (1), we define a scalar observable µ(x(t)), where µ : Rn → R is a differentiable function that returns a measured feature of system (1), such as a displacement coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In order to reconstruct features of the full phase space from the observable, we use delay embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We stack p consecutive measurements separated by a timelag τ > 0 to create an observable space of dimension p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This yields a trajectory in the form y(t) = S(x(t)) ∈ Rp, where we define the sampling map S : Rn → Rp, x �→ � ������ µ(x) µ(F τ(x)) µ(F 2τ(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' µ(F (p−1)τ(x)) � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (9) An important question is how invariant sets of system (1) in Rn are re- produced in the observable space Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In particular, when the full state space trajectory x(t) resides on a d-dimensional invariant manifold M, will y(t) also do so?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Takens’s embedding theorem gives an affirmative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' It states that if µ is generic and no small integer multiple of τ coincides with the period of any possible periodic orbit of (1) lying in M, then for p ≥ 2d + 1, (10) 6 the manifold M will have a diffeomorphic copy ˜ M in Rp via the mapping (9) [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Whereas Takens’s theorem was formulated only for scalar observable functions, this result has since been extended to multi-dimensional µ as long as the total observable space dimension exceeds 2d [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Both the nonlinear geometry and dynamics of M and the observable function influence the geometry of ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' It is therefore difficult to predict its geometry for a general flow map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Around the fixed point q = S(0) ∈ Rp, however, the O(1) expansion of ˜ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', its tangent space Tq ˜ M, can be directly determined, as we will show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Note that since the flow map is the identity at the origin, q lies on the diagonal in the observable space, with each of its identical components given by µ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We start by rewriting (1) in modal coordinates: ˙z = f(z) = Λz + E−1g(Ez), (11) where E = [e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , en] contains the eigenvectors of A and Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , λn) the corresponding eigenvalues, which we assume to be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We define modal coordinates z ∈ Cn by letting z = E−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Whereas the observable function is defined as a function of x, it is notationally convenient to define it as a function of z, as µ(x) = µ(Ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Let M be a d-dimensional invariant manifold of (1) intersecting the origin 0 ∈ Rn, where it is tangent to a set of d eigenvectors e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , ed of A with corresponding eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , λd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We define the Vandermonde matrix V ∈ Cp×d of the d eigenvalues governing the linearized dynamics on M as Vjk = eλkjτ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', V = � ������ 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1 eλ1τ eλ2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' eλdτ e2λ1τ e2λ2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' e2λdτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' e(p−1)λ1τ e(p−1)λ2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' e(p−1)λdτ � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (12) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Under the assumptions of a generic observable function µ : Rn → R and distinct eigenvalues λ1 ̸= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ̸= λd, the tangent space of the observable manifold ˜ M at the fixed point can be written Tq ˜ M = range V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This result is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Note that the observable function must have full rank, as spelled out in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For (13) to hold, we must have ∂µ ∂zk |0 ̸= 0 ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , d}, which defines the genericity of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' If the gradient of the observable function is orthog- onal to any of the eigenvectors e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , ed, the sampling map S will not be an embedding of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7 Manifold in full phase space Manifold in delay embedding space � ���� 1 eλ1τ e2λ1τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' � ���� � ���� 1 eλ2τ e2λ2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' � ���� Sampling map S : Rn → Rp Tangent space embedding DS(T0M) x1 x2 x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=',n y1 y2 y3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=',p M T0M ˜ M Tq ˜ M Figure 1: Delay embedding of the tangent space T0M of an invariant manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The full state space manifold M (left) has a diffeomorphic copy ˜ M in the observable space (right) by Takens’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The shape of the reconstructed manifold ˜ M depends on the flow map, but its tangent space, Tq ˜ M, is directly given by the eigenvalues at the fixed point, independent of the geometry of M and the observable function µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This should be kept in mind particularly when dealing with symmetries of engineering structures, as we will show in our examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The columns of V are eigenvectors of the linearized delay-embedded system at the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Indeed, the dynamics in the observable space can be written ˙y = V ΛV †(y − q) + o(|y − q|) (14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the observable space Rp, the timelag τ and the eigenvalues λk fully determine the tangent space and the linear part of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In particular, the linear dynamics are independent of both the full eigenvectors and the observable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the following, we will demonstrate how this structure can be exploited for parametrizing spectral submanifolds from data, when the corresponding eigen- values are approximately known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Finally, when the observable function is multi-dimensional, the tangent space is influenced by the relative dependency of each component µℓ of the observable function on each modal coordinate zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For a multidimensional observable µ : Rn → Rq with components 8 µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , µq, the tangent space Tq ˜ M ⊂ Rpq can be expressed as Tq ˜ M = range � ���� V diag � ∂µ1 ∂z ��� 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' V diag � ∂µq ∂z ��� 0 � � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' When the observable function is a set of displacements, the linearized multi- dimensional observable function ∂µ ∂z ��� 0 corresponds to the mode shapes of the system in terms of those displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, if the mode shapes and eigenvalues of the observed system are known, we can directly compute the tangent space of ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the special case of a scalar observable, the tangent space is independent of the observable function and we do not need any information about the mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 Delay-embedded spectral submanifold reconstruction These theoretical results can be exploited as a constraint to aid SSM identi- fication from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the case of a scalar signal and with the eigenvalues of interest λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , λd approximately known, we select the matrix representation of the tangent space T appearing in (3) as the Vandermonde matrix (12), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', T := V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (16) We have seen that the gradient of a multi-dimensional observable func- tion µ enters the expression for the tangent space (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While an expres- sion for this gradient is typically not available in experiments, mode shapes ˆE = [ˆe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , ˆed] ∈ Rq×d are often known from theory or obtained experimen- tally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, each mode shape ˆek ∈ Rq describes how the eigenvector ek ∈ Rn is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Specifically, they are related by ˆek = ck ∂µ ∂x ���� 0 ek, (17) where ck ∈ C is a nonzero constant that only rescales the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We select T as the columnwise Kronecker product of the Vandermonde matrix and the observable mode shapes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', T := � �� V diag (ˆe1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' V diag (ˆed) � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (18) A sketch of the geometry is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the case of unknown mode shapes, it may be possible to first project low-amplitude data onto the delay- embedded eigenvectors and then extract the observable mode shapes via SVD, although we do not explore this idea further in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 9 q µ1(x(t + τ)) µ1(x(t)) µ2(x(t)) V ˆE T ˜ M ⊂ Rpq Figure 2: The tangent space Tq ˜ M of the delay-embedded manifold ˜ M for a q-dimensional observable function µ is the range of the matrix T , defined as the columnwise Kronecker product of the Vandermonde matrix V and the mode shapes ˆE in terms of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Prescribing T and projecting the data onto its columns yields modal reduced coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This diagonalization of the system simplifies the learning of the geometry and the reduced dynamics of the SSM via the algorithm outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Choosing proper delay-embedding parameters to reconstruct nonlinear sys- tems can be a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For the linear part of the system, however, our results suggest picking the timelag τ and embedding dimensionality p so as to obtain numerically favorable reduced coordinates along the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We ideally want the columns of the Vandermonde matrix (12) to be orthogonal in order to maxi- mize the signal-to-noise ratio in each of the observed modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To this end, we formulate a minimization problem, (κ⋆, p⋆) = argmin κ,p∈N+ ��V (κ∆t, p)⊤V (κ∆t, p) − I �� F , (19) in which the columns of V are normalized and ∥·∥F denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Since the timelag is an integer multiple of the sampling timestep, τ = κ∆t, (19) defines an optimization over a set of discrete variables which can be solved simply by brute force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Bearing in mind the nonlinear part of the system, however, an optimal choice of delay parameters is not as straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Increasing the timelag and em- bedding dimension tends to curve the SSM, requiring higher orders of approx- 10 imation and in extreme cases folding the manifold, so that it can no longer be parametrized as a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Taking into account the nonlinear part of the SSM reconstruction, therefore, we typically want the total delay embedding to be as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While solving (19) gives some guidance, a suitable choice of τ and p will also depend on the nonlinearity of the system in the data range and the amount of signal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4 Applications We now apply our method to three datasets: two from simulations and one from experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The eigenvalues in these examples are known either from theory or simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We infer the delay-embedded tangent space accordingly before parametrizing the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The examples include an oscillator chain, a clamped- clamped beam and tank sloshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Two-degree-of-freedom oscillator with nonlinear springs As our first example, we consider an oscillator chain of two masses, both at- tached with linear springs to each other and to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In addition, the spring connecting the left mass to the ground has a quadratic softening non- linearity and the spring connecting the masses is of cubic hardening type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The masses and linear spring stiffnesses are set to 1, the softening parameter is −2 and the hardening parameter is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Each of the springs also has a linear damp- ing coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3a shows the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The sampling time is ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' m x1(t) k, κ c m x2(t) k, γ c k c (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 x4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 x1 x3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 M Data Re E1 Im E1 T0M (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 x1(t) x1(t + 30"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 x1(t + 15"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ~ M Data Re V1 Im V1 T0 ~ M (c) Figure 3: (a) Setup for the two-degree-of-freedom oscillator example with two nonlinear springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (b) The slow 2D SSM (gray) in the full state space, along with its tangent space (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (c) The delay-embedded SSM in the observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We compute an initial condition on the slow 2D SSM for the single training trajectory using SSMTool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Our observable function is the first mass displace- ment, µ(x) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The trajectory in the full phase space and the SSM are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The first two eigenvectors span the tangent space of the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 11 Next, we delay embed the trajectory with a timelag τ = 15∆t and embedding dimension p = 5, and seek the 2D SSM in this observable space using fastSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We obtain reduced coordinates by projection of the trajectory data onto the columns of the Vandermonde matrix V as predicted by our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3c shows the SSM in the first three coordinates of the observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Indeed, the tangent space of this observable space is identical to the column space of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 x2(t) x2(t + 30"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 x2(t + 15"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ~ M Data Re V1 Im V1 T0 ~ M (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 x3(t) + x4(t) x3(t + 30"t) + x4(t + 30"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 x3(t + 15"t) + x4(t + 15"t) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ~ M Data Re V1 Im V1 T0 ~ M (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='x1(t + 30"t) + x2(t + 30"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='x1(t) + x2(t) 0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='x1(t + 15"t) + x2(t + 15"t) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 Data (c) Figure 4: (a,b) Changing the observable function leads to different SSM geome- tries, but the tangent space remains the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (c) A nongeneric observable function however, observing only the second mode, does not embed the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Corollary 1 predicts that this tangent space will be independent of the ob- servable function, provided that it is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To illustrate this, we plot the delay-embedded SSM for various observable functions, µ(x) = x1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3c), µ(x) = x2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4a), and µ(x) = ˙x1 + ˙x2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' These different observable functions clearly produce different SSM geometries, but the eigenvectors and tangent spaces of the manifolds all agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' One exception is when we observe the distance between the masses, µ(x) = x2 − x1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In this case, the delay-embedded trajectory no longer lies on an invariant manifold, as is evident by the nonsmooth cusp in the data at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The reason is that this observable is non-generic precisely in the sense of our theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' it coincides with the mode shape of the second, fast mode of the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This means that the observable function acts orthogonally to the slow SSM at the fixed point and thus the delay mapping is not an embedding, by Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Next, we pick µ(x) = x2 and use fastSSM to approximate the cubic order reduced dynamics on the SSM from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Computing the normal form yields � ˙ρ1 ˙θ1 � = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0014 ρ13 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0148 ρ1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0025 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0919 ρ12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (20) The trajectory projected onto the columns of V is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Integrating the obtained normal form and mapping back to the observable space yields a good reconstruction of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Finally, following Theorem 3, we demonstrate how to determine the tangent 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Re(V yy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Im(V yy) (a) 0 500 1000 1500 2000 time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 x2(t) Simulation Prediction (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0 x1(t) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 x2(t + 15"t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 x1(t + 15"t) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ~ M Data Re V1 Im V1 T0 ~ M (c) Figure 5: (a) Projection of the data onto the delay-embedded tangent space predicted by our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (b) fastSSM predicts a model that successfully recon- structs the decay of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (c) A view of the SSM in a delay-embedded space from a multi-dimensional observable, with the tangent space predicted by our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' space of the SSM at the fixed point when the observable is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' When we choose µ(x) = [x1, x2]⊤, unlike for a scalar observable function, the tangent space orientation is influenced not only by the eigenvalues, but also by the shape of the first mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This first mode shape corresponds to the masses moving in unison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ˆe1 = ˆe2 = � 1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (21) Then, by (18), we obtain vectors spanning the tangent space as the columns of the matrix T = � V diag(ˆe1) V diag(ˆe2) � = � V V � , (22) where V is the Vandermonde matrix (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A view of the SSM and its tangent space in this 10-dimensional observable space is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The relation of this mode shape to the observable function is given by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In particular, we can compute the derivative of the observable function with respect to the modal coordinates as � ∂µ1 ∂z1 (0) ∂µ1 ∂z2 (0) ∂µ2 ∂z1 (0) ∂µ2 ∂z2 (0) � = � c1 c2 c1 c2 � , (23) where c1, c2 ∈ C are nonzero constants depending on the scaling of the eigen- vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For simplicity, in (21) we chose c1 = c2 = 1, such that T is the Vandermonde matrix vertically stacked twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 6D SSM in a nonlinear finite-element model of a beam We train an SSM-reduced model with data from numerical simulations of a finite-element (FE) representation of a clamped-clamped von K´arm´an nonlinear beam [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This example was previously studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [31, 37], which identified 13 the slowest 2D SSM in the delay-embedded observable space, predicted the forced response and analyzed the radius of convergence of the analytical normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, thanks to our results on the tangent spaces of delay-embedded SSMs, we can extend the analysis to the six-dimensional SSM emanating from the three slowest modes of the linear part of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Each node in the FE model has three degrees of freedom: axial deformation u, transverse deflection w, and rotation w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The von K´arm´an axial strain is given by ϵ11 = u′(x) + 1 2 (w′(x))2 − zw′′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (24) The axial stress is given by σ = Eϵ11 + c˙ϵ11, (25) where E = 70 GPa denotes the Young’s modulus and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0 × 106 Pa · s the material rate of viscous damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Based on a convergence analysis, we set the number of elements to 12, resulting in a 33-degree of freedom mechanical system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', a 66-dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We set the beam length to 1000 mm, width 50 mm, and thickness 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The sampling time is ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0955 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' w(t) w(t) w(t) Figure 6: von K´arm´an beam: schematic first, second, and third mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The scalar observable must be generic in the sense that it must have contri- butions from all modes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For instance, the midpoint displacement is not excited by the second mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Instead, we choose the shown transverse displacement at 1/4 of the beam length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' By Remark 1, the observable function µ must have significant contributions from all modes zk that we wish to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For example, the midpoint displace- ment chosen as observable function in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [31, 37] was sufficient to model the 2D SSM, but cannot be employed for higher-dimensional SSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This is be- cause the antisymmetric shape of the second mode has zero displacement at the 14 midpoint (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Figure 6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ∂µ ∂z3 (0) = ∂µ ∂z4 (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (26) Instead, we choose the transverse displacement of the beam at one fourth of the total length, µ = w(l/4), as this degree of freedom has nonzero contributions from all three mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For our data-driven modeling objectives, we need training data containing the first three modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To generate initial conditions for such trajectories, we use linear combinations of the mode shapes of the system computed from its linear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Since the SSM is normally attracting, these trajectories will quickly approach it and we can use them to train our reduced-order model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' With this method, we produce three trajectories close to the 6D SSM with different ini- tial conditions, of which we use two as training data and one as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For validation purposes, we also pick the individual mode shapes as initial condi- tions and use as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The individual modal contributions in these initial conditions were chosen as follows: Initial Mode Type condition 1 2 3 1 1 0 0 Test 2 0 1 0 Test 3 0 0 1 Test 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 Train 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 Train 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 Test We choose the delay embedding parameters guided by the observations in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Setting κ = 1 such that the timelag τ = ∆t and the embedding dimension to p = 50 gives a local optimum of the function (19) with the com- puted eigenvalues, while still keeping the maximal delay κp moderate to prevent folding of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7a shows the delay embedding of the single-mode trajectories 1-3, cor- responding to the first three modes, in three of the 50 delay coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' These trajectories visualize the orientations of the corresponding eigenspaces in the observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Indeed, minimization of (19) corresponds to making these planes orthogonal, simplifying their identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7b similarly displays the delay embedding of the first training trajectory along with a visualization of the columns of the Vandermonde matrix as vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Our delay theory predicts that projection of the data onto these vectors yields modal coordinates, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This space will serve as the reduced coordinates of the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' After projection onto these eigenvectors, we approximate the geometry of the 6D SSM with a 3rd order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For the reduced dynamics in fastSSM, we also use a 3rd order approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We compute the normal form of this reduced dynamics up to 7th order and obtain our model for the reduced dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The 15 w(t + 16"t) w(t) w(t + 8"t) Mode 1 Mode 2 Mode 3 (#2) (a) w(t + 16"t) w(t) w(t + 8"t) Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4 Re V1 Re V3 Re V5 (b) (V yy)2 (V yy)3 (V yy)1 Projection of traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4 onto V (c) Figure 7: (a) The trajectories with single modal contributions visualize the modal subspaces in the delay-embedded space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The third mode data has been scaled by a factor 2 to increase visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (b) The same delay-embedded view of the first training trajectory, along with the delay-embedded eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (c) After projection of this trajectory onto the eigenvectors, the modal structure becomes clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' terms up to third order in polar form are found by fastSSM to be of the form � � � � � � � � ˙ρ1 ρ1 ˙θ1 ˙ρ2 ρ2 ˙θ2 ˙ρ3 ρ3 ˙θ3 � � � � � � � � = � � � � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3058 ρ13 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='088ρ1 ρ22 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='091ρ1 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0 ρ13 + 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='70 ρ22ρ1 + 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2ρ1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='705 ρ12ρ2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='723 ρ23 − 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='72ρ2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='64 ρ12ρ2 + 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6 ρ23 + 1812ρ2 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='968ρ3 ρ12 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='27ρ3 ρ22 − 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='47ρ3 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='9 ρ12ρ3 + 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='04 ρ22ρ3 + 3558ρ3 � � � � � � � � + O(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (27) We transform the initial conditions from the observable space to the normal form and integrate our model to predict signal decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This produces a normal- ized mean trajectory error (as defined in [31]) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 % on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Some of the predictions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 time [s] 5 0 5 u [m] #10-3 Training data Reconstruction (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 time [s] 5 0 5 u [m] #10-3 Test data Reconstruction (b) 0 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 1 0 1 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 6 (c) Figure 8: (a,b) Predictions from fastSSM for the decaying trajectories 5 and 6 (c) Phase portrait of the trajectories after transformation to the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We also visualize our reduced-order model by plotting the instantaneous frequency and damping as predicted by the normal form (27) for varying ampli- tudes of mode 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For instance, our model predicts hardening of the first 16 mode with respect to both the first and the second modal amplitudes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 9a), a decrease in the instantaneous damping of mode 1 with respect to mode 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 9b), and independence of the third instantaneous frequency with respect to itself (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The predictions for each of the trajectories are included for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 650 1 700 1 _31 750 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 (a) 3 1 2 1 _;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1=;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 (b) 3550 1 3600 1 _33 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 3650 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 0 0 (c) Figure 9: Visualization of the normal form (27) with the trajectories for (a) instantaneous frequency and (b) damping of mode 1, as well as (c) frequency of mode 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 Multimodal sloshing of water in a tank 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' B¨auerlein and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Avila Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sketch of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A motor (a) drives an eccentric disk which converts the rotary motion of the motor via a pushing rod (b) into a quasi-harmonic horizontal oscillation of the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A positioning sensor (c) directly records the motion of the platform on which the tank (d), two high speed cameras (e) and an USB-camera (f) are mounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For the PIV measurements a light sheet (g) is provided by a laser passing through a cylinder lens (implemented in the stationary laser guiding arm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We find that neither the exact surface shape, nor the frequency spectrum are useful to determine the nonlinear resonance maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The key indicator is the phase-lag between driving and response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We systematically investigate the role of initial conditions, characterise the sloshing amplitude with the motion of the liquid’s centre of mass and directly measure the damping coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The results obtained with our approach are compared to common approaches used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the next section, we describe the experimental methods and in §3 the quantitative characterisation of the sloshing phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In §4 and §5, the Duffing and multimodal model of sloshing are respectively described and briefly compared to our measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Detailed measurements of large-amplitude sloshing are presented in §6 with focus on the nonlinear dynamics of the system, including multiplicity and competition of several flow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The experimental response curves obtained for several amplitudes are presented and compared to the Duffing and multimodal model in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' An assessment of the strengths and weakness of these models in capturing the experimentally measured response is given in §8 before the conclusion in §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Methods Our experiments were performed in a rectangular container subjected to harmonic horizontal excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' As illustrated in figure 1, the flow is quasi-two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Slosh- ing waves reaching from a quasi-planar surface, up to run-up at the tank walls and wave-breaking were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A distinct feature of the sloshing waves in an oscillated (or pitched) tank is their asymmetric shape leading to an oscillation of the liquid’s centre of mass (shown as a red dot in figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Many fundamental studies consider sloshing in wavemaker tanks (Taylor 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fultz 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Chester 1968a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A key difference between oscillated and wavemaker tanks is that in the latter the primary resonant mode is symmetric and the liquid’s centre of mass is steady in the lateral direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Experimental setup A sketch of our experimental setup is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The tank (width w = 500 mm, depth l = 50 mm) is mounted on a platform and filled with water at room temperature (a) 0 100 200 300 400 500 x [mm] Mode 1 Mode 2 Mode 3 Mode 4 (b) Figure 10: (a) Experimental setup for tank sloshing (adapted from [67]) (b) The first four sloshing mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For our final example, we apply our results to sloshing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sloshing models have a wide range of industrial applications, including fluid container interaction with ship motion [68], road transportation of fluids [69], damping devices in towers [70], and fuel tank design in spacecraft [71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A tank partially filled with water exhibits several nonlinear phenomena under horizontal harmonic excitation [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' On the one hand, intensified fluid motion can alter the instantaneous damping and frequency of the first sloshing mode [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' On the other hand, increasing the amplitude further activates several nonlinearly coupled modes of the system and gives rise to a range of different wave motions [75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 17 Our training data comes from experiments described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [67] with a rect- angular tank of width w = 500 mm and thickness 50 mm, partially filled with water up to a height of h = 400 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The tank was attached to a horizon- tally moving platform harmonically excited by a motor at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Then, once the system had reached a steady state, the motor was turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Depending on the forcing frequency, this periodic response exhibited planar, wave-breaking, or three-periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The three-periodic forced state was characterized by an increase in the response amplitude every third forcing cycle, while the wave-breaking response was defined as overturning of the water close to the walls [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A camera detected the surface profile h with the sampling time ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='01 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Figure 10a displays the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While previous work successfully captured the dynamics of the main sloshing mode using a 2D SSM for the center of mass signal [31] and the full surface profile [37], here, we model the decay from a multimodal state by identifying a 6D SSM, corresponding to the nonlinear extension of the three dominant oscillatory modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We train on three decaying measurements: Trajectory 1 and 2 start at a three-periodic state, and Trajectory 3 starts at a wave-breaking state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The observable vector µ is the surface profile measured at 1 771 points along the tank width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Since this function is multi-dimensional, in order to apply our theory on delay-embedded tangent spaces, we need an estimate of the eigen- values and linear mode shapes in our observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The eigenfrequencies can be computed from potential theory [74] as ωk = � gπ w k tanh � πk h w � , (28) which scales approximately with the square root of the mode number k for our configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The first five eigenfrequencies are [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='80, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='7, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6] rad/s, with an approximate 1:2 resonance between frequencies 1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The mode shapes by the same theory are ˆek = cos(kx/w), x ∈ [0, w], (29) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For the tangent space, in principle, we also need the lin- ear damping of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In practice, this real part of the eigenvalues has very little influence on V for limited delay embedding and we pick the values [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='07, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='08, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='09, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1] based on previous fits of the first mode and the assumption of increasing damping with the mode number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Based on (19), we delay-embed the data with timelag τ = 5∆t and dimension p = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A projection of the delay-embedded data onto the eigenvectors T predicted by our theory appears to yield modal coordinates, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Consequently, the norm of these projections can be used as a heuristic mea- sure of the modal content in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This procedure should be used with caution, since it does not take manifold curvature into account, but it can be 18 200 2000 0 2000 (T yy)3 (T yy)1 0 (T yy)2 0 200 2000 2000 Projection of traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2 onto T (a) 0 10 20 30 40 time [s] 0 1000 2000 3000 Projected amplitude Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 (b) 0 5 10 15 time [s] 0 100 200 300 Projected amplitude Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 (c) Figure 11: (a) Projecting one of the trajectories onto the tangent space vec- tors unveils the modal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (b) By projecting the trajectory onto the eigenvectors and taking the absolute value, we can estimate the relative modal contributions in the signal (c) A zoomed-in view of (b) indicates that modes 1, 2, and 4 dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' employed to provide an initial guess for the SSM dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 11b, we plot the absolute value of the projection onto each modal subspace of T over time for Trajectory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This plot shows that the first mode dominates, while the zoomed-in view (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 11c) indicates that the second and fourth modes appear to be the most prevalent of the higher modes throughout the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The third and fifth mode are present at first but quickly die out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' All amplitudes are decaying except for the fourth mode, which instead initially grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Based on this analy- sis, we will identify a 6D SSM emanating from the spectral subspace of modes 1, 2, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This choice also takes SSM theory into account, by which the 1:2 resonance requires that the modal subspace of the fourth mode is included in the spectral subspace of the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We choose to start our training data after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 s, as the third and fifth modal amplitudes are small thereafter and we expect the trajectory to lie sufficiently close to the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' With an SSM parametrization order m = 4, reduced dynamics order r = 3, and normal form order h = 3, we compute the SSM geomety and dynamics and integrate our reduced-order model to predict the decay from the various flow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This yields a normalized mean trajectory error (NMTE) [31] of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' fastSSM successfully detects and accounts for the internal resonance by adding phase-dependent terms to the computed normal form, which reads ˙ρ1 ρ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='056 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0069 sin(ψ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='26)ρ4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0015ρ2 4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='039ρ2 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='023ρ2 1 ˙θ1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='78 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0069 cos(ψ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='26)ρ4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='040ρ2 4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='016ρ2 2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='82ρ2 1 ˙ρ2 ρ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='13 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15ρ2 4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='89ρ2 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='37ρ2 1 ˙θ2 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='57ρ2 4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0085ρ2 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2ρ2 1 ˙ρ4 ρ4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='30 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='29ρ2 4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='67ρ2 2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='27 sin(ψ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4)ρ2 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2ρ2 1 ˙θ4 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='9 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='085ρ2 4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2ρ2 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='27 cos(ψ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4)ρ2 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0ρ2 1 (30) where ψ = θ4 − 2θ1 and the subscripts denote the corresponding mode number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Looking at the linear part, we see that the eigenfrequencies are well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 19 Good agreement between the experimentally measured surface profile eleva- tion at the tank’s leftmost point and the delay-embedded SSM-reduced predic- tion is shown for the first period-three initial state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 12a and the wave- breaking state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Further, our 6D reduced model can accurately predict the full surface profile decay, with snapshots shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 0 5 10 15 20 time 100 0 100 200 hx=0 [mm] Original Reconstruction (a) 0 10 20 30 40 time 100 50 0 50 100 150 hx=0 [mm] Original Reconstruction (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='8 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 0 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2 Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3 (c) Figure 12: The prediction on the 6D SSM for the decay of (a) Trajectory 1 and (b) Trajectory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (c) Phase portrait of the amplitudes of the normal form coor- dinates on the SSM for each of the trajectories shows the modal contributions and development for different intial flow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We project the training trajectories onto the SSM and transform them to the normal form in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The development of the amplitudes in the normal form are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 12c for each trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This plot suggests that (i) the wave-breaking motion (Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3) does not seem to have any significant content of the second mode, (ii) the amplitude of the fourth mode indeed increases after motor detachment, (iii) there is a small oscillation in these signals not captured by our model, which may be due to noise, insufficient separation of the modal subspaces, a mode outside our model, or some other phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 0 100 200 300 400 500 x [mm] 100 0 100 200 Elevation h [mm] t = 1:2 s Experiment Simulation 0 100 200 300 400 500 x [mm] t = 1:77 s 0 100 200 300 400 500 x [mm] t = 14:49 s Figure 13: The experimentally measured surface profile decay agrees with our 6D SSM model prediction for Trajectory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We note that the combined higher modal content in the signal is small - only about 10 % with respect to the first mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This is because the data is decaying from steady states induced by forcing near the first eigenfrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Due to their symmetric shape, isolated forcing of the second and fourth modes is not possible with horizontal harmonic excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nevertheless, we are able 20 to capture these smaller oscillations on the SSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The key technology allowing this enhancement is the enforcement of the delay-embedded tangent space in our SSM reconstruction, based on the theoretical eigenfrequencies and mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Due to the small activation of the higher modes, our model is expected to be sensitive to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' It is, however, stable with respect to changes in starting time, manifold order, and delay parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A more robust model can be obtained by decreasing the manifold dimension to 4, neglecting the relatively minor influence of the fourth mode, resulting in an average NMTE of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='9 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, since our objective was modal analysis of different flow states, we chose the more detailed 6D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5 Conclusions We have shown that for a scalar observation of an invariant manifold tangent to a spectral subspace at a fixed point, the delay-embedded reconstruction of the tangent space is dependent only on the corresponding eigenvalues of the full system linearized at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In particular, we have proven that the columns of a Vandermonde matrix, given by repeated multiplication of the exponential of the eigenvalues times the timelag, are eigenvectors for the linearized system in the observable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, the Vandermonde matrix diagonalizes the linear part of the delay-embedded dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We have also shown that when sev- eral quantities are measured and delay-embedded simultaneously, the tangent space can be expressed given the Vandermonde matrix and the mode shapes expressed in the observable function components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' These results hold for any invariant manifold tangent to a modal subspace with distinct eigenvalues, in- cluding, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', classic stable manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Here, our focus was the application of this result to spectral submanifolds of hyperbolic fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In an attempt to exploit this uncovered structure, we have shown that for data-driven SSM model reduction, when the eigenvalues are approximately known, we can analytically predict the tangent space of the embedded SSM a priori to achieve local modal decomposition and aid the reconstruction of the nonlinear reduced dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We have found that even for small activation of higher modes, this trick helps modeling complex multimodal nonlinear dynam- ics on an SSM, which in turn allows for analysis of modal energy interchange and instantaneous frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Our theory assumes a generic observable function, which we describe in more detail in our first and second example, and distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' While the second assumption is a generic one in a mathematical sense, it is not always satisfied for engineering structures with symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Vibrations in a square plate is an example where our theory would fail, as it has repeated eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Using a vector-valued observable may help in differentiating between the modes in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Further, while technically covered by the theory, possible practical difficulties related to the conditioning of the Vandermonde matrix include highly different or very similar eigenvalues, or eigenvalues of different stability type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 21 In our third example with data from experiments, we also devised a new heuristic scheme for using delay embedding to study modal contents in a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' With this method, that served as an initial guess, we projected the data onto the respective prescribed modal subspaces, thereby implicitly assuming that the SSM of each mode is nearly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' An interesting development of this idea would be its use as a filter, which could remove or keep certain frequencies in a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Another idea would be to use the tangent space condition as a verification or for iterative adjustment of the linear fit of the reduced dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Finally, in analogy with a Fourier analysis, it would be possible to estimate both instanta- neous frequency and damping of a signal by singular value decomposition of the trajectory in delay coordinates followed by analysis of the Vandermonde matrix columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' This ties in with several other observations made in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' for example, for a linear system, these columns agree with the recently proposed notion of principal component trajectories [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Overall, we believe that our findings shed more light on delay-embedding invariant manifolds and selecting delay parameters in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For that reason, we expect these results to be of use for a wide range of data-driven methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Acknowledgements We are grateful to Kerstin Avila and Bastian B¨auerlein (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Bremen) for sharing their experimental surface profile data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [67] with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 Proof of Theorem 1 Let M be a d-dimensional invariant manifold of (1) containing the origin of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The tangent space of M at the origin can be written T0M = span {ek}k∈K, where K ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , n} is an index set labeling the d eigenvectors ek spanning the spectral subspace from which the manifold is emanating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For example, for a stable manifold, K = {k : Re λk < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We also assume that the eigenvalues in question are distinct, i ̸= k ⇔ λi ̸= λk, for i, k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To simplify the notation, we transform the full state space to modal coor- dinates (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We rewrite the observable function on the system x ∈ Rn as an observable on the modal coordinate system z ∈ Cn as ˆµ(z) = µ(Ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We de- note by Φt = E ◦F t ◦E−1 the flow in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Consider the sampling map in modal coordinates ˆS = � ������ ˆµ ˆµ ◦ Φτ ˆµ ◦ Φ2τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ˆµ ◦ Φ(p−1)τ � ������ : Cn → Rp, y = ˆS(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (31) 22 Under the conditions of Takens’s embedding theorem, the delay embedding map Ψ = ˆS|M : M → ˜ M is a smooth embedding with a smooth inverse Ψ−1 : ˜ M → M, and Ψ(0) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In order to derive the tangent space Tq ˜ M, we compute the derivative of the embedding at 0: DΨ(0) = � ���� Dˆµ(0) Dˆµ(Φτ(0)) ◦ DΦτ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dˆµ(Φ(p−1)τ(0)) ◦ DΦ(p−1)τ(0) � ���� = � ���� Dˆµ(0) Dˆµ(0) ◦ eΛτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dˆµ(0) ◦ eΛ(p−1)τ � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (32) Now note that the jth component expressed in modal coordinates is Dˆµ(0) ◦ eΛjτ(z) = � k∈K ∂ˆµ ∂zk ���� 0 eλkjτzk, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' , p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (33) We define the Vandermonde matrix V of the d eigenvalues {λk}k∈K governing the linearized dynamics on M as Vjk = eλkjτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' We conclude that the tangent space of the observable manifold at the fixed point in modal coordinates can be written as Tq ˜ M = {DΨ(0)z, z ∈ Cn} = range � V diag � ∂ˆµ ∂z ���� 0 �� = range V , (34) where the diagonal matrix acts only as a rescaling of each component of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Therefore, one matrix representation of the tangent space of the manifold in the observable space is V , which is independent both of the matrix E of full system eigenvectors and the observable function µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Note that we must have ∂ ˆµ ∂zk |0 ̸= 0 ∀k ∈ K, which defines the genericity of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In practice, this implies that the linearized observable function must contain contributions from all modal coordinates that we wish to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In addition, note that for the embedding of the tangent space itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', the linear system, p = d suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 Proof of Theorem 2 The flow on ˜ M is ˜Φt = Ψ ◦ Φt ◦ Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (35) We compute the ODE on ˜ M, ˙y = ˜f(y), as ˜f = d dt ˜Φt = DΨ ◦ f ◦ Ψ−1, (36) where f is given by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The derivative of ˜f at the fixed point is, therefore, D ˜f(q) = DΨ(0) ◦ Λ ◦ DΨ−1(q) = V diag � ∂ˆµ ∂z ���� 0 � Λ diag � ∂ˆµ ∂z ���� 0 �−1 V † = V ΛV †, (37) 23 where we used the commutative property of multiplication of diagonal matri- ces, which eliminates the linearized observable function terms, and the fact that DΨ−1(q) = diag � ∂ ˆµ ∂z ��� 0 �−1 V † is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' To see this, note that the derivative of the delay embedding map composed with its inverse DΨ(0) ◦ DΨ−1(q) = V diag � ∂ˆµ ∂z ���� 0 � diag � ∂ˆµ ∂z ���� 0 �−1 V † = V V †, (38) maps all points in T0M to themselves, since V has full rank under the assump- tion of distinct eigenvalues {λk}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Taylor-expanding the ODE on ˜ M in the observable space yields ˙y = D ˜f(q)(y − q) + o(|y − q|) = V ΛV †(y − q) + o(|y − q|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (39) Therefore, under the assumptions of a generic observable function and distinct eigenvalues, the tangent space Tq ˜ M and the linearized dynamics D ˜f(q) in the observable space Rp are both fully determined by the timelag τ and the eigenvalues λk, k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' In the special case that M = Rn, if p ≥ 2n + 1, the entire phase space can be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' For a linear system, p = n suffices, and the delay embedding reduces to a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 Proof of Theorem 3 For a vector-valued observable, µ : Rn → Rq, the delay embedding map reads Ψ = � �� Ψµ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ψµq � �� : M → ˜ M ⊂ Rpq, DΨ(0) = � �� DΨµ1(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' DΨµq(0) � �� , (40) where Ψµℓ is the delay embedding map corresponding to the ℓth component of the observable function µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The derivatives are given by DΨµℓ(0) = V diag � ∂ˆµℓ ∂z ���� 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (41) In this case, the tangent space is not independent of the observable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Instead, it is affected by the relative dependency of each component µℓ of the ob- servable function on each modal coordinate zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The tangent space can, however, be expressed as Tq ˜ M = range � ���� V diag � ∂ ˆµ1 ∂z ��� 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' V diag � ∂ ˆµq ∂z ��� 0 � � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (42) 24 References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Lumley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “The Structure of Inhomogeneous Turbulent Flows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Atmo- spheric Turbulence and Radio Wave Propagation (1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 166–177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Awrejcewicz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Krys’ko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Vakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Order Reduction by Proper Orthogonal Decomposition (POD) Analysis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dy- namics of Continuous Elastic Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Springer, Berlin, Heidelberg, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 279–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Dynamic mode decomposition of numerical and experimental data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 656 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dynamic Mode Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Philadelphia, PA: SIAM, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Dynamic Mode Decomposition and Its Variants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Annual Review of Fluid Mechanics 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 225–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1146 / annurev-fluid-030121-015835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Page and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kerswell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Koopman mode expansions between simple invariant solutions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 879 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proctor, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Discovering governing equa- tions from data by sparse identification of nonlinear dynamical systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='15 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3932– 3937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Bertsimas and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Gurnee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='00176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='00176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Parsimony as the ultimate regularizer for physics-informed machine learning”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dynamics 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1801–1817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s11071-021-07118-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s11071-021-07118-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Chen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Billings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Neural networks for nonlinear dynamic system modelling and identification”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' International Journal of Control 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 319–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1080 / 00207179208934317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https : //doi.' metadata={'source': 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Ryckelynck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Model order re- duction assisted by deep neural networks (ROM-net)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' and Simul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' in Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7 (2020), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 105786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Salmela, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Tsipinakis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Foi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Billet, J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 344– 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/s42256-021-00297-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1038/s42256-021-00297-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 25 [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Loiseau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Noack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “From the POD-Galerkin method to sparse manifold models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Model Order Reduction, Volume 3: Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Benner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Grivet-Talocia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Quarteroni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rozza, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Schilders, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Silveira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' De Gruyter, Berlin, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 279–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [14] Cabr´e, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fontich, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' de la Llave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 283–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haro and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' de la Llave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “A parameterization method for the compu- tation of invariant tori and their whiskers in quasi-periodic maps: rigorous results”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Differential Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 530–579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ponsioen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Nonlinear normal modes and spectral sub- manifolds: existence, uniqueness and use in model reduction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1493–1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ponsioen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pedergnana, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Automated computation of autonomous spectral submanifolds for nonlinear modal analysis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sound and Vibration 420 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 269–295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rosenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “The Normal Modes of Nonlinear n-Degree-of-Freedom Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Journal of Applied Mechanics 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1962), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 7–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Vakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Non-linear normal modes (NNMs) and their applications in vibration theory: an overview”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mechanical Systems and Signal Processing 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1997), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3–22.' metadata={'source': 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Vakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Nonlinear nor- mal modes, Part I: A useful framework for the structural dynamicist”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mechanical Systems and Signal Processing 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 170–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='ymssp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Non-linear normal modes and invariant mani- folds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Journal of Sound and Vibration 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 170–173.' metadata={'source': 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+page_content=' Shaw and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Normal modes for non-linear vibratory systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sound and Vibration 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (1993), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 85–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Szalai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Invariant spectral foliations with applications to model order reduction and synthesis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 101 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2645–2669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Thurner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Li, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' SSMTool: Computation of invariant manifolds and their reduced dynamics in high-dimensional me- chanics problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 5281 / zenodo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 4614201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: http : //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='georgehaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ponsioen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pedergnana, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Analytic prediction of iso- lated forced response curves from spectral submanifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 98 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2755–2773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 26 [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ponsioen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Model reduction to spectral sub- manifolds and forced-response calculation in high-dimensional mechanical systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sound and Vibration 488 (2020), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 115640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dynamics 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1417–1450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s11071-021- 06957-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s11071-021-06957-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds – Part I: Pe- riodic response and forced response curve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dynamics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Li and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds – Part II: Bifurcation and quasi-periodic response”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dynamics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s11071-022-07476-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Model reduction for constrained mechanical systems via spectral submanifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 03119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='03119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ax˚as, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' B¨auerlein, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Avila, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Data-driven modeling and prediction of non-linearizable dynamics via spectral sub- manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/s41467-022- 28518-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/s41467-022-28518-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ax˚as, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' SSMLearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' georgehaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Guckenheimer and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Holmes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Oscillations, Dynamical Sys- tems and Bifircation of Vector Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Springer, New York, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ax˚as, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Eriten, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Data-driven nonlinear model reduction to spectral submanifolds in mechanical sys- tems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2229 (June 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1098/ rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1098/rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [35] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kasz´as, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Dynamics-based machine learn- ing of transitions in Couette flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Physical Review Fluids 7 (8 2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L082402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Alora, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Schmerling, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pavone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Data- Driven Spectral Submanifold Reduction for Nonlinear Optimal Control of High-Dimensional Robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='05712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ax˚as, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cenedese, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Fast data-driven model reduction for nonlinear dynamical systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nonlinear Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Williams, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kevrekidis, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rowley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode De- composition”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 9 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1307–1346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 27 [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dylewsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Barajas-Solano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Tartakovsky, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Stochastically Forced Ensemble Dynamic Mode Decomposition for Forecasting and Analysis of Near-Periodic Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' IEEE Access 10 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 33440–33448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='3161438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proctor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kaiser, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Chaos as an intermittently forced linear system”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (May 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/s41467-017-00030-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/s41467-017-00030-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Juang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pappa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “An eigensystem realization algorithm for modal parameter identification and model reduction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Journal of Guid- ance, Control, and Dynamics 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1985), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 620–627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2514/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='20031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2514/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='20031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pan and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Duraisamy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Data-Driven Discovery of Closure Models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' SIAM Journal on Applied Dynamical Systems 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='4 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2381– 2413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1137/18m1177263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1137/ 18m1177263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Pikovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Noise filtering in the discrete time dynamical systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Electron 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 (1986), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 911–914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sugihara and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' May.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Nonlinear forecasting as a way of distin- guishing chaos from measurement error in time series”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Nature 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='6268 (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1990), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 734–741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/344734a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1038/344734a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Crutchfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Prediction and stability in classical mechanics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Senior thesis in physics and mathematics, University of California, Santa Cruz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [46] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Packard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Crutchfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Farmer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Shaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Geometry from a Time Series”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 45 (9 1980), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 712–716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [47] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Takens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Detecting strange attractors in turbulence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dynamical Sys- tems and Turbulence, Warwick 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rand and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Springer Berlin Heidelberg, 1981, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 366–381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [48] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sauer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Yorke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Casdagli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Embedology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 65 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 579–616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Broomhead and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' King.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Eubank, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Farmer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Gibson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “State space recon- struction in the presence of noise”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Physica D: Nonlinear Phenomena 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 52–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' issn: 0167-2789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1016/ 0167 - 2789(91 ) 90222 - U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https : / / www .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' sciencedirect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' com / science/article/pii/016727899190222U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Yap and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rozell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Stable Takens’ Embedding for Linear Dynamical Systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2948–2953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1109/TSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2160629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fraser and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Swinney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Independent coordinates for strange attractors from mutual information”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A 33 (2 Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1986), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1134–1140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' aps.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brown, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Abarbanel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Determining embed- ding dimension for phase-space reconstruction using a geometrical con- struction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A 45 (6 Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3403–3411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1103/ PhysRevA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 45 .' 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“Local false nearest neighbors and dynamical dimensions from observed chaotic data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' E 47 (5 May 1993), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3057–3068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': 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+page_content=' [56] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Bozzo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Carniel, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fasino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Relationship between Singular Spec- trum Analysis and Fourier analysis: Theory and application to the mon- itoring of volcanic activity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Computers and Mathematics with Applica- tions 60.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mezi´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Bagheri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Schlachter, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Henningson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Spectral analysis of nonlinear flows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 641 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 115– 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [59] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Drmaˇc, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mezi´c, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Data Driven Koopman Spectral Anal- ysis in Vandermonde–Cauchy Form via the DFT: Numerical Method and Theoretical Insights”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' SIAM Journal on Scientific Computing 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='5 (2019), A3118–A3151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1137/18M1227688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: 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+page_content=' Brunton, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Time-Delay Ob- servables for Koopman: Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='48550/ ARXIV.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Hirsh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Ichinaga, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Kutz, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Brunton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Structured time-delay models for dynamical systems with connections to Frenet–Serret frame”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2254 (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2021).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 015312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1103/ PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='015312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://link.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='0018295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [66] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Jain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Tiso, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Haller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Exact nonlinear model reduction 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response maxima in liquid-sloshing experiments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 925 (2021), A22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1017/jfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Khoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “A fully coupled ship motion and sloshing analysis in various container geometries”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Journal of Marine Science and Technology 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 139–153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/ s00773-012-0157-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1007/s00773-012- 0157-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [69] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fleissner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Lehnart, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Eberhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Dynamic simulation of slosh- ing fluid and granular cargo in transport vehicles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Vehicle System Dy- namics 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 3–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1080/00423110903042717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1080/00423110903042717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [70] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Hickey, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Broderick, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Fitzgerald, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Moore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Mitigation of wind induced accelerations in tall modular buildings”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Structures 37 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 576–587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='istruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='istruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 30 [71] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Dodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The New ”Dynamic Behavior of Liquids in Moving Contain- ers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Southwest Research Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=', 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https://books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='ch/ books?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='id=RltitwAACAAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [72] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Abramson, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' The dynamic behavior of liquids in moving contain- ers: with applications to space vehicle technology / edited by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Norman Abramson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' NASA SP-106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='C: Scientific, Technical Infor- mation Division, National Aeronautics, and Space Administration, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [73] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “An experimental study of standing waves”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Proceedings of the Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Series A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mathematical and Physical Sciences 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content='1132 (1953), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 44–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [74] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Faltinsen and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Timokha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Sloshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Cambridge University Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' isbn: 9780521881111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https : / / books .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' google .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' ch / books ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' id = 81qkPwAACAAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [75] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Narimanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Movement of a tank partly filled by a fluid: the taking into account of non-smallness of amplitude”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Prikl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Mekh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 21 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' in Russian, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 513–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' [76] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Faltinsen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Rognebakke, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Lukovsky, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Timokha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' “Multidimensional modal analysis of nonlinear sloshing in a rectangu- lar tank with finite water depth”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' Journal of Fluid Mechanics 407 (Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 2000), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 201–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' 1017 / s0022112099007569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE3T4oBgHgl3EQfgQqz/content/2301.04560v1.pdf'} +page_content=' url: https : //doi.' metadata={'source': 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Faria1,2[0000−0003−3825−3954] and Rui Abreu1,3[0000−0003−3734−3157] +1 Faculty of Engineering of the University of Porto, Porto, Portugal +{jpf,rma}@fe.up.pt +2 INESC TEC - Institute for Systems and Computer Engineering, Technology and +Science, Porto, Portugal +3 INESC ID, Lisbon, Portugal +Abstract. Formal verification techniques aim at formally proving the +correctness of a computer program with respect to a formal specification, +but the expertise and effort required for applying formal specification and +verification techniques and scalability issues have limited their practical +application. In recent years, the tremendous progress with SAT and SMT +solvers enabled the construction of a new generation of tools that promise +to make formal verification more accessible for software engineers, by +automating most if not all of the verification process. The Dafny system +is a prominent example of that trend. However, little evidence exists +yet about its accessibility. To help fill this gap, we conducted a set of +10 case studies of developing verified implementations in Dafny of some +real-world algorithms and data structures, to determine its accessibility +for software engineers. We found that, on average, the amount of code +written for specification and verification purposes is of the same order +of magnitude as the traditional code written for implementation and +testing purposes (ratio of 1.14) – an “overhead” that certainly pays off +for high-integrity software. The performance of the Dafny verifier was +impressive, with 2.4 proof obligations generated per line of code written, +and 24 ms spent per proof obligation generated and verified, on average. +However, we also found that the manual work needed in writing auxiliary +verification code may be significant and difficult to predict and master. +Hence, further automation and systematization of verification tasks are +possible directions for future advances in the field. +Keywords: Formal verification · Dafny · Accessibility · Case studies. +1 +Introduction +1.1 +Motivation +Given the increasing dependence of our society on software-based systems, it is +ever more important to assure their correct, secure and safe functioning, partic- +ularly for high-integrity systems [1]. Since software development is a knowledge- +intensive activity and software-based systems are increasingly complex, errors +⋆ This is an extended version, including the source code, of our FSEN 2023 paper. +arXiv:2301.03224v1 [cs.SE] 9 Jan 2023 + +2 +João Pascoal Faria and Rui Abreu +are inevitable, so several techniques need to be applied along the process to +catch and fix defects as early as possible. +Testing and reviews are the most widely applied techniques in the software +industry for defect detection. However, since “program testing can be used to +show the presence of bugs, but never to show their absence” [2], testing alone +cannot be considered sufficient for high-integrity systems. If properly applied [3], +reviews are a cost-effective technique for defect detection and knowledge sharing, +but, like with testing, they cannot be used to show the absence of bugs. +By contrast, formal verification techniques aim at formally proving the cor- +rectness of a computer program, i.e., show the absence of defects. To that end, +we need a formal specification of the program intent and a logic reasoning frame- +work, usually based on Hoare logic [4]. But the expertise and effort required for +applying formal specification and verification techniques and scalability issues +have limited their practical application. In recent years, the tremendous progress +with SAT and SMT solvers [5], such as Z3 [6], enabled the construction of a new +generation of tools that promise to make formal verification accessible for soft- +ware engineers, like Dafny [7], Frama-C [8] and Why3 [9], by automating most if +not all of the verification process. However, little evidence exists yet about their +accessibility, regarding the expertise and effort required to apply them. +The authors have used formal specification languages and automated rea- +soning tools for several years in software engineering research, education, and +practice [10, 11, 12, 13, 14]. E.g., in [11], Alloy [15] was used to automatically +generate unit tests and mock objects in JUnit4 from algebraic specifications of +generic types. Although model-based testing approaches such as this one do not +guarantee the absence of bugs, they provide a higher assurance than manual test +generation and seem to be currently more accessible than formal verification. +From an educational perspective, the authors are also interested in assessing +the feasibility of embedding computer-supported formal specification and veri- +fication techniques in undergraduate programs, namely in courses dedicated to +studying algorithms and data structures. +1.2 +Objectives and Methodology +To help fill the gap in the current state of the art regarding accessibility stud- +ies, we conducted a set of case studies of developing verified implementations +in Dafny of some well-known algorithms and data structures of varying com- +plexity, with the goal of determining its accessibility for software engineering +practitioners, students and researchers, with limited training in formal methods. +Table 1 shows the list of case studies. They explore formal specification and +verification features of increasing complexity. In Sec. 2, we provide some high- +lights for selected features. For each case study, we collected a few metrics and +lessons learned, to help answer our main question, regarding Dafny accessibility. +Those metrics and lessons learned are aggregated and discussed in Sec. 3 ˙The +source code is available in a GitHub repository5 and Appendix A. +4 https://junit.org/ +5 https://github.com/joaopascoalfariafeup/DafnyProjects + +Case studies of development of verified programs with Dafny +3 +Table 1. List of case studies. +Category +Case study +Numerical +algorithms +◦ Integer division (Euclidean division) +◦ Natural power of a number (divide and conquer algorithm) +Searching +& +sorting +algo- +rithms +◦ Binary search +◦ Insertion sort +Collections +◦ Priority queue implemented with a binary heap +◦ Unordered set implemented with a hash table (Hash Set) +◦ Ordered set implemented with a binary search tree (Tree Set) +Matching prob- +lems +◦ Stable marriage problem solved by the Gale-Shapley algorithm +◦ Teachers placement problem reduced to stable marriage +Graph +algo- +rithms +◦ Topological sorting (Khan’s algorithm [16]) +◦ Eulerian circuit (Hierholzer’s algorithm) +1.3 +Structure of the Paper +Sec. 2 presents some highlights about specification and verification features of +increasing complexity in the case studies. Sec. 3 consolidates the metrics collected +and lessons learned, and draws conclusions regarding our research goal. Related +work is discussed in Sec. 4. Conclusions and future work are presented in Sec. 5. +2 +Case Studies Highlights +2.1 +An Introductory Example (Integer Division) +The self-explanatory program in Fig. 1 explores some basic features of Dafny +and serves as our first case study. +Dafny6 [7] is a multi-paradigm programming language and system for the de- +velopment of verified programs. The functional style is typically used for writing +specifications, using value types and side-effect-free expressions, functions, and +predicates. The procedural and object-oriented styles are typically used for writ- +ing implementations, using reference types (arrays, classes, etc.), and methods +and statements with side effects. The Dafny programming system comprises a +verifier (based on Z3), compilers that produce code in several target languages +(C#, Java, JavaScript, Go, and C++), and an extension for Visual Studio Code. +The semantics of a method (div in this case) is formally specified by means +of pre and postconditions, indicated with the requires and ensures clauses, +respectively. The Dafny verifier is in charge of checking (with the help of the +Z3 theorem prover) if such pre and postconditions are satisfied. When the im- +plementation involves a loop, the user has to provide a loop invariant (with +6 https://github.com/dafny-lang/dafny + +4 +João Pascoal Faria and Rui Abreu +// Computes the quotient q and remainder r of the integer division +// of a (non-negative) dividend n by a (positive) divisor d. +method div(n: nat, d: nat) returns (q: nat, r: nat) + requires d > 0 + ensures q * d + r == n && r < d +{ + q := 0; + r := n; + while r >= d + decreases r + invariant q * d + r == n + { + q := q + 1; + r := r - d; + } +} +// Main program, with a simple test case (checked statically!). +method Main() { + var q, r := div(15, 6); + assert q == 2 && r == 3; + print "q=", q, " r=", r, "\n"; +} + + +Fig. 1. A simple program in Dafny for performing integer division. +the invariant clause) and, in some cases, a loop variant (with the decreases +clause), to help the verifier accomplish its job. +The Main method is the entry point of a program in Dafny. In this example, it +exercises the div method for some inputs, and checks (with assert) and prints +the corresponding outputs. Like with pre and postconditions, assert statements +are checked statically by the Dafny verifier. In this example, the verifier will +try to prove the assertion based only on the postcondition of the div method +(i.e., the method body is opaque for this purpose); this makes the verification +modular and scalable. Since assertions are checked statically, test cases such as +the one shown do actually test the specification in pre-compile time, and not the +implementation at run-time; such static test cases are useful to detect problems +in the specification, e.g., incomplete postconditions. +All the specification constructs and assertions mentioned above (indicated +with the requires, ensures, invariant, decreases, and assert clauses) are +used as annotations for verification purposes only (during static analysis), but +are not compiled into the executable program, so do not cause runtime overhead. +2.2 +Lemmas and Automatic Induction (Power of a Number) +In this case study, the goal is to prove the correctness of a well-known O(log n) +divide-and-conquer algorithm to compute the natural power of a real number +(xn). Self-explanatory excerpts are shown in Fig. 2 and the full code is avail- +able in Sec. A.2. It illustrates the usage of lemmas, to specify properties that + +Case studies of development of verified programs with Dafny +5 +Dafny alone cannot deduce, and automatic induction, i.e., the ability of Dafny +to automatically prove some properties by induction (directive :induction a). +// Recursive definition of x^n in functional style. +function power(x: real, n: nat) : real { + if n == 0 then 1.0 else x * power(x, n-1) +} +// Computation of x^n in time and space O(log n). +method powerDC(x: real, n: nat) returns (p : real) + ensures p == power(x, n) +{ ... + if n % 2 == 0 { + productOfPowers(x, n/2, n/2); // recall lemma + var temp := powerDC(x, n/2); + return temp * temp; + } ... +} +// States the property x^a * x^b = x^(a+b), used by 'powerDC'. +// The property is proved by automatic induction on 'a'. +lemma {:induction a} productOfPowers(x: real, a: nat, b: nat) + ensures power(x, a) * power(x, b) == power(x, a + b) +{/*Proof should go here, but is discovered by Dafny!*/} +Fig. 2. Excerpts of a program in Dafny for computing the natural power of a number. +2.3 +Modules, Mutable Objects and Generics (Insertion Sort) +In this case study, we explore Dafny features for working with mutable objects +(in this case, arrays) and generics, and separating specification, implementation, +and test code with modules. Self-explanatory excerpts are shown in Fig. 3. +The array sorting problem is specified by the bodyless sort method in the +abstract module Sorting, resorting to auxiliary predicates. The frame condition +“modifies a” indicates that an implementation may modify the contents ref- +erenced by a. In the postcondition, “old(a[...])” and “a[..]” give the array +contents at the begin and end of method execution, respectively, as mathemat- +ical sequences. Dafny has some support for generic predicates, functions and +methods, but, unfortunately, does not support type parameters that are subject +to operations other than equality (==); so, for demo purposes, we declared the +type of array elements with a specific type definition. +Sorting algorithms may be provided in concrete modules that refine the ab- +stract module, as in the InsertionSort module, inheriting the method contract +and providing the actual algorithm in the body (omitted here). In this case, we +just had to provide the loop invariants for the verifier to successfully check the +correctness of the insertion sort algorithm with respect to the specification. +The module TestSorting shows an example of a test case of the sort +method. For the Dafny verifier to successfully check the test outcome in the + +6 +João Pascoal Faria and Rui Abreu +abstract module Sorting { + type T = int // generics limitation! + method sort(a: array) + modifies a + ensures isSorted(a[..]) && isPermutation(a[..], old(a[..])) +} + +module InsertionSort refines Sorting { + method sort(a: array) {...} +} + +abstract module TestSorting { + import opened Sorting + method testSort () { + var a := new T[] [9, 3, 6, 9]; + assert a[..] == [9, 3, 6, 9]; // proof helper! + sort(a); + SortingUniquenessProp(a[..], [3, 6, 9, 9]); //proof helper! + assert a[..] == [3, 6, 9, 9]; + } + lemma SortingUniquenessProp(a: seq, b: seq) + requires isSorted(a) && isSorted(b) && isPermutation(a, b) + ensures a == b + { /* handwritten proof by induction goes here*/} +} + + + +Fig. 3. Organization of an array sorting program in Dafny using modules. +last assert statement, we had to write an auxiliary lemma implying that the +outcome of sort is unique. Surprisingly, for the code to be checked success- +fully, we also had to provide some further “proof helper” assertions (as the first +assertion) stating trivial facts that we expected to be taken for granted. +2.4 +State Abstraction and Automatic Contracts (Priority Queue) +In this case study, we explore Dafny features for separating specification and +implementation and handling class invariants in object-oriented programs, fol- +lowing design by contract (DbC) principles. Excerpts of the specification of a +priority queue and its implementation with a binary heap are shown in Fig. 3. +The operations’ pre and postconditions of the priority queue (top box in +Fig. 3) are specified independently of the internal state representation (a bi- +nary heap in this case), by resorting to a state abstraction function (elems). +This function gives the priority queue contents as a multiset (allowing repeated +values), and serves only for specification and verification purposes (doesn’t gen- +erate executable code); to keep the specification at a high level of abstraction, +it doesn’t tell the ordering of elements (which is given by deleteMax). +In a subsequent refinement (box at the center of Fig. 3), it is chosen an +internal (concrete) state representation - a binary heap stored in an array. It is + +Case studies of development of verified programs with Dafny +7 +also provided an implementation (body) for each method (box at the bottom of +Fig. 4). The definition and verification of class invariants, stating restrictions on +the internal state to be respected at method boundaries, is facilitated in Dafny +with so-called automatic contracts, using the “:autocontracts” attribute. The +class invariant is specified in a predicate Valid; calls to that predicate, together +with some frame conditions, are automatically injected in the preconditions of +all methods and in the postconditions of all methods and constructors. +class {:autocontracts} PriorityQueue { + function elems(): multiset // State abstraction function + constructor () + ensures isEmpty() + predicate method isEmpty() + ensures isEmpty() <==> elems() == multiset{} + method insert(x : T) + ensures elems() == old(elems()) + multiset{x} + method deleteMax() returns (x: T) + requires ! isEmpty() + ensures isMax(x,old(elems())) && elems()==old(elems())-multiset{x} +} + +// Concrete state representation +var heap: array; +var size : nat; +// State abstraction function +function elems(): multiset { multiset(heap[..size]) } +// Class invariant (heap invariant) +predicate Valid() { + // valid size && each node is less or equal than its parent + size<=heap.Length && forall i :: 1<=i heap[i]<=heap[(i-1)/2] +} + +// Inserts a value x in the heap. +method insert(x : T) + ensures elems() == old(elems()) + multiset{x} +{ + // if needed, grows the array + if size == heap.Length { grow(); } + // Place at the bottom + heap[size] := x; + size := size + 1; + // Move up as needed in the heap + heapifyUp(); +} + + +Fig. 4. Excerpts of a specification (top) of a priority queue and its implementation +(center and bottom) with a binary heap in Dafny. +Thanks to the state abstraction function and the class invariant, the Dafny +verifier is able to automatically check the conformity of the methods’ imple- + +8 +João Pascoal Faria and Rui Abreu +mentation (defined in terms of the concrete state) against the methods’ pre and +postconditons (defined in terms of the abstract state), without further burden +from the user! We only had to define an auxiliary lemma, showing that the heap +invariant (indicated by the predicate Valid in Fig. 4) implies that the maximum +is at the top (array index 0). +2.5 +Proof Techniques (Topological Sorting, Eulerian Circuit) +Not surprisingly, simple algorithms may require complex proofs, as illustrated +in the topological sorting case study. In fact, the Kahn’s algorithm [16] can be +encoded in just 6 lines of code (at a high level of abstraction), but, to prove its +correctness, we had to write 7 auxiliary lemmas, sketched in Fig. 5. Fortunately, +Dafny supports a rich variety of proof techniques and is able to fill in most (if +not all) of the proof steps, so we only had to provide key intermediate steps, +making the handwritten proof of each lemma rather short. +a non-empty acyclic graph must +have at least one vertex without +incoming edges (by contradiction) +Topological sorting of an acyclic +directed graph (Kahn’s algorithm) +removing a vertex v from an +acyclic graph G produces an +acyclic graph (by contradiction) +it is possible to generate a path of +any length n in a non-empty graph +G in which all vertices have +incoming edges (by construction) +given a path p in a non-empty +graph G, if the length of p +exceeds the number of vertices, +then G has cycles (by deduction) +the length of a sequence p +of distinct elements from a set +s cannot exceed the cardinality +of the set (by induction) +given a complex path p in a graph G, +there exists a simple path (without +repeated edges) in G from the first to +the last vertex in the p (by induction) +if there is a path from u to v in a +graph G then a path from u to v +also exists in any super-graph +G' of G (by induction) +Fig. 5. Lemmas and proof techniques used to prove the correctness of Kahn’s algorithm. +However, the way the proof steps are written may have a significant impact +on the verification time. E.g., in the Eulerian circuit case study, approximately +20 seconds were spent in the verification of a lemma stating that, if an Euler +trail r exists in a graph G (i.e., a path that traverses each edge of G exactly +once), then each vertex of G has an even number of adjacent vertices, except +for the first and last vertex in r in case they are different. The proof is done +by induction. By rewriting the inductive step so that the first edge is removed +from r and G instead of the last one (possibly better matching the structure of +recursive definitions needed in the proof), the verification time was reduced to +less than 1 second! + +Case studies of development of verified programs with Dafny +9 +3 +Results and Discussion +In this section, we summarize the metrics collected and lessons learned from the +case studies conducted, and draw some conclusions regarding our research goal. +3.1 +Metrics Collected +Table 2 summarizes the metrics collected in the case studies. Size of the code +categories described in Table 3 is measured in physical lines of code (LOC), +ignoring blank lines and comments. +The execution times were measured in an Intel(R) Core(TM) i7-8750H CPU +@ 2.20GHz laptop with 6 cores and 16 GB RAM running Windows 10 Enterprise. +We used v2.1.1 of the Dafny extension for VS Code and version 3.3.0 of the Dafny +server and, in some cases, version 2.3.0 due to a bug with Z3 and Dafny v3 7. +Table 2. Results of the case studies (size, time and proof obligations). +Program +Impl. +LOC +Test +LOC +Spec. +LOC +Verif. +LOC +Total +LOC +(S+V)/ +(I+T) +Proof +Oblig. +Ver.Time +(sec.) +Integer Division +10 +5 +2 +2 +19 +0.27 +15 +0.5 +Power of a Number +17 +7 +4 +5 +33 +0.38 +45 +0.5 +Binary Search +15 +7 +7 +3 +32 +0.45 +51 +0.5 +Insertion Sort +13 +13 +10 +21 +57 +1.19 +90 +1 +Priority Queue +74 +13 +30 +35 +152 +0.75 +483 +3 +Hash Set +86 +16 +57 +38 +197 +0.93 +656 +16 +Tree Set +87 +13 +39 +38 +177 +0.77 +809 +18 +Stable Marriage +50 +66 +54 +10 +180 +0.55 +209 +7 +Topological Sorting +19 +18 +21 +94 +152 +3.11 +157 +3 +Eulerian Circuit +32 +10 +66 +115 +223 +4.31 +407 +19 +Total +403 +168 +290 +361 +1222 +1.14 +2922 +69 +Table 3. Code categories. +Category +Description +Implemen- +tation +“Traditional”, compilable, implementation code (method signatures, +method bodies, data definitions, etc.). +Test +Test code (checked statically or dynamically), including assertions. +Specification Specification of contracts, including requires and ensures clauses, class +invariants, frame conditions, and auxiliary definitions used in them. +Verification +Verification helper code, such as, lemmas and all non-compilable code +inside method bodies (loop variants, loop invariants, assertions, invo- +cation of lemmas, manipulation of ghost variables, etc.). +7 https://github.com/dafny-lang/dafny/issues/1498 + +10 +João Pascoal Faria and Rui Abreu +Impl. LOC +33% +Test LOC +14% +Verif. LOC +29% +Spec. LOC +24% +Code size (LOC) distribution +Fig. 6. Code size (LOC) distribution. +On average, the amount of code written for formal specification (S) and +verification (V) purposes is of the same order of magnitude as the “traditional” +code written for implementation (I) and testing (T) purposes – an “overhead” +that certainly pays off, at least for high-integrity software. The average ratio is +(S+V)/(I+T)=1.14, ranging from 0.27 in the simplest case to 4.31 in the most +complex case. The pie chart of Fig. 6 shows a balanced size distribution, on +average, between the different code categories. +The overhead on user time is difficult to measure as it depends heavily on the +user experience. A fair assessment should be done in a different context (in the +case studies, the algorithms were known, but the verification strategies had to be +discovered in many cases). We believe that, with proper training, in cases where +new algorithms have to be designed, the specification and verification effort can +be of the same order of magnitude as the design, implementation, and test effort. +The number of proof obligations (POs) generated and checked by the Dafny +verifier is impressive, with 2.4 POs generated on average per LOC written (2922 +POs/1222 LOC in Table 2), and 7.3 per implementation LOC (2922 POs/403 +LOC in Table 2), in the case studies. The performance of the Dafny verifier was +also impressive, with 24 ms spent on average per PO generated and verified (69 +sec/292 POs in Table 2), in this set of case studies. +However, based on the experience of the case studies, it is important to note +that the verification of some POs may be significantly higher, in the order of +minutes, or not even terminate. When that happens, with careful debugging +and refactoring (of assertions, verification code, etc.), one may usually reduce +the verification time drastically (as illustrated in the Euler Circuit case study). +3.2 +Lessons Learned +The lessons learned from the case studies are summarized in Tables 4 and 5, +using a color scheme to highlight strengths and weaknesses. Overall, the Dafny +language and verifier proved to be very powerful, automating most of the ver- +ification work, with minor language limitations (regarding generics, automatic +contracts, and other aspects). Regarding our main research question, the major +difficulty we found is that the manual verification work may be significant and +difficult to predict and master in non-trivial programs. + +Case studies of development of verified programs with Dafny +11 +Table 4. Lessons learned from the case studies (Part I). +Category Lessons learned (strengths and weaknesses) +Dafny +Lan- +guage +– Integrated language for writing specifications (methods’ pre and +postconditions), implementations (methods’ bodies), and verifica- +tion helper code (e.g., loop invariants)[ex: Integer Division]. +– Rich set of logical quantifiers (forall, exists, etc.) and mathe- +matical collections (sequences, sets, multisets, maps, etc.), for writ- +ing specifications and assertions and describing complex algorithms at a +high level of abstraction [ex: Binary Search, Stable Marriage]. +– Inductive data types and pattern matching expressions may be +used to keep the code at a high level of abstraction [ex: Hash Set]. +– Null safety: reference types are not nullable unless they are marked +with the “?” suffix. [ex: Tree Set] +– Constructs to specify frame conditions and query the old object +state, when working with mutable objects [ex: Insertion Sort]. +– Modules enable a clear separation between specification, implementa- +tion, and test code [ex: Insertion Sort]. +– Limited support for generics: lack of support for type parameters that +are subject to operations other than equality [ex: Binary Search]. +– The support for explicitly separating specification and implementation +and hiding implementation details in object-oriented programs has room +for improvement (e.g., there are no visibility modifiers) [ex: Tree Set]. +Dafny +Com- +piler +– The Dafny compiler is able to generate executable code in multiple +target languages (in this case, only C# is explored). +– Assertions and other constructs used for specification & verification pur- +poses are not compiled, so they imply no runtime overhead. +Dafny +Verifier +– In many cases, the verifier is able to automatically check that the +implementation conforms to the specification, with minimal user +help (that may only have to write loop invariants) [ex: Integer Division]. +– Dafny is frequently able to discover loop variants [ex: Binary Search]. +– Outside of a method, the method body is opaque for verification pur- +poses (only the pre and postconditions matter), making the verification +process modular and scalable. +Manual +Verifi- +cation +Work +– Dafny effectively supports a rich variety of proof techniques (by de- +duction, by induction, by contradiction, by construction, calcu- +lational[17]) [ex: Topological Sorting, Tree Set] +– Auxiliary properties may need to be defined by the user (as lemmas) to +help the verifier, but the proof itself may be greatly or totally automated, +with many details automatically filled in; discovering what properties +need to be defined is not trivial, though [ex: Power, Top. Sort.]. +– It is difficult to predict when and what manual work will be +needed (beyond writing loop invariants) for a successful verification +[ex: Insertion Sort, Topological Sorting]. + +12 +João Pascoal Faria and Rui Abreu +Table 5. Lessons learned from the case studies (Part II). +Category Lessons learned (strengths and weaknesses) +Auto- +matic +con- +tracts +– Dafny supports the definition and enforcement of class invariants, +especially using the ”:autocontracts“ attribute, also taking care of the +generation of appropriate frame conditions [ex: Priority Queue]. +– Automatic contracts have room for improvement; in some cases, the user +may need to resort to lower level features [ex: Tree Set, Hash Set]. +– Getting the contracts right in classes that represent self-referencing data +structures may be rather tricky [ex: Tree Set]. +– There are apparent conflicts between inheritance and automatic con- +tracts [ex: Priority Queue]. +State +Abstrac- +tion +– State abstraction functions (ghost functions) allow specifying the +semantics (pre/postconditions) of the services provided by a class inde- +pendently from the implementation (method bodies and internal state +representation) [ex: Priority Queue]. +– State abstraction may also be accomplished through abstract state +variables (ghost variables), whose abstraction relation to the concrete +state variables is specified in the class invariant [ex: Hash Set]. +Testing +– Testing is still relevant, but mainly for statically testing the specifi- +cation, and not dynamically testing the implementation (proved to be +correct with respect to the specification) [ex: Integer division, Ins. Sort]. +– Test cases that allow multiple outputs can be easily specified and checked +[ex: Insertion Sort]. +Debug- +ging and +Profiling +– When verification fails, the Dafny language and the Dafny verifier pro- +vide several convenient features for debugging purposes, such as the +assume statement and the “/tracePOs” option [ex: Eulerian Circuit]. +– When the verification time is high, most of the time may be concentrated +on one or two assertions. By identifying and rewriting such assertions, +the verification time may be drastically reduced [ex: Eulerian Circuit]. +3.3 +Accessibility assessment +We distinguish three levels of competencies required for the development of ver- +ified programs in Dafny, with decreasing accessibility: +– basic: writing implementation and test code; +– intermediate: writing specifications (pre/post-conditions, frame conditions, +class invariants, and related predicates and functions), and loop variants and +invariants; +– advanced: identifying and writing the needed verification code, besides loop +variants and invariants (auxiliary lemmas, assertions, ghost variables, etc.). +Lessons learned and metrics collected in the case studies suggest that, even +in seemingly simple problems, the user may need to be skilled in advanced veri- +fication features and techniques. + +Case studies of development of verified programs with Dafny +13 +Hence, despite the impressive improvements in automated program verifi- +cation provided by Dafny, we claim that “we are very close to, but not there +yet” regarding the goal of making the development of verified programs acces- +sible for software engineering practitioners and students. Further automation +and systematization of verification tasks (including reusable libraries of common +properties and “how to” guides), and integration in mainstream languages, are +possible directions for further work in the field. +Our assessment is corroborated by our experience in teaching a course on +“Formal Methods in Software Engineering”8 with 151 master students enrolled +in the 2020/21 academic year, with a very positive students feedback (average +score of 6 out of 7). Students with a high grade (≥ 85%) in a midterm exam +were invited to develop a project in Dafny, consisting in the development of a +verified implementation of an algorithm or data structure of medium complexity +(hash set, tree set, stable marriage, topological sorting, Eulerian circuit, and text +compression). Out of 28 students eligible, 14 picked the challenge, but only 9 +delivered, and none met the goals fully. We should note that the classes on formal +specification and verification (4 hours per week during 6 weeks) only superficially +addressed advanced verification techniques, and the students had a relatively +short time to do the project (1 month). This experience led us to conclude that +more advanced training is required to prepare interested students to handle non- +trivial specification and verification problems using Dafny or similar systems. +4 +Related Work +In [18], the authors report their experience of using Dafny at the VerifyThis +2021 program verification competition, which aims to evaluate the usability of +logic-based program verification tools in a controlled experiment, challenging +both the verification tools and the users of those tools. They tackled two of the +proposed challenges, and, as a result, identify strengths and weaknesses of Dafny +in the verification of relatively complex algorithms. Some strengths mentioned +are: Dafny’s ability to prove termination and memory safety with little input; +built-in value types, such as sets, sequences, multisets, and maps; predicates and +lemmas for more concise specifications; automatic induction; ghost variables and +functions. They found it difficult to verify properties of possibly null objects, +among other difficulties, impeding them from completing all the tasks on time. +In [19] the authors argue that formal verification tools are often developed +by experts for experts; as a result, their usability by programmers with little for- +mal methods experience may be severely limited. They present their experiences +with AutoProof (a tool that can verify the functional correctness of object- +oriented software in Eiffel) in two contexts representative of non-expert usage. +First, they discuss its usability by students in a graduate course on software +verification, who were tasked with verifying implementations of various sorting +algorithms. Second, they evaluate its usability in verifying code developed for +8 https://sigarra.up.pt/feup/en/UCURR_GERAL.FICHA_UC_VIEW?pv_ocorrencia_i +d=459493 + +14 +João Pascoal Faria and Rui Abreu +programming assignments of an undergraduate course. They report their experi- +ences and lessons learned, from which they derive some suggestions for improving +the usability of verification tools. They report an average 1.3 ratio between the +number of tokens in specification and verification annotations and implemen- +tation code, in two small programs. In spite of the differences in context and +measurement units, that ratio is of the same order of magnitude as ours. +In [20] the authors refer that formal methods are often resisted by students +due to perceived difficulty, mathematicity, and practical irrelevance. They re- +developed their software correctness course by taking a programming intensive +approach, using Dafny to provide instant formative feedback via automated as- +sessment, which resulted in increased student retention and course evaluation. +Although very positive overall, their students found Dafny difficult to learn and +use, and the informal observations of the authors are that many of those diffi- +culties stem from “accidental” complexity introduced by the Dafny tool. They +propose some changes to Dafny’s design to tackle some issues found related to +program testing, verification debugging, and class invariants, among others. +5 +Conclusions and Future Work +We conducted a set of case studies of developing verified implementations in +Dafny of some real-world and well-known algorithms and data structures, with +the goal of determining its accessibility for software engineering students, practi- +tioners and researchers. We concluded that, despite the impressive improvements +in automated program verification provided by Dafny, the manual work needed in +writing auxiliary verification code may be significant and difficult to predict and +master. Further automation and systematization of verification tasks (including +reusable libraries of common properties and “how to” guides), and integration in +mainstream languages, are possible directions for further work in the field. We +also intend to conduct further studies with other verifiers and problems. +Acknowledgements +This work is financed by National Funds through the Portuguese funding agency, +FCT — Fundação para a Ciência e a Tecnologia within project EXPL/CCI- +COM/1637/2021. +References +[1] +Barry Boehm. “Some future trends and implications for systems and soft- +ware engineering processes”. In: Systems Engineering 9.1 (2006), pp. 1– +19. +[2] +Edsger Wybe Dijkstra et al. Notes on structured programming. 1970. +[3] +Watts S Humphrey. Introduction to the team software process (sm). Addison- +Wesley Professional, 2000. + +Case studies of development of verified programs with Dafny +15 +[4] +Charles Antony Richard Hoare. “An axiomatic basis for computer pro- +gramming”. In: Communications of the ACM 12.10 (1969), pp. 576–580. +[5] +Moshe Y Vardi. “The automated-reasoning revolution: from theory to prac- +tice and back”. In: Distinguished Lecture at NSF CISE, Spring (2016). +[6] +Leonardo de Moura and Nikolaj Bjørner. “Z3: An efficient SMT solver”. +In: Int. Conf. on Tools and Algorithms for the Construction and Analysis +of Systems. Springer. 2008, pp. 337–340. +[7] +K Rustan M Leino. “Accessible software verification with Dafny”. In: IEEE +Software 34.6 (2017), pp. 94–97. +[8] +Pascal Cuoq et al. “Frama-c”. In: Int. conf. on software engineering and +formal methods. Springer. 2012, pp. 233–247. +[9] +Jean-Christophe Filliâtre and Andrei Paskevich. “Why3—where programs +meet provers”. In: European symposium on programming. Springer. 2013, +pp. 125–128. +[10] +Rui Abreu et al. “Using constraints to diagnose faulty spreadsheets”. In: +Software Quality Journal 23.2 (2015), pp. 297–322. +[11] +Francisco Rebello de Andrade et al. “Specification-driven unit test genera- +tion for java generic classes”. In: Int. Conf. on Integrated Formal Methods. +Springer. 2012, pp. 296–311. +[12] +José Campos and Rui Abreu. “Encoding test requirements as constraints +for test suite minimization”. In: 2013 10th Int. Conf. on Information Tech- +nology: New Generations. IEEE. 2013, pp. 317–322. +[13] +Alexander Diedrich et al. “Applying simulated annealing to problems in +model-based diagnosis”. In: Int. Workshop on Principles of Diagnosis: DX- +2016. ARC-E-DAA-TN35662. ebook DX conference series. 2016. +[14] +Bruno Lima, João Pascoal Faria, and Robert Hierons. “Local observability +and controllability analysis and enforcement in distributed testing with +time constraints”. In: IEEE Access 8 (2020), pp. 167172–167191. +[15] +Daniel Jackson. Software Abstractions: logic, language, and analysis. MIT +press, 2012. +[16] +Arthur B Kahn. “Topological sorting of large networks”. In: Communica- +tions of the ACM 5.11 (1962), pp. 558–562. +[17] +K Rustan M Leino and Nadia Polikarpova. “Verified calculations”. In: +Working Conf. on Verified Software: Theories, Tools, and Experiments. +Springer. 2013, pp. 170–190. +[18] +Marie Farrell, Conor Reynolds, and Rosemary Monahan. “Using dafny +to solve the VerifyThis 2021 challenges”. In: Proc. of the 23rd ACM Int. +Workshop on Formal Techniques for Java-like Programs. 2021, pp. 32–38. +[19] +Carlo A Furia, Christopher M Poskitt, and Julian Tschannen. “The Auto- +Proof verifier: Usability by non-experts and on standard code”. In: arXiv +preprint arXiv:1508.03895 (2015). +[20] +James Noble et al. “More Programming Than Programming: Teaching +Formal Methods in a Software Engineering Programme”. In: NASA Formal +Methods Symposium. Springer. 2022, pp. 431–450. + +16 +João Pascoal Faria and Rui Abreu +A +Code of the Case Studies +A.1 +Integer Division +/∗ +∗ The Dafny " Hello , +World ! " : +a simple +program +f o r +performing +∗ +i n t e g e r +d i v i s i o n . +∗/ +// Computes the +quotient +’q ’ +and remainder +’ r ’ +of +the +i n t e g e r +// +d i v i s i o n +of +a (non−negative ) +dividend +’n ’ +by a ( p o s i t i v e ) +// +d i v i s o r +’d ’ . +method div (n : +nat , +d : +nat ) +returns +(q : +nat , +r : +nat ) +r e q u i r e s d > 0 +ensures q ∗ d + r == n && r < d +{ +q := +0; +r := n ; +while +r >= d +decreases +r +invariant +q ∗ d + r == n +{ +q := q + 1; +r := r − d ; +} +} +// Main program , +with a simple +t e s t +case +( checked +s t a t i c a l l y ! ) +method Main () +{ +var q , +r := div (15 , +6 ) ; +a s s e r t +q == 2 && r == 3; +print +"q = " , q , +" r =", r , +"\n "; +} +A.2 +Power of a Number +/∗ +∗ Formal +v e r i f i c a t i o n +of +an O( log n) +algorithm +to +c a l c u l a t e +∗ the +natural +power +of +a +r e a l +number (x^n ) , +i l l u s t r a t i n g +the +∗ usage +of lemmas and automatic +induction +in +Dafny . +∗/ +// +Recursive +d e f i n i t i o n +of x^n in +f u n c t i o n a l +style , +// with +time and space +complexity O(n ) . +function +power (x : +real , +n : +nat ) +: +r e a l +{ +i f +n == 0 then +1.0 +e l s e +x ∗ power (x , n−1) +} + +Case studies of development of verified programs with Dafny +17 +// Computation +of x^n in +time and space O( log n ) . +method powerDC(x : +real , +n : +nat ) +returns +(p +: +r e a l ) +ensures p == power (x , +n) +{ +i f +n == 0 { +return +1 . 0 ; +} +e l s e +i f +n == 1 { +return x ; +} +e l s e +i f +n % 2 == 0 { +productOfPowers (x , +n/2 , n / 2 ) ; +// +r e c a l l +lemma +var temp := powerDC(x , +n / 2 ) ; +return temp ∗ temp ; +} +e l s e +{ +productOfPowers (x , +(n−1)/2 , +(n−1)/2); // +r e c a l l +lemma +var temp := powerDC(x , +(n−1)/2); +return temp ∗ temp ∗ x ; +} +} +// +States +the +property x^a ∗ x^b = x^(a+b ) . +// The property +i s +proved by automatic +induction on +’a ’ . +lemma {: induction +a} productOfPowers (x : +real , +a : +nat , +b : +nat ) +ensures +power (x , +a ) ∗ power (x , +b) +== power (x , +a + b) +{ } +// A few +t e s t +cases +( checked +s t a t i c a l l y +by Dafny ) . +method testPowerDC () +{ +var p1 := powerDC( +2.0 , +5 ) ; +a s s e r t +p1 == 3 2 . 0 ; +var p2 := powerDC( −2.0 , +2 ) ; +a s s e r t +p2 == 4 . 0 ; +var p3 := powerDC( −2.0 , +1 ) ; +a s s e r t +p3 == −2.0; +var p4 := powerDC( −2.0 , +0 ) ; +a s s e r t +p4 == 1 . 0 ; +var p5 := powerDC( +0.0 , +0 ) ; +a s s e r t +p5 == 1 . 0 ; +} +A.3 +Binary Search +/∗ +∗ Formal +v e r i f i c a t i o n +of +the +binary +search +algorithm +with +∗ Dafny . +∗/ +type T = int +// +f o r demo purposes , +but +could be another +type +// Checks +i f +array +’a ’ +i s +sorted . + +18 +João Pascoal Faria and Rui Abreu +predicate +isSorted ( a : +array) +reads a +{ +f o r a l l +i , +j +: : +0 <= i < j < a . Length ==> a [ i ] <= a [ j ] +} +// Finds a value +’x ’ +in a +sorted +array +’a ’ , +and +returns +// +i t s +index , +or −1 +i f +not found . +method binarySearch ( a : +array, x : T) +returns +( index : +int ) +r e q u i r e s +isSorted ( a ) +ensures +(0 <= index < a . Length && a [ index ] == x) +| | +( index == −1 && x +! in a [ . . ] ) +{ +var low , +high := 0 , a . Length ; +while +low < high +invariant +0 <= low <= high <= a . Length +invariant +x +! in a [ . . low ] && x +! in a [ high . . ] +{ +var mid := low + ( high − low ) / +2; +i f +{ +case a [ mid ] +< x => low := mid + 1; +case a [ mid ] +> x => high := mid ; +case a [ mid ] == x => return mid ; +} +} +return +−1; +} +// Simple +t e s t +cases +to +check +the +post−condition . +method testBinarySearch () +{ +var a := new int [ 5 ] +[ 1 , +4 , +4 , +6 , +8 ] ; +a s s e r t +a [ . . ] +== [ 1 , +4 , +4 , +6 , +8 ] ; +// Proof +helper +var +id1 := binarySearch (a , +6 ) ; +a s s e r t +id1 == 3; +var +id2 := binarySearch (a , +3 ) ; +a s s e r t +id2 == −1; +var +id3 := binarySearch (a , +4 ) ; +a s s e r t +id3 +in +{1 , +2}; +} +A.4 +Insertion Sort +/∗ +∗ Formal +v e r i f i c a t i o n +of +the +i n s e r t i o n +sort +algorithm +∗ with Dafny . +∗/ +// Contract +f o r +s o r t i n g +algorithms . +abstract +module +Sorting +{ +type T = int +// +f o r demo purposes , +but +could be another +type + +Case studies of development of verified programs with Dafny +19 +// Abstract method +d e f i n i n g +the +contract +( semantics ) +of +// +array +s o r t i n g . +method +sort ( a : +array) +modifies +a +ensures +isSorted ( a [ . . ] ) +ensures +multiset ( a [ . . ] ) == multiset ( old ( a [ . . ] ) ) +// +Auxiliary +predicate +that +checks +i f +a sequence +’a ’ +// +i s +sorted . +predicate +isSorted ( s : +seq) { +f o r a l l +i , +j +: : +0 <= i < j < | s | ==> s [ i ] <= s [ j ] +} +} +// +S t a t i c +t e s t s +of +the +Sorting +contract +abstract +module +TestSorting { +import opened +Sorting +method +testSortSimple () +{ +var a := new T [ ] +[ 9 , +4 , +6 , +3 , +8 ] ; +a s s e r t +a [ . . ] == [ 9 , +4 , +6 , +3 , +8 ] ; +// +prover +helper ! +sort ( a ) ; +a s s e r t +a [ . . ] == [ 3 , +4 , +6 , +8 , +9 ] ; +} +method testSortWithDups () +{ +var a := new T [ ] +[ 9 , +3 , +6 , +9 ] ; +a s s e r t +a [ . . ] == [ 9 , +3 , +6 , +9 ] ; +// +prover +helper +sort ( a ) ; +SortingUniquenessProp ( a [ . . ] , +[ 3 , +6 , +9 , +9 ] ) ; +a s s e r t +a [ . . ] == +[ 3 , +6 , +9 , +9 ] ; +// +a s s e r t i o n +v i o l a t i o n +( ! ? ) +} +// +State and prove by induction +the +property +that , +i f +two +// +sequences +are +sorted +and have the same +multiset +of +// elements , +then +they must be +i d e n t i c a l +( so +s o r t i n g +has a +// unique +s o l u t i o n ) . +lemma SortingUniquenessProp ( a : +seq, b : +seq) +r e q u i r e s +isSorted ( a ) && isSorted (b) +&& multiset ( a ) == multiset (b) +ensures a == b +{ +// +r e c a l l s +u s e f u l +p r o p e r t i e s +about +sequences and +t h e i r +// +multisets +seqProps ( a ) ; +seqProps (b ) ; +// key +steps +of +proof by induction on +’a ’ +and +’b ’ +// ( the +r e s t +i s +f i l l e d +in by Dafny ) +i f +| a | > 0 { +SortingUniquenessProp ( a [ 1 . . ] , +b [ 1 . . ] ) ; + +20 +João Pascoal Faria and Rui Abreu +} +} +// +States +two +p r o p e r t i e s +about +sequences +( proved by Dafny +// +alone ) : +// − sequence +concatenation +r e v e r t s +s p l i t t i n g +in +head and +// +t a i l ; +// − elements +of +a sequence +belong +to +i t s +multiset . +lemma seqProps ( a : +seq) +ensures +| a | > 0 ==> a == [ a [ 0 ] ] + a [ 1 . . ] +ensures +f o r a l l +i +: : +0 <= i < | a | ==> a [ i ] +in +multiset ( a ) +{} +} +module +I n s e r t i o n S o r t +r e f i n e s +Sorting +{ +// +Sorts +array +’a ’ +using +the +i n s e r t i o n +sort +algorithm . +// +I n h e r i t s +the +contract +from +Sorting . +method +sort ( a : +array) { +f o r +i +:= 0 to a . Length +invariant +isSorted ( a [ . . i ] ) +invariant +multiset ( a [ . . ] ) == multiset ( old ( a [ . . ] ) ) +{ +var +j +:= +i ; +while +j > 0 && a [ j −1] > a [ j ] +invariant +f o r a l l +l , +r +: : +0 <= l < r <= i && r != +j +==> a [ l ] <= a [ r ] +invariant +multiset ( a [ . . ] ) == multiset ( old ( a [ . . ] ) ) +{ +a [ j −1] , a [ j ] +:= a [ j ] , +a [ j −1]; +//swap ( p a r a l l e l +assign . ) +j +:= +j − 1; +} +} +} +} +A.5 +Priority Queue +/∗ +∗ Formal +s p e c i f i c a t i o n +and +v e r i f i c a t i o n +of +a +P r i o r i t y +Queue +∗ implemented +as a heap . A heap +i s +a +p a r t i a l l y +ordered +set +∗ +represented +in an array , +suited +to +implement +p r i o r i t y +∗ queues +operations +i n s e r t +and deleteMax +in O( heapSize ) . +∗ +I l l u s t r a t e s +the +v e r i f i c a t i o n +of +object −oriented +programs +∗/ +type T = int +// +f o r demo purposes , +but +could be +real , +etc . + +Case studies of development of verified programs with Dafny +21 +c l a s s +{: autocontracts } PriorityQueue { +// Concrete +s t a t e +representation +var heap : +array; +var +s i z e +: +nat ; +// +Configuration +parameters +s t a t i c +const +i n i t i a l C a p a c i t y +:= +10; +// +Class +invariant +( heap +invariant + automatic +things +// +generated by +: autocontracts ) +predicate +Valid () +{ +heapInv () +} +// Heap invariant +predicate +{: autocontracts +f a l s e } heapInv () +reads +this , +heap +{ +// +valid +s i z e +s i z e <= heap . Length +// each node +i s +l e s s +or +equal +than +i t s +parent +&& f o r a l l +i +: : +1 <= i < s i z e ==> heap [ i ] <= heap [ ( i −1)/2] +} +// +State +abstraction +function : +gets +the heap +contents +as a +// +multiset . +function +elems ( ) : +multiset +{ +multiset ( heap [ . . s i z e ] ) +} +// +I n i t i a l i z e s +the heap as empty . +constructor () +ensures +isEmpty () +{ +heap := new T[ i n i t i a l C a p a c i t y ] ; +s i z e +:= +0; +} +// Checks +i f +the heap +i s +empty +predicate +method isEmpty () +ensures +isEmpty () <==> elems () == multiset {} +{ +// to +help +proving +the +post−condition +a s s e r t +elems () == multiset {} <==> | elems ( ) | == 0; +// +actual +expression +s i z e == 0 +} +// +I n s e r t s +a value x in +the heap . + +22 +João Pascoal Faria and Rui Abreu +method +i n s e r t (x +: T) +ensures +elems () == old ( elems ( ) ) + multiset {x} +{ +i f +s i z e == heap . Length { +grow ( ) ; +} +// Place +at +the bottom +heap [ s i z e ] +:= x ; +s i z e +:= +s i z e + 1; +// Move up as +needed +in +the heap +heapifyUp ( ) ; +} +// Method used +i n t e r n a l l y +to grow the heap +capacity +method grow () +r e q u i r e s +s i z e == heap . Length +ensures +heap . Length > s i z e +ensures +heap [ . . s i z e ] == old ( heap [ . . s i z e ] ) +{ +var oldHeap := heap ; +heap := new T[ i f +s i z e == 0 then +i n i t i a l C a p a c i t y +e l s e +2 ∗ +s i z e ] ; +f o r a l l +i +| +0 <= i < oldHeap . Length { +heap [ i ] +:= oldHeap [ i ] ; +} +} +// +Auxiliary method to move a +dirty +node from the bottom +// upwards +in +the heap +method +{: autocontracts +f a l s e } heapifyUp () +r e q u i r e s +s i z e > 0 && heapifyUpInv ( size −1) +modifies +heap +ensures +heapInv () +ensures +multiset ( heap [ . . s i z e ])== old ( multiset ( heap [ . . s i z e ] ) ) +{ +var k := +s i z e − 1; +while k > 0 && heap [ k ] > heap [ ( k − 1) / +2] +invariant +0 <= k < s i z e +invariant +heapifyUpInv (k) +invariant +multiset ( heap [ . . s i z e ] ) == +old ( multiset ( heap [ . . s i z e ] ) ) +{ +heap [ k ] , +heap [ ( k−1) / +2] +:= heap [ ( k − 1) / +2 ] , +heap [ k ] ; +k := (k − 1) / +2; +} +} +// During heapifyUp , +while moving a node up at +index k , +// +there +are some +d i f f e r e n c e s : + +Case studies of development of verified programs with Dafny +23 +// +children +of k are +sorted +wrt +parent +of k , +and k +i s +not +// +sorted +wrt +i t s +parent . +predicate +{: autocontracts +f a l s e } heapifyUpInv (k : +nat ) +reads +this , +heap +{ +s i z e <= heap . Length +&& ( f o r a l l +i +: : +1 <= i < s i z e +&& i +!= k ==> heap [ i ] <= heap [ ( i − 1 ) / 2 ] ) +&& (k > 0 ==> f o r a l l +i +: : +1 <= i < s i z e && ( i −1)/2 == k +==> heap [ i ] <= heap [ ( ( i − 1)/2 − 1 ) / 2 ] ) +} +// +Deletes and +r e t r i e v e s +the maximum value +in +the heap +// ( assumed not empty ) . +method deleteMax () +returns +(x : T) +r e q u i r e s +! +isEmpty () +ensures +isMax (x , +old ( elems ( ) ) ) +ensures +elems () == old ( elems ( ) ) − multiset {x} +{ +// +r e c a l l +the lemma +. . . +maxIsAtTop ( ) ; +// +pick +the maximum from the +top +x := heap [ 0 ] ; +// reduce +the +s i z e +s i z e +:= +s i z e − 1; +i f +s i z e > 0 { +// move +l a s t +element +to +top +heap [ 0 ] +:= heap [ s i z e ] ; +// move down as +needed +in +the heap +heapifyDown ( ) ; +} +} +// +Deletes and +r e t r i e v e s +the maximum value +in +the heap +// ( assumed not empty ) . +method geteMax () +returns +(x : T) +r e q u i r e s +! +isEmpty () +ensures +isMax (x , elems ( ) ) +{ +maxIsAtTop ( ) ; +return +heap [ 0 ] ; +} +// +Auxiliary +predicate +to +check +i f +a value +i s +a maximum in +// a +multiset . +predicate +isMax (x : T, m: +multiset ) { +x in m && f o r a l l +y +: : +y in m ==> y <= x +} + +24 +João Pascoal Faria and Rui Abreu +// +Auxiliary method to move a +dirty +node from the +top down +// +in +the heap +method +{: autocontracts +f a l s e } heapifyDown () +r e q u i r e s +s i z e > 0 && heapifyDownInv (0) +modifies +heap +ensures +heapInv () +ensures +multiset ( heap [ . . s i z e ] ) == +old ( multiset ( heap [ . . s i z e ] ) ) +{ +var k := +0; +while +true +decreases +s i z e − k +invariant +0 <= k < s i z e +invariant +heapifyDownInv (k) +invariant +multiset ( heap [ . . s i z e ] ) == +old ( multiset ( heap [ . . s i z e ] ) ) +{ +var +l e f t C h i l d +:= 2 ∗ k + 1; +// index +of +l e f t +c h i l d +var +rightChild +:= 2 ∗ k + 2; +i f +l e f t C h i l d >= s i z e +{ +return ; +// reached +the bottom +} +var maxChild := +i f +rightChild < s i z e +&& heap [ rightChild ] > heap [ l e f t C h i l d ] +then +rightChild +e l s e +l e f t C h i l d ; +i f +heap [ k ] > heap [ maxChild ] +{ +return ; +// +already +sorted +} +// move up and continue +heap [ k ] , +heap [ maxChild ] +:= heap [ maxChild ] , +heap [ k ] ; +k := maxChild ; +} +} +// During heapifyDown , +while +moving a node down at +index k , +// +there +are some +d i f f e r e n c e s : +// +children +of k are +sorted +wrt +parent +of k , +and k +i s +not +// +sorted +wrt +i t s +children . +predicate +{: autocontracts +f a l s e } heapifyDownInv (k : +nat ) +reads +this , +heap +{ +s i z e <= heap . Length +&& ( f o r a l l +i +: : +1 <= i < s i z e && ( i −1)/2 != k +==> heap [ i ] <= heap [ ( i − 1 ) / 2 ] ) +&& (k > 0 ==> f o r a l l +i +: : +1 <= i < s i z e && ( i −1)/2 == k +==> heap [ i ] <= heap [ ( ( i − 1)/2 − 1 ) / 2 ] ) +} +// Lemma s t a t i n g +that +the maximum i s +at +the +top +of +the heap +// ( p o s i t i o n +0 ) . +This +property +i s +assumed by deleteMax and + +Case studies of development of verified programs with Dafny +25 +// +f o l l o w s +from the heap +invariant . +// Proved by induction on the +s i z e +of +the heap , +reason why +// +i t +r e c e i v e s +a parameter +with +the +s i z e +to +consider . +lemma {: induction n} maxIsAtTop (n : +nat := +s i z e ) +r e q u i r e s n <= s i z e +ensures +f o r a l l +i +: : +0 <= i < n ==> heap [ i ] <= heap [ 0 ] +{} +} +// A simple +t e s t +scenario . +method testPriorityQueue () +{ +var h := new PriorityQueue ( ) ; +a s s e r t h . isEmpty ( ) ; +h . i n s e r t ( 2 ) ; +h . i n s e r t ( 5 ) ; +h . i n s e r t ( 1 ) ; +h . i n s e r t ( 1 ) ; +var x := h . deleteMax ( ) ; +a s s e r t +x == 5; +x := h . deleteMax ( ) ; +a s s e r t +x == 2; +x := h . deleteMax ( ) ; +a s s e r t +x == 1; +x := h . deleteMax ( ) ; +a s s e r t +x == 1; +a s s e r t h . isEmpty ( ) ; +} +A.6 +Hash Set +/∗ +∗ +V e r i f i e d +implementation +of +a hash +set +with open +addressing +∗ and +l i n e a r +probing +in +Dafny . +∗ Provides +the +fundamental +set +operations +( contains , +insert , +∗ +d e l e t e ) , +s p e c i f i e d +at an +abstract +l e v e l , +r e s o r t i n g +to an +∗ +abstract +s t a t e +v a r i a b l e s +’ elems ’ +with +the +set +contents . +∗/ +// Datatype +f o r +the +content +of +each +c e l l +of +the +hash +table . +// +I t +s t o r e s +a value +of +type T, +Nil +( no value ) +or +Deleted +// ( c e l l +marked as +deleted ) . +datatype +Cell = Nil +| +Deleted +| Some( value : T) +// Function +type +f o r +hash +functions +type HashFunction = (T) −> nat +// +Represents a hash +set +of +elements +of +type T ( comparable +f o r +// +equality ) , +i . e . , +a +set +stored +in a hash +table . +// Uses +the " autocontracts " +a t t r i b u t e +to +automatically +i n j e c t +// +c l a s s +invariant +checking and frame +conditions +// +in +methods ’ +pre and post−conditions . +c l a s s +{: autocontracts } HashSet { + +26 +João Pascoal Faria and Rui Abreu +// Ghost +v a r i a b l e +( abstract +s t a t e +v a r i a b l e ) +used +f o r +// +s p e c i f i c a t i o n +purposes +only . +ghost +var +elems +: +set; +// Concrete +s t a t e +v a r i a b l e +with +i n t e r n a l +representation . +var +hashTable : +array>; +// Hash function +to be used +( provided +to +the +constructor ) . +const +hash : +HashFunction; +// Number of +p o s i t i o n s +used +( with some value ) and marked as +// +deleted +in +the +hash +table . +var +used : +nat ; +var +deleted : +nat ; +// +I n i t i a l +capacity +of +the +hash +table . +s t a t i c +const +i n i t i a l C a p a c i t y +:= 101; +// Ghost +predicate +that +f o r m a l i z e s +the +c l a s s +invariant . +predicate +Valid () +{ +// +Constraint +that +defin e +the +abstraction +r e l a t i o n +between +// +abstract +and +concrete +s t a t e +v a r i a b l e s +elems == valSet ( hashTable , +hashTable . Length ) +// +Constraints on the +i n t e r n a l +s t a t e +representation +&& hashTable . Length > 0 +&& hashTableInv ( hashTable ) +&& used == | valSet ( hashTable , +hashTable . Length ) | +&& deleted == | delSet ( hashTable , +hashTable . Length ) | +} +// Ghost +predicate +that +checks +the +consistency +of +a hash +// +table +’ t ’ . +predicate +{: autocontracts +f a l s e } +hashTableInv ( t : +array>) +reads +t +{ +f o r a l l +i +: : +0 <= i < t . Length && t [ i ] . Some? +==> validPos ( t [ i ] . value , +i , +t ) +} +// Ghost +predicate +that +checks +that +’ i ’ +i s +a +valid +p o s i t i o n +// +f o r +value +’x ’ +in +hash +table +’ t ’ . +// +( ’ x ’ may be or +not +currently +stored +in +that +p o s i t i o n ) +predicate +{: autocontracts +f a l s e } +validPos (x : T, +i : +nat , +t : +array>) +r e q u i r e s +0 <= i < t . Length +reads +t +{ +var h := hash (x) % t . Length ; + +Case studies of development of verified programs with Dafny +27 +h == i +| | +(h < i && f o r a l l +j +: : +h <= j < i +==> t [ j ] +!= Nil && t [ j ] +!= Some(x )) +| | +(h > i && f o r a l l +j +: : +h <= j < t . Length +| | +0 <= j < i +==> t [ j ] +!= Nil && t [ j ] +!= Some(x )) +} +// Ghost +function +that +r e t r i e v e s +the +set +of +values +stored +in +// the +f i r s t +’n ’ +p o s i t i o n s +of +hash +table +’ t ’ . +function +{: autocontracts +f a l s e } +valSet ( t : +array>, n : +nat ) : +set +r e q u i r e s +0 <= n <= t . Length +reads +t +{ +set +i +| +0 <= i < n && t [ i ] . Some? +: : +t [ i ] . value } +// Ghost +function +that +r e t r i e v e s +the +set +of +p o s i t i o n s +marked +// as +Deleted +in +the +f i r s t +’n ’ +p o s i t i o n s +of +hash +table +’ t ’ . +function +{: autocontracts +f a l s e } +delSet ( t : +array>, n : +nat ) : +set +r e q u i r e s +0 <= n <= t . Length +reads +t +{ +set +i +| +0 <= i < n && t [ i ] . Deleted ? } +// Ghost +function +that +r e t r i e v e s +the +set +of +p o s i t i o n s +marked +// as +Nil +in +the +f i r s t +’n ’ +p o s i t i o n s +of +hash +table +’ t ’ . +function +{: autocontracts +f a l s e } +n i l S e t ( t : +array>, n : +nat ) : +set +r e q u i r e s +0 <= n <= t . Length +reads +t +{ +set +i +| +0 <= i < n && t [ i ] . Nil ? } +// +Auxiliary lemma that +s t a t e s +the +f o l l o w i n g +property : +the +// sum of +the +s i z e s +of +valSet , +delSet +and +n i l S e t +// +of +a +valid +hash +table +i s +equal +to +the +length +of +the +hash +// +table +( array ) . +// This +i s +true +because +the +hash +table +invariant +implies +// +that +there +are no +duplicate +values +stored . +// The proof +i s +done by induction on the +length +of +the +table +// ( omitting +steps +f i l l e d +in by Dafny ) . +lemma {: autocontracts +f a l s e } +countingLemma ( ht : +array>, len : +nat , +v : +nat , +d : +nat , +n : +nat ) +r e q u i r e s +0 <= len <= ht . Length +r e q u i r e s +hashTableInv ( ht ) +r e q u i r e s +v == | valSet ( ht , +len ) | +&& d == | delSet ( ht , +len ) | +&& n == | n i l S e t ( ht , +len ) | +ensures v + d + n == len +{ +i f +len > 0 { + +28 +João Pascoal Faria and Rui Abreu +var +vs := valSet ( ht , +len ) ; +var +ds := +delSet ( ht , +len ) ; +var +ns := +n i l S e t ( ht , +len ) ; +var +vs1 := valSet ( ht , +len −1); +var +ds1 := +delSet ( ht , +len −1); +var +ns1 := +n i l S e t ( ht , +len −1); +// +r e c u r s i v e +part +countingLemma ( ht , +len −1, +| vs1 | , +| ds1 | , +| ns1 | ) ; +// +incremental +part +match ht [ len −1] { +case +Deleted => a s s e r t +vs == vs1 +&& ds == ds1 + { len −1} && ns == ns1 ; +case +Nil +=> a s s e r t +vs == vs1 +&& ds == ds1 && ns == ns1 + { len −1}; +case Some(x) => a s s e r t +vs == vs1 + {x} +&& ds == ds1 && ns == ns1 ; +} +} +} +// +I n t e r n a l +predicate +that +checks +i f +the +hash +table +i s +// +’ f u l l ’ , +in +the +sense +that +a l l +p o s i t i o n s +are +occupied +with +// a value +or +are marked as +deleted +( i . e . , +there +are no +// +p o s i t i o n s +with +Nil ) . +In +that +case , +i n s e r t i n g +a new value +// might not be +p o s s i b l e . +predicate +method +f u l l () +ensures +f u l l () <==> n i l S e t ( hashTable , hashTable . Length)=={} +{ +// to +help +proving +the +post−condition +( equivalence ) : +countingLemma ( hashTable , +hashTable . Length , +used , +deleted , +| n i l S e t ( hashTable , +hashTable . Length ) | ) ; +// the +actual +function +value +used + deleted == hashTable . Length +} +// +Public method that +checks +i f +t h i s +set +contains +a value x . +method contains (x : T) +returns +( res : +bool ) +ensures +res <==> x in +elems +{ +var +pos := +l o c a t e (x ) ; +return +pos != −1 && hashTable [ pos ] == Some(x ) ; +} +// +I n t e r n a l +method that +determines +the +l o c a t i o n +( ’ pos ’ ) +f o r +// a value +’x ’ +( e x i s t e n t +or +to be +i n s e r t e d ) . +// +I f +such a +l o c a t i o n +cannot be found +( because +the +table +i s +// +f u l l ) , +returns +−1. +// In +the +case +of +a new value , +t r i e s +to +reuse +p o s i t i o n s +// marked as +deleted . +method +l o c a t e (x : T) +returns +( pos : +int ) + +Case studies of development of verified programs with Dafny +29 +r e q u i r e s +Valid () +ensures x in +elems ==> 0 <= pos < hashTable . Length +&& hashTable [ pos ] == Some(x) +ensures x +! in +elems ==> ( pos == −1 && f u l l ( ) ) +| | +(0 <= pos < hashTable . Length +&& ! hashTable [ pos ] . Some? +&& validPos (x , +pos , +hashTable )) +{ +var h := hash (x) % hashTable . Length ; +var +reuse := −1; +f o r +i +:= h to +hashTable . Length +invariant +f o r a l l +j +: : +h <= j < i +==> hashTable [ j ] +!= Nil && hashTable [ j ] +!= Some(x) +invariant +reuse == −1 +| | +(h <= reuse < i && hashTable [ reuse ] == Deleted ) +{ +i f +hashTable [ i ] == Nil +| | +hashTable [ i ] == Some(x) { +return +i ; +} +i f +hashTable [ i ] == Deleted && reuse == −1 { +reuse := +i ; +} +} +f o r +i +:= 0 to h +invariant +f o r a l l +j +: : 0<=j hashTable [ j ] +!= Nil && hashTable [ j ] +!= Some(x) +invariant +reuse == −1 +| | +((0 <= reuse < i +| | +h <= reuse < hashTable . Length ) +&& hashTable [ reuse ] == Deleted ) +{ +i f +hashTable [ i ] == Nil +| | +hashTable [ i ] == Some(x) { +return +i ; +} +i f +hashTable [ i ] == Deleted && reuse == −1 { +reuse := +i ; +} +} +return +reuse ; +} +// +Public +constructor +that +r e c e i v e s +the +hash +function +to be +// used and +i n i t i a l i z e s +the +set +as empty . +constructor +( hash : +HashFunction) +ensures +elems == {} +{ +// +i n i t i a l i z e +concrete +s t a t e +v a r i a b l e s +t h i s . hash := hash ; +hashTable := new Cell[ i n i t i a l C a p a c i t y ] +(_ => Nil ) ; +used := +0; +deleted +:= +0; + +30 +João Pascoal Faria and Rui Abreu +// +i n i t i a l i z e +ghost / abstract +s t a t e +v a r i a b l e s +elems := +{}; +} +// +I n t e r n a l +method that +i n s e r t s +a new value +’x ’ +into +the +// hash +set , +guaranteed +to be not +f u l l . +method insertAux (x +: T) +r e q u i r e s +x +! in +elems +r e q u i r e s +! +f u l l () +ensures +elems == old ( elems ) + {x} +ensures +deleted <= old ( deleted ) // +u s e f u l l +f o r +rehash ? +ensures +hashTable == old ( hashTable ) // +u s e f u l l +f o r +rehash ? +{ +var +i +:= +l o c a t e (x ) ; +i f +hashTable [ i ] == Deleted { +// to +help +proving +that +deleted > 0 +a s s e r t +i +in +delSet ( hashTable , +hashTable . Length ) ; +// now , +can decrement +deleted +:= +deleted − 1; +} +hashTable [ i ] +:= Some(x ) ; +used := used + 1; +elems := elems + {x }; +// to +help +proving +that +elems == valSet () +a s s e r t +f o r a l l +k +: : +0 <= k < hashTable . Length && k != +i +==> hashTable [ k ] == old ( hashTable [ k ] ) ; +// to +help +proving +that +deleted == | delSet ( ) | +a s s e r t +delSet ( hashTable , +hashTable . Length ) == +old ( delSet ( hashTable , +hashTable . Length )) − { i }; +} +// +I n t e r n a l +method that +grows and +cleans up the +hash +table . +method rehash () +ensures +! +f u l l () +ensures +elems == old ( elems ) +{ +var +oldTable := hashTable ; +hashTable:= new Cell[hashTable . Length ∗2+1] +(_ => Nil ) ; +deleted +:= +0; +used := +0; +elems := +{}; +// need +also +to +update +ghost +v a r i a b l e +’ Repr ’ +generated by +// +: autocontracts , +to be +able +to +c a l l +InsertAux +in a +valid +// +s t a t e +Repr := { this , +hashTable }; + +Case studies of development of verified programs with Dafny +31 +f o r +i +:= 0 to +oldTable . Length +invariant +elems == valSet ( oldTable , +i ) +invariant +deleted == 0 // to +prove +! f u l l () +invariant +oldTable +! in Repr +// to +assure +i s +not changed by insertAux +invariant +Valid () +// to +ensure +consistency +i s +maintained +invariant +oldTable . Length < hashTable . Length +// to +prove +! f u l l () +invariant +f r e s h ( Repr − old ( Repr )) +// to +enable +insertAux +to +modify +the new hashTable +invariant +oldTable . Length < hashTable . Length +// to +prove +! f u l l () +invariant +f o r a l l +k +: : +0 <= k < oldTable . Length +==> oldTable [ k ] == old ( hashTable [ k ] ) +// to +prove +post−condition +{ +// to +help +proving +! +f u l l () +countingLemma ( oldTable , +i , +| valSet ( oldTable , +i ) | , +| delSet ( oldTable , +i ) | , +| n i l S e t ( oldTable , +i ) | ) ; +i f +( oldTable [ i ] . Some?) { +insertAux ( oldTable [ i ] . value ) ; +} +} +// to +help +proving +! +f u l l () +var +i +:= oldTable . Length ; +countingLemma ( oldTable , +i , +| valSet ( oldTable , +i ) | , +| delSet ( oldTable , +i ) | , +| n i l S e t ( oldTable , +i ) | ) ; +} +// +I n s e r t s +a new value +’x ’ +into +t h i s +hash +set . +method +i n s e r t (x +: T) +r e q u i r e s +x +! in +elems +ensures +elems == old ( elems ) + {x} +{ +i f +f u l l () +{ +rehash ( ) ; +} +insertAux (x ) ; +} +// +Deletes +an +e x i s t e n t +value +’x ’ +from +t h i s +hash +set . +method +d e l e t e (x +: T) +r e q u i r e s +x in +elems +ensures +elems == old ( elems ) − {x} +{ +var h := hash (x) % hashTable . Length ; +var +i +:= +l o c a t e (x ) ; + +32 +João Pascoal Faria and Rui Abreu +elems := elems − {x }; +hashTable [ i ] +:= Deleted ; +deleted +:= +deleted + 1; +used := used − 1; +// to +help +proving +that +elems == valSet ( hashTable , +// hashTable . Length ) +a s s e r t +f o r a l l +k +: : +0 <= k < hashTable . Length && k != +i +==> hashTable [ k ] == old ( hashTable [ k ] ) ; +// to +help +proving +that +deleted == | delSet ( ) | +a s s e r t +delSet ( hashTable , +hashTable . Length ) == +old ( delSet ( hashTable , +hashTable . Length )) + { i }; +} +} +method testHashSet () +{ +var h := new HashSet(x => | x | ) ; +a s s e r t h . elems == {}; +h . i n s e r t (" Hello " ) ; +a s s e r t h . elems == {" Hello "}; +h . i n s e r t ("World " ) ; +a s s e r t h . elems == {" Hello " , "World "}; +var +found := h . contains (" Hello " ) ; +a s s e r t +found ; +found := h . contains ("ANSI " ) ; +a s s e r t +! found ; +h . d e l e t e (" Hello " ) ; +a s s e r t h . elems == {"World "}; +found := h . contains (" Hello " ) ; +a s s e r t +! found ; +} +A.7 +Tree Set +/∗ +∗ +S p e c i f i c a t i o n +and +v e r i f i c a t i o n +of +a +sorted +set +implemented +∗ with a binary +search +t r e e +(BST) . +∗ +I l l u s t r a t e s +the +usage +of +ghost +v a r i a b l e s +f o r +data +∗ +abstraction +and +separation +of +s p e c i f i c a t i o n +and +∗ implementation . +∗ Uses +the +{: autocontracts } +a t t r i b u t e +to +take +care +of +c l a s s +∗ +invariant +enforcement and frame +generation +( read / modifies ) . +∗/ +type T = int +// +f o r demo purposes +// Sequence +without +du p l i c a t e s + +Case studies of development of verified programs with Dafny +33 +type +useq = +s : +seq | +! hasDuplicates ( s ) +// Checks +i f +a sequence +’ s ’ +has +d u p l i c a te s . +predicate +hasDuplicates(s : +seq) +{ +e x i s t s +i , +j +: : +0 <= i < j < | s | && s [ i ] == s [ j ] +} +// Node of +a binary +search +t r e e +c l a s s +{: autocontracts } BSTNode { +// Concrete +implementation +v a r i a b l e s +var +value : T // +value +in +t h i s +node +var +l e f t : BSTNode? +// +elements +smaller +than +’ value ’ +var +r i g h t : BSTNode? // +elements +greater +than +’ value ’ +// +(? − may be +n u l l ) +// Abstract +v a r i a b l e +used +f o r +s p e c i f i c a t i o n & v e r i f i c a t i o n +// +purposes +ghost +var +elems : +set // +set +of +values +in +the +subtree +// +s t a r t i n g +in +t h i s +node ( including +t h i s +value ) +// +Class +invariant +with +the +i n t e g r i t y +c o n s t r a i n t s +f o r +the +// above +v a r i a b l e s +predicate +Valid () +{ +( elems == { value } ++ ( i f +l e f t == n u l l +then {} +e l s e +l e f t . elems ) ++ ( i f +r i g h t== n u l l +then {} +e l s e +r i g h t . elems )) +&& ( l e f t +!= +n ul l ==> f o r a l l +x +: : +x in +l e f t . elems +==> x < value ) +&& ( r i g h t +!= +n u l l ==> f o r a l l +x +: : +x in +r i g h t . elems +==> x > value ) +&& ( r i g h t +!= +n u l l ==> f o r a l l +x +: : +x in +r i g h t . elems +==> x > value ) +&& ( l e f t +!= +n ul l ==> l e f t . Valid ( ) ) +&& ( r i g h t +!= +n u l l ==> r i g h t . Valid ( ) ) +&& ( l e f t +!= +n ul l && r i g h t +!= +n u l l +==> l e f t . Repr +! ! +r i g h t . Repr ) +// +d i s j o i n t s +s e t s +of +o b je c t s +in +l e f t +and +r i g h t +// +subtree , +needed +to make sure +that +changing +nodes +// +in +one +subtree +doesn ’ t +a f f e c t +the +other ! +} +// +I n i t i a l i z e s +a new node with +value +’x ’ +and empty +l e f t +// and +r i g h t +subtrees . +constructor +(x : T) +ensures +elems == {x} +{ +value := x ; +l e f t +:= +n u ll ; +r i g h t +:= +n u l l ; +elems := {x }; + +34 +João Pascoal Faria and Rui Abreu +} +// Checks +i f +the +subtree +s t a r t i n g +in +t h i s +node +contains +a +// +value +’x ’ . +Runs in +time O( log h ) , +where +’h ’ +i s +the +// +height +of +the +subtree . +predicate +method contains (x : T) +ensures +contains (x) <==> x in +elems +{ +i f +x == value +then +true +e l s e +i f +x < value && l e f t != n u l l +then +l e f t . contains (x) +e l s e +i f +x > value && r i g h t != n u l l +then +r i g h t . contains (x) +e l s e +f a l s e +} +// +I n s e r t s +a value +’x ’ +in +the +subtree +s t a r t i n g +in +t h i s +// node . +I f +the +value +already +e x i s t s , +does +nothing . +// Runs in +time O( log h ) , +where h +i s +the +subtree +height . +method +i n s e r t (x : T) +ensures +elems == old ( elems ) + {x} +decreases +elems +{ +i f +x == value { +return ; +} +e l s e +i f +x < value { +i f +l e f t == n u l l +{ +l e f t +:= new BSTNode(x ) ; +} +e l s e +{ +l e f t . i n s e r t (x ) ; +} +} +e l s e +{ +i f +r i g h t == n u l l +{ +r i g h t +:= new BSTNode(x ) ; +} +e l s e +{ +r i g h t . i n s e r t (x ) ; +} +} +elems := elems + {x }; +} +// +Public +function +to +find +the maximum value +in +t h i s +// +subtree . +Runs in +time O( log h ) , +where +’h ’ +i s +the +// +height +of +the +subtree . +function +method max() +: T +ensures max() +in +elems +&& f o r a l l +x +: : +x in +elems ==> x <= max() +{ + +Case studies of development of verified programs with Dafny +35 +i f +r i g h t == n u l l +then +value +e l s e +r i g h t . max() +} +// +Public +function +to +find +the minimum value +in +t h i s +// +subtree . +Runs in +time O( log h ) , +where +’h ’ +i s +the +// +height +of +the +subtree . +function +method min () +: T +ensures min () +in +elems +&& f o r a l l +x +: : +x in +elems ==> x >= min () +{ +i f +l e f t +== n u l l +then +value +e l s e +l e f t . min () +} +// +Deletes +a value +’x ’ +from the +subtree +s t a r t i n g +in +t h i s +// node , +and +returns +the new head +of +the +subtree +( which +// +w i l l +be +n u ll +i f +’x ’ +was the +only +value +in +the +subtree ) . +// +I f +the +value +doesn ’ t +exist , +does +nothing . +// Currently , +seems +to run +in +time O( log h ∗ +log h ) , +where +// +’h ’ +i s +the +height +of +the +subtree . +method +d e l e t e (x : T) +returns ( res : BSTNode?) +ensures +i f +old ( elems ) == {x} then +res == n u l l +e l s e +res +!= +n u l l && res . elems == old ( elems)−{x} +&& res . Valid () +// not added +by autocontracts +. . . +&& res . Repr <= old ( Repr ) +// to +preserve +d i s j o i n t n e s s +. . . +decreases +elems +{ +i f +x == value { +i f +l e f t == n u l l +{ +res +:= +r i g h t ; +// +j u s t +changes +the head +return ; +} +e l s e +i f +r i g h t == n u l l +{ +res +:= +l e f t ; +// +j u s t +changes +the head +return ; +} +e l s e +{ +i f +∗ /∗ non +d e t e r m i n i s t i c +choice +∗/ { +value := +l e f t . max ( ) ; +l e f t +:= +l e f t . d e l e t e ( value ) ; +} +e l s e +{ +value := +r i g h t . min ( ) ; +r i g h t +:= +r i g h t . d e l e t e ( value ) ; +} +} +} +e l s e +i f +x > value && r i g h t +!= +n u l l +{ +r i g h t +:= +r i g h t . d e l e t e (x ) ; + +36 +João Pascoal Faria and Rui Abreu +} +e l s e +i f +x < value && l e f t +!= +n u l l +{ +l e f t +:= +l e f t . d e l e t e (x ) ; +} +res +:= +t h i s ; +elems := elems − {x }; +} +method asSeq () +returns +( s : +useq) +ensures +isSorted ( s ) && asSet ( s ) == elems +decreases +elems +{ +var +l , m, +r := +[ ] , +[ value ] , +[ ] ; +i f +l e f t +!= +n u l l +{ +l +:= +l e f t . asSeq ( ) ; +} +i f +r i g h t +!= +n u l l +{ r := +r i g h t . asSeq ( ) ; +} +asSetProp ( l , m) ; +// +r e c a l l +lemma +asSetProp ( l + m, +r ) ; +// +r e c a l l +lemma +return +l + m + r ; +} +} +// +Auxiliary +function +that +obtains +the +set +of +elements +in a +// sequence . +function +asSet ( s : +seq) : +set +ensures +f o r a l l +i +: : +0 <= i < | s | ==> s [ i ] +in +asSet ( s ) +ensures +f o r a l l +x +: : +x in +asSet ( s ) ==> x in +s +{ +i f +| s | == 0 then {} +e l s e +{ s [ 0 ] } + asSet ( s [ 1 . . ] ) +} +// +Auxiliary +predicate +that +checks +i f +a sequence +i s +s t r i c t l y +// +sorted . +predicate +isSorted ( s : +useq) { +f o r a l l +i , +j +: : +0 <= i < j < | s | ==> s [ i ] < s [ j ] +} +// Lemma that +s t a t e s +and proves by induction +the +f o l l o w i n g +// +property : +the +set +of +elements +of +sequence +concatenation +i s +// the +union +of +the +i n d i v i d u a l +s e t s +of +elements . +lemma asSetProp ( s1 : +seq, s2 : +seq) +ensures +asSet ( s1 + s2 ) == asSet ( s1 ) + asSet ( s2 ) +{ +i f +| s1 | > 0 { +a s s e r t +s1 == s1 [ . . 1 ] + s1 [ 1 . . ] ; +a s s e r t +( s1 [ . . 1 ] + s1 [ 1 . . ] ) + s2 == s1 [ . . 1 ] + ( s1 [ 1 . . ] + s2 ) ; +asSetProp ( s1 [ 1 . . ] , +s2 ) ; +} +e l s e +{ +a s s e r t +[ ] + s2 == s2 ; +} + +Case studies of development of verified programs with Dafny +37 +} +// Lemma that +s t a t e s +and proves by induction +the +f o l l o w i n g +// +property : +i f +two sequences +without +d u p l i c a t e s +are +sorted +// and have the same +set +of +elements , +then +they must be +// +i d e n t i c a l . +lemma sortingUniqueness ( a : +useq, b : +useq) +r e q u i r e s +isSorted ( a ) && isSorted (b) && asSet ( a ) == asSet (b) +ensures a == b +{ +i f +| a | > 0 { +sortingUniqueness ( a [ 1 . . ] , +b [ 1 . . ] ) ; +} +} +// A simple +t e s t +case . +method +testSortedSet () +{ +var +s := new BSTNode ( 2 ) ; +s . i n s e r t ( 5 ) ; +s . i n s e r t ( 1 ) ; +s . i n s e r t ( 4 ) ; +s . i n s e r t ( 4 ) ; +var +t := s . asSeq ( ) ; +sortingUniqueness ( t , +[ 1 , +2 , +4 , +5 ] ) ; +// to +help +prove +next +a s s e r t i o n +a s s e r t +t == [ 1 , +2 , +4 , +5 ] ; +a s s e r t +s . min () == 1; +a s s e r t +s .max() == 5; +var +s2 := s . d e l e t e ( 5 ) ; +a s s e r t +s2 . elems == {1 , +2 , +4}; +} +A.8 +Stable Marriage +/∗ +∗ Formal +v e r i f i c a t i o n +with Dafny +of +the Gale−Shapley +algorithm +∗ to +solve +the +s t a b l e +marriage +problem , +both +described +in +∗ https :// en . wikipedia . org / wiki /Stable_marriage_problem . +∗ Then , +t h i s +algorithm +i s +applied +to +solve +the +teachers +∗ placement problem +that +caused +s e r i o u s +trouble +in +Portugal +∗ in +2004. +∗/ +// Sequence +without +du p l i c a t e s +type +useq = +s : +seq | +! hasDuplicates ( s ) +// +I n j e c t i v e map +type +inmap = +m: map | +i s I n j e c t i v e (m) + +38 +João Pascoal Faria and Rui Abreu +// Checks +i f +a sequence +’ s ’ +has +d u p l i c a te s . +predicate +hasDuplicates(s : +seq) +{ +e x i s t s +i , +j +: : +0 <= i < j < | s | && s [ i ] == s [ j ] +} +// Checks +i f +a map +’m’ +i s +i n j e c t i v e , +i . e . , +d i s t i n c t +keys +are +// mapped to +d i s t i n c t +values . +predicate +i s I n j e c t i v e (m: map) { +f o r a l l +i , +j +: : +i +in m && j +in m && i +!= +j ==> m[ i ] +!= m[ j ] +} +// Checks +i f +element +’ e1 ’ +precedes +’ e2 ’ +in +sequence +’ s ’ . +predicate +method precedes(e1 : T, +e2 : T, +s : +seq) { +e x i s t s +i , +j +: : +0 <= i < j < | s | && s [ i ] == e1 && s [ j ] == e2 +} +// Obtains +the +set +of +elements +in a sequence +function +elems(s : +useq): +set +ensures +f o r a l l +x +: : +x in +elems ( s ) ==> x in +s +ensures +f o r a l l +x +: : +x in +s ==> x in +elems ( s ) +{ +set +i +| +0 <= i < | s | +: : +s [ i ] +} +// Checks +i f +a matching +of +couples +i s +valid , +i . e . , men and +// women can be engaged +only +i f +they +are +mentioned +in +each +// +others +p r e f e r e n c e s +predicate +isValid (couples : +inmap , +menPrefs : map>, +womenPrefs : map >) +{ +f o r a l l m : : m in +couples ==> var w := couples [m] ; +m in +menPrefs && w in +womenPrefs +&& w in +menPrefs [m] && m in +womenPrefs [w] +} +// Checks +i f +a matching +of +couples +i s +stable , +i . e . , +there +i s +// no +pair +(m, w) +that +p r e f e r +each +other +as compared to +t h e i r +// +current +s i t u a t i o n . +predicate +isStable (couples : +inmap , +menPrefs : map>, +womenPrefs : map >) +{ +! +e x i s t s m, w : : m in +menPrefs . Keys +&& w in +womenPrefs . Keys +&& unstable (m, w, +couples , +menPrefs , +womenPrefs ) +} + +Case studies of development of verified programs with Dafny +39 +predicate +unstable(m: Man, w: Woman, +couples : +inmap , +menPrefs : map>, +womenPrefs : map >) +r e q u i r e s m in +menPrefs . Keys && w in +womenPrefs . Keys +{ +w in +menPrefs [m] && m in +womenPrefs [w] && +(m in +couples ==> precedes (w, +couples [m] , +menPrefs [m] ) ) +&& ( f o r a l l m’ +: : m’ +in +couples && couples [m’ ] == w +==> precedes (m, m’ , +womenPrefs [w] ) ) +} +// +Stable +matching by the Gale−Shapley +algorithm +with +// +incomplete +l i s t s +and no +t i e s . +// +Receives +the +l i s t s +of +p r e f e r e n c e s +of men and women and +// +returns +the +couples +created . +// Time complexity +( with +proper +data +s t r u c t u r e s ) +i s +// O( |M| ∗ |W| ) , +where W i s +the +set +of women and M the +set +of +// men . +// The types Man and Woman are +defined +as +type +parameters +// because +t h e i r +i n t e r n a l +structure +i s +not +relevant +here . +method stableMatching( +menPrefs : map>, +womenPrefs : map >) +returns ( couples : +inmap ) +// P1 : women referenced +in men p r e f e r e n c e s +must +e x i s t +r e q u i r e s +f o r a l l m : : m in +menPrefs +==> f o r a l l w : : w in +menPrefs [m] ==> w in +womenPrefs +// P2 : man referenced +in women p r e f e r e n c e s +must +e x i s t +r e q u i r e s +f o r a l l w : : w in +womenPrefs +==> f o r a l l m : : m in +womenPrefs [w] ==> m in +menPrefs +// Q1: men and women can be engaged +only +i f +they +are +// mentioned +in +each +other ’ s +p r e f e r e n c e s +ensures +isV alid ( couples , +menPrefs , +womenPrefs ) +// Q2: +s t a b l e +marriage +( and maximality ) +ensures +i s S t a b l e ( couples , +menPrefs , +womenPrefs ) +{ +// +I n i t i a t e +the +r e s u l t +as empty +couples +:= map [ ] ; +// +I n i t i a l i z e +the men p r e f e r e n c e s +already +explored empty +var +menPrefsExplored +:= map m | m in +menPrefs +: : +[ ] ; +// Ghost +v a r i a b l e +used +f o r +proving +termination +with Dafny +// ( instead +of +menPrefsExplored , +that +has a too +complex +// +structure ) +ghost +var +unexploredPairs := +set m, w | m in +menPrefs +&& w in +menPrefs [m] +: : +(m, w) ; +// +while +e x i s t s +a +f r e e man m who +s t i l l +has a woman w to + +40 +João Pascoal Faria and Rui Abreu +// propose +to +while +e x i s t s m : : m in +menPrefs && m ! in +couples +&& menPrefsExplored [m] < menPrefs [m] +decreases +unexploredPairs +// +I1 : +menPrefsExplored +has +the same keys +(men) +as +// menPrefs +invariant +menPrefs . Keys == menPrefsExplored . Keys +// +I2 : +l i s t s +in +menPrefsExplored must be +s u b l i s t s +// ( p r e f i x e s ) +in +menPrefs +invariant +f o r a l l m : : m in +menPrefsExplored +==> menPrefsExplored [m] <= menPrefs [m] +// +I3 : +to +assure Q1 incrementally , +with +menPrefsExplored +// +instead +of +menPrefs +invariant +isV alid ( couples , +menPrefsExplored , +womenPrefs ) +// +I4 : +to +assure Q2 incrementally , +with +menPrefsExplored +// +instead +of +menPrefs +invariant +i s S t a b l e ( couples , menPrefsExplored , womenPrefs ) +// +I5 : +while +engaged , men do not +propose +to +f u r t h e r +// women ( needed +to +preserve +i s S t a b l e ) +invariant +f o r a l l m : : m in +couples +==> couples [m] == l a s t ( menPrefsExplored [m] ) +// +I6 : +inv . +d e f i n i n g +the +contents +of +unexploredPairs +invariant +unexploredPairs == set m, w | m in +menPrefs +&& w in +menPrefs [m] +&& w ! in +menPrefsExplored [m] +: : +(m, w) +{ +// +s e l e c t +a man in +such +condition +( f r e e man m who +// +s t i l l +has a woman w to +propose +to ) +var m : | m in +menPrefs && m ! in +couples +&& menPrefsExplored [m] < menPrefs [m] ; +// +s e l e c t +the +next woman on m’ s +l i s t +( using +a u x i l i a r y +// +function +to +circumvent Dafny +l i m i t a t i o n ) +var w := nth ( menPrefs [m] , +| menPrefsExplored [m] | ) ; +// +i f w isn ’ t +f r e e +( i . e . , +some +pair +(m’ ,w) +e x i s t s +yet ) +i f m’ +: | m’ +in +couples && couples [m’ ] == w +{ +// +i f w p r e f e r s m to m’ +i f m in +womenPrefs [w] +&& precedes (m, m’ , +womenPrefs [w] ) +{ +// m’ +becomes +f r e e +couples := map x +| +x in +couples +&& x != m’ +: : +couples [ x ] ; +// (m, w) become engaged +couples := couples [m := w ] ; +} +} +e l s e +// w i s +f r e e + +Case studies of development of verified programs with Dafny +41 +{ +// +i f w i s +i n t e r e s t e d +in m +i f m in +womenPrefs [w] +{ +// (m, w) become engaged +couples := couples [m := w ] ; +} +} +// mark +t h i s +pair +as +explored +menPrefsExplored := +menPrefsExplored [m := menPrefsExplored [m] + [w ] ] ; +unexploredPairs := unexploredPairs − {(m, w) } ; +} +} +/∗ +∗ Some t e s t +cases +f o r +the +s t a b l e +marriage +problem . +∗/ +method testStableMatching1 () +{ +var +menPrefs := map [1 +:= +[ 1 , +2 ] , +2 := +[ 1 , +2 ] ] ; +var womenPrefs := map [1 +:= +[ 1 ] , +2 := +[ 2 ] ] ; +var +expectedCouples := map[1 +:= 1 , 2 := +2 ] ; +var +actualCouples := stableMatching ( menPrefs , +womenPrefs ) ; +a s s e r t +isV alid ( expectedCouples , +menPrefs , +womenPrefs ) ; +// +proof +helper . . . +a s s e r t +actualCouples == expectedCouples ; +} +method testStableMatching2 () +{ +var +menPrefs := map [1 +:= +[ 2 , +1 ] , +2 := +[ 1 , +2 ] ] ; +var womenPrefs := map [1 +:= +[ 1 , +2 ] , +2 := +[ 2 , +1 ] ] ; +var +expectedCouples1 := map[1 +:= 2 , 2 := +1 ] ; +var +expectedCouples2 := map[1 +:= 1 , 2 := +2 ] ; +var +actualCouples := stableMatching ( menPrefs , +womenPrefs ) ; +a s s e r t +isV alid ( expectedCouples1 , +menPrefs , +womenPrefs ) ; +// +proof +helper . . . +a s s e r t +actualCouples == expectedCouples1 +| | +actualCouples == expectedCouples2 ; +} +method testStableMatching3 () +{ +var +menPrefs := map [1 +:= +[ 1 , +2 ] , +2 := +[ 1 ] ] ; +var womenPrefs := map [1 +:= +[ 1 , +2 ] , +2 := +[ 2 , +1 ] ] ; +var +expectedCouples1 := map[1 +:= 2 , 2 := +1 ] ; +var +expectedCouples2 := map[1 +:= +1 ] ; +var +actualCouples := stableMatching ( menPrefs , +womenPrefs ) ; +a s s e r t +isV alid ( expectedCouples1 , +menPrefs , +womenPrefs ) ; +// +proof +helper . . . + +42 +João Pascoal Faria and Rui Abreu +a s s e r t +actualCouples == expectedCouples1 +| | +actualCouples == expectedCouples2 ; +} +/∗ +∗ Application +to +solve +the +teachers +placement problem . +∗/ +type +Teacher = nat +type Vacancy = nat +// +Auxiliary +function +to move an element +’x ’ +in a sequence +’ s ’ +// ( without +duplicates ) +to +the head +of +the +sequence . +function +method moveToHead(s : +useq, x : T) +: +useq +r e q u i r e s +x in +s +ensures +f o r a l l +y +: : +y in +s ==> y in moveToHead( s , +x) +{ +var +i +: | +0 <= i < | s | && s [ i ] == x ; +[ s [ i ]]+ s [ . . i ]+ s [ i + 1 . . ] +} +// Gets +the +l a s t +element +in a sequence +function +last (s : +seq): T +r e q u i r e s +| s | > 0 +{ s [ | s | −1] } +// Gets +the n−th +element +in a sequence +function +method nth(s : +seq, n : +nat ) : T +r e q u i r e s +0 <= n < | s | +{ s [ n ] +} +// +Auxiliary +predicate +that +checks +i f +a +teacher +’ t ’ +has +// +precedence +over +the +current +teacher +that +occupies +vacancy +// +’v ’ , +i f +any , +knwowing the +ranked +l i s t +of +teachers , +t h e i r +// +i n i t i a l +placement , +and the +f i n a l +placement . +// A teacher +that +i n i t i a l l y +occupied +’v ’ +has +p r i o r i t y +over +a l l +// +others ; +otherwise , +p r i o r i t y +i s +given +to +teachers +with +// +higher +rank . +predicate +method teacherHasPrecedenceForVacancy ( t : +Teacher , +v : +Vacancy , +finalPlacement : +inmap, +teachers : +useq, +initialPlacement : +inmap) +{ +i f +t2 +: | +t2 +in +finalPlacement && finalPlacement [ t2 ] == v +then +t != t2 +&& (( t , +v) +in +i ni ti al P la ce me nt . Items +| | +(( t2 , +v) +! in +in i ti al Pl a ce me nt . Items +&& precedes ( t , +t2 , +teachers ) ) ) +e l s e +true +// the +vacancy +i s +s t i l l +free , +so any +teacher +i s +// +better +than +remaining +f r e e +} + +Case studies of development of verified programs with Dafny +43 +// +Solution +f o r +teachers +placement problem , +by reducing +i t +to +// the +s t a b l e +marriage +problem . +// Input +parameters : +// +vacancies − set +of +vacancies +a v a i l a b l e +( includes +// +p o s i t i o n s +currently +occupied by teachers +that +// +want to +change +p o s i t i o n ) +// +teachers − ordered +set +of +teachers , +ordered by +t h e i r +// +ranking +( represented +as a sequence +without +// +duplica t e s ) +// +p r e f e r e n c e s − map that +i n d i c a t e s +f o r +each +teacher +the +// +ordered +l i s t +of +vacancies +wanted +// +initialPlacement − map that +i n d i c a t e s +the +i n i t i a l +// +placement +of +teachers +with +i n i t i a l +placement +// Output parameters : +// +finalPlacement − f i n a l +teachers +placement +method teachersPlacement ( vacancies : +set, +teachers : +useq, +p r e f e r e n c e s : map>, +initialPlaceme nt : +inmap ) +returns ( finalPlacement : +inmap) +// P1 : +the +teachers +in +the +ranked +sequence and the +teachers +// with +preferences , +are +the same +r e q u i r e s +elems ( teachers ) == p r e f e r e n c e s . Keys +// P2 : +the +vacancies +mentioned +in +teachers +p r e f e r e n c e s +must +// +e x i s t +in +the +set +of +vacancies +r e q u i r e s +f o r a l l +t +: : +t +in +p r e f e r e n c e s +==> +elems ( p r e f e r e n c e s [ t ] ) <= vacancies +// P3 : +the +teachers +and vacancies +mentioned +in +the +i n i t i a l +// placement must +e x i s t +in +the +s e t s +of +teachers +and +// +vacancies +r e q u i r e s +f o r a l l +t +: : +t +in +i ni ti al P la ce me nt +==> t +in +teachers && i ni ti al P la ce me nt [ t ] +in +vacancies +// P4 : +the +i n i t i a l +placement +of +a +teacher must be mentioned +// +in +his +l i s t +of +p r e f e r e n c e s +as +the +l a s t +preference +r e q u i r e s +f o r a l l +t +: : +t +in +i ni ti al P la ce me nt ==> +initialPlace me nt [ t ] +in +p r e f e r e n c e s [ t ] +&& initialPl ac em en t [ t ] == l a s t ( p r e f e r e n c e s [ t ] ) +// Q1: +the +teachers +mentioned +in +the +f i n a l +placement must +// +e x i s t +in +the +set +of +teachers +ensures +finalPlacement . Keys <= elems ( teachers ) +// Q2: +a +teacher may only be +placed +in a vacancy mentioned +// +in +his / her +l i s t +of +p r e f e r e n c e s +ensures +f o r a l l +t +: : +t +in +finalPlacement +==> finalPlacement [ t ] +in +p r e f e r e n c e s [ t ] +// Q3: +the +assignment +i s +stable , +i . +e . , +there +i s +no +pair +of +// +teacher +t and vacancy v in +his +l i s t +of +preferences , +// such +that +t +p r e f e r s +v over +his +current +s i t u a t i o n +( e i t h e r +// because +t +i s +free , +or +because +t +p r e f e r s +v over +the +// +assigned +p o s i t i o n ) , +and v +p r e f e r s +t +over +i t s +current + +44 +João Pascoal Faria and Rui Abreu +// +s i t u a t i o n +( e i t h e r +because v +i s +f r e e +and so +p r e f e r s +any +// +teacher +as compared to +remaining +free , +or +t +i s +the +// +teacher +i n i t i a l l y +placed and +i s +not +occupying v , +or +the +// +teacher +t ’ +that +currently +occupies v was not +i n i t i a l l y +// +placed +there and has a lower +rank than +t ) +ensures +! +e x i s t s +t , +v +: : +t +in +teachers +&& v in +p r e f e r e n c e s [ t ] +&& ( t +in +finalPlacement ==> precedes (v , +finalPlacement [ t ] , +p r e f e r e n c e s [ t ] ) ) / / t +p r e f e r s +v +&& teacherHasPrecedenceForVacancy ( t , +v , +finalPlacement , +teachers , +in it i al Pl ac em en t ) +// Q4: +teachers +that +have an +i n i t i a l +p o s i t i o n +must +a ls o +have +// a +f i n a l +p o s i t i o n +ensures +f o r a l l +t : : +t +in +i n it ia lP l ac em en t +==> t +in +finalPlacement +{ +// Reduction +to +the +problem +of +s t a b l e +marriage , +with +// +teachers +as men ( with +the +given +p r e f e r e n c e s ) , +// +vacancies +as women, +and the +p r e f e r e n c e s +of +each +vacancy +// +given by the +ranked +l i s t +of +teachers +with +the +teacher +// +i n i t i a l l y +placed +there +( i f +any ) moved to +the head +finalPlacement := stableMatching ( preferences , +vacanciesPrefs ( vacancies , +teachers , +in i ti al Pl ac e me nt ) ) ; +} +// +p r e f e r e n c e s +of +each vacancy +given by the +ranked +l i s t +of +// +teachers +with +the +teacher +i n i t i a l l y +placed +there +( i f +any ) +// moved to +the head +function +method +vacanciesPrefs ( vacancies : +set, +teachers : +useq, +initialPlace me nt : +inmap ): +map> +r e q u i r e s +f o r a l l +t +: : +t +in +i ni ti al P la ce me nt ==> t +in +teachers +&& i n it ia lP l ac em en t [ t ] +in +vacancies +{ +map v +| +v in +vacancies +: : +i f +t +: | +t +in +initia l Pl ac em en t && in it i al Pl ac em e nt [ t ] == v +then moveToHead( teachers , +t ) +e l s e +teachers +} +/∗ +∗ Some t e s t +cases +f o r +the +teachers +placement problem . +∗/ +method test1TP () +{ +var +vacancies +:= {1 , +2}; +var +teachers +:= +[ 1 , +2 , +3 ] ; +var +p r e f e r e n c e s +:= map [1 +:= +[ 2 , +1 ] , +2 := +[ 1 , +2 ] , +3 := +[ 2 ] ] ; +var +initialPlacement +:= +map [1 +:= +1 ] ; + +Case studies of development of verified programs with Dafny +45 +var +expectedVacanciesPrefs := map[1 +:= +[ 1 , 2 , 3 ] , +2 := +[ 1 , 2 , 3 ] ] ; +var +expectedFinalPlacement := map[1 +:= 2 , 2 := +1 ] ; +var +actualFinalPlacement := teachersPlacement ( vacancies , +teachers , +preferences , +in it ia l Pl ac em en t ) ; +a s s e r t +isV alid ( expectedFinalPlacement , +preferences , +expectedVacanciesPrefs ) ; +// +proof +helper . . . +a s s e r t +! unstable (1 , +1 , +expectedFinalPlacement , +preferences , +expectedVacanciesPrefs ) ; +// +proof +helper . . . +a s s e r t +actualFinalPlacement == expectedFinalPlacement ; +} +method test2TP () +{ +var +vacancies +:= {1 , +2}; +var +teachers +:= +[ 1 , +2 , +3 ] ; +var +p r e f e r e n c e s +:= map [1 +:= +[ 2 , +1 ] , +2 := +[ 1 , +2 ] , +3 := +[ 2 , +1 ] ] ; +var +initialPlacement +:= +map [3 +:= +1 ] ; +var +expectedVacanciesPrefs := map[1 +:= +[ 3 , +1 , +2 ] , +2 := +[ 1 , +2 , +3 ] ] ; +var +expectedFinalPlacement := map[1 +:= 2 , 3 := +1 ] ; +a s s e r t moveToHead( teachers , +3) == [ 3 , +1 , +2 ] ; +// +proof +helper +var +actualFinalPlacement := teachersPlacement ( vacancies , +teachers , +preferences , +in it i al Pl ac em en t ) ; +a s s e r t +isV alid ( expectedFinalPlacement , +preferences , +expectedVacanciesPrefs ) ; +// +proof +helper . . . +a s s e r t +! unstable (1 , +1 , +expectedFinalPlacement , +preferences , +expectedVacanciesPrefs ) ; +// +proof +helper . . . +a s s e r t +actualFinalPlacement == expectedFinalPlacement ; +} +A.9 +Topological Sorting +/∗ +∗ Proof +of +c o r r e c t n e s s +of +the +c l a s s i c +t o p o l o g i c a l +s o r t i n g +∗ algorithm +(Kahn ’ s +algorithm ) , +s i m p l i f i e d , +in +Dafny . +∗/ +// +Defines a +directed +graph with +v e r t i c e s +of +any type T as a +// +pair +(V, E) , +where V i s +the +vertex−set +and E i s +the +// edge−set . +// Each +directed +edge +i s +represented by a +pair +of +v e r t i c e s . +datatype Graph = Graph(V: +set, E: +set <(T,T)>) +// Checks +i f G d e f i n e s +a +valid +graph +( checks +that E i s +a +// +subset +of V∗V) . +predicate +validGraph(G: +Graph) { +f o r a l l +e +: : +e +in G.E ==> e .0 +in G.V && e .1 +in G.V + +46 +João Pascoal Faria and Rui Abreu +} +// Checks +i f +a graph +i s +a c y c l i c . +predicate +acyclic (G: +Graph) { +! +e x i s t s +v +: : +v in G.V && existsSimplePath (G, +v , +v) +} +// Check +i f +there +i s +a non−empty simple +path from +vertex u to +// +vertex v in +graph G. +( Currently , +’ simple ’ +means without +// +repeated +edges , +but +could be without +repeated +v e r t i c e s ) . +predicate +existsSimplePath(G: +Graph, u : T, +v : T) +decreases G.E +{ +(u , +v) +in G.E +| | +e x i s t s +e +: : +e +in G.E && e .0 == u +&& existsSimplePath (Graph(G.V, G.E−{e }) , +e . 1 , +v) +} +// Removes a vertex v and +i t s +incident +edges +from a graph G. +function +method removeVertex(v : T, G: +Graph) : +Graph +{ +Graph(G.V − {v} , +set +e +| +e +in G.E && e .0 +!= v && e .1 +!= v) +} +// Checks +i f +a sequence +s +of +v e r t i c e s +i s +a +t o p o l o g i c a l +// +ordering +of +the +v e r t i c e s +of +a graph G. +predicate +isTopSorting(s : +seq, G: +Graph) +r e q u i r e s +validGraph (G) +{ +multiset ( s ) == multiset (G.V) +&& f o r a l l +i , +j : : +0 <= i <= j < | s | ==> ( s [ j ] , +s [ i ] ) +! in G.E +} +// Checks +i f +a vertex v in a graph G has +incoming +edges . +predicate +method hasIncomingEdges(G: +Graph, v : T) +r e q u i r e s +v in G.V +{ +e x i s t s +u +: : +u in G.V && (u , +v) +in G.E +} +// +Topological +s o r t i n g +of +the +v e r t i c e s +of +an +a c y c l i c +directed +// graph . +Returns a sequence +( l i n e a r +ordering ) +of +the +v e r t i c e s +// +in +t o p o l o g i c a l +ordering . +method topsort (G: +Graph) returns +( s : +seq) +r e q u i r e s +validGraph (G) && a c y c l i c (G) +ensures +isTopSorting ( s , G) +{ +s := +[ ] ; +var R := G; +// remaining +graph + +Case studies of development of verified programs with Dafny +47 +while R.V != {} +// +r e l a t i o n +between +s and R ( b a s i c a l l y , R = G − s ) +invariant R == Graph( set +v +| +v in G.V && v +! in +s , +set +e +| +e +in G.E && e .0 +! in +s && e .1 +! in +s ) +// s +i s +a +t o p o l o g i c a l +s o r t i n g +of G − R +invariant +multiset ( s ) == multiset (G.V − R.V) +invariant +f o r a l l +i , +j : : +0 <= i <= j < | s | +==> ( s [ j ] , +s [ i ] ) +! in G.E +// +there +are no edges +from +v e r t i c e s +in R to +v e r t i c e s +in +s +invariant +f o r a l l +i : : +0 <= i < | s | +==> f o r a l l +v +: : +v in R.V ==> (v , +s [ i ] ) +! in G.E +decreases R.V +{ +// +r e c a l l +property : +a subgraph +of +an +a y c l i c +graph +i s +al so +// +a c y c l i c +lemmaAcyclicSubgraph (R, G) ; +// +r e c a l l +property : +a vertex +without +incoming +edges must +// +e x i s t +in a non−empty +a c y c l i c +graph +lemmaAcyclicIndegrees (R) ; +// +pick a vertex +without +incoming +edges +var v +: | +v in R.V && ! hasIncomingEdges (R, +v ) ; +// append to +the +r e s u l t +s := s + [ v ] ; +// remove +that +vertex and +i t s +outgoing +edges +from the +// graph +R := removeVertex (v , R) ; +} +} +/∗∗ SECOND LEMMA ∗∗∗/ +// +States +and proves by +contradiction +the +f o l l o w i n g +property : +// a non−empty +a c y c l i c +graph +must have +at +l e a s t +one +vertex +// without +incoming +edges +(0 +indegree ) . +lemma lemmaAcyclicIndegrees(G: +Graph) +r e q u i r e s +validGraph (G) && G.V != {} && a c y c l i c (G) +ensures +e x i s t s +v +: : +v in G.V && ! hasIncomingEdges (G, +v) +{ +// For the +sake +of +contradiction , +assume +that +a l l +v e r t i c e s +// have incoming +edges . +i f +f o r a l l +v +: : +v in G.V ==> hasIncomingEdges (G, +v) { +// Then a path +of +any +length +can be +built , +p o s s i b l y +with +// +repeated +edges and +v e r t i c e s , +namely a path with +length +// +|G.V| + 1 +var p := genPath (G, +|G.V| + 1 ) ; +// Such a path must have +repeated +v e r t i c e s , +and + +48 +João Pascoal Faria and Rui Abreu +// +consequently +at +l e a s t +one +cycle +lemmaPathLen (G, +p ) ; +// So the +graph +i s +not +acyclic , +which +c o n t r a d i c t s +the +// pre−condition +a s s e r t +! +a c y c l i c (G) ; +} +} +// Generates a +valid +path +of +a +s p e c i f i e d +length n in a non +// empty graph G in +which +a l l +v e r t i c e s +have incoming +edges . +// Because +of +t h i s +property , +a path with any +length may be +// +constructed +( pos sib l y +with +repeated +edges and +v e r t i c e s ) . +lemma genPath (G: +Graph, n : +nat ) +returns +(p : +seq) +r e q u i r e s +validGraph (G) && G.V != {} +r e q u i r e s +f o r a l l +v +: : +v in G.V ==> hasIncomingEdges (G, +v) +ensures +| p | == n && validPath (p , G) +{ +p := +[ ] ; +while +| p | < n +invariant +| p | <= n && validPath (p , G) +{ +var u +: | +u in G.V && (p == [ ] +| | +(u , +p [ 0 ] ) +in G.E) ; +p := +[ u ] + p ; +} +} +// Checks +i f +a sequence p of +v e r t i c e s +d e f i n e s +a +valid +path +// ( allowing +repeated +v e r t i c e s +and edges ) +in a graph G. +predicate +method validPath(p : +seq, G: +Graph) { +f o r a l l +i +: : +0 <= i < | p | ==> p [ i ] +in G.V +&& ( i < | p | − 1 ==> (p [ i ] , +p [ i +1]) +in G.E) +} +// +States +and proves +the +property : +given a graph G and a path +// p in G, +i f +the +length +of p exceeds +the number of +v e r t i c e s +// then G has +c y c l e s . +lemma lemmaPathLen(G: +Graph, p : +seq) +r e q u i r e s +validGraph (G) && validPath (p , G) && | p | > |G.V| +ensures +! a c y c l i c (G) +{ +// +f i r s t +notice +that , +i f +a l l +v e r t i c e s +are +d i s t i n c t , +the +// +length +of +the +path cannot +exceed +the number of +v e r t i c e s +// +in G +i f +nodups (p) { +lemmaSeqLen (p , G.V) ; +} +// +consequently , +there must +e x i s t +repeated +v e r t i c e s +in p , +so +// we pick a +c y c l i c +( complex ) +subpath +var +i , +j +: | +0 <= i < j < | p | && p [ i ] == p [ j ] ; + +Case studies of development of verified programs with Dafny +49 +var p ’ +:= p [ i +. . +j +1]; +// +r e c a l l +a u x i l i a r y lemma that +assures +that a simple +cycle +// +also +e x i s t +lemmaComplexPath (G, p ’ ) ; +} +// +States +and proves +(by induction ) +the +property : +given any +// +valid +complex path p ( p o s s i b l y +with +repeated +edges and/ or +// +v e r t i c e s ) +in a graph G, +there +e x i s t s +a simple +path +( without +// +repeated +edges ) +in G from the +f i r s t +to +the +l a s t +vertex +in +// the +complex path . +lemma lemmaComplexPath(G: +Graph, p : +seq) +r e q u i r e s G.V != {} && validPath (p , G) && | p | > 1 +ensures +existsSimplePath (G, +p [ 0 ] , +p [ | p| −1]) +decreases p +{ +// +handles +case +of +f i r s t +vertex +repeated +in +the +middle +i f +i +: | +1 <= i < | p|−1 && p [ i ] == p [ 0 ] +{ +lemmaComplexPath (G, +p [ i . . ] ) ; +} +// +handles +r e c u r s i v e +case +of +proof by induction +e l s e +i f +| p | > 2 { +lemmaComplexPath (Graph(G.V, G.E − {(p [ 0 ] , p [ 1 ] ) } ) , +p [ 1 . . ] ) ; +} +} +function +elems(s : +seq): +set { +set +x +| +x in +s +} +predicate +nodups(s : +seq) { +f o r a l l +i , +j +: : +0 <= i < j < | s | ==> s [ i ] +!= s [ j ] +} +// +States +and proves +(by induction ) +the +f o l l o w i n g +property : +// the +length +of +a sequence p of +d i s t i n c t +elements +from a +set +// s +cannot +exceed +the +c a r d i n a l i t y +of +the +set . +lemma lemmaSeqLen(p : +seq, s : +set) +r e q u i r e s +nodups (p) && elems (p) <= s +ensures +| p | <= | s | +{ +i f +p != +[ ] +{ +lemmaSeqLen (p [ 1 . . ] , +s − {p [ 0 ] } ) ; +} +} +/∗∗ FIRST LEMMA ∗∗∗/ +// +States +and proves +(by +contradiction ) +the +f o l l o w i n g + +50 +João Pascoal Faria and Rui Abreu +// +property : +a subgraph G of +an +a c y c l i c +graph G’ +i s +al s o +// +a c y c l i c . +lemma lemmaAcyclicSubgraph(G: +Graph, G’ : +Graph) +r e q u i r e s +validGraph (G) && validGraph (G’ ) +&& a c y c l i c (G’ ) && isSubGraph (G, G’ ) +ensures +a c y c l i c (G) +{ +i f +! +a c y c l i c (G) { +var u +: | +u in G.V && existsSimplePath (G, u , +u ) ; +// +ex is ts , +by the +d e f i n i t i o n +of +a c y c l i c +lemmaExistsSimplePath (G, G’ , u , +u ) ; +// +r e c a l l +lemma implying +that +such a path +al so +e x i s t s +// +in G +a s s e r t +! a c y c l i c (G’ ) ; +// so G would not be +acyclic , +contradicting +the +precond . +} +} +// Checks +i f +a graph G i s +a subgraph +of +another +graph G’ . +predicate +isSubGraph(G: +Graph, G’ : +Graph) { +G.E <= G’ . E && G.V <= G’ .V +} +// +States +and proves +(by induction ) +the +f o l l o w i n g +property : +i f +// +there +i s +a ( simple ) +path u−−>v in a graph G and G i s +a +// subgraph +of G’ , +then a path u−−>v +a ls o +e x i s t s +in G’ . +lemma lemmaExistsSimplePath(G: +Graph, G’ : +Graph, +u : T, +v : T) +r e q u i r e s +validGraph (G) && validGraph (G’ ) +&& isSubGraph (G, G’ ) && existsSimplePath (G, u , +v) +ensures +existsSimplePath (G’ , u , +v) +decreases G.E +{ +i f +(u , +v) +! in G.E { // +r e c u r s i v e +case +var e +: | +e +in G.E && e .0 == u +&& existsSimplePath (Graph(G.V, G.E−{e }) , +e . 1 , +v ) ; +// must +e x i s t +by +d e f i n i t i o n +of +existsPath +lemmaExistsSimplePath (Graph(G.V, G.E−{e }) , +Graph(G’ . V, G’ . E−{e }) , +e . 1 , +v ) ; +// +t h i s lemma implies +that +’ e ’ +a ls o +e x i s t +in G’ +} +} +/∗∗ Test +cases +∗∗∗/ +method +testTopSortingSingleSolution () +{ +var G: +Graph := Graph ({1 , +2 , +3} , +{(1 , +2) , +(2 , +3 ) } ) ; +a s s e r t +validGraph (G) && a c y c l i c (G) ; +var +s +: +seq := +[ 1 , +2 , +3 ] ; +a s s e r t +isTopSorting ( s , G) ; +var +t := +topsort (G) ; + +Case studies of development of verified programs with Dafny +51 +a s s e r t +t == s ; +} +method +testTopSortingMultipleSolutions () +{ +var G: +Graph := Graph ({1 , +2 , +3} , +{(1 , +2) , +(1 , +3 ) } ) ; +a s s e r t +validGraph (G) && a c y c l i c (G) ; +var +s1 +: +seq := +[ 1 , +2 , +3 ] ; +var +s2 +: +seq := +[ 1 , +3 , +2 ] ; +a s s e r t +isTopSorting ( s1 , G) ; +a s s e r t +isTopSorting ( s2 , G) ; +var +t := +topsort (G) ; +a s s e r t +t == s1 +| | +t == s2 ; +} +A.10 +Eulerian Circuit +/∗ +∗ Proof +of +c o r r e c t n e s s +of +the +Hierholzer +algorithm +(1873) +to +∗ +find +an Eulerian +c i r c u i t +in an Eulerian +graph +( method +∗ +f i n d E u l e r C i r c u i t ) . +∗ Reference : +https :// en . wikipedia . org / wiki / Eulerian_path . +∗/ +/∗∗∗∗ Graph +representation +and +v a l i d i t y +∗∗∗∗/ +// +Vertices +can be +of +any type +that +supports +equality . +type +Vertex = nat // or +other +type +// Graph represented +as a mapping from +v e r t i c e s +to +s e t s +of +// +adjacent +v e r t i c e s . +type Graph = m: map> | +definesValidGraph (m) +// The mapping must be +anti−r e f l e x i v e +and symmetric . +predicate +definesValidGraph (m: map>) { +f o r a l l +v , w : : +v in m && w in m[ v ] +==> w != v && w in m && v in m[w] +} +/∗∗∗∗ Graph modification +operations +∗∗∗∗/ +// Removes a vertex v from a graph G ( i f +e x i s t e n t ) . +function +rmvVertex (v : +Vertex , G: +Graph ) : +Graph { +map u +| +u in G && u != v +: : G[ u ] − {v} +} +// Removes an edge +(u , +v) +from a graph G ( i f +e x i s t e n t ) . +function +method rmvEdge(u : +Vertex , +v : +Vertex , G: +Graph ) : +Graph + +52 +João Pascoal Faria and Rui Abreu +ensures +var G’ +: +Graph := rmvEdge(u , +v , G) ; +u in G && v in G && v in G[ u ] ==> +hasEvenCard (G’ [ v ] ) +!= hasEvenCard (G[ v ] ) +&& hasEvenCard (G’ [ u ] ) +!= hasEvenCard (G[ u ] ) +{ +map x +| +x in G : : +i f +x == u then G[ x ] − {v} +e l s e +i f +x == v then G[ x ] − {u} +e l s e G[ x ] +} +// Adds and edge +(u , +v) +to a graph G. +function +addEdge (u : +Vertex , +v : +Vertex , G: +Graph ) : +Graph +r e q u i r e s u in G && v in G && u != v +{ +map x +| +x in G : : +i f +x == u then G[ x ] + {v} +e l s e +i f +x == v then G[ x ] + {u} +e l s e G[ x ] +} +/∗∗∗∗ Subgraphs ∗∗∗∗/ +// Check +i f G1 i s +a subgraph +of G2 in +terms +of +edges , +but with +// the same vertex−set . +predicate +isSubgraphE (G1: +Graph , G2: +Graph) { +G1. Keys == G2. Keys && f o r a l l +x +: : +x in G1 ==> G1[ x ] <= G2[ x ] +} +/∗∗∗∗ +Connectivity +∗∗∗∗/ +// Checks +i f +a given +graph +i s +connected , +i . e . , +there +i s +a path +// between +every two +v e r t i c e s . +predicate +isConnected (G: +Graph) { +f o r a l l +u , +v +: : +u in G && v in G +==> connectedVertices (u , +v , G) +} +// Checks +i f +v e r t i c e s +u and v are +connected +in a graph G, +// +i . e . , +there +i s +a path +connecting them ( without +repeated +// +v e r t i c e s ) . +predicate +connectedVertices (u : +Vertex , +v : +Vertex , G: +Graph) +r e q u i r e s u in G && v in G +decreases G +{ +u == v +| | +e x i s t s w : : w in G[ u ] +&& connectedVertices (w, +v , +rmvVertex (u , G)) +} +// Proves by induction +that +i f +a vertex u belongs +to a +closed +// vertex−set C ( under +adjacency ) +in a graph G and v does not , +// then +they must be +disconnected . +lemma unconnectedVerticesLemma (u : +Vertex , +v : +Vertex , G: +Graph , + +Case studies of development of verified programs with Dafny +53 +C: +set) +r e q u i r e s u in G && v in G && u in C && v +! in C +r e q u i r e s +f o r a l l +x +: : +x in C && x in G ==> G[ x ] <= C +// C i s +a +closed +vertex−set +decreases G +ensures +! connectedVertices (u , +v , G) +{ +// mimics +the +structure +of +connectedVertices +f o r a l l w | w in G[ u ] +{ +unconnectedVerticesLemma (w, +v , +rmvVertex (u , G) , C) ; +} +} +/∗∗∗∗ Vertex +degrees +∗∗∗∗/ +// Checks +i f +a l l +v e r t i c e s +in a graph G have even +degree +( even +// number of +incident +edges ) . +predicate +hasEvenDegrees (G: +Graph) { +f o r a l l +v +: : +v in G ==> hasEvenCard (G[ v ] ) +} +// Checks +i f +a +set +s +has an even number of +elements +( even +// +cardinal ) . +predicate +hasEvenCard(s : +set) { +| s | % 2 == 0 +} +// +I f +we remove from a graph G with even +vertex +degrees +the +// edges +of +a subgraph T with even +vertex +degrees , we obtain a +// subgraph R with even +vertex +degrees . +lemma evenDegreesLemma (G: +Graph , R: +Graph , T: +Graph) +r e q u i r e s G. Keys == T. Keys == R. Keys +r e q u i r e s +f o r a l l +x +: : +x in G +==> T[ x ] <= G[ x ] && R[ x ] == G[ x ] − T[ x ] +r e q u i r e s +hasEvenDegrees (G) && hasEvenDegrees (T) +ensures +hasEvenDegrees (R) +{ // Thanks Dafny +} +/∗∗∗ Walks , +t r a i l s , +c i r c u i t s +and augmentation +p r o p e r t i e s +∗∗∗∗/ +// Checks +i f +a sequence +s +of +v e r t i c e s +d e f i n e s +a +valid +walk +in +// a graph G. +predicate +isValidWalk ( s : +seq, G: +Graph) +{ +( f o r a l l +x +: : +x in +s ==> x in G) +&& ( f o r a l l +i +: : +0 <= i < | s | − 1 ==> s [ i +1] +in G[ s [ i ] ] ) +} +// +U s e f u l l +augmentation +property +of +valid +walks . +lemma walkAugmentationLemma ( s : +seq, G: +Graph , + +54 +João Pascoal Faria and Rui Abreu +u : +Vertex ) +r e q u i r e s +isValidWalk ( s , G) && u in G +&& ( | s | == 0 +| | +u in G[ s [ | s | −1]]) +ensures +isValidWalk ( s + [ u ] , G) +{ /∗ Thanks Dafny ∗/ } +// Checks +i f +a sequence +s +of +v e r t i c e s +t r a v e r s e s +an edge +// (u , +v ) . +predicate +traversesEdge ( s : +seq, u : +Vertex , +v : +Vertex ) +r e q u i r e s u != v +{ +e x i s t s +i +: : +1 <= i < | s | && { s [ i −1] , +s [ i ]} == {u , +v} +} +// +Useful +augmentation +property +of +traversed +edges . +lemma traversesEdgeProp ( s : +seq, v : +Vertex ) +r e q u i r e s +| s | > 0 && v != s [ | s | −1] +ensures +traversesEdge ( s + [ v ] , +s [ | s | −1] , v ) ; +{ +// +i t +seems +the +only +thing +Dafny needs +i s +to show the +// "de−concatenation " +a s s e r t +var s ’ +:= s + [ v ] ; +s ’ [ . . | s | ] == s && s ’ [ | s | ] == v ; +} +// Checks +i f +a sequence +s +of +v e r t i c e s +d e f i n e s +a +valid +t r a i l +// +in a graph G, +i . e . , +a +valid +walk without +repeated +edges . +predicate +i s V a l i d T r a i l ( s : +seq, G: +Graph) { +isValidWalk ( s , G) +&& f o r a l l +i +: : +1 <= i < | s | +==> ! +traversesEdge ( s [ . . i ] , +s [ i −1] , +s [ i ] ) +} +// +U s e f u l l +aumentation +property +of +valid +t r a i l s . +lemma trailAugmentationLemma ( s : +seq, G: +Graph , +u : +Vertex ) +r e q u i r e s +i s V a l i d T r a i l ( s , G) && u in G +r e q u i r e s +| s | > 0 ==> u in G[ s [ | s | −1]] +// to make valid +walk +r e q u i r e s +| s | > 0 ==> ! traversesEdge ( s , +s [ | s | −1] , u) +// to make valid +t r a i l +ensures +i s V a l i d T r a i l ( s + [ u ] , G) +{ /∗ Thanks Dafny ∗/ } +// Checks +i f +a sequence +s +of +v e r t i c e s +d e f i n e s +a +valid +c i r c u i t +// +in a graph G, +i . e . , +a non−empty +t r a i l +in +which the +f i r s t +// and +l a s t +v e r t i c e s +are +i d e n t i c a l . +predicate +i s V a l i d C i r c u i t ( s : +seq, G: +Graph) { +i s V a l i d T r a i l ( s , G) && | s | > 0 && s [ | s | −1] == s [ 0 ] +} + +Case studies of development of verified programs with Dafny +55 +// Shows that +c i r c u i t +augmentation +(by embedding ) +implies +the +// union +of +traversed +edges . +lemma circuitAugmentationLemma1 ( s1 : +seq, +i : +int , +s2 : +seq, s3 : +seq, G: +Graph) +r e q u i r e s +i s V a l i d C i r c u i t ( s1 , G) && i s V a l i d C i r c u i t ( s2 , G) +r e q u i r e s +0 <= i < | s1 | && s2 [ 0 ] == s1 [ i ] +&& s3 == s1 [ . . i ] + s2 + s1 [ i + 1 . . ] +ensures +f o r a l l +x , +y +: : +x in G && y in G[ x ] ==> +( traversesEdge ( s3 , +x , +y) <==> traversesEdge ( s1 , +x , +y) +| | +traversesEdge ( s2 , +x , +y )) +{ +// +i t +seems +the +only +thing +Dafny needs +i s +to show the +// "de−concatenation " ( s l i c i n g ) +a s s e r t +s1 [ . . i ] == s3 [ +. . +i ] ; +a s s e r t +s2 == s3 [ i +. . +i + | s2 | ] ; +a s s e r t +s1 [ i + 1 +. . ] == s3 [ i + | s2 | . . ] ; +} +// Proves by deduction +that +c i r c u i t +augmentation , +by embedding +// another +c i r c u i t +with +d i s j o i n t +edges , +r e s u l t s +in a +valid +// +c i r c u i t , +without +repeated +edges . +lemma circuitAugmentationLemma2 ( s1 : +seq, +i : +int , +s2 : +seq, s3 : +seq, G: +Graph) +r e q u i r e s +i s V a l i d C i r c u i t ( s1 , G) && i s V a l i d C i r c u i t ( s2 , G) +r e q u i r e s +0 <= i < | s1 | && s2 [ 0 ] == s1 [ i ] +&& s3 == s1 [ . . i ] + s2 + s1 [ i + 1 . . ] +r e q u i r e s +f o r a l l +k +: : +1 <= k < | s1 | +==> ! traversesEdge ( s2 , +s1 [ k−1] , +s1 [ k ] ) +r e q u i r e s +f o r a l l +k +: : +1 <= k < | s2 | +==> ! traversesEdge ( s1 , +s2 [ k−1] , +s2 [ k ] ) +ensures +i s V a l i d C i r c u i t ( s3 , G) +{ +// mimics +the +procedure +f o r +checking +e x i s t e n c e +of +duplicate +// edges +f o r a l l +j , +k +| +1 <= j < k < | s3 | +ensures { s3 [ k−1] , +s3 [ k ]} +!= { s3 [ j −1] , +s3 [ j ]} +{ +// map to +i n d i c e s +in +o r i g i n a l +sequences +var +( j ’ , +s j ) := +i f +j <= i +then +( j , +s1 ) +e l s e +i f +j < i +| s2 | +then +( j−i , +s2 ) +e l s e +( j −(| s2 | −1) , +s1 ) ; +var +(k ’ , +sk ) := +i f +k <= i +then +(k , +s1 ) +e l s e +i f +k < i +| s2 | +then +(k−i , +s2 ) +e l s e +(k−(| s2 | −1) , +s1 ) ; +// +r e c a l l +that +edges +are +d i s t i n c t +in +o r i g i n a l +sequences +// ( from pre−conditions ) +a s s e r t +{sk [ k ’ −1] , +sk [ k ’ ] } +!= { s j [ j ’ −1] , +s j [ j ’ ] } ; +} + +56 +João Pascoal Faria and Rui Abreu +} +/∗∗∗ +Euler +t r a i l s +and +c i r c u i t s +∗∗∗∗/ +// Checks +i f +a sequence +s +of +v e r t i c e s +d e f i n e s +an Euler +c i r c u i t +// +in a graph G, +i . e . , +a +c i r c u i t +that +t r a v e r s e s +each +edge +of G +// +exactly +once . +predicate +i s E u l e r C i r c u i t ( s : +seq, G: +Graph) { +i s V a l i d C i r c u i t ( s ,G) // +ensures no +duplicate +edge +c r o s s i n g +&& f o r a l l +u , v +: : +u in G && v in G[ u ] +==> traversesEdge ( s , u , +v) +} +// Proves by +contradiction +that a non−augmentable +c i r c u i t +r +// +in a connected +graph G must cover +a l l +edges , +i . e . , +must be +// an Euler +c i r c u i t . +lemma nonAugmentableCircuitLemma (G: +Graph , +r : +seq) +r e q u i r e s +isConnected (G) && i s V a l i d C i r c u i t ( r , G) +r e q u i r e s +f o r a l l +x , +y +: : +x in +r && y in G[ x ] +==> traversesEdge ( r , +x , +y) +ensures +i s E u l e r C i r c u i t ( r , G) +{ +a s s e r t +f o r a l l +x , +y +: : +x in +r && y in G[ x ] ==> y in +r ; +// +implied by 2nd precondition +i f +v +: | +v in G && v +! in +r { +unconnectedVerticesLemma ( r [ 0 ] , +v , G, +set +x +| +x in +r ) ; +// +t h i s +co nt r ad ic t s +the +hypothesis +that G i s +connected , +// so v cannot +e x i s t ; +hence +a l l +v e r t i c e s +of G are +covered +// by r , +and hence +t h e i r +incident +edges +(by 2nd pre ) . +} +} +// Checks +i f +a sequence +s +of +v e r t i c e s +d e f i n e s +an Euler +t r a i l +// +in a graph G, +i . e . , +a +t r a i l +that +t r a v e r s e s +each +edge +of G +// +exactly +once . +predicate +i s E u l e r T r a i l ( s : +seq, G: +Graph) { +| s | > 0 && i s V a l i d T r a i l ( s , G) +&& f o r a l l +x , +y +: : +x in G && y in G[ x ] +==> traversesEdge ( s , +x , +y) +} +// Property +of +vertex +degrees +in an Euler +t r a i l : +the number of +// +incident +edges on each +vertex +i s +even +except +f o r +the +f i r s t +// and +l a s t +vertex +i f +d i f f e r e n t . +predicate +EulerTrailDegrees (G: +Graph , +r : +seq) +r e q u i r e s +i s E u l e r T r a i l ( r , G) +{ +var +f i r s t , +l a s t +:= r [ 0 ] , +r [ | r | −1]; +f o r a l l +x +: : +x in G ==> +(( x==f i r s t ) != (x==l a s t ) /∗ xor ∗/ <==> ! hasEvenCard (G[ x ] ) ) + +Case studies of development of verified programs with Dafny +57 +} +// Proves by induction +the +above +property +about +the +vertex +// +degrees +in an Euler +t r a i l . +lemma EulerTrailLemma (G: +Graph , +r : +seq) +r e q u i r e s +i s E u l e r T r a i l ( r , G) +ensures +EulerTrailDegrees (G, +r ) +{ +i f +| r | > 1 { +EulerTrailLemma (rmvEdge( r [ 0 ] , +r [ 1 ] , G) , +r [ 1 . . ] ) ; +} +} +/∗∗∗∗ Main algorithms +∗∗∗∗/ +// +Hierholzer +algorithm +to +find +an Euler +c i r c u i t +in a +// non−empty Eulerian +graph G based on depth−f i r s t +search . +method +f i n d E u l e r C i r c u i t (G: +Graph) +returns +( r : +seq) +r e q u i r e s +isConnected (G) && hasEvenDegrees (G) && |G| > 0 +ensures +i s E u l e r C i r c u i t ( r , G) +{ +// +build +i n i t i a l +c i r c u i t , +s t a r t i n g +in an +a r b i t r a r y +vertex , +// and obtain +remaining +graph +var v +: | +v in G; +var R : +Graph ; +r , R := +dfs (v , G) ; +// Ghost +v a r i a b l e +to +help +proving +termination +( v e r t i c e s +to +// +explore ) +ghost +var V := +set +x | +x in R && R[ x ] +!= +{}; +// augment r +as +p o s s i b l e +while +e x i s t s +i +: : +0 <= i < | r | && R[ r [ i ] ] +!= {} +// r +i s +a +valid +c i r c u i t +in G s t a r t i n g +in v +invariant +i s V a l i d C i r c u i t ( r , G) && | r | > 0 && r [ 0 ] == v +// R i s +a subgraph +of G with even +vertex +degrees +invariant +hasEvenDegrees (R) && isSubgraphE (R, G) +// R contains +the +edges +not +traversed by r +in G +invariant +f o r a l l +x , y : : x in G && y in G[ x ] +==> (y +! in R[ x ] <==> traversesEdge ( r , x , y )) +// V ( variant ) +i s +the +set +of +v e r t i c e s +that +have +adjacent +// +v e r t i c e s +not +yet +explored +invariant V == set +x +| +x in R && R[ x ] +!= {} +decreases V +{ +// +s e l e c t +a vertex +in +r +with +outgoing +edges +to +explore + +58 +João Pascoal Faria and Rui Abreu +var +i +: | +0 <= i < | r | && R[ r [ i ] ] +!= +{}; +var u := r [ i ] ; +// memmorize old +values +needed +l a t e r +ghost +var +oldr , +oldV := r , V; +// do a DFS from +t h i s +vertex +in +the +remaining graph , +// +obtaining +a new s u b c i r c u i t +and remaining +graph +var c +: +seq; +c , R +:= +dfs (u , R) ; +// +i n s e r t +the +s u b c i r c u i t +in +the main +c i r c u i t +r := r [ . . +i ] + c + r [ i + 1 . . ] ; +// +r e c a l l +c i r c u i t +augmentation +p r o p e r t i e s +to make sure +// +i n v a r i a n t s +are +maintained +circuitAugmentationLemma1 ( oldr , +i , +c , +r , G) ; +// union +of +traversed +edges +circuitAugmentationLemma2 ( oldr , +i , +c , +r , G) ; +// no +duplicate +edges +// prove +that +the +variant +decreases +V := +set +x +| +x in R && R[ x ] +!= +{}; +a s s e r t u in +oldV && u +! in V; +a s s e r t V < oldV ; +} +// show that +a l l +edges +of G have been +traversed , +because G +// +i s +connected +nonAugmentableCircuitLemma (G, +r ) ; +} +// By performing a depth−f i r s t +search , +produces a complete +// +valid +t r a i l +in a graph G s t a r t i n g +in a vertex v . +// Assuming +a l l +v e r t i c e s +in +the +graph have even +degree , +the +// produced +t r a i l +i s +c i r c u l a r . +// Returns +the +c i r c u l a r +t r a i l +( r ) and the +// remaining +graph R ( with +unexplored +edges ) . +method +dfs (v : +Vertex , G: +Graph) +returns +( r : +seq, +R: +Graph) +r e q u i r e s +hasEvenDegrees (G) && v in G +ensures +i s V a l i d C i r c u i t ( r , G) && | r | > 0 && r [ 0 ] == v +ensures +isSubgraphE (R, G) && hasEvenDegrees (R) +ensures +f o r a l l +x , +y +: : +x in G && y in G[ x ] +==> (y +! in R[ x ] <==> traversesEdge ( r , +x , +y )) +ensures R[ v ] == {} // +a l l +s u c c e s s o r s +of v have been +explored +{ +R := G; +// subgraph +with +edges +remaining +to be +v i s i t e d +ghost +var T: +Graph := map x +| +x in G : : +{}; +// subgraph +with +edges +already +traversed + +Case studies of development of verified programs with Dafny +59 +// Ghost +v a r i a b l e +to +help +proving +termination +( edges +// remaining ) +ghost +var E := +set x , +y +| +x in G && y in G[ x ] +: : +(x , +y ) ; +// +i n i t i a t e +the +r e s u l t +with +the +i n i t i a l +vertex +( al so +l a s t +// +vertex +at +t h i s +point ) +r := +[ v ] ; +var u := v ; +// augment r +as +p o s s i b l e +while R[ u ] +!= {} +// R i s +a subgraph +of G ( with +the same vertex−set ) +invariant +isSubgraphE (R, G) +// T i s +a subgraph +of G with +edges G − R +invariant T. Keys == G. Keys +&& f o r a l l +x +: : +x in G ==> T[ x ] == G[ x ] − R[ x ] +// E i s +the +set +of +edges +in R +invariant +f o r a l l +x , +y +: : +x in R && y in R[ x ] +<==> (x , +y) +in E +// r +i s +a sequence +of +v e r t i c e s +s t a r t i n g +in v and +// ending +in u +invariant +| r | > 0 && r [ 0 ] == v && r [ | r | −1] == u +// r +travers +exactly +the +edges +in T ( without +repetions ) +invariant +i s V a l i d T r a i l ( r , T) +invariant +f o r a l l +x , +y +: : +x in G && y in G[ x ] +==> (y in T[ x ] <==> traversesEdge ( r , +x , +y )) +// +variant +to +ensure +termination +decreases E +{ +// +s e l e c t +an adjacent +vertex +f o l l o w i n g +an edge +not +// +previously +v i s i t e d +var w : | w in R[ u ] ; +// +r e c a l l +some +t r a i l +augmentation +p r o p e r t i e s +trailAugmentationLemma ( r , G, w) ; +// +valid +t r a i l +traversesEdgeProp ( r , w) ; +// +traversed +edges +// augment the +t r a i l +and update +v i s i t e d +and +// non−v i s i t e d +edges and +l a s t +vertex +r := r + [w ] ; +R := rmvEdge(u , w, R) ; +T := addEdge (u , w, T) ; +E := E − {(u , w) , +(w, +u ) } ; +u := w; + +60 +João Pascoal Faria and Rui Abreu +} +// shows +that +the +obtained +t r a i l +( Euler +t r a i l +in T) +ends +// +in +the +s t a r t +vertex +a s s e r t T[ u ] == G[ u ] ; +// because R[ u ] == {} +a s s e r t +hasEvenCard (T[ u ] ) ; +// because hasEvenCard (G[ u ] ) +EulerTrailLemma (T, +r ) ; +a s s e r t u == v ; +// shows +that +in +the +remaining +graph +(R) +a l l +v e r t i c e s +have +// even +degrees +a s s e r t +hasEvenDegrees (T) ; +evenDegreesLemma (G, R, T) ; +a s s e r t +hasEvenDegrees (R) ; +} +method +t e s t E u l e r C i r c u i t () +{ +var G : +Graph := map[1 +:= {2 , +3} , 2 := {1 , +3} , +3 := {1 , +2 , +4 , +5} , 4 := {3 , +5} , 5 := {3 , +4 } ] ; +var c +: +seq := +[ 1 , +2 , +3 , +4 , +5 , +3 , +1 ] ; +a s s e r t +c == [ c [ 0 ] , +c [ 1 ] , +c [ 2 ] , +c [ 3 ] , +c [ 4 ] , +c [ 5 ] , +c [ 6 ] ] ; +// +helper +. . . +a s s e r t +i s E u l e r C i r c u i t ( c , G) ; +} +method +t e s t E u l e r T r a i l () +{ +var G : +Graph := map[1 +:= {2 , +3} , 2 := {1 , +3} , +3 := {1 , +2 , +4} , 4 := {3 , +5} , 5 := +{ 4 } ] ; +var c +: +seq := +[ 3 , +2 , +1 , +3 , +4 , +5 ] ; +a s s e r t +c == [ c [ 0 ] , +c [ 1 ] , +c [ 2 ] , +c [ 3 ] , +c [ 4 ] , +c [ 5 ] ] ; +// +helper +. . . +a s s e r t +i s E u l e r T r a i l ( c , G) ; +} + diff --git a/jNE1T4oBgHgl3EQfgARW/content/tmp_files/load_file.txt b/jNE1T4oBgHgl3EQfgARW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2773370c275e9a1c9fcb8e5776158c3701fcd53a --- /dev/null +++ b/jNE1T4oBgHgl3EQfgARW/content/tmp_files/load_file.txt @@ -0,0 +1,1775 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf,len=1774 +page_content='Case studies of development of verified programs with Dafny for accessibility assessment⋆ João Pascoal Faria1,2[0000−0003−3825−3954] and Rui Abreu1,3[0000−0003−3734−3157] 1 Faculty of Engineering of the University of Porto, Porto, Portugal {jpf,rma}@fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='pt 2 INESC TEC - Institute for Systems and Computer Engineering, Technology and Science, Porto, Portugal 3 INESC ID, Lisbon, Portugal Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Formal verification techniques aim at formally proving the correctness of a computer program with respect to a formal specification, but the expertise and effort required for applying formal specification and verification techniques and scalability issues have limited their practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In recent years, the tremendous progress with SAT and SMT solvers enabled the construction of a new generation of tools that promise to make formal verification more accessible for software engineers, by automating most if not all of the verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The Dafny system is a prominent example of that trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, little evidence exists yet about its accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' To help fill this gap, we conducted a set of 10 case studies of developing verified implementations in Dafny of some real-world algorithms and data structures, to determine its accessibility for software engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We found that, on average, the amount of code written for specification and verification purposes is of the same order of magnitude as the traditional code written for implementation and testing purposes (ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='14) – an “overhead” that certainly pays off for high-integrity software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The performance of the Dafny verifier was impressive, with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='4 proof obligations generated per line of code written, and 24 ms spent per proof obligation generated and verified, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, we also found that the manual work needed in writing auxiliary verification code may be significant and difficult to predict and master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Hence, further automation and systematization of verification tasks are possible directions for future advances in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Keywords: Formal verification · Dafny · Accessibility · Case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 Motivation Given the increasing dependence of our society on software-based systems, it is ever more important to assure their correct, secure and safe functioning, partic- ularly for high-integrity systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Since software development is a knowledge- intensive activity and software-based systems are increasingly complex, errors ⋆ This is an extended version, including the source code, of our FSEN 2023 paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='03224v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='SE] 9 Jan 2023 2 João Pascoal Faria and Rui Abreu are inevitable, so several techniques need to be applied along the process to catch and fix defects as early as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Testing and reviews are the most widely applied techniques in the software industry for defect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, since “program testing can be used to show the presence of bugs, but never to show their absence” [2], testing alone cannot be considered sufficient for high-integrity systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' If properly applied [3], reviews are a cost-effective technique for defect detection and knowledge sharing, but, like with testing, they cannot be used to show the absence of bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' By contrast, formal verification techniques aim at formally proving the cor- rectness of a computer program, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', show the absence of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' To that end, we need a formal specification of the program intent and a logic reasoning frame- work, usually based on Hoare logic [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' But the expertise and effort required for applying formal specification and verification techniques and scalability issues have limited their practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In recent years, the tremendous progress with SAT and SMT solvers [5], such as Z3 [6], enabled the construction of a new generation of tools that promise to make formal verification accessible for soft- ware engineers, like Dafny [7], Frama-C [8] and Why3 [9], by automating most if not all of the verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, little evidence exists yet about their accessibility, regarding the expertise and effort required to apply them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The authors have used formal specification languages and automated rea- soning tools for several years in software engineering research, education, and practice [10, 11, 12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', in [11], Alloy [15] was used to automatically generate unit tests and mock objects in JUnit4 from algebraic specifications of generic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Although model-based testing approaches such as this one do not guarantee the absence of bugs, they provide a higher assurance than manual test generation and seem to be currently more accessible than formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' From an educational perspective, the authors are also interested in assessing the feasibility of embedding computer-supported formal specification and veri- fication techniques in undergraduate programs, namely in courses dedicated to studying algorithms and data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2 Objectives and Methodology To help fill the gap in the current state of the art regarding accessibility stud- ies, we conducted a set of case studies of developing verified implementations in Dafny of some well-known algorithms and data structures of varying com- plexity, with the goal of determining its accessibility for software engineering practitioners, students and researchers, with limited training in formal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Table 1 shows the list of case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They explore formal specification and verification features of increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2, we provide some high- lights for selected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' For each case study, we collected a few metrics and lessons learned, to help answer our main question, regarding Dafny accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Those metrics and lessons learned are aggregated and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3 ˙The source code is available in a GitHub repository5 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4 https://junit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='org/ 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='com/joaopascoalfariafeup/DafnyProjects Case studies of development of verified programs with Dafny 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' List of case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Category ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Case study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Numerical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='algorithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Integer division (Euclidean division) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Natural power of a number (divide and conquer algorithm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Searching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='sorting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='algo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='rithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Binary search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Insertion sort ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Collections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Priority queue implemented with a binary heap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Unordered set implemented with a hash table (Hash Set) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Ordered set implemented with a binary search tree (Tree Set) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Matching prob- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='lems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Stable marriage problem solved by the Gale-Shapley algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Teachers placement problem reduced to stable marriage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='algo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='rithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Topological sorting (Khan’s algorithm [16]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Eulerian circuit (Hierholzer’s algorithm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 Structure of the Paper Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2 presents some highlights about specification and verification features of increasing complexity in the case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3 consolidates the metrics collected and lessons learned, and draws conclusions regarding our research goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Related work is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Conclusions and future work are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2 Case Studies Highlights 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 An Introductory Example (Integer Division) The self-explanatory program in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1 explores some basic features of Dafny and serves as our first case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Dafny6 [7] is a multi-paradigm programming language and system for the de- velopment of verified programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The functional style is typically used for writing specifications, using value types and side-effect-free expressions, functions, and predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The procedural and object-oriented styles are typically used for writ- ing implementations, using reference types (arrays, classes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ), and methods and statements with side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The Dafny programming system comprises a verifier (based on Z3), compilers that produce code in several target languages (C#, Java, JavaScript, Go, and C++), and an extension for Visual Studio Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The semantics of a method (div in this case) is formally specified by means of pre and postconditions, indicated with the requires and ensures clauses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The Dafny verifier is in charge of checking (with the help of the Z3 theorem prover) if such pre and postconditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' When the im- plementation involves a loop, the user has to provide a loop invariant (with 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='com/dafny-lang/dafny 4 João Pascoal Faria and Rui Abreu // Computes the quotient q and remainder r of the integer division // of a (non-negative) dividend n by a (positive) divisor d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method div(n: nat, d: nat) returns (q: nat, r: nat) requires d > 0 ensures q * d + r == n && r < d { q := 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' r := n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while r >= d decreases r invariant q * d + r == n { q := q + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' r := r - d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // Main program, with a simple test case (checked statically!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method Main() { var q, r := div(15, 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' assert q == 2 && r == 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' print "q=", q, " r=", r, "\\n";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' A simple program in Dafny for performing integer division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' the invariant clause) and, in some cases, a loop variant (with the decreases clause), to help the verifier accomplish its job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The Main method is the entry point of a program in Dafny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In this example, it exercises the div method for some inputs, and checks (with assert) and prints the corresponding outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Like with pre and postconditions, assert statements are checked statically by the Dafny verifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In this example, the verifier will try to prove the assertion based only on the postcondition of the div method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', the method body is opaque for this purpose);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' this makes the verification modular and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Since assertions are checked statically, test cases such as the one shown do actually test the specification in pre-compile time, and not the implementation at run-time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' such static test cases are useful to detect problems in the specification, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', incomplete postconditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' All the specification constructs and assertions mentioned above (indicated with the requires, ensures, invariant, decreases, and assert clauses) are used as annotations for verification purposes only (during static analysis), but are not compiled into the executable program, so do not cause runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2 Lemmas and Automatic Induction (Power of a Number) In this case study, the goal is to prove the correctness of a well-known O(log n) divide-and-conquer algorithm to compute the natural power of a real number (xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Self-explanatory excerpts are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2 and the full code is avail- able in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' It illustrates the usage of lemmas, to specify properties that Case studies of development of verified programs with Dafny 5 Dafny alone cannot deduce, and automatic induction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', the ability of Dafny to automatically prove some properties by induction (directive :induction a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Recursive definition of x^n in functional style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function power(x: real, n: nat) : real { if n == 0 then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 else x * power(x, n-1) } // Computation of x^n in time and space O(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method powerDC(x: real, n: nat) returns (p : real) ensures p == power(x, n) { .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' if n % 2 == 0 { productOfPowers(x, n/2, n/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // recall lemma var temp := powerDC(x, n/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' return temp * temp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=" } // States the property x^a * x^b = x^(a+b), used by 'powerDC'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=" // The property is proved by automatic induction on 'a'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma {:induction a} productOfPowers(x: real, a: nat, b: nat) ensures power(x, a) * power(x, b) == power(x, a + b) {/*Proof should go here, but is discovered by Dafny!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' */} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Excerpts of a program in Dafny for computing the natural power of a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 Modules, Mutable Objects and Generics (Insertion Sort) In this case study, we explore Dafny features for working with mutable objects (in this case, arrays) and generics, and separating specification, implementation, and test code with modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Self-explanatory excerpts are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The array sorting problem is specified by the bodyless sort method in the abstract module Sorting, resorting to auxiliary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The frame condition “modifies a” indicates that an implementation may modify the contents ref- erenced by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In the postcondition, “old(a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='])” and “a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.]” give the array contents at the begin and end of method execution, respectively, as mathemat- ical sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Dafny has some support for generic predicates, functions and methods, but, unfortunately, does not support type parameters that are subject to operations other than equality (==);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' so, for demo purposes, we declared the type of array elements with a specific type definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Sorting algorithms may be provided in concrete modules that refine the ab- stract module, as in the InsertionSort module, inheriting the method contract and providing the actual algorithm in the body (omitted here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In this case, we just had to provide the loop invariants for the verifier to successfully check the correctness of the insertion sort algorithm with respect to the specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The module TestSorting shows an example of a test case of the sort method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' For the Dafny verifier to successfully check the test outcome in the 6 João Pascoal Faria and Rui Abreu abstract module Sorting { type T = int // generics limitation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method sort(a: array) modifies a ensures isSorted(a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.]) && isPermutation(a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.], old(a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.])) } module InsertionSort refines Sorting { method sort(a: array) {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='} } abstract module TestSorting { import opened Sorting method testSort () { var a := new T[] [9, 3, 6, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' assert a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.] == [9, 3, 6, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // proof helper!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' sort(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' SortingUniquenessProp(a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.], [3, 6, 9, 9]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' //proof helper!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' assert a[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.] == [3, 6, 9, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } lemma SortingUniquenessProp(a: seq, b: seq) requires isSorted(a) && isSorted(b) && isPermutation(a, b) ensures a == b { /* handwritten proof by induction goes here*/} } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Organization of an array sorting program in Dafny using modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' last assert statement, we had to write an auxiliary lemma implying that the outcome of sort is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Surprisingly, for the code to be checked success- fully, we also had to provide some further “proof helper” assertions (as the first assertion) stating trivial facts that we expected to be taken for granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='4 State Abstraction and Automatic Contracts (Priority Queue) In this case study, we explore Dafny features for separating specification and implementation and handling class invariants in object-oriented programs, fol- lowing design by contract (DbC) principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Excerpts of the specification of a priority queue and its implementation with a binary heap are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The operations’ pre and postconditions of the priority queue (top box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3) are specified independently of the internal state representation (a bi- nary heap in this case), by resorting to a state abstraction function (elems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' This function gives the priority queue contents as a multiset (allowing repeated values), and serves only for specification and verification purposes (doesn’t gen- erate executable code);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' to keep the specification at a high level of abstraction, it doesn’t tell the ordering of elements (which is given by deleteMax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In a subsequent refinement (box at the center of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3), it is chosen an internal (concrete) state representation - a binary heap stored in an array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' It is Case studies of development of verified programs with Dafny 7 also provided an implementation (body) for each method (box at the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The definition and verification of class invariants, stating restrictions on the internal state to be respected at method boundaries, is facilitated in Dafny with so-called automatic contracts, using the “:autocontracts” attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The class invariant is specified in a predicate Valid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' calls to that predicate, together with some frame conditions, are automatically injected in the preconditions of all methods and in the postconditions of all methods and constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' class {:autocontracts} PriorityQueue { function elems(): multiset // State abstraction function constructor () ensures isEmpty() predicate method isEmpty() ensures isEmpty() <==> elems() == multiset{} method insert(x : T) ensures elems() == old(elems()) + multiset{x} method deleteMax() returns (x: T) requires !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' isEmpty() ensures isMax(x,old(elems())) && elems()==old(elems())-multiset{x} } // Concrete state representation var heap: array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var size : nat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // State abstraction function function elems(): multiset { multiset(heap[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='.size]) } // Class invariant (heap invariant) predicate Valid() { // valid size && each node is less or equal than its parent size<=heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Length && forall i :: 1<=i heap[i]<=heap[(i-1)/2] } // Inserts a value x in the heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method insert(x : T) ensures elems() == old(elems()) + multiset{x} { // if needed, grows the array if size == heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Length { grow();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Place at the bottom heap[size] := x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' size := size + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Move up as needed in the heap heapifyUp();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Excerpts of a specification (top) of a priority queue and its implementation (center and bottom) with a binary heap in Dafny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Thanks to the state abstraction function and the class invariant, the Dafny verifier is able to automatically check the conformity of the methods’ imple- 8 João Pascoal Faria and Rui Abreu mentation (defined in terms of the concrete state) against the methods’ pre and postconditons (defined in terms of the abstract state), without further burden from the user!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We only had to define an auxiliary lemma, showing that the heap invariant (indicated by the predicate Valid in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4) implies that the maximum is at the top (array index 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='5 Proof Techniques (Topological Sorting, Eulerian Circuit) Not surprisingly, simple algorithms may require complex proofs, as illustrated in the topological sorting case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In fact, the Kahn’s algorithm [16] can be encoded in just 6 lines of code (at a high level of abstraction), but, to prove its correctness, we had to write 7 auxiliary lemmas, sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Fortunately, Dafny supports a rich variety of proof techniques and is able to fill in most (if not all) of the proof steps, so we only had to provide key intermediate steps, making the handwritten proof of each lemma rather short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a non-empty acyclic graph must have at least one vertex without incoming edges (by contradiction) Topological sorting of an acyclic directed graph (Kahn’s algorithm) removing a vertex v from an acyclic graph G produces an acyclic graph (by contradiction) it is possible to generate a path of any length n in a non-empty graph G in which all vertices have incoming edges (by construction) given a path p in a non-empty graph G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' if the length of p exceeds the number of vertices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' then G has cycles (by deduction) the length of a sequence p of distinct elements from a set s cannot exceed the cardinality of the set (by induction) given a complex path p in a graph G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=" there exists a simple path (without repeated edges) in G from the first to the last vertex in the p (by induction) if there is a path from u to v in a graph G then a path from u to v also exists in any super-graph G' of G (by induction) Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Lemmas and proof techniques used to prove the correctness of Kahn’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, the way the proof steps are written may have a significant impact on the verification time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', in the Eulerian circuit case study, approximately 20 seconds were spent in the verification of a lemma stating that, if an Euler trail r exists in a graph G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', a path that traverses each edge of G exactly once), then each vertex of G has an even number of adjacent vertices, except for the first and last vertex in r in case they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The proof is done by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' By rewriting the inductive step so that the first edge is removed from r and G instead of the last one (possibly better matching the structure of recursive definitions needed in the proof), the verification time was reduced to less than 1 second!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 9 3 Results and Discussion In this section, we summarize the metrics collected and lessons learned from the case studies conducted, and draw some conclusions regarding our research goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 Metrics Collected Table 2 summarizes the metrics collected in the case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Size of the code categories described in Table 3 is measured in physical lines of code (LOC), ignoring blank lines and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The execution times were measured in an Intel(R) Core(TM) i7-8750H CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='20GHz laptop with 6 cores and 16 GB RAM running Windows 10 Enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We used v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 of the Dafny extension for VS Code and version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 of the Dafny server and, in some cases, version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 due to a bug with Z3 and Dafny v3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Results of the case studies (size, time and proof obligations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Program Impl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC Test LOC Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC Verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC Total LOC (S+V)/ (I+T) Proof Oblig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='Time (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=') Integer Division 10 5 2 2 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='27 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='5 Power of a Number 17 7 4 5 33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='38 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='5 Binary Search 15 7 7 3 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='45 51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='5 Insertion Sort 13 13 10 21 57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='19 90 1 Priority Queue 74 13 30 35 152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='75 483 3 Hash Set 86 16 57 38 197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='93 656 16 Tree Set 87 13 39 38 177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='77 809 18 Stable Marriage 50 66 54 10 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='55 209 7 Topological Sorting 19 18 21 94 152 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='11 157 3 Eulerian Circuit 32 10 66 115 223 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='31 407 19 Total 403 168 290 361 1222 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='14 2922 69 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Code categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Category Description Implemen- tation “Traditional”, compilable, implementation code (method signatures, method bodies, data definitions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Test Test code (checked statically or dynamically), including assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Specification Specification of contracts, including requires and ensures clauses, class invariants, frame conditions, and auxiliary definitions used in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Verification Verification helper code, such as, lemmas and all non-compilable code inside method bodies (loop variants, loop invariants, assertions, invo- cation of lemmas, manipulation of ghost variables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='com/dafny-lang/dafny/issues/1498 10 João Pascoal Faria and Rui Abreu Impl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC 33% Test LOC 14% Verif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC 29% Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' LOC 24% Code size (LOC) distribution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Code size (LOC) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' On average, the amount of code written for formal specification (S) and verification (V) purposes is of the same order of magnitude as the “traditional” code written for implementation (I) and testing (T) purposes – an “overhead” that certainly pays off, at least for high-integrity software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The average ratio is (S+V)/(I+T)=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='14, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='27 in the simplest case to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='31 in the most complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The pie chart of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 6 shows a balanced size distribution, on average, between the different code categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The overhead on user time is difficult to measure as it depends heavily on the user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' A fair assessment should be done in a different context (in the case studies, the algorithms were known, but the verification strategies had to be discovered in many cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We believe that, with proper training, in cases where new algorithms have to be designed, the specification and verification effort can be of the same order of magnitude as the design, implementation, and test effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The number of proof obligations (POs) generated and checked by the Dafny verifier is impressive, with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='4 POs generated on average per LOC written (2922 POs/1222 LOC in Table 2), and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 per implementation LOC (2922 POs/403 LOC in Table 2), in the case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' The performance of the Dafny verifier was also impressive, with 24 ms spent on average per PO generated and verified (69 sec/292 POs in Table 2), in this set of case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' However, based on the experience of the case studies, it is important to note that the verification of some POs may be significantly higher, in the order of minutes, or not even terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' When that happens, with careful debugging and refactoring (of assertions, verification code, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ), one may usually reduce the verification time drastically (as illustrated in the Euler Circuit case study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2 Lessons Learned The lessons learned from the case studies are summarized in Tables 4 and 5, using a color scheme to highlight strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Overall, the Dafny language and verifier proved to be very powerful, automating most of the ver- ification work, with minor language limitations (regarding generics, automatic contracts, and other aspects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Regarding our main research question, the major difficulty we found is that the manual verification work may be significant and difficult to predict and master in non-trivial programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 11 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Lessons learned from the case studies (Part I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Category Lessons learned (strengths and weaknesses) Dafny Lan- guage – Integrated language for writing specifications (methods’ pre and postconditions), implementations (methods’ bodies), and verifica- tion helper code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', loop invariants)[ex: Integer Division].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Rich set of logical quantifiers (forall, exists, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=') and mathe- matical collections (sequences, sets, multisets, maps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ), for writ- ing specifications and assertions and describing complex algorithms at a high level of abstraction [ex: Binary Search, Stable Marriage].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Inductive data types and pattern matching expressions may be used to keep the code at a high level of abstraction [ex: Hash Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Null safety: reference types are not nullable unless they are marked with the “?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' suffix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [ex: Tree Set] – Constructs to specify frame conditions and query the old object state, when working with mutable objects [ex: Insertion Sort].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Modules enable a clear separation between specification, implementa- tion, and test code [ex: Insertion Sort].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Limited support for generics: lack of support for type parameters that are subject to operations other than equality [ex: Binary Search].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – The support for explicitly separating specification and implementation and hiding implementation details in object-oriented programs has room for improvement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=', there are no visibility modifiers) [ex: Tree Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Dafny Com- piler – The Dafny compiler is able to generate executable code in multiple target languages (in this case, only C# is explored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Assertions and other constructs used for specification & verification pur- poses are not compiled, so they imply no runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Dafny Verifier – In many cases, the verifier is able to automatically check that the implementation conforms to the specification, with minimal user help (that may only have to write loop invariants) [ex: Integer Division].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Dafny is frequently able to discover loop variants [ex: Binary Search].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Outside of a method, the method body is opaque for verification pur- poses (only the pre and postconditions matter), making the verification process modular and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Manual Verifi- cation Work – Dafny effectively supports a rich variety of proof techniques (by de- duction, by induction, by contradiction, by construction, calcu- lational[17]) [ex: Topological Sorting, Tree Set] – Auxiliary properties may need to be defined by the user (as lemmas) to help the verifier, but the proof itself may be greatly or totally automated, with many details automatically filled in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' discovering what properties need to be defined is not trivial, though [ex: Power, Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – It is difficult to predict when and what manual work will be needed (beyond writing loop invariants) for a successful verification [ex: Insertion Sort, Topological Sorting].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 12 João Pascoal Faria and Rui Abreu Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Lessons learned from the case studies (Part II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Category Lessons learned (strengths and weaknesses) Auto- matic con- tracts – Dafny supports the definition and enforcement of class invariants, especially using the ”:autocontracts“ attribute, also taking care of the generation of appropriate frame conditions [ex: Priority Queue].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Automatic contracts have room for improvement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' in some cases, the user may need to resort to lower level features [ex: Tree Set, Hash Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Getting the contracts right in classes that represent self-referencing data structures may be rather tricky [ex: Tree Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – There are apparent conflicts between inheritance and automatic con- tracts [ex: Priority Queue].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' State Abstrac- tion – State abstraction functions (ghost functions) allow specifying the semantics (pre/postconditions) of the services provided by a class inde- pendently from the implementation (method bodies and internal state representation) [ex: Priority Queue].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – State abstraction may also be accomplished through abstract state variables (ghost variables), whose abstraction relation to the concrete state variables is specified in the class invariant [ex: Hash Set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Testing – Testing is still relevant, but mainly for statically testing the specifi- cation, and not dynamically testing the implementation (proved to be correct with respect to the specification) [ex: Integer division, Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Sort].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – Test cases that allow multiple outputs can be easily specified and checked [ex: Insertion Sort].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Debug- ging and Profiling – When verification fails, the Dafny language and the Dafny verifier pro- vide several convenient features for debugging purposes, such as the assume statement and the “/tracePOs” option [ex: Eulerian Circuit].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – When the verification time is high, most of the time may be concentrated on one or two assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' By identifying and rewriting such assertions, the verification time may be drastically reduced [ex: Eulerian Circuit].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 Accessibility assessment We distinguish three levels of competencies required for the development of ver- ified programs in Dafny, with decreasing accessibility: – basic: writing implementation and test code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – intermediate: writing specifications (pre/post-conditions, frame conditions, class invariants, and related predicates and functions), and loop variants and invariants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' – advanced: identifying and writing the needed verification code, besides loop variants and invariants (auxiliary lemmas, assertions, ghost variables, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Lessons learned and metrics collected in the case studies suggest that, even in seemingly simple problems, the user may need to be skilled in advanced veri- fication features and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 13 Hence, despite the impressive improvements in automated program verifi- cation provided by Dafny, we claim that “we are very close to, but not there yet” regarding the goal of making the development of verified programs acces- sible for software engineering practitioners and students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Further automation and systematization of verification tasks (including reusable libraries of common properties and “how to” guides), and integration in mainstream languages, are possible directions for further work in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Our assessment is corroborated by our experience in teaching a course on “Formal Methods in Software Engineering”8 with 151 master students enrolled in the 2020/21 academic year, with a very positive students feedback (average score of 6 out of 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Students with a high grade (≥ 85%) in a midterm exam were invited to develop a project in Dafny, consisting in the development of a verified implementation of an algorithm or data structure of medium complexity (hash set, tree set, stable marriage, topological sorting, Eulerian circuit, and text compression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Out of 28 students eligible, 14 picked the challenge, but only 9 delivered, and none met the goals fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We should note that the classes on formal specification and verification (4 hours per week during 6 weeks) only superficially addressed advanced verification techniques, and the students had a relatively short time to do the project (1 month).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' This experience led us to conclude that more advanced training is required to prepare interested students to handle non- trivial specification and verification problems using Dafny or similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 4 Related Work In [18], the authors report their experience of using Dafny at the VerifyThis 2021 program verification competition, which aims to evaluate the usability of logic-based program verification tools in a controlled experiment, challenging both the verification tools and the users of those tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They tackled two of the proposed challenges, and, as a result, identify strengths and weaknesses of Dafny in the verification of relatively complex algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Some strengths mentioned are: Dafny’s ability to prove termination and memory safety with little input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' built-in value types, such as sets, sequences, multisets, and maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicates and lemmas for more concise specifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' automatic induction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ghost variables and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They found it difficult to verify properties of possibly null objects, among other difficulties, impeding them from completing all the tasks on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In [19] the authors argue that formal verification tools are often developed by experts for experts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' as a result, their usability by programmers with little for- mal methods experience may be severely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They present their experiences with AutoProof (a tool that can verify the functional correctness of object- oriented software in Eiffel) in two contexts representative of non-expert usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' First, they discuss its usability by students in a graduate course on software verification, who were tasked with verifying implementations of various sorting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Second, they evaluate its usability in verifying code developed for 8 https://sigarra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='pt/feup/en/UCURR_GERAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='FICHA_UC_VIEW?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='pv_ocorrencia_i d=459493 14 João Pascoal Faria and Rui Abreu programming assignments of an undergraduate course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They report their experi- ences and lessons learned, from which they derive some suggestions for improving the usability of verification tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They report an average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 ratio between the number of tokens in specification and verification annotations and implemen- tation code, in two small programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In spite of the differences in context and measurement units, that ratio is of the same order of magnitude as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In [20] the authors refer that formal methods are often resisted by students due to perceived difficulty, mathematicity, and practical irrelevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They re- developed their software correctness course by taking a programming intensive approach, using Dafny to provide instant formative feedback via automated as- sessment, which resulted in increased student retention and course evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Although very positive overall, their students found Dafny difficult to learn and use, and the informal observations of the authors are that many of those diffi- culties stem from “accidental” complexity introduced by the Dafny tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' They propose some changes to Dafny’s design to tackle some issues found related to program testing, verification debugging, and class invariants, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 5 Conclusions and Future Work We conducted a set of case studies of developing verified implementations in Dafny of some real-world and well-known algorithms and data structures, with the goal of determining its accessibility for software engineering students, practi- tioners and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We concluded that, despite the impressive improvements in automated program verification provided by Dafny, the manual work needed in writing auxiliary verification code may be significant and difficult to predict and master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Further automation and systematization of verification tasks (including reusable libraries of common properties and “how to” guides), and integration in mainstream languages, are possible directions for further work in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' We also intend to conduct further studies with other verifiers and problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Acknowledgements This work is financed by National Funds through the Portuguese funding agency, FCT — Fundação para a Ciência e a Tecnologia within project EXPL/CCI- COM/1637/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' References [1] Barry Boehm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Some future trends and implications for systems and soft- ware engineering processes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Systems Engineering 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1– 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [2] Edsger Wybe Dijkstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Notes on structured programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [3] Watts S Humphrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Introduction to the team software process (sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Addison- Wesley Professional, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 15 [4] Charles Antony Richard Hoare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “An axiomatic basis for computer pro- gramming”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Communications of the ACM 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='10 (1969), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 576–580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [5] Moshe Y Vardi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “The automated-reasoning revolution: from theory to prac- tice and back”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Distinguished Lecture at NSF CISE, Spring (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [6] Leonardo de Moura and Nikolaj Bjørner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Z3: An efficient SMT solver”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' on Tools and Algorithms for the Construction and Analysis of Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 337–340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [7] K Rustan M Leino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Accessible software verification with Dafny”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: IEEE Software 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='6 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 94–97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [8] Pascal Cuoq et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Frama-c”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' on software engineering and formal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 233–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [9] Jean-Christophe Filliâtre and Andrei Paskevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Why3—where programs meet provers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: European symposium on programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 125–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [10] Rui Abreu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Using constraints to diagnose faulty spreadsheets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Software Quality Journal 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 297–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [11] Francisco Rebello de Andrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Specification-driven unit test genera- tion for java generic classes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' on Integrated Formal Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 296–311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [12] José Campos and Rui Abreu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Encoding test requirements as constraints for test suite minimization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: 2013 10th Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' on Information Tech- nology: New Generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 317–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [13] Alexander Diedrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Applying simulated annealing to problems in model-based diagnosis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Workshop on Principles of Diagnosis: DX- 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ARC-E-DAA-TN35662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ebook DX conference series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [14] Bruno Lima, João Pascoal Faria, and Robert Hierons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Local observability and controllability analysis and enforcement in distributed testing with time constraints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: IEEE Access 8 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 167172–167191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [15] Daniel Jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Software Abstractions: logic, language, and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' MIT press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [16] Arthur B Kahn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Topological sorting of large networks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Communica- tions of the ACM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='11 (1962), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 558–562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [17] K Rustan M Leino and Nadia Polikarpova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Verified calculations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Working Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' on Verified Software: Theories, Tools, and Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 170–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [18] Marie Farrell, Conor Reynolds, and Rosemary Monahan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “Using dafny to solve the VerifyThis 2021 challenges”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' of the 23rd ACM Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Workshop on Formal Techniques for Java-like Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 32–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [19] Carlo A Furia, Christopher M Poskitt, and Julian Tschannen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “The Auto- Proof verifier: Usability by non-experts and on standard code”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: arXiv preprint arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='03895 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' [20] James Noble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' “More Programming Than Programming: Teaching Formal Methods in a Software Engineering Programme”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In: NASA Formal Methods Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 431–450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 16 João Pascoal Faria and Rui Abreu A Code of the Case Studies A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='1 Integer Division /∗ ∗ The Dafny " Hello , World !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' " : a simple program f o r performing ∗ i n t e g e r d i v i s i o n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗/ // Computes the quotient ’q ’ and remainder ’ r ’ of the i n t e g e r // d i v i s i o n of a (non−negative ) dividend ’n ’ by a ( p o s i t i v e ) // d i v i s o r ’d ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method div (n : nat , d : nat ) returns (q : nat , r : nat ) r e q u i r e s d > 0 ensures q ∗ d + r == n && r < d { q := 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' r := n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while r >= d decreases r invariant q ∗ d + r == n { q := q + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' r := r − d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // Main program , with a simple t e s t case ( checked s t a t i c a l l y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ) method Main () { var q , r := div (15 , 6 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t q == 2 && r == 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' print "q = " , q , " r =", r , "\\n ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='2 Power of a Number /∗ ∗ Formal v e r i f i c a t i o n of an O( log n) algorithm to c a l c u l a t e ∗ the natural power of a r e a l number (x^n ) , i l l u s t r a t i n g the ∗ usage of lemmas and automatic induction in Dafny .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗/ // Recursive d e f i n i t i o n of x^n in f u n c t i o n a l style , // with time and space complexity O(n ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function power (x : real , n : nat ) : r e a l { i f n == 0 then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 e l s e x ∗ power (x , n−1) } Case studies of development of verified programs with Dafny 17 // Computation of x^n in time and space O( log n ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method powerDC(x : real , n : nat ) returns (p : r e a l ) ensures p == power (x , n) { i f n == 0 { return 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } e l s e i f n == 1 { return x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } e l s e i f n % 2 == 0 { productOfPowers (x , n/2 , n / 2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // r e c a l l lemma var temp := powerDC(x , n / 2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' return temp ∗ temp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } e l s e { productOfPowers (x , (n−1)/2 , (n−1)/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // r e c a l l lemma var temp := powerDC(x , (n−1)/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' return temp ∗ temp ∗ x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // States the property x^a ∗ x^b = x^(a+b ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // The property i s proved by automatic induction on ’a ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma {: induction a} productOfPowers (x : real , a : nat , b : nat ) ensures power (x , a ) ∗ power (x , b) == power (x , a + b) { } // A few t e s t cases ( checked s t a t i c a l l y by Dafny ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method testPowerDC () { var p1 := powerDC( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 , 5 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t p1 == 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var p2 := powerDC( −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 , 2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t p2 == 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var p3 := powerDC( −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 , 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t p3 == −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var p4 := powerDC( −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 , 0 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t p4 == 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var p5 := powerDC( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='0 , 0 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t p5 == 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='3 Binary Search /∗ ∗ Formal v e r i f i c a t i o n of the binary search algorithm with ∗ Dafny .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗/ type T = int // f o r demo purposes , but could be another type // Checks i f array ’a ’ i s sorted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 18 João Pascoal Faria and Rui Abreu predicate isSorted ( a : array) reads a { f o r a l l i , j : : 0 <= i < j < a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ==> a [ i ] <= a [ j ] } // Finds a value ’x ’ in a sorted array ’a ’ , and returns // i t s index , or −1 i f not found .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method binarySearch ( a : array, x : T) returns ( index : int ) r e q u i r e s isSorted ( a ) ensures (0 <= index < a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && a [ index ] == x) | | ( index == −1 && x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' in a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) { var low , high := 0 , a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while low < high invariant 0 <= low <= high <= a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length invariant x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' in a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' low ] && x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' in a [ high .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] { var mid := low + ( high − low ) / 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i f { case a [ mid ] < x => low := mid + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' case a [ mid ] > x => high := mid ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' case a [ mid ] == x => return mid ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } return −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Simple t e s t cases to check the post−condition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method testBinarySearch () { var a := new int [ 5 ] [ 1 , 4 , 4 , 6 , 8 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] == [ 1 , 4 , 4 , 6 , 8 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Proof helper var id1 := binarySearch (a , 6 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t id1 == 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var id2 := binarySearch (a , 3 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t id2 == −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var id3 := binarySearch (a , 4 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t id3 in {1 , 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='4 Insertion Sort /∗ ∗ Formal v e r i f i c a t i o n of the i n s e r t i o n sort algorithm ∗ with Dafny .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗/ // Contract f o r s o r t i n g algorithms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' abstract module Sorting { type T = int // f o r demo purposes , but could be another type Case studies of development of verified programs with Dafny 19 // Abstract method d e f i n i n g the contract ( semantics ) of // array s o r t i n g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method sort ( a : array) modifies a ensures isSorted ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) ensures multiset ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) == multiset ( old ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) ) // Auxiliary predicate that checks i f a sequence ’a ’ // i s sorted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate isSorted ( s : seq) { f o r a l l i , j : : 0 <= i < j < | s | ==> s [ i ] <= s [ j ] } } // S t a t i c t e s t s of the Sorting contract abstract module TestSorting { import opened Sorting method testSortSimple () { var a := new T [ ] [ 9 , 4 , 6 , 3 , 8 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] == [ 9 , 4 , 6 , 3 , 8 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // prover helper !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' sort ( a ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] == [ 3 , 4 , 6 , 8 , 9 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } method testSortWithDups () { var a := new T [ ] [ 9 , 3 , 6 , 9 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] == [ 9 , 3 , 6 , 9 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // prover helper sort ( a ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' SortingUniquenessProp ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] , [ 3 , 6 , 9 , 9 ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] == [ 3 , 6 , 9 , 9 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // a s s e r t i o n v i o l a t i o n ( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ) } // State and prove by induction the property that , i f two // sequences are sorted and have the same multiset of // elements , then they must be i d e n t i c a l ( so s o r t i n g has a // unique s o l u t i o n ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma SortingUniquenessProp ( a : seq, b : seq) r e q u i r e s isSorted ( a ) && isSorted (b) && multiset ( a ) == multiset (b) ensures a == b { // r e c a l l s u s e f u l p r o p e r t i e s about sequences and t h e i r // multisets seqProps ( a ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' seqProps (b ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // key steps of proof by induction on ’a ’ and ’b ’ // ( the r e s t i s f i l l e d in by Dafny ) i f | a | > 0 { SortingUniquenessProp ( a [ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] , b [ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 20 João Pascoal Faria and Rui Abreu } } // States two p r o p e r t i e s about sequences ( proved by Dafny // alone ) : // − sequence concatenation r e v e r t s s p l i t t i n g in head and // t a i l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // − elements of a sequence belong to i t s multiset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma seqProps ( a : seq) ensures | a | > 0 ==> a == [ a [ 0 ] ] + a [ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ensures f o r a l l i : : 0 <= i < | a | ==> a [ i ] in multiset ( a ) {} } module I n s e r t i o n S o r t r e f i n e s Sorting { // Sorts array ’a ’ using the i n s e r t i o n sort algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // I n h e r i t s the contract from Sorting .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method sort ( a : array) { f o r i := 0 to a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length invariant isSorted ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i ] ) invariant multiset ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) == multiset ( old ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) ) { var j := i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while j > 0 && a [ j −1] > a [ j ] invariant f o r a l l l , r : : 0 <= l < r <= i && r !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= j ==> a [ l ] <= a [ r ] invariant multiset ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) == multiset ( old ( a [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ] ) ) { a [ j −1] , a [ j ] := a [ j ] , a [ j −1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' //swap ( p a r a l l e l assign .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ) j := j − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } } } A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='5 Priority Queue /∗ ∗ Formal s p e c i f i c a t i o n and v e r i f i c a t i o n of a P r i o r i t y Queue ∗ implemented as a heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' A heap i s a p a r t i a l l y ordered set ∗ represented in an array , suited to implement p r i o r i t y ∗ queues operations i n s e r t and deleteMax in O( heapSize ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗ I l l u s t r a t e s the v e r i f i c a t i o n of object −oriented programs ∗/ type T = int // f o r demo purposes , but could be real , etc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 21 c l a s s {: autocontracts } PriorityQueue { // Concrete s t a t e representation var heap : array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var s i z e : nat ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Configuration parameters s t a t i c const i n i t i a l C a p a c i t y := 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Class invariant ( heap invariant + automatic things // generated by : autocontracts ) predicate Valid () { heapInv () } // Heap invariant predicate {: autocontracts f a l s e } heapInv () reads this , heap { // valid s i z e s i z e <= heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length // each node i s l e s s or equal than i t s parent && f o r a l l i : : 1 <= i < s i z e ==> heap [ i ] <= heap [ ( i −1)/2] } // State abstraction function : gets the heap contents as a // multiset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function elems ( ) : multiset { multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) } // I n i t i a l i z e s the heap as empty .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' constructor () ensures isEmpty () { heap := new T[ i n i t i a l C a p a c i t y ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e := 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Checks i f the heap i s empty predicate method isEmpty () ensures isEmpty () <==> elems () == multiset {} { // to help proving the post−condition a s s e r t elems () == multiset {} <==> | elems ( ) | == 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // actual expression s i z e == 0 } // I n s e r t s a value x in the heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' 22 João Pascoal Faria and Rui Abreu method i n s e r t (x : T) ensures elems () == old ( elems ( ) ) + multiset {x} { i f s i z e == heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length { grow ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Place at the bottom heap [ s i z e ] := x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e := s i z e + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Move up as needed in the heap heapifyUp ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Method used i n t e r n a l l y to grow the heap capacity method grow () r e q u i r e s s i z e == heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ensures heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length > s i z e ensures heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] == old ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) { var oldHeap := heap ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' heap := new T[ i f s i z e == 0 then i n i t i a l C a p a c i t y e l s e 2 ∗ s i z e ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' f o r a l l i | 0 <= i < oldHeap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length { heap [ i ] := oldHeap [ i ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // Auxiliary method to move a dirty node from the bottom // upwards in the heap method {: autocontracts f a l s e } heapifyUp () r e q u i r e s s i z e > 0 && heapifyUpInv ( size −1) modifies heap ensures heapInv () ensures multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ])== old ( multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) ) { var k := s i z e − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while k > 0 && heap [ k ] > heap [ ( k − 1) / 2] invariant 0 <= k < s i z e invariant heapifyUpInv (k) invariant multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) == old ( multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) ) { heap [ k ] , heap [ ( k−1) / 2] := heap [ ( k − 1) / 2 ] , heap [ k ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' k := (k − 1) / 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // During heapifyUp , while moving a node up at index k , // there are some d i f f e r e n c e s : Case studies of development of verified programs with Dafny 23 // children of k are sorted wrt parent of k , and k i s not // sorted wrt i t s parent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate {: autocontracts f a l s e } heapifyUpInv (k : nat ) reads this , heap { s i z e <= heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && ( f o r a l l i : : 1 <= i < s i z e && i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= k ==> heap [ i ] <= heap [ ( i − 1 ) / 2 ] ) && (k > 0 ==> f o r a l l i : : 1 <= i < s i z e && ( i −1)/2 == k ==> heap [ i ] <= heap [ ( ( i − 1)/2 − 1 ) / 2 ] ) } // Deletes and r e t r i e v e s the maximum value in the heap // ( assumed not empty ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method deleteMax () returns (x : T) r e q u i r e s !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' isEmpty () ensures isMax (x , old ( elems ( ) ) ) ensures elems () == old ( elems ( ) ) − multiset {x} { // r e c a l l the lemma .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' maxIsAtTop ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // pick the maximum from the top x := heap [ 0 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // reduce the s i z e s i z e := s i z e − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i f s i z e > 0 { // move l a s t element to top heap [ 0 ] := heap [ s i z e ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // move down as needed in the heap heapifyDown ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // Deletes and r e t r i e v e s the maximum value in the heap // ( assumed not empty ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method geteMax () returns (x : T) r e q u i r e s !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' isEmpty () ensures isMax (x , elems ( ) ) { maxIsAtTop ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' return heap [ 0 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Auxiliary predicate to check i f a value i s a maximum in // a multiset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate isMax (x : T, m: multiset ) { x in m && f o r a l l y : : y in m ==> y <= x } 24 João Pascoal Faria and Rui Abreu // Auxiliary method to move a dirty node from the top down // in the heap method {: autocontracts f a l s e } heapifyDown () r e q u i r e s s i z e > 0 && heapifyDownInv (0) modifies heap ensures heapInv () ensures multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) == old ( multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) ) { var k := 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' while true decreases s i z e − k invariant 0 <= k < s i z e invariant heapifyDownInv (k) invariant multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) == old ( multiset ( heap [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s i z e ] ) ) { var l e f t C h i l d := 2 ∗ k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // index of l e f t c h i l d var rightChild := 2 ∗ k + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i f l e f t C h i l d >= s i z e { return ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // reached the bottom } var maxChild := i f rightChild < s i z e && heap [ rightChild ] > heap [ l e f t C h i l d ] then rightChild e l s e l e f t C h i l d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i f heap [ k ] > heap [ maxChild ] { return ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // already sorted } // move up and continue heap [ k ] , heap [ maxChild ] := heap [ maxChild ] , heap [ k ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' k := maxChild ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } // During heapifyDown , while moving a node down at index k , // there are some d i f f e r e n c e s : // children of k are sorted wrt parent of k , and k i s not // sorted wrt i t s children .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate {: autocontracts f a l s e } heapifyDownInv (k : nat ) reads this , heap { s i z e <= heap .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && ( f o r a l l i : : 1 <= i < s i z e && ( i −1)/2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= k ==> heap [ i ] <= heap [ ( i − 1 ) / 2 ] ) && (k > 0 ==> f o r a l l i : : 1 <= i < s i z e && ( i −1)/2 == k ==> heap [ i ] <= heap [ ( ( i − 1)/2 − 1 ) / 2 ] ) } // Lemma s t a t i n g that the maximum i s at the top of the heap // ( p o s i t i o n 0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' This property i s assumed by deleteMax and Case studies of development of verified programs with Dafny 25 // f o l l o w s from the heap invariant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Proved by induction on the s i z e of the heap , reason why // i t r e c e i v e s a parameter with the s i z e to consider .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma {: induction n} maxIsAtTop (n : nat := s i z e ) r e q u i r e s n <= s i z e ensures f o r a l l i : : 0 <= i < n ==> heap [ i ] <= heap [ 0 ] {} } // A simple t e s t scenario .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method testPriorityQueue () { var h := new PriorityQueue ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' isEmpty ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i n s e r t ( 2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i n s e r t ( 5 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i n s e r t ( 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' i n s e r t ( 1 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var x := h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' deleteMax ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t x == 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' x := h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' deleteMax ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t x == 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' x := h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' deleteMax ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t x == 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' x := h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' deleteMax ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t x == 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' a s s e r t h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' isEmpty ( ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='6 Hash Set /∗ ∗ V e r i f i e d implementation of a hash set with open addressing ∗ and l i n e a r probing in Dafny .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗ Provides the fundamental set operations ( contains , insert , ∗ d e l e t e ) , s p e c i f i e d at an abstract l e v e l , r e s o r t i n g to an ∗ abstract s t a t e v a r i a b l e s ’ elems ’ with the set contents .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ∗/ // Datatype f o r the content of each c e l l of the hash table .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // I t s t o r e s a value of type T, Nil ( no value ) or Deleted // ( c e l l marked as deleted ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' datatype Cell = Nil | Deleted | Some( value : T) // Function type f o r hash functions type HashFunction = (T) −> nat // Represents a hash set of elements of type T ( comparable f o r // equality ) , i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' , a set stored in a hash table .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Uses the " autocontracts " a t t r i b u t e to automatically i n j e c t // c l a s s invariant checking and frame conditions // in methods ’ pre and post−conditions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' c l a s s {: autocontracts } HashSet { 26 João Pascoal Faria and Rui Abreu // Ghost v a r i a b l e ( abstract s t a t e v a r i a b l e ) used f o r // s p e c i f i c a t i o n purposes only .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ghost var elems : set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Concrete s t a t e v a r i a b l e with i n t e r n a l representation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var hashTable : array>;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Hash function to be used ( provided to the constructor ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' const hash : HashFunction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Number of p o s i t i o n s used ( with some value ) and marked as // deleted in the hash table .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var used : nat ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var deleted : nat ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // I n i t i a l capacity of the hash table .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' s t a t i c const i n i t i a l C a p a c i t y := 101;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // Ghost predicate that f o r m a l i z e s the c l a s s invariant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate Valid () { // Constraint that defin e the abstraction r e l a t i o n between // abstract and concrete s t a t e v a r i a b l e s elems == valSet ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ) // Constraints on the i n t e r n a l s t a t e representation && hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length > 0 && hashTableInv ( hashTable ) && used == | valSet ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ) | && deleted == | delSet ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ) | } // Ghost predicate that checks the consistency of a hash // table ’ t ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate {: autocontracts f a l s e } hashTableInv ( t : array>) reads t { f o r a l l i : : 0 <= i < t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Some?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' ==> validPos ( t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' value , i , t ) } // Ghost predicate that checks that ’ i ’ i s a valid p o s i t i o n // f o r value ’x ’ in hash table ’ t ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // ( ’ x ’ may be or not currently stored in that p o s i t i o n ) predicate {: autocontracts f a l s e } validPos (x : T, i : nat , t : array>) r e q u i r e s 0 <= i < t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length reads t { var h := hash (x) % t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Case studies of development of verified programs with Dafny 27 h == i | | (h < i && f o r a l l j : : h <= j < i ==> t [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Nil && t [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Some(x )) | | (h > i && f o r a l l j : : h <= j < t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length | | 0 <= j < i ==> t [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Nil && t [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Some(x )) } // Ghost function that r e t r i e v e s the set of values stored in // the f i r s t ’n ’ p o s i t i o n s of hash table ’ t ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function {: autocontracts f a l s e } valSet ( t : array>, n : nat ) : set r e q u i r e s 0 <= n <= t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length reads t { set i | 0 <= i < n && t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Some?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' : : t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' value } // Ghost function that r e t r i e v e s the set of p o s i t i o n s marked // as Deleted in the f i r s t ’n ’ p o s i t i o n s of hash table ’ t ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function {: autocontracts f a l s e } delSet ( t : array>, n : nat ) : set r e q u i r e s 0 <= n <= t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length reads t { set i | 0 <= i < n && t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Deleted ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Ghost function that r e t r i e v e s the set of p o s i t i o n s marked // as Nil in the f i r s t ’n ’ p o s i t i o n s of hash table ’ t ’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' function {: autocontracts f a l s e } n i l S e t ( t : array>, n : nat ) : set r e q u i r e s 0 <= n <= t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length reads t { set i | 0 <= i < n && t [ i ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Nil ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // Auxiliary lemma that s t a t e s the f o l l o w i n g property : the // sum of the s i z e s of valSet , delSet and n i l S e t // of a valid hash table i s equal to the length of the hash // table ( array ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // This i s true because the hash table invariant implies // that there are no duplicate values stored .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // The proof i s done by induction on the length of the table // ( omitting steps f i l l e d in by Dafny ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' lemma {: autocontracts f a l s e } countingLemma ( ht : array>, len : nat , v : nat , d : nat , n : nat ) r e q u i r e s 0 <= len <= ht .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length r e q u i r e s hashTableInv ( ht ) r e q u i r e s v == | valSet ( ht , len ) | && d == | delSet ( ht , len ) | && n == | n i l S e t ( ht , len ) | ensures v + d + n == len { i f len > 0 { 28 João Pascoal Faria and Rui Abreu var vs := valSet ( ht , len ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var ds := delSet ( ht , len ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var ns := n i l S e t ( ht , len ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var vs1 := valSet ( ht , len −1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var ds1 := delSet ( ht , len −1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var ns1 := n i l S e t ( ht , len −1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // r e c u r s i v e part countingLemma ( ht , len −1, | vs1 | , | ds1 | , | ns1 | ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // incremental part match ht [ len −1] { case Deleted => a s s e r t vs == vs1 && ds == ds1 + { len −1} && ns == ns1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' case Nil => a s s e r t vs == vs1 && ds == ds1 && ns == ns1 + { len −1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' case Some(x) => a s s e r t vs == vs1 + {x} && ds == ds1 && ns == ns1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } } // I n t e r n a l predicate that checks i f the hash table i s // ’ f u l l ’ , in the sense that a l l p o s i t i o n s are occupied with // a value or are marked as deleted ( i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' , there are no // p o s i t i o n s with Nil ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' In that case , i n s e r t i n g a new value // might not be p o s s i b l e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' predicate method f u l l () ensures f u l l () <==> n i l S e t ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length)=={} { // to help proving the post−condition ( equivalence ) : countingLemma ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length , used , deleted , | n i l S e t ( hashTable , hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ) | ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // the actual function value used + deleted == hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length } // Public method that checks i f t h i s set contains a value x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method contains (x : T) returns ( res : bool ) ensures res <==> x in elems { var pos := l o c a t e (x ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' return pos !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= −1 && hashTable [ pos ] == Some(x ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } // I n t e r n a l method that determines the l o c a t i o n ( ’ pos ’ ) f o r // a value ’x ’ ( e x i s t e n t or to be i n s e r t e d ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // I f such a l o c a t i o n cannot be found ( because the table i s // f u l l ) , returns −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' // In the case of a new value , t r i e s to reuse p o s i t i o n s // marked as deleted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' method l o c a t e (x : T) returns ( pos : int ) Case studies of development of verified programs with Dafny 29 r e q u i r e s Valid () ensures x in elems ==> 0 <= pos < hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && hashTable [ pos ] == Some(x) ensures x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' in elems ==> ( pos == −1 && f u l l ( ) ) | | (0 <= pos < hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length && !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' hashTable [ pos ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Some?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' && validPos (x , pos , hashTable )) { var h := hash (x) % hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' var reuse := −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' f o r i := h to hashTable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' Length invariant f o r a l l j : : h <= j < i ==> hashTable [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Nil && hashTable [ j ] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content='= Some(x) invariant reuse == −1 | | (h <= reuse < i && hashTable [ reuse ] == Deleted ) { i f hashTable [ i ] == Nil | | hashTable [ i ] == Some(x) { return i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } i f hashTable [ i ] == Deleted && reuse == −1 { reuse := i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE1T4oBgHgl3EQfgARW/content/2301.03224v1.pdf'} +page_content=' } } f o r i := 0 to h invariant f o r a l l j : : 0<=j 10−25 eV) but will probe fully non-linear scales. +Ref. [39] searched for axions as the only DM species in DES-Y1 galaxy shear data using +an axion halo model that analytically captures the effect of axions on the formation and +clustering of DM halos [70]. Ref. [71] extended this model to the case of mixed axion and +cold DM. A complementary approach is to capture non-linear modes using machine learning +models called emulators which are trained on the outputs of cosmological simulations [e. g., +72–79]. Emulators have been used successfully to set DM constraints, e. g., with the Lyman- +alpha forest [28, 31, 80], where astrophysical effects can be captured in training simulations. +Accurate emulator predictions rely on accurate input simulations. There is much progress +in our ability to simulate axion structure formation using fluid approximations [81] and by +solving the full axion field equations [32–34, 82–86]. +There are discrepancies between CMB, galaxy clustering and galaxy shear inferences on +the amplitude of matter density fluctuations [see 87, for a recent review]. This is typically +characterised by the matter clustering parameter σ8, the amplitude at redshift z = 0 when +averaged over 8 h−1 Mpc scales, or by the degenerate combination S8 ≡ +� +Ωm +0.3 σ8 (where Ωm +is the matter energy density), which is well constrained by large-scale structure experiments. +The statistical significance of the so-called S8 tension ranges from 2 to 3 σ depending on +the data considered; galaxy shear, in particular, drives the largest discrepancies with CMB +data. Notwithstanding undetected systematic errors in the data, the S8 tension has proposed +solutions based on physics beyond ΛCDM typically by introducing either a time-dependence +or a scale-dependence in the DM dynamics. This can be achieved by, e. g., coupling DM +to DE [88, 89], a complex dark sector (e. g., atomic DM [90–92]), decaying DM [93, 94], or +baryon-DM interactions [95]; Ref. [96] more generally considers modifications to non-linear +clustering including the effects of baryonic feedback. +Ultra-light axions form a component of the dark sector with a scale-dependent growth +factor. We therefore hypothesise that axions could alleviate the S8 tension, by behaving like +standard cold DM at the scales probed by current CMB surveys, while suppressing the growth +of structure at the smaller scales to which galaxy surveys are sensitive. We investigate this +hypothesis by jointly analysing CMB and galaxy clustering data. The inclusion of galaxy +shear measurements is left for future work. Another discrepancy in the ΛCDM model is the +H0 tension, the ∼ 5σ difference in the Hubble expansion rate today H0 as inferred from +different direct and indirect distance ladders [see, e. g., 87]. +Many proposed solutions to +the H0 tension based on new physics, however, exacerbate the discrepancy in S8 [e. g., 97]. +Models of ultra-light axions, with ma ∼ (10−27 − 10−26) eV, combined with modifications +to the dynamics of the DE component [98–100] are invoked to alleviate simultaneously both +parameter tensions. In this work, we stress the importance of assessing tension in the full +parameter space. In testing the extent to which ultra-light axions can improve consistency +between CMB and large-scale structure data, we therefore use metrics of tension that account +for the full non-Gaussian posterior distribution. +In § 2, we introduce our model for axion structure formation: the linear theory in § 2.1 +and the EFT of LSS that we use as our non-linear theory in § 2.2. We discuss our data in § 3: +CMB in § 3.1, baryon acoustic oscillations (BAO) and supernovae in § 3.2, full-shape BOSS +galaxy clustering in § 3.3 and our parameter inference methods in § 3.4. We present results +from the CMB, BAO and supernovae in § 4.1 and from BOSS galaxy clustering in § 4.2. In +– 3 – + +§ 5, we discuss these results and draw conclusions in § 6. +2 +Axion structure formation model +2.1 +Linear theory +2.1.1 +Axion cosmology +In order to model the effect of ultra-light axions (ULAs) on the cosmic microwave back- +ground (CMB), we calculate linear-order perturbations using the Einstein-Boltzmann solver +axionCAMB2 [22, 57]. The fundamental equation governing the axion field φ is the Klein- +Gordon equation: +□φ − m2 +aφ = 0, +(2.1) +where □ is the d’Alembert operator. We consider a temperature-independent axion mass, +which is appropriate for string theory axions, where the mass switches on at a high energy +scale (typically the geometric mean of the supersymmetry scale and the Planck scale [19]). +We ignore self-interactions of the axion (valid for initial field misalignment angles that are not +tuned close to π [101–104]). The axion-photon coupling can affect CMB polarisation if it is +large (see e. g., Refs. [105–108]), but does not back-react significantly on the axion DM density +(although see Ref. [109]). Cosmologically, all other axion couplings can lead only to a small +thermal population of axions, which is negligible for couplings consistent with astrophysical +limits, current constraints on the effective number of relativistic species Neff, and in the mass +range that we consider (for related discussion, see, e. g., Refs. [110, 111]). Thus, we set the +axion couplings to zero and consider only gravitational effects. The gravitational couplings +of the axion are contained in the metric dependence of □. +At early times, defined as when the Hubble expansion rate H ≫ ma, axionCAMB solves +fluid equations, equivalent to the full Klein-Gordon equation (Eq. (2.1)) at linear order in +spatial fluctuations of φ and metric perturbations, in the synchronous gauge. The homoge- +neous axion field begins to oscillate when H ≈ ma. After this time, axionCAMB uses the WKB +approximation to adopt an effective fluid description [112–115]. The fluid model is equivalent +to the Madelung formulation [116] and is accurate up to shell crossing. For further discussion +of the accuracy of the adopted approximations, see Refs. [22, 59, 60]. +The main physical features that distinguish ULAs from standard ΛCDM components, +as pertains to cosmological observables, are two-fold [22]. First, the slow roll of the axion +field when H ≫ ma leads to a distinctive background evolution equivalent to a fraction of the +matter component behaving like an early form of dark energy. This leads to differences in +the diffusion damping and Sachs-Wolfe contributions to the CMB, changes the sound horizon, +and changes the distance to the surface of last scattering (if axions begin their oscillation after +matter-radiation equality, i. e., for ma ≤ 10−28 eV). Second, the gradient terms in the Klein- +Gordon equation appear as an effective pressure, opposing gravitational collapse, leading to +a Jeans scale for the ULAs and, consequently, a suppression in the amplitude of density +perturbations on small scales [21, 113, 117, 118]. +2.1.2 +S8 and the linear matter power spectrum +Figure 1 illustrates this Jeans scale (for wavenumbers k above the Jeans wavenumber, the +linear matter power spectrum P linear(k) is suppressed relative to the ΛCDM limit) and how +2https://github.com/dgrin1/axionCAMB. +– 4 – + +−1.5 +−1.0 +−0.5 +0.0 +1.0 +1.5 +2.0 +2.5 +log +� +kP linear(k) +� +h−2 Mpc2�� +ΛCDM, S8 = 0.834 +ma = 10−28 eV, Ωah2 = 0.001, S8 = 0.811 +ma = 10−27 eV, Ωah2 = 0.002, S8 = 0.798 +ma = 10−26 eV, Ωah2 = 0.007, S8 = 0.773 +ma = 10−25 eV, Ωah2 = 0.033, S8 = 0.793 +ma = 10−24 eV, Ωah2 = 0.109, S8 = 0.828 +−1.5 +−1.0 +−0.5 +0.0 +log +� +k +� +h Mpc−1�� +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P linear +ULA DM(k) +P linear +ΛCDM(k) +S8 integral kernel +kPlanck +max +kBOSS +max +Figure 1. The effect of ultra-light axions on the linear matter power spectrum (top panel) and the +ratio to the ΛCDM limit (bottom panel). Power spectra are shown at the 95% upper limit on the +axion energy density Ωah2 given Planck CMB and BOSS galaxy clustering data and thus reflect the +tightening density constraint at lower axion mass ma (see Table 2; all parameters fixed apart from +ma, Ωah2, the cold DM density Ωch2 and the dark energy fraction ΩΛ). In the bottom panel, the +shaded area shows the Fourier-space filter k2W 2(k) (in arbitrary units) of kP linear(k) in the integral +calculation of the matter clumping factor S8 (see Eq. (2.2)), where W(k) is the Fourier transform of +a top-hat filter in real space with radius 8 h−1 Mpc and Ωm is the (fixed) total matter energy density. +The shaded area thus indicates the wavenumbers to which S8 is most sensitive; for ma ≤ 10−25 eV, +axions suppress power and thus lower S8; for ma ≥ 10−24 eV, the axion-induced power suppression +is at too large a wavenumber to change S8 significantly. The dashed line indicates the maximum +wavenumber which we probe in the Planck likelihood (see § 3.1.1); the dotted line indicates the +maximum wavenumber which we model in the BOSS galaxy power spectrum (see § 3.3). +– 5 – + +it depends on axion mass ma: the lighter the axion, the larger the power suppression scale +(the smaller the suppression wavenumber). In Fig. 1, we show linear matter power spectra +at the 95 % upper limits on the axion energy density Ωah2 given a combination of Planck +CMB and BOSS galaxy clustering data (see § 4.2). As will be expanded later, these data set +stronger constraints on the amount of axions at lower mass; Fig. 1 thereby illustrates how a +lower Ωah2 reduces the strength of the power suppression. The amplitude of the linear matter +power spectrum today (redshift z = 0) is often summarised by the cosmological parameter +σ8 = +� +dlnk k3 +2πW 2(k)P linear(k), +(2.2) +where W(k) is the Fourier transform of a top-hat filter in real space with radius 8 h−1 Mpc; +large-scale structure (LSS) data are then typically used to constrain the parameter combina- +tion S8 = +� +Ωm +0.3 σ8, where Ωm is the total matter energy density3. S8 is therefore sensitive to +a filtered integral of the linear matter power spectrum; the bottom panel of Fig. 1 shows this +filter and indicates the scales to which S8 is sensitive. It can be seen that, for ma ≥ 10−24 eV, +the power suppression is on too small scales to lower significantly S8. For ma ≤ 10−28 eV, +the data constraint is too strong to leave an appreciable amount of axions that significantly +lowers S8. However, there is a window for ma ∈ [10−27, 10−25] eV, where the presence of +axions significantly lowers S8 and is allowed by the data we consider. We therefore discuss +the prospects of axions resolving discrepancies in the inferred values of S8 given CMB and +LSS data in § 5. +2.1.3 +Cosmic microwave background +In modelling CMB anisotropies, we consider only adiabatic initial perturbations; we defer a +search for isocurvature perturbations to future work [see 36, 119, for consequences on the +energy scale of inflation]. Fig. 2 illustrates how axions change the CMB temperature TT, +polarisation EE, cross TE and lensing potential φφ angular power spectra Cℓ as a function +of multipole ℓ. For the multipoles probed by current data (Planck, ACT-DR4, SPT-3G; see +§ 3), the impact of axions for ma ≥ 10−25 eV on these data is small compared to statistical +uncertainties. +For axions that become a dark matter component before matter-radiation +equality (ma ≥ 10−27 eV), much of the data constraint comes from the change in the relative +heights of acoustic peaks arising from the change in the matter-to-radiation ratio (see Fig. 2). +For axions that still behave like dark energy after matter-radiation equality (ma ≤ 10−28 eV), +much of the data constraint (once the angular size of the sound horizon is constrained) +comes from the change in the integrated Sachs-Wolfe effect at the smallest multipoles. The +lensing power spectrum is sensitive to all axion masses through a scale (multipole)-dependent +suppression arising from the matter power spectrum (see Fig. 1; see Refs. [22, 36, 120] for +summaries of the effects of axions on the CMB). +Since we conservatively ignore small-scale CMB lensing anisotropies (we use only mul- +tipoles 8 ≤ L ≤ 400 as recommended by the Planck Collaboration; see § 3.1.1), we ignore +non-linear effects in the lensing power spectrum. Refs. [48, 71, 120] indicated that this is a +good approximation as, for axion mass ma < 10−25 eV, axions with observationally-allowed +3The parameter combination S8 was historically optimised to project away parameter degeneracies in +galaxy weak lensing experiments. +We typically use this parameter combination in this work with galaxy +clustering since it still does a good job of projecting away degeneracies and it simplifies comparisons to the +literature. +– 6 – + +0 +1000 +2000 +3000 +4000 +0 +2500 +5000 +ℓ(ℓ+1) +2π CTT +ℓ +� +µK2� +Max post (CMB+BAO+SNe): ma = 10−25 eV, Ωah2 = 0.03 +ma = 10−25 eV, Ωah2 = 0.12 +ma = 10−27 eV, Ωah2 = 0.12 +Planck +ACT-DR4 +SPT-3G +0 +1000 +2000 +3000 +4000 +−100 +0 +100 +ℓ(ℓ+1) +2π CTE +ℓ +� +µK2� +0 +1000 +2000 +3000 +4000 +0 +50 +ℓ(ℓ+1) +2π CEE +ℓ +� +µK2� +0 +100 +200 +300 +ℓ +0 +1 +107[ℓ(ℓ+1)]2 +2π +Cφφ +ℓ +Figure 2. The effect of ultra-light axions on (from top to bottom) the cosmic microwave background +(CMB) TT, TE, EE and φφ angular power spectra, compared to data from Planck (blue), the +Atacama Cosmology Telescope (ACT-DR4; red) and the South Pole Telescope (SPT-3G; orange). +We show the maximum posterior model (solid) given Planck, ACT and SPT CMB, galaxy baryon +acoustic oscillation (BAO) and supernovae (SNe) data for axion mass ma = 10−25 eV. We compare +this to the cases where the axion energy density Ωah2 = 0.12 for ma = 10−25, 10−27 eV (all other +parameters fixed to their maximum posterior values). At the multipoles currently probed, axions +for ma ≥ 10−25 eV are poorly constrained by these data; upcoming high-resolution CMB lensing +measurements will increase sensitivity to heavier axions. Data are shown as points with 68% c.l. +errorbars. +energy densities do not non-linearly cluster at observationally-relevant wavenumbers and red- +– 7 – + +200 +400 +600 +800 +1000 +L +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +∆Cφφ +L /Cφφ +L +ma = 10−25 eV +Halo model − linear +Planck +CMB − S4 +Figure 3. Fractional difference in the cosmic microwave background (CMB) lensing potential angular +power spectrum Cφφ +L +as a function of multipole L between a non-linear axion halo model [71] and the +linear theory prediction from axionCAMB [22, 57] (black line). We show the difference in the theoretical +prediction for an axion of mass ma = 10−25 eV which constitutes 50% of the total dark matter energy +density. There is a negligible difference for L ≤ 400 compared to the error in the Planck data that we +use (orange; see § 3.1.1). The inclusion of non-linearities will however be necessary for future CMB +surveys such as CMB-S4 (purple; forecasted data error from Ref. [36]). +shifts. For ma ≥ 10−25 eV, the non-linear effects are only significant for larger multipoles than +we use. A halo model was developed to capture the effects of ultra-light axions on non-linear +scales in CMB and galaxy lensing [71]. Using this halo model, we reconsider the impact of +ignoring non-linear effects in our Planck CMB lensing analysis (Fig. 3). We conclude that +this observable is well captured using linear theory for the L range considered in this work. +Forthcoming measurements of small-scale lensing anisotropies from ground-based CMB (and +future galaxy weak lensing) experiments will increase sensitivity to smaller axion suppression +scales and, hence, larger ma. We anticipate that non-linear modelling will become necessary +in this regime. +2.2 +Galaxy clustering and the effective field theory of large-scale structure +2.2.1 +Galaxy power spectrum and bispectrum multipoles +In order to capture the anisotropic clustering in the galaxy distribution arising from redshift- +space effects, we model galaxy power spectrum multipoles [e. g., 67]: +Pℓ(k, z) ≡ 2ℓ + 1 +2 +� 1 +−1 +dµ Lℓ(µ)Pg(k, µ, z). +(2.3) +Here, Pg(k, µ, z) is the full anisotropic galaxy power spectrum depending on wavenumber +k, the cosine of the angle between the wavenumber and the line-of-sight µ, and redshift z; +– 8 – + +Lℓ(µ) are Legendre polynomials indexed by multipole ℓ. Non-linear redshift-space distortions +(“fingers of God” [121]) are non-trivial to model with accuracy on small scales, while the power +suppression effect of axions is stronger as wavenumber increases. Therefore, to increase the +constraining power from galaxy data and following Ref. [122], we estimate the reconstructed +real-space4 galaxy power spectrum Q0(k, z) ≡ P0(k, z) − 1 +2P2(k, z) + 3 +8P4(k, z) [see also 123]. +Ref. [122] demonstrates that this estimator effectively down-weights information in line-of- +sight modes (µ > 0.3) that are heavily contaminated by redshift-space distortions. Further, +we extract information on the post-reconstructed baryon acoustic oscillation feature using the +Alcock-Paczynski (AP) parameters: +α∥(z) ≡ Hfid(z)rfid +s (zd) +H(z)rs(zd) +, α⊥(z) ≡ DA(z)rfid +s (zd) +Dfid +A (z)rs(zd) . +(2.4) +Here, H(z) is the Hubble parameter, rs(zd) is the sound horizon at the redshift of decoupling, +DA(z) is the angular diameter distance, and “fid” indicates a fiducial cosmology. For the +first time, in order to extract information beyond the two-point statistics described above, +we model the effect of axions on the galaxy bispectrum (Fourier transform of the three-point +correlation function). In this work, we consider only the angle-averaged bispectrum monopole +(ℓ = 0) B0(k1, k2, k3). +2.2.2 +Introduction to the effective field theory of large-scale structure +In order to model the effect of ULAs on the redshift-space galaxy power spectrum and bis- +pectrum, we calculate mildly non-linear perturbations using the effective field theory of +large-scale structure [EFT of LSS; 61–63, 68] as implemented in the Einstein-Boltzmann +solver CLASS-PT5 [67]. Following the effective field theory of large-scale structure, this is a +schematic view of our power spectrum model [65, 67]: +Pℓ(k, z) = P tree +ℓ +(k, z) + P one−loop +ℓ +(k, z) + P counterterms +ℓ +(k, z) + P stochastic +ℓ +(k, z). +(2.5) +Here, P tree +ℓ +(k, z) captures linear bias and redshift-space distortions (∝ P linear(k, z), which +is the linear matter power spectrum) [124]; P one−loop +ℓ +(k, z) captures perturbative corrections +up to one loop in order (∝ k2P linear(k, z) on large scales); P counterterms +ℓ +(k, z) captures ultra- +violet counterterms that consistently account for small-scale physics (∝ k2P linear(k, z)); and +P stochastic +ℓ +(k, z) captures stochastic effects including shot noise and fingers of God (∝ constant, +plus corrections). +We also include infrared resummation to account for non-perturbative +long-wavelength displacements [67, 125–127], and account for the so-called Alcock-Paczynski +distortion [128] (i. e., the effects of an incorrect fiducial cosmology above) by wavevector +rescalings [see e. g., 55, 104]. +The bispectrum model can be given schematically in 3D space [129]: +B(k1, k2, z) = Btree(k1, k2, z) + Bcounterterms(k1, k2, z) + Bstochastic(k1, k2, z), +(2.6) +for wavevectors k1, k2. Here, Btree(k1, k2, z) captures tree-level perturbations (∝ P 2 +linear(k, z)); +Bcounterterms(k1, k2, z) is a proxy for ultraviolet fingers-of-God counterterms (∝ k2 +∥P 2 +linear(k, z), +where k∥ is the wavenumber for modes parallel to the line of sight); and Bstochastic(k1, k2, z) +4I. e., without redshift-space distortions. +5https://github.com/Michalychforever/CLASS-PT. Background evolution and linear matter power spec- +trum calculations are done in axionCAMB, which are passed to CLASS-PT for non-linear corrections. +– 9 – + +captures stochastic effects (∝ Plinear(k, z) + constant). +For the bispectrum, the tree-level +model is sufficiently accurate for the small wavenumbers that we consider in our data (k < +0.08 h Mpc−1; see § 3.3). We again account for infrared resummation [127] and the Alcock- +Paczynski distortion as above. We then integrate over external angles to calculate the bis- +pectrum monopole B0(k1, k2, k3), which can be compared to data without additional window +convolutions. In comparing to BOSS data, we multiply B0 by a discreteness weight vector to +account for the finite resoltuion of the Fourier grid6. +2.2.3 +Axions and the effective field theory of large-scale structure +The EFT of LSS was originally developed with the assumption of cold, collisionless dark +matter (CDM). However, we follow Ref. [38] in noting that the effect of ultra-light axion +dark matter (not to cluster below a characteristic scale) is qualitatively the same as for +free-streaming neutrinos (although the physical reason is different). Ref. [69] found that, to +first order, the additional counterterms needed to account for the effect of neutrinos have the +same functional form as existing CDM counterterms. We therefore assume that the additional +axion-induced counterterms will also have the same functional form, although with different +constants of proportionality which must be marginalised. Further, Ref. [38] demonstrated +that linear-order axion-wave corrections are negligible since they manifest on scales that are +already heavily suppressed in the linear matter power spectrum. Ref. [38] also demonstrated +that axion and cosmological parameters can be inferred without bias from a simulated BOSS +galaxy catalogue in the presence of axions, when marginalising over the EFT of LSS model +presented above. It follows that phenomenology of axions can be captured by only modifying +the background evolution and linear matter power spectrum (presented in § 2.1 and calculated +using axionCAMB) as input to the EFT of LSS model presented in Eqs. (2.5) and (2.6). +We marginalise over a full set of EFT of LSS nuisance parameters: linear b1, quadratic +b2, tidal bG2 and third-order bΓ3 galaxy biases; monopole c0, quadrupole c2, hexadecapole c4, +fingers-of-God ˜c and bispectrum c1 counterterms; power spectrum Pshot and bispectrum Bshot +shot noise parameters; and power spectrum scale-dependent stochastic parameters a0 and a2 +[more details are given in 129]. +Figure 4 shows the effect of ultra-light axions on the galaxy power spectrum, for ma = +10−25 eV. The solid line shows the power spectrum for axion energy density Ωah2 = 0.03, +while the dashed line shows the power spectrum for Ωah2 = 0.09 (all other parameters fixed to +the same values as for the solid line). There are two main effects. The first is a scale-dependent +suppression in the power spectrum, which gets stronger on smaller scales (and so is most +significantly seen in the Q0 statistic). This is qualitatively similar to the effect in the linear +matter power spectrum. The effect is physically caused by axions not clustering on scales +below their Jeans wavelength at matter-radiation equality. The second effect is a small-scale +enhancement in the galaxy quadrupole (and, to a much-lesser extent, hexadecapole). This +effect is caused by a reduction in the fingers of God effect owing to lower peculiar velocities at +weaker matter over-densities, although this is degenerate with the EFT of LSS counterterm +parameter that controls the fingers of God amplitude. These effects are discussed in further +detail in Ref. [38]. The dotted lines lower the primordial power spectrum amplitude As with +respect to the solid lines and thus suppress the galaxy power spectrum at all wavenumbers. +Fig. 5 shows the effect of ultra-light axions on the galaxy bispectrum monopole, for +ma = 10−25 eV. As above, the dashed line shows the effect of increasing the axion density +6BOSS discreteness weight vectors can be found at https://github.com/oliverphilcox/full_shape_ +likelihoods. +– 10 – + +0.0 +0.1 +0.2 +0.3 +0.4 +k [h Mpc−1] +−1000 +0 +1000 +2000 +kPℓ(k) +� +h−2 Mpc2� +Monopole P0(k) +Quadrupole P2(k) +Hexadecapole P4(k) +Real-space Q0(k) +Max like (BOSS+Planck): Ωah2 = 0.03 +Ωah2 = 0.09 +As = 1.5 × 10−9 +Figure 4. The effect of ultra-light axions (mass ma = 10−25 eV, axion energy density Ωah2 = 0.09, +dashed lines) on the Baryon Oscillation Spectroscopic Survey (BOSS) galaxy power spectrum, com- +pared to maximum-likelihood model parameters (with Ωah2 = 0.03, solid lines; all other parameters +fixed to maximum-likelihood values). +The solid line shows the maximum likelihood given BOSS +galaxy power spectrum and Planck cosmic microwave background (CMB) data; here, we maximise +the likelihood with respect to all cosmological and EFT of LSS parameters, including those that are +usually analytically marginalised. We also show the case (dotted lines) where we lower the primor- +dial power spectrum amplitude from its best-fit value given Planck + BOSS (As = 2.15 × 10−9) to +its best-fit value given only BOSS galaxy power spectra (As = 1.53 × 10−9). This illustrates the +lack of degeneracy with heavier axions (ma = 10−25 eV); nonetheless, a good fit to BOSS data is +maintained given the addition of Planck data by reducing the best-fit linear galaxy bias (see also +Fig. 16). We anticipate degeneracy between high As and high Ωah2 for ma ≤ 10−28 eV since the large +Jeans scale suppresses all BOSS wavenumbers in a similar way as reducing As. We show the galaxy +power spectrum monopole P0(k) (blue), quadrupole P2(k) (red), hexadecapole P4(k) (orange), and +the reconstructed real-space galaxy power spectrum Q0(k) (green), as a function of wavenumber k. +BOSS data are shown as points with 68% c.l. errorbars. +while keeping all other parameters fixed. The effect is a small scale-dependent suppression +in the bispectrum, which gets stronger for smaller-scale triangles. However, at the current +statistical precision of BOSS data and on the relatively large scales modelled here in the +bispectrum (k < 0.08 h Mpc−1), the effect is negligible. We anticipate that axions will impact +the smaller-scale, one-loop bispectrum [130, 131] more strongly; we will investigate this in +future work. +– 11 – + +0 +10 +20 +30 +40 +50 +60 +Triangle index +−40 +−20 +0 +20 +40 +60 +80 +100 +10−4k1k2k3B0(k1, k2, k3) +� +h−3 Mpc3� +BOSS bispectrum data +Max post (BOSS+Planck): Ωah2 = 0.003 +Ωah2 = 0.09 +Figure 5. The effect of ultra-light axions (ma = 10−25 eV, Ωah2 = 0.09, dashed line) on the BOSS +galaxy bispectrum monopole B0(k1, k2, k3), compared to maximum-posterior model parameters (with +negligible axion densities, solid line; all other parameters fixed to their maximum posterior values). +The solid line shows the maximum posterior given BOSS power spectrum and bispectrum and Planck +CMB data. We show B0 as a function of k1, k2, k3 wavenumber triangles, where triangle index increases +first with k1, then with k2, and then with k3, for [k1, k2, k3] ∈ [0.01, 0.08] h Mpc−1; i. e., wavenumber +triangles with smaller sides on the left and larger sides on the right. BOSS data are shown as points +with 68% c.l. errorbars. +Data +Description +Nuisance pars. +§ +Refs. +Planck 2018 +CTT,TE,EE,φφ +ℓ +: ℓ ≤ 2508, L ≤ 400 +aPlanck +3.1.1 +[2] +ACT-DR4 +CTT,TE,EE +ℓ +: 326 ≤ ℓ ≤ 4325 +yp +3.1.2 +[53] +SPT-3G +CTE,EE +ℓ +: 300 ≤ ℓ ≤ 2999 +Fixed +3.1.3 +[54] +BAO + SNe +6dFGS, MGS, BOSS DR12, JLA +α, β, δM +3.2 +[132–135] +BOSS full-shape +P0,2,4(k, z), Q0(k, z), B0(k, z), AP +EFT of LSS +3.3 +[55] +Table 1. A summary of the data used in this work. +3 +Data +In Table 1, we give a summary of the data used in this work. In § 3.1 to 3.3, we give more +details about the data and, in § 3.4, we give details on our parameter inference method +including the prior distribution. +– 12 – + +3.1 +Cosmic microwave background +3.1.1 +Planck +We consider baseline Planck 2018 CMB temperature, polarisation and lensing angular +power spectra [2]. +We use: +the low-multipole (2 ≤ ℓ ≤ 29) temperature TT auto- +spectrum likelihood commander_dx12_v3_2_29; the low-multipole (2 ≤ ℓ ≤ 29) polari- +sation EE auto-spectrum likelihood simall_100x143_offlike5_EE_Aplanck_B; the high- +multipole, nuisance-marginalised, TT (30 ≤ ℓ ≤ 2508), TE and EE (30 ≤ ℓ ≤ 1996) +likelihood plik_lite_v22_TTTEEE; and the lensing φφ auto-spectrum likelihood (8 ≤ L ≤ +400) smicadx12_Dec5_ftl_mv2_ndlcpp_p_teb_consext8. As we use compressed, nuisance- +marginalised likelihoods, we have remaining a single nuisance calibration parameter aPlanck. +Ref. [22] demonstrated that there is no statistically-significant effect on axion parameter infer- +ence if re-marginalising nuisance foreground parameters in an axion model with Planck data. +The use of Planck 2018 data to constrain Ωah2 is an update from Refs. [36, 38], which used +Planck 2015 data [52], and from Ref. [39], which used Planck 2018 data to constrain only ma +in the case where axions are all the dark matter. The main differences from 2015 data are +a new low-ℓ polarisation likelihood and a larger-scale cut in the lensing likelihood, i. e., Lmin +goes from 40 to 8. We anticipate improved bounds on the lightest axions from this additional +large-scale information (see Fig. 7 for a breakdown of how 2018 data improves axion limits). +3.1.2 +Atacama Cosmology Telescope +We consider Atacama Cosmology Telescope (ACT) data release 4 (DR4) temperature and +polarisation angular power spectra [53]. We use the baseline nuisance-marginalised (“CMB- +only”) likelihood actpollite. +This includes TT power spectra for 576 ≤ ℓ ≤ 4325 and +TE and EE power spectra for 326 ≤ ℓ ≤ 4325. The foreground marginalisation leaves a +single nuisance parameter yp, which is an overall polarisation efficiency that re-scales the +TE and EE spectra. Our baseline analysis combines ACT and Planck (see § 3.1.1) data. +However, the cross-covariance between these data has not yet been released. +Therefore, +to reduce the amount of cross-covariance that we ignore, we follow Ref. [53] in setting the +minimum multipole in the ACT TT spectrum ℓmin = 1800, with no cut on the TE and EE +spectra. Ref. [53] found that these approximations are sufficient to keep the underestimation +of parameter uncertainties to less than 5%, including for one-parameter extensions of the +standard cosmological model. +3.1.3 +South Pole Telescope +We consider South Pole Telescope (SPT-3G) TE and EE angular power spectra for 300 ≤ ℓ ≤ +29997 [54]. We use the baseline spt3g_2020 likelihood that has twenty nuisance foreground +and calibration parameters8. In order to reduce the dimensionality of the parameter space, we +fix the nuisance parameters to fiducial values9. We confirm that fixing these parameters makes +no difference to the inferred cosmological posterior distribution by comparing to the case where +all nuisance parameters are marginalised. Our baseline analyses combine SPT with Planck +7In the latter stages of manuscript preparation, SPT-3G TT angular power spectra were released [136]; we +will include these data in a future analysis, although we do not anticipate a significant change to our results +since the multipoles contained in this release are already covered by the ACT data that we use (§ 3.1.2). +8We write a CosmoSIS [137] wrapper to the Cobaya [138] likelihood available at https://github.com/ +xgarrido/spt_likelihoods. +9We take fiducial values to be the maximum prior probability values from the fiducial SPT-3G analysis: +https://github.com/xgarrido/spt_likelihoods/blob/master/spt3g_2020/TEEE.yaml. +– 13 – + +(see § 3.1.1) and ACT (see § 3.1.2) data. We follow Ref. [54] in ignoring the cross-covariance +between SPT and Planck since the survey sky overlap is small; we ignore cross-covariance +between SPT and ACT for the same reason [see e. g., 139, for similar assumptions]. +3.2 +Baryon acoustic oscillations & supernovae +We consider a compendium of galaxy baryon acoustic oscillation (BAO) data from: the +6dF Galaxy Survey (6dFGS) at z = 0.106 [132]; the Sloan Digital Sky Survey data release +7 Main Galaxy Sample (SDSS DR7 MGS) at z = 0.15 [133]; and the Baryon Oscillation +Spectroscopic Survey data release 12 (BOSS DR12) at z = [0.38, 0.51, 0.61] [134]. +These +galaxy samples are largely independent. +In § 3.3, we will consider instead the full-shape +galaxy power spectrum and bispectrum as measured from the BOSS DR12 galaxy sample, +which captures the BOSS BAO information plus additional information in the full power +spectrum and bispectrum. In the data in § 3.3, the BAO information is extracted in the +Alcock-Paczynski parameters defined in Eq. (2.4) from the reconstructed power spectrum, +taking into account their covariance with the full-shape information [140]. We never combine +the full-shape BOSS likelihood in § 3.3 with the “standard” BOSS BAO likelihood (used in +this section) as they contain identical BAO information. +We consider a compendium of type Ia supernovae (SNe) data from the Joint Light-curve +Analysis (JLA) [135]. We marginalise over the shape parameter α, the colour parameter β +and the magnitude parameter δM. +3.3 +Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispec- +trum +We use the twelfth data release of the Baryon Oscillation Spectrosopic Survey (BOSS DR12) +[56, 134], which is part of the Sloan Digital Sky Survey (SDSS) [141]. +This data release +contains ∼ 8 × 105 galaxies across two redshift slices (LOWZ sample: 0.2 < z < 0.5; CMASS +sample: 0.5 < z < 0.75) and across both the north and south Galactic cap (NGC/SGC) +sky cuts. +We use the window-free galaxy power spectrum and bispectrum measurements +described in Ref. [55], which are measured respectively with the approaches of Refs. [142] and +[143]. We bin in k with ∆k = 0.005 h Mpc−1 for power spectra and ∆k = 0.01 h Mpc−1 for +bispectra, with minimum wavenumber kmin = 0.01 h Mpc−1 to avoid large-scale systematics +[e. g., 144]. As discussed in § 2.2, we fix the maximum wavenumber kmax = 0.2 h Mpc−1 for +the power spectrum (using the ℓ = 0, 2, 4 multipoles) Pℓ(k, z) [55, 145], kmax = 0.08 h Mpc−1 +for the bispectrum monopole [55, 129] B0(k, z), and we include the Q0(k, z) statistic for +k ∈ [0.2, 0.4] h Mpc−1 following Ref. [122]. We extract from the BOSS galaxy power spectrum +information on the post-reconstructed BAO feature using the AP parameters (Eq. (2.4)). +We use power spectrum, bispectrum and AP measurements for each of the four redshift/sky +cuts (NGC and SGC, both in redshift samples with central redshifts 0.38 and 0.61). The +BOSS power spectrum and bispectrum data for the North Galactic cap at z = 0.38 are +respectively shown in Figs. 4 and 5. These data are analysed with the EFT of LSS model +presented in § 2.2; we use the existing public BOSS likelihood10, with the covariance estimated +from 2048 MultiDark-Patchy simulations [146, 147]. The EFT of LSS nuisance parameters +(see § 2.2) are allowed to differ independently for each of the four BOSS data cuts due to +their different redshifts and calibrations. Only b1, b2 and bG2 enter the model non-linearly: +the others are analytically marginalized and only this partially-marginalised likelihood is +numerically sampled (see § 3.4 for details about our numerical sampling approach). +10https://github.com/oliverphilcox/full_shape_likelihoods. +– 14 – + +3.4 +Parameter inference +All the likelihoods presented in § 3.1 to 3.3 are implemented in the cosmological parameter +estimation code CosmoSIS [137]. We infer the posterior distribution for an axion cosmolog- +ical model with uniform prior distributions on: the Hubble parameter h; the baryon energy +density Ωbh2; the cold dark matter energy density Ωbh2; the axion energy density Ωah2; the +primordial power spectrum amplitude As; the primordial power spectrum tilt ns; and the +reionisation optical depth τ 11. We fix the neutrino energy density Ωνh2 = 0.0006442 with +one massive neutrino at its minimally-allowed mass; Refs. [71, 120, 148] discuss degeneracies +between axion and neutrino density parameters. We consistently calculate the helium abun- +dance given the baryon density and number of neutrinos using the bbn_consistency module +[149]. We often show the posterior for derived cosmological parameters: the total matter +energy density today Ωm (to which axions always contribute); and the matter clumping fac- +tor S8 ≡ +� +Ωm +0.3 σ8, where σ8 is the amplitude of the linear matter power spectrum averaged +over 8 h−1 Mpc scales. We do not vary the axion mass ma, but rather follow Refs. [36, 38] +by inferring the posterior for fixed values of ma ∈ [10−32 eV, 10−24 eV]. This is because the +full posterior projected in the ma - Ωah2 plane has a highly non-trivial degeneracy, which is +difficult to sample numerically in a converged manner. We defer solving this sampling prob- +lem to future work. We consider neither lighter axions as these are indistinguishable from a +cosmological constant, nor heavier axions as these are indistinguishable from cold dark matter +on the scales probed by the data we use [see 39, for more discussion and tests on axion prior +choices]. +We use a uniform prior on the ACT calibration parameter yp and the three supernovae +standardisation parameters [α, β, δM]. We use a Gaussian prior on the Planck calibration +parameter aPlanck ∼ N(1, 0.0025). For the EFT of LSS nuisance parameters (see § 2.2), we +use the following priors (from Ref. [55]): +b1 ∼ U(0, 4), +b2 ∼ N(0, 12), +bG2 ∼ N(0, 12), +bΓ3 ∼ N +�23 +42(b1 − 1), 12 +� +c0 +[h−1 Mpc]2 ∼ N(0, 302), +c2 +[h−1 Mpc]2 ∼ N(30, 302), +c4 +[h−1 Mpc]2 ∼ N(0, 302), +˜c +[h−1 Mpc]4 ∼ N(500, 5002), +c1 +[h−1 Mpc]2 ∼ N(0, 52), +Pshot ∼ N(0, ¯n−2), +Bshot ∼ N(1, ¯n−2), +a0 ∼ N(0, ¯n−2), +a2 ∼ N(0, ¯n−2), +(3.1) +where the inverse galaxy number density ¯n−1 = 5000 [h−1 Mpc]3 for the high-z samples and +¯n−1 = 3500 [h−1 Mpc]3 for the low-z samples. For discussion on these priors and a comparison +to the choices made in the PyBird implementation of the EFT of LSS model (whose parameters +are a linear combination of the above and which was used in a previous axion analysis [38]), +see § 5 and Refs. [150] and [151]. +We numerically sample posterior distributions using the importance nested sampling +algorithm MultiNest [152, 153]. We use 480 live points and we stop chains when posterior +weights reach a tolerance of 1% of their maximum, thus ensuring that we sample the bulk of +the posterior weight. We check that our chains are converged with respect to the number of +live points and tolerance by running test chains with 3600 live points and a tolerance of 10−5 +and determining no shift in inferred distributions. +11When considering BOSS data alone, we do not vary τ since large-scale structure data are insensitive to +this parameter. +– 15 – + +−32 +−30 +−28 +−26 +−24 +log [Axion mass ma (eV)] +−3.0 +−2.5 +−2.0 +−1.5 +−1.0 +log +� +Axion energy density Ωah2� +BOSS +Planck 2015 +Planck 2018 +Planck 2018 + BOSS +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +log +� +Ωah2 +ΩDMh2=0.12 +� +Figure 6. 95% credible upper limits on axion energy density Ωah2, as a function of axion mass ma, +as inferred: from BOSS galaxy clustering data (blue; see § 4.2); from Planck 2015 CMB data (red; +[36]); from Planck 2018 CMB data (orange; see § 4.1); and as jointly inferred from Planck 2018 CMB +and BOSS galaxy clustering data (green; see § 4.2). On the right-hand side, we show the 95% upper +limit on the ratio of the axion energy density to the best-fit dark matter (DM) energy density as +inferred from Planck in the ΛCDM model ΩDMh2 = 0.12. The black horizontal dashed and dotted +lines respectively indicate the energy densities at which axions form 10% and 1% of the DM today. +4 +Results +4.1 +Cosmic microwave background, baryon acoustic oscillations & supernovae +4.1.1 +Planck +We first search for ultra-light axions (ULAs) in baseline Planck 2018 CMB temperature, +polarisation and lensing data (see § 3.1.1 for a description of the data). We note that this is +an update from Refs. [36, 38] which considered older Planck 2015 data. We now have access +to a more robust measurement of the large-scale polarisation signal, and large-scale lensing +anisotropies not previously released (Lmin goes from 40 to 8). We anticipate improved bounds +on the lightest axions that we consider since their effect is strong on large scales. Fig. 6 shows +the 95% upper limit on the axion energy density allowed by Planck 2018 as a function of +mass (it also compares to joint constraints from Planck CMB and BOSS galaxy clustering +data and constraints from BOSS alone, which we discuss in § 4.2; these results are also +shown in Table 2). We see the typical “u”-shaped constraints [154] where axions in the “belly” +(10−30 eV ≤ ma ≤ 10−28 eV) are heavily constrained, but dark energy (DE)-like axions for +ma < 10−30 eV and dark matter (DM)-like axions for ma ≥ 10−27 eV can still be a significant +cosmological component. In particular, Planck data lose sensitivity for ma ≥ 10−25 eV as +– 16 – + +ma +Ωah2 (Planck) +S8 (Planck) +Ωah2 (Planck+BOSS) +S8 (Planck+BOSS) +ΛCDM +– +0.834+0.014 +−0.013 +– +0.827 ± 0.011 +10−24 eV +< 0.11399 +0.831 ± 0.014 +< 0.10858 +0.826+0.011 +−0.012 +10−25 eV +< 0.09667 +0.811+0.025 +−0.039 +< 0.03306 +0.818+0.015 +−0.017 +10−26 eV +< 0.00615 +0.819 ± 0.020 +< 0.00689 +0.804+0.020 +−0.024 +10−27 eV +< 0.00344 +0.822+0.016 +−0.020 +< 0.00181 +0.819+0.013 +−0.014 +10−28 eV +< 0.00163 +0.831+0.014 +−0.012 +< 0.00095 +0.824 ± 0.011 +10−29 eV +< 0.00136 +0.836 ± 0.014 +< 0.00097 +0.826 ± 0.011 +10−30 eV +< 0.00145 +0.837+0.014 +−0.013 +< 0.00099 +0.827 ± 0.011 +10−31 eV +< 0.00247 +0.838+0.015 +−0.014 +< 0.00140 +0.827 ± 0.011 +10−32 eV +< 0.00833 +0.843+0.019 +−0.016 +< 0.00321 +0.829+0.012 +−0.011 +Table 2. Constraints on axion energy density Ωah2 and the matter clumping factor S8, as a function +of axion mass ma (top to bottom), as inferred from Planck CMB data (left; see § 4.1) and as jointly +inferred from Planck CMB and BOSS galaxy clustering data (right; see § 4.2). For Ωah2, we give the +95% upper c.l.; for S8, we give the maximum marginalised posterior with the asymmetric 68% c.l. +the scale-dependent suppression is on scales smaller than those that Planck probes and the +background evolution is the same as ΛCDM deep into the radiation epoch. +Our Planck 2018 results are consistent with previous Planck 2015 limits [36] for ma ≥ +10−27 eV, but stronger for ma ≤ 10−28 eV (see Fig. 6). Fig. 7 investigates which parts of +the updated 2018 data are most important in improving axion constraints. In systematically +replacing parts of the 2015 likelihood12 with 2018 updates, we find that it is the inclusion +of the 2018 low-ℓ likelihood that accounts for the vast majority of the improvement in the +axion energy density bound. +The main difference at low ℓ in 2018 data, arising from an +analysis of high (electromagnetic) frequency polarisation modes, is a stronger and slightly +lower constraint on the reionisation optical depth τ thanks to measurement of the large-scale +reionisation bump in the EE power spectrum. +This τ measurement breaks degeneracies +with the primordial power spectrum amplitude As and the axion energy density Ωah2. We +show results for ma = 10−30 eV. These are indicative for all ma ≤ 10−28 eV, since the effect +of the lightest DE-like axions in CMB data is restricted to the largest scales (through the +integrated Sachs-Wolfe effect) that are degenerate with the primordial amplitude [22]. Hence, +the improved τ measurement does not improve axion constraints for heavier DM-like axions +(ma ≥ 10−27 eV), whose effect is restricted to smaller scales. +Figure 8 shows the Planck constraints on other cosmological parameters. DE-like axions +(ma < 10−27 eV) are consistent with lower values of h as they drive accelerated expansion +after matter-radiation equality [22]13. As DE-like axions have all started oscillating (and so +behave like DM) by today, they count towards the total matter energy density Ωm, but are +not degenerate with cold DM in the CMB. Thus, for ma ≤ 10−27 eV, larger values of Ωm +are allowed. This also drives compatibility with larger values of the matter clumping factor +12We consider the same Planck +2015 CMB likelihood as used in Ref. [36]: +the low-ℓ likelihood +lowl_SMW_70_dx11d_2014_10_03_v5c_Ap for 2 ≤ ℓ ≤ 29, the high-ℓ likelihood plik_lite_v18_TTTEEE for +30 ≤ ℓ ≤ 2508 (TT power spectrum) and 30 ≤ ℓ ≤ 1996 (TE and EE power spectra), and the lensing +likelihood smica_g30_ftl_full_pp for 40 ≤ L ≤ 400. +13We are considering different axion models than those that are typically invoked to increase h (and so +address the Hubble parameter tension). These so-called “early dark energy” axions are contrived to induce +a burst of accelerated expansion before recombination and typically require non-trivial axion potentials [see +e. g., 155]. +– 17 – + +Planck 2015 +Planck 2015 low-ℓ & lensing + 2018 high-ℓ +Planck 2015 lensing + 2018 low-ℓ & high-ℓ +Planck 2018 +2.0 +2.2 +2.4 +As [×10−9] +0.05 +0.10 +τ +2.5 +5.0 +Ωah2 [×10−3] +2.0 +2.2 +2.4 +As [×10−9] +2.5 +5.0 +Ωah2 [×10−3] +Figure 7. +The effect of updating from Planck 2015 (blue) to Planck 2018 CMB data on axion +constraints for ma = 10−30 eV. We systematically update parts of the 2015 data with 2018 results: +first the high-multipole ℓ likelihood (red), then also the low-ℓ likelihood (orange), and then finally +also the lensing likelihood (green). +We find that the vast majority of improvement in the axion +energy density bound comes from 2018 low-ℓ information, i. e., the measurement of the large-scale +reionisation bump breaks degeneracies between the reionisation optical depth τ, the primordial power +spectrum amplitude As and the axion energy density Ωah2. For each set, the inner and outer contours +respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, +with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded. +since S8 ∝ √Ωm. Conversely, DM-like axions (ma > 10−27 eV) are degenerate with cold +DM (CDM) in the CMB and so, in the high-mass limit where axions are poorly constrained, +the CDM density is also poorly constrained. Further, DM-like axions suppress the matter +power spectrum on scales below their de Broglie wavelength. Thus, when DM-like axions can +– 18 – + +Planck (ΛCDM) +h +Ωbh2 +Ωch2 +As +Planck (ma = 10−24 eV) +Planck (ma = 10−25 eV) +Planck (ma = 10−26 eV) +Planck (ma = 10−27 eV) +Planck (ma = 10−28 eV) +Planck (ma = 10−29 eV) +Planck (ma = 10−30 eV) +Planck (ma = 10−31 eV) +Planck (ma = 10−32 eV) +0.64 +0.66 +0.68 0.0222 +0.0224 +0.0226 +0.08 +0.10 +0.12 +2.10 +2.15 +×10−9 +Planck (ΛCDM) +ns +τ +Ωm +S8 +aPlanck +Planck (ma = 10−24 eV) +Planck (ma = 10−25 eV) +Planck (ma = 10−26 eV) +Planck (ma = 10−27 eV) +Planck (ma = 10−28 eV) +Planck (ma = 10−29 eV) +Planck (ma = 10−30 eV) +Planck (ma = 10−31 eV) +Planck (ma = 10−32 eV) +0.96 +0.97 +0.05 +0.06 +0.325 +0.350 +0.375 +0.80 +0.85 +1.0000 +1.0025 +Figure 8. The effect of axion mass ma on cosmological parameter constraints from Planck CMB +data. We see how dark energy-like axions (ma < 10−27 eV) have degeneracy with lower values of +the Hubble parameter h, while dark matter (DM)-like axions (ma ≥ 10−27 eV) have degeneracy with +the cold DM density Ωch2 and lower values of the matter clumping factor S8. These degeneracies +are explored further in Fig. 9. Each point indicates the maximum marginalised posterior, while the +errorbar indicates the marginalised 68% c.l. As is in units of 10−9. +comprise a significant fraction (≳ 2%) of the total DM budget (10−27 eV ≤ ma ≤ 10−25 eV), +Planck data are compatible with lower values of S8 than in the ΛCDM model, since S8 +integrates over lower-amplitude modes. For ma > 10−25 eV, the power spectrum suppression +is on scales smaller than those to which the S8 parameter is most sensitive and so ΛCDM +values of S8 are returned. This suggests that axions with ma ∈ [10−27, 10−25] eV could help +to resolve the so-called S8 tension by bringing CMB data into compatibility with the lower S8 +values inferred from large-scale structure data. Fig. 9 explicitly illustrates that it is degeneracy +with the axion energy density that allows lower values of h and higher values of Ωm for DE-like +axions (ma ∼ 10−30 eV), and lower values of S8 for DM-like axions (ma ∼ [10−26−10−25] eV). +Although axions with ma = 10−25 eV can comprise the dark matter according to Planck +data, there is no preference for such a model compared to ΛCDM according to the Bayesian +evidence. The log-ratio of model evidences (or Bayes factor) given Planck data is 1.8 in favour +of ΛCDM (see Table 3). This amounts to “positive” evidence in favour of ΛCDM according +to the Jeffreys scale as given by Ref. [156]. This lack of preference for extended cosmological +models is consistent with previous studies [e. g., 157], and there is no improvement in the +maximum likelihood (or minimum chi-squared). +– 19 – + +Planck CMB (ΛCDM) +Planck CMB (ma = 10−24 eV) +Planck CMB (ma = 10−25 eV) +Planck CMB (ma = 10−26 eV) +Planck CMB (ma = 10−30 eV) +0.32 +0.36 +Ωm +0.650 +0.675 +h +0.72 +0.80 +0.88 +S8 +10−3 +10−1 +Ωah2 +0.32 +0.36 +Ωm +0.650 +0.675 +h +10−3 +10−1 +Ωah2 +Figure 9. The effect of ultra-light axions on the matter clumping factor S8, matter energy density Ωm +and Hubble parameter h, inferred from Planck, as a function of axion mass ma. Dark matter (DM)- +like axions for ma ∈ [10−26, 10−25] eV give lower S8 values by a scale-dependent power spectrum +suppression, while dark energy-like axions (e. g., ma = 10−30 eV) give lower h values by causing +accelerated expansion after matter-radiation equality. DM-like axions with ma = 10−24 eV have a +negligible effect on S8 as the power spectrum suppression is on scales smaller than those to which +S8 is sensitive. For each set, the inner and outer contours respectively indicate the 68% and 95% +credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on +the diagonal, where 68% credible regions are shaded. +4.1.2 +All CMB, BAO & supernovae +For the first time in a ULA search, we consider the addition of higher-resolution CMB data +from the ACT and SPT experiments. We defer a systematic search of the axion mass param- +eter space to future work, in anticipation of upcoming high-resolution lensing data. In this +– 20 – + +Data +ma +Bayes factor relative to ΛCDM +Planck +10−25 eV +-1.8 +Planck + ACT-DR4 +-0.6 +Planck + SPT-3G +-1.8 +All CMB + BAO + SNe +-0.4 +10−24 eV +-1.5 +10−25 eV +-2.6 +10−26 eV +-1.6 +10−27 eV +-4.0 +Planck + BOSS +10−28 eV +-3.1 +10−29 eV +-3.1 +10−30 eV +-3.2 +10−31 eV +-2.5 +10−32 eV +-2.6 +Table 3. Bayes factor (log-ratio of model evidences; right column) for the indicated data (left column) +given the indicated axion model (middle column) relative to the ΛCDM model. For all the data +combinations shown, the Bayesian evidence favours the ΛCDM model; although we find that axions +can improve consistency between datasets (see § 4.2), there is no preference for an extension beyond +ΛCDM given these data. +Data +S8 (ΛCDM) +Ωah2 (ma = 10−25 eV) +S8 (ma = 10−25 eV) +Planck +0.834+0.014 +−0.013 +< 0.09667 +0.811+0.025 +−0.039 +Planck + ACT-DR4 +0.835+0.013 +−0.012 +< 0.10745 +0.789+0.027 +−0.041 +Planck + SPT-3G +0.828+0.014 +−0.011 +< 0.10580 +0.799+0.027 +−0.046 +All CMB + BAO + SNe +0.827 ± 0.010 +< 0.10610 +0.774+0.032 +−0.037 +Table 4. Constraints on axion energy density Ωah2 and the matter clumping factor S8, for different +CMB, galaxy BAO and supernovae data combinations (see § 3.1 for a description of the data). For +Ωah2, we give the 95% upper c.l.; for S8, we give the maximum marginalised posterior with the +asymmetric 68% c.l. +study, we focus on the impact of current ACT (see § 3.1.2) and SPT (see § 3.1.3) data (and +a compendium of low-z galaxy BAO and supernovae; see § 3.2) on DM-like axions that most +significantly increase compatibility with low values of S8: ma = 10−25 eV. Fig. 10 illustrates +the effect on the S8 - Ωm - Ωah2 planes from adding these data to the Planck data considered +above. The posterior shifts with respect to Planck alone are small. There is a ∼ 0.5σ decrease +in Ωm when adding BAO and SNe data (∼ 1σ decrease seen in the ΛCDM case; see Fig. 11). +In particular, the axion energy density bounds at ma = 10−25 eV are slightly weakened with +the addition of these data (see also Table 4). Correspondingly, there is a shift to even lower +values of S8 driven by its parameter degeneracy with Ωah2. This weakening of constraints +is consistent with previous searches for massive neutrinos in high-resolution CMB data [e. g., +53]. Similarly to neutrinos, DM-like axions are constrained in primary CMB anisotropy power +spectra through the lensing-induced smoothing of acoustic peaks. Here, gravitational lensing +by lower-redshift (mostly z < 2) large-scale structure dampens the amplitude of peaks in +angular power spectra. It follows that the amount of lensing-induced smoothing is sensitive +to the presence of ultra-light axions or neutrinos which suppress the growth of structure and +thus reduce the amount of smoothing. The amount of lensing relative to the best-fit ΛCDM +– 21 – + +Planck (ma = 10−25 eV) +Planck + ACT-DR4 (ma = 10−25 eV) +Planck + SPT-3G (ma = 10−25 eV) +All CMB + BAO + SNe (ma = 10−25 eV) +0.30 +0.32 +0.34 +Ωm +0.72 +0.80 +S8 +0.05 +0.10 +Ωah2 +0.30 +0.32 +0.34 +Ωm +0.05 +0.10 +Ωah2 +Figure 10. The effect of current higher-resolution CMB data (Planck and ACT-DR4 in red; Planck +and SPT-3G in orange), galaxy baryon acoustic oscillations (BAO) and supernovae (SNe) (all com- +bined with Planck in green; Planck only in blue) on axion constraints for ma = 10−25 eV. For each +set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D +marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% +credible regions are shaded. From left to right, S8 is the matter clumping factor, Ωm is the matter +energy density and Ωah2 is the physical axion energy density. +expectation is quantified by the multiplicative correction to the theoretical expectation AL. +In particular, both ACT-DR4 (AL = 1.01 ± 0.11) [53] and SPT-3G (AL = 0.98 ± 0.12) [54] +prefer lower values of AL compared to Planck (AL = 1.180 ± 0.065) [2]. This means that +when adding ACT or SPT data to Planck, constraints on models that suppress structure +and lower the lensing signal are weakened, e. g., massive neutrinos [53] or ultra-light axions. +Fig. 11 shows marginalised constraints on all other cosmological parameters, also comparing +– 22 – + +Planck (ΛCDM) +Ωah2 +h +Ωbh2 +Ωch2 +As +Planck (ULADM) +Planck+ACT-DR4 (ΛCDM) +Planck+ACT-DR4 (ULADM) +Planck+SPT-3G (ΛCDM) +Planck+SPT-3G (ULADM) +CMB+BAO+SNe (ΛCDM) +CMB+BAO+SNe (ULADM) +0.00 +0.05 +0.67 +0.68 +0.0222 +0.0224 +0.0226 +0.05 +0.10 +2.10 +2.15 +×10−9 +Planck (ΛCDM) +ns +τ +Ωm +S8 +aPlanck +Planck (ULADM) +Planck+ACT-DR4 (ΛCDM) +Planck+ACT-DR4 (ULADM) +Planck+SPT-3G (ΛCDM) +Planck+SPT-3G (ULADM) +CMB+BAO+SNe (ΛCDM) +CMB+BAO+SNe (ULADM) +0.96 +0.97 +0.05 +0.06 +0.07 +0.31 +0.32 +0.33 +0.75 +0.80 +0.85 +1.000 +1.005 +Figure 11. The effect of current higher-resolution CMB data (ACT-DR4, SPT-3G), galaxy BAO and +supernovae (SNe) on ultra-light axion and cosmological constraints for ma = 10−25 eV (ULA DM), +also comparing to ΛCDM and Planck-only constraints. Each point indicates the marginalised mean, +while the errorbar indicates the marginalised 68% c.l. As is in units of 10−9. +to the ΛCDM case. We see the typical degeneracy for DM-like axions with standard cold DM +meaning that weakened constraints on Ωah2 lead to correspondingly-weakened constraints on +Ωch2. We note a ∼ 1σ increase in the Planck calibration parameter aPlanck when adding ACT +data which is seen in both ΛCDM and axion models. Similarly as for Planck data, there is no +preference given these combined datasets for an axion model compared to ΛCDM according +to the Bayesian evidence (see Table 3). The Bayes factors amount to evidence in favour of +ΛCDM that ranges from “positive” to “not worth more than a bare mention” according to the +Jeffreys scale [156]. +We demonstrate above that, in an axion model with ma = 10−25 eV, the combination +of Planck, ACT-DR4 and SPT-3G CMB, galaxy BAO and supernovae data are compatible +with lower values of the matter clumping factor (S8 = 0.774+0.032 +−0.037) than in ΛCDM (S8 = +0.827 ± 0.010). Fig. 12 compares this result to fiducial ΛCDM constraints from combined +galaxy weak lensing and clustering (3 × 2) data. We consider ΛCDM constraints (with fixed +neutrino energy density) from the combination of galaxy clustering, galaxy lensing shear +and galaxy – galaxy lensing two-point correlation functions (3 × 2) as measured by the Dark +Energy Survey (DES) [158]14. We also consider ΛCDM constraints (with fixed neutrino energy +density) from the same combination of three × two-point correlation functions as measured +by the Kilo-Degree Survey (KiDS) [159], which includes redshift-space galaxy clustering data +from BOSS [160] and galaxy – galaxy lensing data from the survey overlap between KiDS, +BOSS and the spectroscopic 2-degree Field Lensing Survey (2dFLenS) [161]15. We note that +the KiDS 3 × 2 constraints are therefore not entirely independent of the CMB + BAO + SNe +14This is the publicly-released posterior chain chain_3x2pt_fixednu_lcdm. +15This is the publicly-released posterior chain samples_multinest_blindC_EE_nE_w. +– 23 – + +All CMB + BAO + SNe (ΛCDM) +All CMB + BAO + SNe (ma = 10−25 eV) +DES-Y3 3 × 2 (ΛCDM) +KiDS 3 × 2 (ΛCDM) +0.24 +0.32 +0.40 +Ωm +0.72 +0.78 +0.84 +S8 +0.04 +0.08 +Ωah2 +0.24 +0.32 +0.40 +Ωm +0.04 +0.08 +Ωah2 +Figure 12. Comparison of CMB (Planck, ACT-DR4, SPT-3G), galaxy BAO and supernovae (SNe) +constraints with fiducial galaxy weak lensing and clustering (3 × 2) ΛCDM constraints from the Dark +Energy Survey (DES) and the Kilo-Degree Survey (KiDS) (all with fixed neutrino mass). In ΛCDM, +CMB, BAO and SNe data prefer systematically higher values of the matter clumping factor S8 than is +inferred from fiducial 3 × 2 analyses. When axions of ma = 10−25 eV contribute to the energy budget +with energy density Ωah2, CMB, BAO and SNe data are consistent with lower values of S8. In order +to assess consistency between data in an axion model, it is necessary to re-analyse the 3 × 2 data in +the presence of axions; in § 4.2, we consider the first part with galaxy clustering from BOSS. For each +set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D +marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% +credible regions are shaded. From left to right, S8 is the matter clumping factor, Ωm is the matter +energy density and Ωah2 is the physical axion energy density. +compendium we consider, since part of the BAO measurements we use is derived from the +same BOSS data (see § 3.2) as goes into the KiDS 3 × 2 measurement. However, we note +– 24 – + +that the addition of BAO and SNe data makes only a small difference to the S8 constraint +from Planck + ACT and Planck + SPT (see e. g., Fig. 10). We anticipate that, in § 4.2, +we will consider the full-shape galaxy clustering power spectrum from BOSS, which will be +much more significantly correlated with the KiDS 3×2 analysis. Despite this proviso, Fig. 12 +illustrates how, in the ΛCDM model, 3 × 2 analyses from both DES (S8 = 0.783 ± 0.020) +and KiDS (S8 = 0.765±0.017) prefer systematically lower values of S8 than the compendium +of CMB, BAO and SNe data (S8 = 0.827 ± 0.010). This is a manifestation of the so-called +“S8 tension”, where many galaxy clustering, weak lensing and galaxy cluster observations +prefer lower values of S8 than is inferred from CMB observations, with statistical significance +ranging from 2 to 3 σ depending on the data comparison [see e. g., 87, for a recent review]. +However, when axions of ma = 10−25 eV contribute to the energy budget, the CMB, BAO +and SNe compendium is compatible with the low S8 values preferred by DES and KiDS in the +ΛCDM model. We therefore hypothesise that axions could resolve the S8 tension. In order +to assess this, we must reanalyse the 3 × 2 data in the axion model. In this work, we consider +the first part of this in analysing full-shape galaxy clustering information from BOSS. We +present these results in § 4.2. +4.2 +Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispec- +trum +We now consider the effect on ultra-light axion constraints from the galaxy power spectrum +and bispectrum as measured from the Baryon Oscillation Spectroscopic Survey (BOSS; see +§ 3.3 for a description of the data). +4.2.1 +ΛCDM +Before studying the combination of Planck CMB and BOSS galaxy clustering data, we assess +constraints independently from each dataset. Fig. 13 shows ΛCDM cosmological constraints +from the BOSS galaxy power spectrum only (P0, P2, P4, Q0 and the post-reconstructed BAO +Alcock-Paczynski parameters), the BOSS galaxy power spectrum and bispectrum monopole +(additionally B0), and Planck CMB data (previously shown in § 4.1). +In particular, we +consider BOSS constraints without a prior on the baryon energy density Ωbh2 or any other +cosmological parameters. It is striking how much more constraining is Planck data on the +full cosmological model than BOSS data alone. However, as is typical of large-scale structure +experiments, BOSS provides more competitive constraints when projected onto the plane of +derived parameters Ωm and S8. +4.2.2 +Parameter tension metrics +In order to assess consistency between datasets in their cosmological constraints, we consider +three metrics of parameter tension (the difference in S8 only, the difference in the S8 - Ωm +plane, and the difference in the full posterior distribution). We now describe these metrics in +more detail. The first metric, which is most widely quoted in the literature, is the discrepancy +in the marginalised S8 constraint from two datasets (labelled 1 and 2), defined as +∆S8 +σS8 += +µ1 − µ2 +� +σ2 +1 + σ2 +2 +. +(4.1) +Here, µi and σi are respectively the parameter posterior mean and standard deviation given +experiment i. This metric is given in the third column of Table 5. We also consider a second +– 25 – + +BOSS galaxy power spectrum (ΛCDM) +BOSS galaxy power spectrum + bispectrum (ΛCDM) +Planck cosmic microwave background (ΛCDM) +0.02 +0.03 +Ωbh2 +0.12 +0.18 +Ωch2 +1.2 +1.8 +2.4 +As +0.8 +1.0 +ns +0.30 +0.35 +Ωm +0.60 +0.75 +h +0.60 +0.75 +0.90 +S8 +0.02 +0.03 +Ωbh2 +0.12 +0.18 +Ωch2 +1.2 +1.8 +2.4 +As +0.8 +1.0 +ns +0.30 +0.35 +Ωm +0.60 +0.75 +0.90 +S8 +Figure 13. Comparison of BOSS galaxy clustering and Planck CMB constraints on ΛCDM cosmo- +logical parameters. BOSS data alone (in particular without an Ωbh2 prior) are much less constraining +than Planck data on the standard cosmological model. For each set, the darker and lighter shaded +contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior +distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are +shaded. As is in units of 10−9. +metric, which is an extension of Eq. (4.1) to higher dimensions: +χ2 = (µ1 − µ2)T(C1 + C2)−1(µ1 − µ2). +(4.2) +Here, µi is now the vector of parameter posterior means and Ci is the posterior covariance, +both given experiment i. We then calculate the probability p to exceed χ2 (for a χ2 distri- +bution with degrees of freedom equal to the number of parameters) and convert this to a +– 26 – + +number N of σ using the standard Gaussian interpretation16. +Both Eqs. (4.1) and (4.2) are good measures of parameter discrepancy in the limit of +Gaussian posterior distributions. We therefore give, in the fourth column of Table 5, the +metric defined in Eq. (4.2) evaluated for the marginalised posterior in the S8 − Ωm plane. +In this plane, the BOSS data are most constraining and the distribution is reasonably Gaus- +sian. It is important nonetheless also to consider consistency in the full set of parameters +constrained by both datasets. However, the full BOSS posterior distribution appears highly +non-Gaussian and so the metrics defined above will not be a good measure of consistency in +the full parameter space. We therefore elect to calculate the full posterior distribution of the +parameter difference ∆θ (marginalised over the parameters θ) [162]: +P(∆θ) = +� +dθP1(θ)P2(θ − ∆θ). +(4.3) +Here, Pi(θ) is the posterior distribution given experiment i. +We can then calculate the +significance of the inferred parameter shift (relative to none) by integrating P(∆θ) above +the iso-probability contour that goes through ∆θ = 0 (this probability to exceed can be +converted to a number of σ as above)17. In this way, this third tension metric accounts for non- +Gaussianities in the parameter posterior distribution. We therefore give, in the final column +of Table 5, the metric derived from Eq. (4.3) as evaluated in the volume of all parameters +constrained by both Planck and BOSS [h, Ωbh2, Ωch2, As, ns, Ωm, S8 and Ωah2 when part of +the model]. +There are many proposed approaches to evaluating parameter consistency in high- +dimensional and non-Gaussian distributions. Although these different approaches tend to +agree in terms of trend (i. e., they typically agree with respect to an increasing or decreasing +tension) [see e. g., 163], they typically disagree as to the particular value of tension. We there- +fore urge caution when interpreting Table 5 that it is most useful as a measure of relative +tension given different models. All three metrics considered in Table 5 (and Fig. 13) illus- +trate that the addition of BOSS bispectrum data B0 increases the discrepancy with respect to +Planck mostly by preferring slightly lower values of the primordial power spectrum amplitude +As. This in turn pushes S8 to slightly lower values. +4.2.3 +BOSS-only axion constraints +Figure 14 shows the same set of posterior contours as in Fig. 13 but for an axion model with +ma = 10−25 eV. Although BOSS is less constraining than Planck on ΛCDM parameters, it +is significantly more constraining on the axion energy density. This is driven by the addition +of smaller-scale data in the reconstructed real-space galaxy power spectrum Q0 (see Fig. 19 +and discussion below). However, since Planck alone is unconstraining on the axion energy +density at this mass (see also § 4.1), it is more consistent with the lower values of S8 that +BOSS (and indeed other large-scale structure experiments) prefer. This means that there +is more posterior overlap in the S8 - Ωm plane and this is reflected in the improved tension +metrics in Table 5. Notably, the tension in S8 when comparing Planck to full BOSS data is +reduced from 2.70 σ (ΛCDM) to 1.63 σ (for ma = 10−25 eV). However, there is no degeneracy +between Ωah2 and As at ma = 10−25 eV. This is because Ωah2 largely affects the small-scale +power spectrum, while As is constrained by the overall normalisation at all wavenumbers (see +16N = +√ +2 erf−1(p). +17We numerically evaluate this integral using the tensiometer package: https://github.com/mraveri/ +tensiometer. +– 27 – + +BOSS galaxy power spectrum (ma = 10−25 eV) +BOSS galaxy power spectrum + bispectrum (ma = 10−25 eV) +Planck cosmic microwave background (ma = 10−25 eV) +0.60 +0.75 +h +0.015 +0.030 +Ωbh2 +0.08 +0.16 +Ωch2 +1.2 +1.8 +2.4 +As +0.8 +1.0 +ns +0.30 +0.35 +Ωm +0.05 +0.10 +Ωah2 +0.60 +0.75 +S8 +0.60 +0.75 +h +0.015 +0.030 +Ωbh2 +0.08 +0.16 +Ωch2 +1.2 +1.8 +2.4 +As +0.8 +1.0 +ns +0.30 +0.35 +Ωm +0.60 +0.75 +S8 +Figure 14. +Comparison of BOSS galaxy clustering and Planck CMB constraints on axion and +cosmological parameters, for axion mass ma = 10−25 eV. BOSS data alone are more constraining +than Planck data on axion energy density Ωah2 since BOSS probes smaller scales (k < 0.4 h Mpc−1); +in the extended axion model, there is more posterior overlap in the S8 - Ωm plane than in ΛCDM (see +Fig. 13). For each set, the darker and lighter shaded contours respectively indicate the 68% and 95% +credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on +the diagonal, where 68% credible regions are shaded. As is in units of 10−9. +Fig. 4). This means there is no improvement in the As discrepancy between Planck and BOSS +even in the presence of axions at ma = 10−25 eV and this is reflected in the full parameter +space tension metric given in Table 5. Indeed, when not including bispectrum data, the full +parameter tension increases slightly. +In Fig. 6, we show the 95 % upper limits on Ωah2 derived from BOSS data across the +full mass range that we consider (see also Table 6). For ma < 10−25 eV, the BOSS-only +– 28 – + +Data +Model +S8 (σ) +S8 − Ωm (σ) +All parameters (σ) +Planck, BOSS [no B0] +ΛCDM +2.1 +2.02 +1.77 +ma = 10−25 eV +1.32 +1.14 +2.14 +Planck, BOSS +ΛCDM +2.70 +2.82 +4.36 +ma = 10−25 eV +1.63 +1.57 +3.70 +ma = 10−26 eV +3.63 +3.81 +5.38 +ma = 10−27 eV +2.28 +2.11 +3.63 +ma = 10−28 eV +1.78 +1.76 +3.31 +ma = 10−29 eV +1.74 +2.44 +3.19 +ma = 10−30 eV +2.22 +2.82 +4.11 +ma = 10−31 eV +2.24 +2.73 +2.95 +ma = 10−32 eV +2.58 +2.78 +3.19 +Table 5. Discrepancy in parameters (given in the top row) as inferred from the two datasets given +in the first column, for the model given in the second column. The third column is the discrepancy in +the marginalised S8 constraint, and the fourth column is the discrepancy in the marginalised S8 −Ωm +plane, both with the reasonable approximation of a Gaussian posterior distribution. The final column +is the discrepancy in the marginalised constraint on all cosmological (and axion) parameters, where +we account for non-Gaussianity by calculating the full parameter difference posterior. The full details +of the tension metrics that we use are given in § 4.2. +ma +Ωah2 (BOSS) +S8 (BOSS) +ΛCDM +– +0.723+0.041 +−0.037 +10−24 eV +< 0.15539 +0.718+0.038 +−0.039 +10−25 eV +< 0.04174 +0.709+0.043 +−0.037 +10−26 eV +< 0.01717 +0.653 ± 0.040 +10−27 eV +< 0.00542 +0.719+0.040 +−0.038 +10−28 eV +< 0.00842 +0.742+0.050 +−0.040 +10−29 eV +< 0.02259 +0.759+0.044 +−0.043 +10−30 eV +< 0.02771 +0.745+0.041 +−0.040 +10−31 eV +< 0.02706 +0.744+0.040 +−0.042 +10−32 eV +< 0.03126 +0.737+0.040 +−0.038 +Table 6. Constraints on axion energy density Ωah2 and the matter clumping factor S8, as a function +of axion mass ma (top to bottom), as inferred from BOSS galaxy clustering data. For Ωah2, we give +the 95% upper c.l.; for S8, we give the maximum marginalised posterior with the asymmetric 68% +c.l. For consistency with other masses, at ma = 10−26 eV, we give the upper limit on the axion +density; nonetheless, Ωah2 = 0 is disfavoured at ∼ 2.7σ, i. e., the maximum marginalised posterior +Ωah2 = 0.0100+0.0048 +−0.0037. +constraints are weaker than Planck, although the BOSS data are crucial in strengthening the +combined CMB and galaxy clustering limit at nearly all masses (see below). Nonetheless, we +see the typical “u”-shaped constraints (that we see with CMB data) also given BOSS alone: +at higher mass, BOSS loses sensitivity since the scale-dependent suppression manifests at +larger wavenumbers than those we model in BOSS (crucially, BOSS probes smaller scales +than Planck and so we have improved sensitivity for ma = 10−25 eV); at lower mass, BOSS +loses sensitivity owing to degeneracy with As. The degeneracy with As for ma ≤ 10−28 eV is +illustrated in Fig. 15, where we show how axions impact BOSS constraints on all cosmological +– 29 – + +BOSS (ΛCDM) +Ωah2 +h +Ωbh2 +Ωch2 +As +BOSS (ma = 10−24 eV) +BOSS (ma = 10−25 eV) +BOSS (ma = 10−26 eV) +BOSS (ma = 10−27 eV) +BOSS (ma = 10−28 eV) +BOSS (ma = 10−29 eV) +BOSS (ma = 10−30 eV) +BOSS (ma = 10−31 eV) +BOSS (ma = 10−32 eV) +0.00 +0.05 +0.10 +0.7 +0.8 +0.02 +0.03 +0.05 +0.10 +0.15 +1.0 +1.5 +2.0 +×10−9 +BOSS (ΛCDM) +ns +Ωm +S8 +b1 +BOSS (ma = 10−24 eV) +BOSS (ma = 10−25 eV) +BOSS (ma = 10−26 eV) +BOSS (ma = 10−27 eV) +BOSS (ma = 10−28 eV) +BOSS (ma = 10−29 eV) +BOSS (ma = 10−30 eV) +BOSS (ma = 10−31 eV) +BOSS (ma = 10−32 eV) +0.8 +1.0 +0.325 +0.350 +0.375 +0.6 +0.7 +0.8 +2.2 +2.4 +2.6 +Figure 15. The effect of axion mass ma on cosmological parameter constraints from BOSS galaxy +clustering data. We stress that even the ΛCDM constraints differ from those reported in Ref. [55] as, +in this work, we do not use a Big Bang nucleosynthesis (BBN) prior on the baryon energy density +Ωbh2. Each point indicates the marginalised mean, while the errorbar indicates the marginalised 68% +c.l. As is in units of 10−9; b1 is the linear galaxy bias at z = 0.61 in the north Galactic cap (NGC; +similar values are found in all four redshift/sky samples). +parameters. This degeneracy arises at low mass since the axion Jeans wavenumber is then +smaller than the smallest wavenumber that we model in BOSS. This means that axions +suppress all BOSS wavenumbers, which is degenerate with lowering As and so lowering the +overall power amplitude. BOSS data are therefore compatible with higher values of S8 (driven +by higher As and also higher Ωm) than for ΛCDM for ma ≤ 10−28 eV (the effects of higher As +and Ωah2 do not cancel perfectly at the scales to which S8 is sensitive). This drives an increase +in compatibility between BOSS and Planck around ma ∼ 10−29 eV, including (unlike with +heavier axions) with regards to the As discrepancy (see Table 5). Beyond degeneracy with +As, we also see the typical degeneracy with Ωch2 for (heavier) DM-like axions and degeneracy +with Ωm for (lighter) DE-like axions since they additionally count as matter by today. Fig. 15 +also reveals that at ma = 10−26 eV, rather than an upper limit on the axion density, BOSS +data alone disfavour no axions at ∼ 2.7σ significance; we discuss this in more detail below +(see Fig. 18 and surrounding discussion). +– 30 – + +BOSS galaxy power spectrum (ma = 10−25 eV) +Planck (ma = 10−25 eV) +Planck + BOSS galaxy power spectrum (ma = 10−25 eV) +0.30 +0.35 +Ωm +0.60 +0.75 +S8 +0.05 +0.10 +Ωah2 +2.0 +2.4 +2.8 +b1(z = 0.61; NGC) +0.30 +0.35 +Ωm +0.60 +0.75 +S8 +2.0 +2.4 +2.8 +b1(z = 0.61; NGC) +Figure 16. Comparison of BOSS galaxy power spectrum (blue), Planck CMB (orange) and joint +(black) constraints on axion and cosmological parameters, for axion mass ma = 10−25 eV. +The +strongest bound on the axion energy density Ωah2 comes from combining the datasets; in order to +maintain a good fit to the galaxy data in the joint constraint, lower (though still physically plausible) +values of the linear galaxy bias b1 are preferred. For each set, the darker and lighter shaded contours +respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, +with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded. From +left to right, Ωah2 is the physical axion energy density, Ωm is the matter energy density, S8 is the +matter clumping factor and b1(z = 0.61; NGC) is the linear galaxy bias at redshift z = 0.61 in the +north Galactic cap (NGC; similar values are found in all four redshift/sky samples). +4.2.4 +Joint Planck and BOSS axion constraints +In Fig. 16, we show the joint constraint from Planck and the BOSS galaxy power spectrum +on axions for ma = 10−25 eV. The strongest limit on the axion energy density comes from +– 31 – + +Planck (ΛCDM) +h +Ωbh2 +Ωch2 +As +Planck+BOSS (ΛCDM) +Planck+BOSS (ma = 10−24 eV) +Planck+BOSS (ma = 10−25 eV) +Planck+BOSS (ma = 10−26 eV) +Planck+BOSS (ma = 10−27 eV) +Planck+BOSS (ma = 10−28 eV) +Planck+BOSS (ma = 10−29 eV) +Planck+BOSS (ma = 10−30 eV) +Planck+BOSS (ma = 10−31 eV) +Planck+BOSS (ma = 10−32 eV) +0.67 +0.68 0.0222 +0.0224 +0.0226 +0.11 +0.12 +2.075 +2.100 +2.125 +2.150 +×10−9 +Planck (ΛCDM) +ns +τ +Ωm +S8 +b1 +Planck+BOSS (ΛCDM) +Planck+BOSS (ma = 10−24 eV) +Planck+BOSS (ma = 10−25 eV) +Planck+BOSS (ma = 10−26 eV) +Planck+BOSS (ma = 10−27 eV) +Planck+BOSS (ma = 10−28 eV) +Planck+BOSS (ma = 10−29 eV) +Planck+BOSS (ma = 10−30 eV) +Planck+BOSS (ma = 10−31 eV) +Planck+BOSS (ma = 10−32 eV) +0.96 +0.97 +0.05 +0.06 +0.31 +0.32 +0.33 +0.80 +0.85 +2.00 +2.05 +2.10 +Figure 17. The effect of axion mass ma on cosmological parameter constraints from the joint inference +of Planck CMB and BOSS galaxy clustering data, and a comparison to the Planck ΛCDM inference. +Since Planck is much more constraining than BOSS alone on ΛCDM cosmological parameters, the +joint constraints on these parameters are broadly consistent with the Planck ΛCDM case. Each point +indicates the marginalised mean, while the errorbar indicates the marginalised 68% c.l. As is in units +of 10−9; b1 is the linear galaxy bias at z = 0.61 in the north Galactic cap (NGC; similar values are +found in all four redshift/sky samples). The Ωch2 constraint at ma = 10−24 eV extends to 0.05; we +zoom-in for clarity at other masses. +combining the datasets. Since Planck is significantly more constraining than BOSS alone on +ΛCDM parameters, the joint constraint on those parameters is largely driven by Planck (see +also Fig. 17). BOSS (and galaxy clustering data in general) are constraining on a degenerate +combination b1S8 of the power spectrum amplitude S8 and the linear galaxy bias b1, since this +combination scales the large-scale galaxy power spectrum (see § 2.2; although this degeneracy +is partly broken by the quadrupole’s sensitivity to fσ8, where f is the growth rate). +It +follows that, in the joint constraint, since Planck drives higher values of the power spectrum +amplitude (even in the presence of axions) that a good fit to BOSS data is maintained by +preferring a lower value of b1. This is illustrated in Fig. 16, where the joint constraint on b1 +is lower than for BOSS alone (moving along the b1S8 degeneracy), but still has a value b1 ∼ 2 +that is consistent with previous findings. This behaviour is observed at other axion masses +and in the ΛCDM case (see Fig. 17). +Figure 6 shows the joint limit from Planck and BOSS on the axion energy density +across the full axion mass range to which we are sensitive (10−32 eV ≤ ma ≤ 10−24 eV; +– 32 – + +BOSS (ma = 10−26 eV) +Planck (ma = 10−26 eV) +Planck + BOSS (ma = 10−26 eV) +0.32 +0.36 +Ωm +0.60 +0.75 +S8 +0.01 +0.02 +Ωah2 +0.8 +1.0 +ns +0.32 +0.36 +Ωm +0.60 +0.75 +S8 +0.8 +1.0 +ns +Figure 18. Comparison of BOSS galaxy clustering (all data; blue), Planck CMB (orange) and joint +(black) constraints on axion and cosmological parameters, for axion mass ma = 10−26 eV. Although +BOSS data give a hint of a significant axion energy density at this mass, Planck data disfavour +this scenario. The consequence is that the joint axion constraint is weaker than for Planck data +alone. However, we note that, unlike axions at other masses, axions with ma = 10−26 eV increase the +discrepancy between Planck and BOSS data with respect to ΛCDM (see Table 5) and so the joint +constraint should be considered with caution. For each set, the darker and lighter shaded contours +respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, +with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded. From +left to right, Ωah2 is the physical axion energy density, Ωm is the matter energy density, S8 is the +matter clumping factor and ns is the primordial power spectrum tilt. +see also Table 2). +At nearly all masses, the strongest bound comes from combining the +datasets. Fig. 17 shows the joint constraints on the other cosmological parameters and the +– 33 – + +linear galaxy bias. As discussed above (Fig. 16), since Planck is much more constraining on +ΛCDM parameters, the joint Planck + BOSS constraints on these parameters is largely driven +by Planck. Nonetheless, we note the typical degeneracy of Ωah2 with, for DE-like axions, lower +values of h and higher values of Ωm, and for DM-like axions, with lower values of Ωch2 (see +also Planck data in § 4.1). BOSS, in general, strengthens the limit on the amount of axions. +However, in the DM-like mass range to which we are sensitive (10−27 eV ≤ ma ≤ 10−25 eV), +the joint bound leaves enough axions still to drive consistency with lower values of S8 (see +also Table 2). Below, we consider which parts of the BOSS data are most responsible for +improving constraints (Fig. 19) and discuss further the implications of these results for the +S8 tension (Figs. 20 and 21). Similarly as for the CMB data considered in § 4.1, with the +addition of BOSS data, there remains no preference for axion models according to the Bayesian +evidence (see Table 3). The Bayes factors amount to evidence in favour of ΛCDM ranging +from “positive” to “strong” [156]. +It is striking that the addition of BOSS data strengthens axion bounds at all masses apart +from ma = 10−26 eV, where in fact the bound is weakened. Fig. 18 breaks down the constraint +at this mass into its constituent parts. While Planck alone sets a 95% credible upper limit +Ωah2 < 0.00615, BOSS alone actually favours a contribution of axions Ωah2 = 0.0100+0.0048 +−0.0037, +which excludes no axions at ∼ 2.7σ (the best-fit model with respect to BOSS data has a +chi-squared reduced by ∆χ2 = −7.7). This discrepancy in the axion constraint, however, +increases the tension between all parameters inferred from Planck and BOSS, as seen in all +three tension metrics shown in Table 5. In particular, the preference in BOSS data for axions +of ma = 10−26 eV increases the discrepancy in S8 from 2.70 σ in the ΛCDM case to 3.63 σ. +This is because the power suppression of axions combines with the already-low value of As +(Ωah2 and As are constrained from different parts of the galaxy power spectrum; see, e. g., +Fig. 4) to lower further the power spectrum amplitude S8 that is inferred from BOSS. For +completeness, we show the joint constraint although we caution that it derives from two +datasets that are in more discrepancy than in the ΛCDM case. As before, Planck dominates +the constraint on ΛCDM parameters, while the joint limit on Ωah2 is slightly weaker than for +Planck alone. Ref. [38] in their analysis of previous BOSS data do not report a preference for +axions at this mass. There are a number of differences with respect to this study (summarised +at the start of § 4.2). However, in particular, previously, the primordial power spectrum tilt +was fixed: ns = 0.9611. Fig. 18 illustrates that fixing ns at this value will break degeneracy +with Ωah2 such that the preference for a non-zero contribution is removed (this degeneracy +with ns in this mass range is also seen in Planck data; see Fig. 8)18. This preference for axions +is not seen at any other mass. At all other masses, BOSS data strengthen the axion limit and +also increase consistency between Planck and BOSS datasets (see Table 5). The fixed axion +masses which we consider are arbitrary. Thus, this result means that there is a preference in +BOSS data alone for a contribution of axions with a mass in a window ma ∈ [10−27, 10−25] eV, +which motivates future work where we additionally sample ma. +Notwithstanding ma = 10−26 eV, BOSS data otherwise always improve axion limits +with respect to Planck alone, and axions improve consistency between the datasets. Fig. 19 +illustrates which parts of the BOSS data are most constraining at ma = 10−25 eV. +We +18Using a Big Bang nucleosynthesis (BBN) prior on the baryon energy density Ωbh2 ∼ N(0.02268, 0.00038) +[? ] reduces the significance for an axion component at ma = 10−26 eV to 2.1σ; further adding a Planck- +motivated prior ns ∼ N(0.9649, 0.0042) [2] reduces the significance to 1.7σ. Weakening the prior on the EFT +of LSS bias and counterterm parameters (see § 3.3 and 3.4) (by doubling the standard deviation in Gaussian +prior distributions and doubling the width in uniform prior distributions) increases the significance to 3.2σ. +– 34 – + +Planck +Planck + BOSS [BAO, P0, P2, P4] +Planck + BOSS [BAO, P0, P2, P4, Q0] +Planck + BOSS [BAO, P0, P2, P4, Q0, B0] +0.30 +0.32 +0.34 +≠m +0.72 +0.80 +S8 +0.05 +0.10 +≠ah2 +0.30 +0.32 +0.34 +≠m +0.05 +0.10 +≠ah2 +Planck +Planck + BOSS [BAO, P0, P2, P4] +Planck + BOSS [BAO, P0, P2, P4, Q0] +Planck + BOSS [BAO, P0, P2, P4, Q0, B0] +0.30 +0.32 +0.34 +≠m +0.72 +0.80 +S8 +0.05 +0.10 +≠ah2 +0.30 +0.32 +0.34 +≠m +0.05 +0.10 +≠ah2 +Figure 19. The effect of adding different parts of the BOSS galaxy clustering data on axion energy +density Ωah2 constraints for ma = 10−25 eV. We systematically add to the Planck CMB likelihood +(blue) different parts of the BOSS likelihood: first, BAO and power spectrum multipoles [P0, P2, P4] +up to maximum wavenumber kmax = 0.2 h Mpc−1 (red); then, also the reconstructed real-space power +spectrum Q0 for k ∈ [0.2, 0.4] h Mpc−1 (orange); and finally, also the bispectrum monopole B0 (green). +We find that the vast majority of the improvement in the bound comes from the addition of smaller- +scale information in the Q0 likelihood, since the suppression effect of axions is stronger on smaller +scales (see Fig. 4). For each data cut, we show the 1D marginalised posterior for Ωah2, where the +68% credible region is shaded. +systematically add different parts of the BOSS data to a joint constraint with Planck. We +find that it is the addition of the small-scale reconstructed real-space galaxy power spectrum +Q0 for 0.2 h Mpc−1 < k < 0.4 h Mpc−1 which drives the vast majority of the improvement in +the bound. This arises because the power suppression effect of axions is always stronger on +smaller scales. +Figure 20 illustrates the degeneracy between Ωah2 and S8 within the 95% credible upper +limits on Ωah2 that are allowed by the joint analysis of Planck and BOSS. As we saw above, +there is no such degeneracy for DE-like axions (ma < 10−28 eV) or for DM-like axions where +the power suppression scale is too small (ma ≥ 10−24 eV). In the mass window (10−28 eV ≤ +ma ≤ 10−25 eV) however, the joint constraint still allows enough axions to drive consistency +with lower values of S8. Although more axions are allowed at higher masses (in the DM- +like regime), since the suppression scale is smaller at higher mass, there is less total power +suppression at the wavenumbers to which S8 is sensitive. The lowest values of S8 are in fact +found at ma = 10−26 eV (S8 = 0.804+0.020 +−0.024; see also Table 2 and Fig. 17). However we caution +that this constraint arises from two datasets that are in stronger tension than the ΛCDM case. +Nonetheless, at ma = 10−25 eV (S8 = 0.818+0.015 +−0.017) and ma = 10−27 eV (S8 = 0.819+0.013 +−0.014), the +parameter discrepancy between Planck and BOSS is reduced and the joint constraint on S8 +is shifted to lower values than the ΛCDM case (S8 = 0.827 ± 0.011). Fig. 21 updates Fig. 12 +with the joint Planck + BOSS constraints (see § 4.1 for details about the DES and KiDS +ΛCDM contours that we show). In comparison to Fig. 12, we note how the addition of BOSS +data more strongly constrains the axion energy density and in turn reduces the extent to +which low values of S8 are allowed. Nonetheless, there remains a tail in the posterior to lower +values of S8 in the presence of axions with ma = 10−25 eV. Fully assessing the consistency +with galaxy weak lensing experiments like DES and KiDS requires re-analysing these data in +– 35 – + +Figure 20. 95% credible upper limits on axion energy density Ωah2, as a function of axion mass ma, +as jointly inferred from Planck CMB and BOSS galaxy clustering data. We illustrate the degeneracy +with the matter clumping factor S8 by colouring (unweighted) posterior samples according to their +S8 value. The lowest values of S8 are allowed for dark matter-like axions with ma ∈ [10−27, 10−25] eV. +On the right-hand side, we show the 95% upper limit on the ratio of the axion energy density to the +best-fit dark matter (DM) energy density as inferred from Planck ΩDMh2 = 0.12. +the axion models we consider here. We discuss the prospects for this in § 5. +5 +Discussion +In § 4, we present several new results in searching for ultra-light axions in a compendium +of CMB and large-scale structure data. In § 4.1, we present legacy constraints on the axion +energy density from Planck 2018 CMB temperature, polarisation and lensing anisotropies. We +find that, compared to previous Planck 2015 results [36], a new measurement of the optical +depth to reionisation (through large-scale polarisation) breaks parameter degeneracies and +improves energy density bounds for DE-like axions (ma ≤ 10−28 eV; see Fig. 7). Further, +we search for axions in a compendium of higher-resolution CMB data (ACT-DR4, SPT-3G), +galaxy BAO and supernovae data. We find that the addition of these data marginally weakens +the axion energy density bound for ma = 10−25 eV (see Table 4). +– 36 – + +S: +0.78 +0.80 +0.82 +0.84 +0.0 +-1.0 +-0.5 +1.5 +-1.0 +.2.0 +-1.5 +2.5 +log +2.0 +-3.0 +32 +-30 +-28 +-26 +-24 +log [Axion mass ma (eV)]Planck + BOSS (ΛCDM) +Planck + BOSS (ma = 10−25 eV) +DES-Y3 3 × 2 (ΛCDM) +KiDS 3 × 2 (ΛCDM) +0.24 +0.32 +0.40 +Ωm +0.72 +0.78 +0.84 +S8 +0.025 +0.050 +Ωah2 +0.24 +0.32 +0.40 +Ωm +0.025 +0.050 +Ωah2 +Figure 21. Comparison of joint Planck CMB and BOSS galaxy clustering constraints (in both axion +and ΛCDM models) with fiducial galaxy weak lensing and clustering (3 × 2) ΛCDM constraints from +the Dark Energy Survey (DES) and the Kilo-Degree Survey (KiDS) (all with fixed neutrino mass). +Planck and BOSS data are consistent with lower values of S8 in the presence of axions with mass +ma = 10−25 eV compared to the ΛCDM case. In order to assess consistency between all data in an +axion model, it is necessary to re-analyse the 3×2 data in the presence of axions; we discuss the future +analysis of cosmic shear data in § 5. We note caution in assessing parameter tension by eye, especially +as the Planck + BOSS and KiDS datasets are not independent, since KiDS uses BOSS clustering +information in their 3 × 2 measurement. +For each set, the inner and outer contours respectively +indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D +marginalised posteriors on the diagonal, where 68% credible regions are shaded. From left to right, S8 +is the matter clumping factor, Ωm is the matter energy density and Ωah2 is the physical axion energy +density. +– 37 – + +In § 4.2, we present axion constraints from BOSS galaxy clustering data. +We find +that the addition of BOSS to Planck improves axion energy density bounds at nearly all +axion masses that we consider (10−32 eV ≤ ma ≤ 10−25 eV). Crucially, we find that the +inclusion of new small-scale modes (Q0 for k ∈ [0.2, 0.4] h Mpc−1) strengthens the constraint +at ma = 10−25 eV with respect to Planck only (see Fig. 19). +This is driven by gaining +sensitivity to larger wavenumbers where the power suppression of heavier axions manifests. +Gains in sensitivity from BOSS data to lighter, DE-like axions (ma ≤ 10−28 eV) are limited +by degeneracy between Ωah2 and As at those masses. This arises since the axion-induced +power suppression occurs at wavenumbers smaller than we model in BOSS data and so the +axion effect is degenerate with an overall re-scaling of the galaxy power spectrum amplitude +(e. g., see Fig. 15). This suggests that robustly modelling larger-volume galaxy surveys can +improve sensitivity to DE-like axions. Robustly modelling smaller-scale correlations in galaxy +positions will be extremely challenging owing to the non-trivial way that galaxies trace dark +matter on small scales (i. e., non-linear galaxy bias). We therefore suggest alternative probes +like galaxy and CMB weak lensing (that are insensitive to galaxy bias) to increase sensitivity +at ma ≥ 10−24 eV (see above and below for more discussion about probes complementary to +galaxy correlations). Nonetheless, our results demonstrate the power in combining CMB and +large-scale structure data when constraining dark matter models beyond standard CDM. +5.1 +Comparison to previous work +There are a number of differences between this study and a previous BOSS analysis presented +in Ref. [38]. First, as discussed above, we model more of the BOSS data, in particular, addi- +tionally, the galaxy power spectrum hexadecapole P4, the small-scale real-space galaxy power +spectrum Q0 (where we conservatively project away hard-to-model non-linear redshift-space +distortions) and the galaxy bispectrum monopole B0. Further, we choose less informative +priors on cosmological parameters, i. e., we do not use BBN information to place a prior on +the baryon energy density Ωbh2 and, importantly, we do not fix the primordial power spec- +trum tilt ns. The latter is important as we do in general observe degeneracy between Ωah2 +and ns (e. g., see Fig. 15) and this degeneracy will be broken by fixing ns. We thus find that +our bounds from BOSS alone are weaker than those reported in Ref. [38]. Ref. [38] combined +Planck and BOSS through a Planck-motivated prior on cosmological and axion parameters +(except ns which remained fixed) combined with the BOSS likelihood. Instead, in this work, +for the first time, we jointly sample the Planck and BOSS likelihoods in a full axion and +cosmological model in setting axion constraints. We find in general that our combined con- +straints are stronger than those reported in Ref. [38]. We attribute a large degree of this to +the information gained by updating to Planck 2018 data (Planck 2015 data was previously +considered) for low masses (see above) and using the small-scale Q0 statistic for higher masses. +The results in Ref. [38] are affected by an error in the BOSS data weights, which has since +been corrected and does not affect the results presented here. +There are further pipeline differences between the two analyses. In particular, beyond +the different and more complete compression of the BOSS data discussed above, we use +different implementations of the BOSS likelihood and EFT of LSS theory calculations (namely, +CLASS-PT/full_shape_likelihoods and, previously, PyBird) and, correspondingly, different +EFT of LSS parameter priors (namely, so-called “East Coast” and, previously, “West Coast” +priors). In general, the different prior choices will lead to differences in parameter inference +given the same set of BOSS data (in ΛCDM, the cosmological constraints are consistent +within ∼ 1σ; see Ref. [151]). Importantly, Refs. [150] and [151] demonstrated that, with +– 38 – + +external CMB information from Planck, the prior sensitivity is significantly reduced, while +future larger-volume galaxy surveys will have sufficient constraining power also to lose prior +sensitivity. We defer to future work a detailed study of the effect of EFT of LSS priors on +BOSS axion constraints since, in this work, it is non-trivial to disentangle the other analysis +differences. +5.2 +ma = 10−26 eV +A striking difference between this work and Ref. [38] is the axion constraint at ma = 10−26 eV. +Unlike at other axion masses that we consider, at ma = 10−26 eV, rather than setting an upper +limit on the axion energy density, we find, given BOSS data only, Ωah2 = 0.0100+0.0048 +−0.0037 that +excludes no axions at ∼ 2.7σ significance. However, such a large contribution of axions at this +mass is disfavoured by Planck (Ωah2 < 0.00615) and so the tension in parameter inference +between these datasets is increased at this axion mass with respect to ΛCDM (at all other +masses, the tension is reduced; see more discussion below). For completeness, we consider +the joint constraint (which is thus weaker than Planck alone) although we caution that this +derives from two datasets in greater discrepancy than in the standard CDM model. +We +find that the preference for axions at this mass opens up degeneracy with other cosmological +parameters, in particular ns and b1 (although in an opposite sense as at other masses; e. g., +see Figs. 15 and 18). This explains why this preference was not observed in Ref. [38] where ns +was fixed; indeed, when giving a BBN prior on Ωbh2 and a Planck prior on ns, the significance +of the axion preference is reduced to only 1.7σ. Further, if an inflation-motivated prior that +excluded low values of ns was imposed, we anticipate that the significance would also be +reduced. +The anomalous (with respect to other masses) degeneracy between higher Ωah2 and lower +ns and b1 suggests an effect from marginalisation over other EFT of LSS parameters; indeed, +weakening the prior on EFT of LSS bias and counterterm parameters increases the axion pref- +erence to 3.2σ. As discussed in § 3.3, in order to reduce the dimensions of the sampling task, +we analytically marginalise over a number of bias and counterterm parameters. We therefore +leave for future work a detailed study of the effect of nuisance parameter marginalisation by +numerically sampling the full joint cosmological and EFT of LSS posterior distribution. We +stress, nonetheless, owing to the way that we consider axion masses only at a number of fixed +values, that there is an element of the look-elsewhere effect where it is not surprising to find +one of the nine axion masses has a mild preference unlike the others. Future galaxy data (e. g., +from the Dark Energy Spectroscopic Instrument [165] or the Rubin Observatory [166]) will be +crucial in determining if this preference is only a statistical anomaly or otherwise. Reconcili- +ation with the Planck bound may be connected to the AL anomaly. As discussed above, we +find that CMB datasets with lower (and theoretically-consistent) amounts of lensing weaken +axion bounds and so we hypothesise that the AL anomaly in Planck is strengthening the +bound and increasing the discrepancy with BOSS at ma = 10−26 eV. We will investigate this +hypothesis in future work. +5.3 +Comparison to other axion probes and future prospects +Notwithstanding the mild preference for axions at ma = 10−26 eV given BOSS data only, our +Planck and joint Planck and BOSS analyses set strong limits on the axion energy density for +ma ≤ 10−25 eV. Axions are well-motivated in a range of particle masses and can be produced +in a mixture with other axions (the so-called “axiverse” [e. g., 21]) and/or with other DM and +DE particle candidates. This motivates a search for axions across a wide range of masses and +– 39 – + +10−32 +10−30 +10−28 +10−26 +10−24 +10−22 +10−20 +10−18 +Axion mass ma (eV) +10−4 +10−3 +10−2 +10−1 +100 +Axion energy density Ωa +CMB-S4 +GUT-scale fa +Lyα BOSS+Hi-Res +Lyα DESI+Hi-Res ++MW-Rubin +PTA ++WL-DES ++BOSS +CMB-Planck 2018 ++MW-DES +IM-SKA +CMB-HD +Figure 22. +95% c.l. +axion energy density Ωa bounds presented in this work from Planck 2018 +CMB data (top left) and from a joint analysis of Planck CMB and BOSS galaxy clustering data +(+BOSS) compared to other cosmological bounds (shaded solid) and projected bounds (thick lines). +Our results are complementary to existing bounds at higher masses derived from probes of smaller- +scale structure: a joint analysis of Planck CMB and galaxy weak lensing data from the Dark Energy +Survey (+WL-DES) [39]; the Milky Way sub-halo mass function from DES (+MW-DES) [29]; the +strongest lower limit for axions being all the DM (ma > 2 × 10−20 eV) comes from high-resolution +Lyman-alpha forest data (Lyα Hi-Res) [28, 31], while Ref. [42] considered a sub-dominant axion +contribution (see also Ref. [43] for a BOSS Lyman-alpha forest analysis). We show projected bounds +for: the CMB-S4 experiment [120]; the CMB-HD experiment using the Ostriker-Vishniac signal [37]; +the Rubin Observatory using the MW sub-halo mass function [167]; future Lyman-alpha forest data +from the Dark Energy Spectroscopic Instrument (DESI) and high-resolution quasar spectra (Hi-Res) +[168]; intensity mapping from the Square Kilometre Array (IM-SKA) [48]; and pulsar timing array +(PTA) residuals [168]. We indicate (between the black dotted lines) the parameter space where the +axion decay constant fa is at the Grand Unified Theory (GUT) scale. Where bounds exclude axions +being all the DM, we additionally exclude higher energy densities (up to Ωa = 1) by enforcing that the +Universe is not over-closed. The projections rely on different assumptions and have varying degrees +of rigour and so are only indicative of future progress. +for sub-dominant energy densities so that an axion of a particular mass is not prematurely +excluded by assuming that it constitutes the entirety of the DM or DE. Fig. 2219 compares +our new bounds for ma ≤ 10−25 eV to other cosmological bounds across the mass range where +the gravitational effect of axions is distinguishable in the large-scale structure from standard +cold DM (10−32 eV ≤ ma ≤ 10−18 eV)20. The details of each experiment and current and +projected bounds are given in the caption. We stress that, to probe across the parameter +space, it is necessary to use complementary probes of large- and small-scale structure to +search for, respectively, lighter and heavier axions. +We anticipate progress in this regard +from ongoing, upcoming and proposed CMB (e. g., ACT, SPT, Simons Observatory, CMB-S4 +[169], CMB-HD), large-scale structure (e. g., Rubin, DESI), intensity mapping (e. g., SKA) +and pulsar timing array observations. +19An up-to-date version of Fig. 22 is maintained at https://keirkwame.github.io/DM_limits. +20There are many ongoing and proposed experimental efforts to probe axions at and above this mass range; +see Refs. [50, 51] for recent reviews; Fig. 22 shows only cosmological probes. +– 40 – + +5.4 +Axions as a resolution to the S8 parameter tension +A key aim of this study is not only to search for axions as a DM and DE candidate, but also +to consider the extent to which axions can improve consistency between CMB and large-scale +structure datasets in their parameter inference, in particular with respect to the so-called +S8 tension. The cosmological parameter S8, through its dependence on the matter power +spectrum amplitude σ8, is a measure of the clustering of matter at z = 0 when averaged +over 8 h−1 Mpc scales. CMB experiments prefer higher values of S8 than various large-scale +structure analyses with statistical significance ranging from 2 to 3 σ depending on the data +comparison [see e. g., 87, for a recent review]. Since CMB experiments generally probe struc- +ture at higher redshift (even CMB lensing is more sensitive to structure at earlier times than +current galaxy surveys), most concrete model solutions to the S8 tension invoke a redshift- +dependent suppression in the growth of structure, e. g., decaying dark matter [e. g., 93]. In +this way, it is argued that this explains why probes of later-time structure have lower ampli- +tude. In this work, we investigate the hypothesis that the S8 tension is a discrepancy between +probes of larger- and smaller-scale structure, with axions as a concrete model, and with no +need to invoke a late-time decay in the nature of DM. +Fig. 1 illustrates our hypothesis by showing how Planck CMB data lose sensitivity to +small-scale modes to which S8 is sensitive. It follows that it is possible to invoke a scale- +dependent suppression in the matter power spectrum that is consistent with current CMB +data on large scales, while lowering the value of S8 to improve compatibility with galaxy +surveys (in particular, galaxy weak lensing) as a more direct probe of the scales to which S8 +is sensitive. Indeed, we find (in § 4.1) axions with ma ∈ [10−28, 10−25] eV as a good candidate +where they are compatible with a compendium of “large-scale” probes (CMB, galaxy BAO +and supernovae) and the lower values of S8 that are inferred from fiducial galaxy clustering +and weak lensing (3 × 2) analyses (see Fig. 12). Lighter axions are largely incompatible with +Planck data (except as a highly sub-dominant contribution that does little to S8), while +heavier axions are unconstrained by these data but suppress wavenumbers larger than those +to which S8 is sensitive. +In order to assess whether axions can resolve parameter tensions between CMB and +large-scale structure data, it is necessary also to analyse large-scale structure data in an +axion model. In § 4.2, we analyse BOSS galaxy clustering data and model the effect of axions +in the mildly non-linear regime using the effective field theory of large-scale structure (see +§ 2.2 for details). We find two regimes in which the S8 discrepancy between Planck and +BOSS is reduced. The first is for ma ∼ 10−25 eV, where, as discussed above, large axion +contributions are unconstrained by CMB data and so bring CMB data into compatibility +with lower values of S8: the S8 discrepancy is reduced from 2.70σ in ΛCDM to 1.63σ. The +second regime is for ma ∼ 10−28 eV, where, instead, the effect of axions in BOSS data is +partly degenerate with the overall amplitude of the galaxy power spectrum since all BOSS +wavenumbers are suppressed by axions of this mass. This weakens BOSS constraints on the +lightest axions, while allowing higher values of the primordial power spectrum amplitude As +and also S8 (the effects of higher As and higher Ωah2 not cancelling exactly). Thus, the S8 +discrepancy is reduced to 1.78σ, but, importantly, values of As (which are low in this BOSS +analysis; see Ref. [151] for a discussion on the effect of EFT of LSS priors on cosmological +parameter inference) are brought into greater compatibility with the higher values inferred +from Planck. +Indeed, although S8 is a reasonably good compression of the information contained in +large-scale structure data (though not necessarily optimal for all experiments), it is necessary +– 41 – + +to assess tension in the full posterior, in particular accounting for non-Gaussianity in the +distribution (which one-parameter tension metrics do not capture). In this work, in order +better to capture tension in the full parameter space, we estimate the posterior distribution +of the parameter difference [162] inferred given the two experiments (Planck and BOSS). If +the two experiments are in perfect agreement, the parameter difference posterior will peak +at zero; we assign the significance of the discrepancy between experiments to the amount +of shift from perfect agreement (see § 4.2 for more details). We find that this measure of +tension improves from ΛCDM at nearly all axion masses apart from ma = 10−26 eV (as +discussed above). There is no single tension metric on which the community has converged; +different metrics tend to disagree in terms of absolute value though they agree with regards +to increasing or decreasing tension [see e. g., 163]. The metrics we use in this work (and +quite generally) depend on the parameterisation of the model. Since S8 and σ8 may not be +optimal measures of the matter clustering information directly probed by CMB and even +many large-scale structure experiments, we defer to future work studies of the agreement +between datasets directly in the linear matter power spectrum using Bayesian metrics like +the posterior predictive distribution. +Nonetheless, our results suggest that axions with masses in a window [10−28, 10−25] eV +can be a promising candidate to improve consistency between CMB and large-scale structure +observations, in particular by bringing Planck CMB and BOSS galaxy clustering data into +consistency with lower values of S8 that are preferred by galaxy weak lensing data (see e. g., +Figs. 20 and 21). +We stress, though, that this is achieved through only upper limits on +the axion energy density and there is no preference for model extensions beyond ΛCDM +given these data according to the Bayesian evidence in any of our analysis (see Table 3). A +more stringent test of the ability for axions to address cosmological parameter tension is the +inclusion of galaxy weak lensing data. It is common in the literature to include the effect +of weak lensing through a prior on S8 derived from ΛCDM S8 constraints, see e.g. [97, 170]. +This is a good measure of the information content in the ΛCDM model, but we caution that +this may not be the case in extended models like axions which affect in a non-trivial way the +non-linear modes probed by galaxy shear (indeed, we see with BOSS how the S8 constraint +changes with axions). We therefore leave for future work an analysis of galaxy shear and 3×2 +clustering and shear data using a fully non-linear halo model of axion structure formation +[e. g., 71]. This will build on initial studies of DES cosmic shear in the limited case that axions +comprise the entirety of the DM [39]. +6 +Conclusions +We present a comprehensive search for ultra-light axions as a well-motivated dark matter +and dark energy particle candidate using a compendium of CMB and large-scale structure +data. We set the strongest bounds to date on the axion energy density for axion masses ma ∈ +[10−32, 10−25] eV through a joint analysis of Planck 2018 CMB and BOSS full-shape galaxy +power spectrum and bispectrum data, modelling the effect of axions in the mildly non-linear +regime using the effective field theory of large-scale structure. We exclude axions being more +than 10% of the DM today for ma ≤ 10−26 eV and more than 1% for ma ∈ [10−30, 10−28] eV. +We give legacy constraints from Planck 2018 CMB data and find that measurements of +the optical depth to reionisation break parameter degeneracies and improve bounds for DE- +like axions (ma ≤ 10−28 eV). +For the first time, we consider high-resolution CMB data +from the Atacama Cosmology Telescope and the South Pole Telescope (in combination with +– 42 – + +galaxy BAO and supernovae data), which we find to weaken marginally axion bounds at +ma = 10−25 eV. Similarly to the effect of massive neutrinos, we attribute this weakening +to the lower (and theoretically-consistent) amounts of lensing observed in ACT and SPT +angular power spectra, which allow more structure suppression arising from axions. In the +first full joint analysis of Planck 2018 and BOSS full-shape data, we find that galaxy clustering +information strengthens axion energy density limits at nearly all masses that we consider. The +exception is at ma = 10−26 eV, where BOSS data alone have a mild preference for a non-zero +axion contribution, excluding no axions at ∼ 2.7σ. The significance is reduced to only 1.7σ +when including Big Bang nucleosynthesis constraints on the baryon energy density Ωbh2 and +Planck constraints on the primordial power spectrum tilt ns. Such an axion contribution is, +further, disfavoured by Planck and we caution that the look-elsewhere effect applies owing to +the large number of axion masses that we consider. Future galaxy data (e. g., DESI, Rubin) +will be crucial in assessing the significance of this result. +We propose axions as a candidate to address the so-called “S8 tension”, where CMB +experiments infer systematically higher values of S8 (which is sensitive to the matter power +spectrum amplitude at z = 0) than various large-scale structure datasets, with significance +ranging from 2 to 3 σ [e. g., 87]. We hypothesise that the scale-dependent power spectrum +suppression (relative to standard cold DM) arising from axion DM can reconcile current CMB +data (which probe larger scales and prefer higher amplitude) with the more direct probes of +smaller-scale structure in galaxy clustering and weak lensing that prefer lower amplitude. We +indeed find that a compendium of “large-scale” data (CMB, galaxy BAO and supernovae) +are compatible with lower values of S8 = 0.774+0.032 +−0.037 for ma = 10−25 eV than in ΛCDM +(S8 = 0.827 ± 0.010). This is achieved since this data combination is not sensitive to the +small-scale suppression arising from axions of this mass, while the axion suppression still +occurs at wavenumbers to which S8 is sensitive, thus lowering its value. Although BOSS +full-shape data, in general, strengthen axion density bounds (apart from at ma = 10−26 eV), +we find that axions can improve inferred parameter consistency between Planck and BOSS +and that the joint Planck and BOSS constraint is still consistent with lower values of S8 than +ΛCDM in a window of masses [10−28, 10−25] eV. In future work, we will assess consistency +with upcoming CMB and galaxy weak lensing data using a fully non-linear (halo) model of +axion structure formation [e. g., 39, 71]. +Acknowledgments +The authors thank Daniel Grin for valuable discussions. +The Dunlap Institute is funded +through an endowment established by the David Dunlap family and the University of Toronto. +RH is a CIFAR Azrieli Global Scholar (Gravity & the Extreme Universe Program 2019) and +a 2020 Alfred P. Sloan Research Fellow; and is supported by the Natural Sciences and En- +gineering Research Council of Canada Discovery Grant Program and the Connaught Fund. +MMI is supported by the National Aeronautics and Space Administration (NASA) through +the NASA Hubble Fellowship grant #HST-HF2-51483.001-A awarded by the Space Telescope +Science Institute, which is operated by the Association of Universities for Research in Astron- +omy, Incorporated, under NASA contract NAS5-26555. +OHEP is a Junior Fellow of the +Simons Society of Fellows and thanks the Institute for Advanced Study for their hospitality +and abundance of baked goods. KA is supported by Japan Society for the Promotion of +Science (JSPS) Overseas Research Fellowships. DJEM is supported by an Ernest Rutherford +Fellowship from the Science and Technologies Facilities Council in the United Kingdom. +– 43 – + +References +[1] D. Brout, D. Scolnic, B. Popovic, A. G. Riess, A. Carr, J. Zuntz et al., The Pantheon+ +Analysis: Cosmological Constraints, ApJ 938 (2022) 110 [2202.04077]. +[2] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi +et al., Planck 2018 results. VI. Cosmological parameters, A&A 641 (2020) A6 [1807.06209]. +[3] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard et al., Nine-year +Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results, +ApJS 208 (2013) 20 [1212.5225]. +[4] Z. Hou, C. L. Reichardt, K. T. Story, B. Follin, R. Keisler, K. A. Aird et al., Constraints on +Cosmology from the Cosmic Microwave Background Power Spectrum of the 2500 deg2 SPT-SZ +Survey, ApJ 782 (2014) 74 [1212.6267]. +[5] M. S. Madhavacheril, N. Sehgal and T. R. Slatyer, Current dark matter annihilation +constraints from CMB and low-redshift data, Phys. Rev. D 89 (2014) 103508 [1310.3815]. +[6] J. L. Sievers, R. A. Hlozek, M. R. Nolta, V. Acquaviva, G. E. Addison, P. A. R. Ade et al., +The Atacama Cosmology Telescope: cosmological parameters from three seasons of data, J. +Cosmology Astropart. Phys. 2013 (2013) 060 [1301.0824]. +[7] M. B. Wise, H. Georgi and S. L. Glashow, SU(5) and the Invisible Axion, Phys. Rev. Lett. 47 +(1981) 402. +[8] M. Dine, W. Fischler and M. Srednicki, A simple solution to the strong CP problem with a +harmless axion, Physics Letters B 104 (1981) 199. +[9] M. Dine, Axions: Visible and Invisible, in Novel Results in Particle Physics - 1982: Fifth +International Conference on Particle Physics, vol. 93 of American Institute of Physics +Conference Series, pp. 66–76, Nov., 1982, DOI. +[10] L. F. Abbott, Axion Cosmology, in Relativity, Cosmology, Topological Mass and Supergravity; +SILARG IV, p. 100, Jan., 1983. +[11] J. Preskill, M. B. Wise and F. Wilczek, Cosmology of the invisible axion, Physics Letters B +120 (1983) 127. +[12] P. J. Steinhardt and M. S. Turner, Saving the invisible axion, Phys. Lett. B 129 (1983) 51. +[13] J. E. Kim, Light pseudoscalars, particle physics and cosmology., Phys. Rep. 150 (1987) 1. +[14] Z. G. Berezhiani, A. S. Sakharov and M. Y. Khlopov, Primordial background of cosmological +axions., Soviet Journal of Nuclear Physics 55 (1992) 1063. +[15] R. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. +38 (1977) 1440. +[16] S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40 (1978) 223. +[17] F. Wilczek, Problem of Strong p and t Invariance in the Presence of Instantons, +Phys. Rev. Lett. 40 (1978) 279. +[18] E. Witten, Some properties of O(32) superstrings, Physics Letters B 149 (1984) 351. +[19] P. Svrcek and E. Witten, Axions In String Theory, JHEP 06 (2006) 051 [hep-th/0605206]. +[20] S. Weinberg, A new light boson?, Phys. Rev. Lett. 40 (1978) 223. +[21] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, String +Axiverse, Phys. Rev. D 81 (2010) 123530 [0905.4720]. +[22] R. Hlozek, D. Grin, D. J. E. Marsh and P. G. Ferreira, A search for ultralight axions using +precision cosmological data, Phys. Rev. D 91 (2015) 103512 [1410.2896]. +– 44 – + +[23] J. S. Bullock and M. Boylan-Kolchin, Small-Scale Challenges to the ΛCDM Paradigm, +ARA&A 55 (2017) 343 [1707.04256]. +[24] D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray and A. H. G. Peter, Cold +dark matter: Controversies on small scales, Proceedings of the National Academy of Science +112 (2015) 12249 [1306.0913]. +[25] L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, Ultralight scalars as cosmological dark +matter, Phys. Rev. D 95 (2017) 043541 [1610.08297]. +[26] A. Pontzen and F. Governato, Cold dark matter heats up, Nature 506 (2014) 171 [1402.1764]. +[27] A. Drlica-Wagner, K. Bechtol, S. Mau, M. McNanna, E. O. Nadler, A. B. Pace et al., Milky +Way Satellite Census. I. The Observational Selection Function for Milky Way Satellites in +DES Y3 and Pan-STARRS DR1, ApJ 893 (2020) 47 [1912.03302]. +[28] K. K. Rogers and H. V. Peiris, Strong Bound on Canonical Ultralight Axion Dark Matter from +the Lyman-Alpha Forest, Phys. Rev. Lett. 126 (2021) 071302 [2007.12705]. +[29] E. O. Nadler, A. Drlica-Wagner, K. Bechtol, S. Mau, R. H. Wechsler, V. Gluscevic et al., +Constraints on Dark Matter Properties from Observations of Milky Way Satellite Galaxies, +Phys. Rev. Lett. 126 (2021) 091101 [2008.00022]. +[30] A. Laguë, J. R. Bond, R. Hložek, D. J. E. Marsh and L. Söding, Evolving ultralight scalars into +non-linearity with Lagrangian perturbation theory, MNRAS 504 (2021) 2391 [2004.08482]. +[31] K. K. Rogers and H. V. Peiris, General framework for cosmological dark matter bounds using +N -body simulations, Phys. Rev. D 103 (2021) 043526 [2007.13751]. +[32] T. Dome, A. Fialkov, P. Mocz, B. M. Schäfer, M. Boylan-Kolchin and M. Vogelsberger, On +the Cosmic Web Elongation in Fuzzy Dark Matter Cosmologies, arXiv e-prints (2022) +arXiv:2208.03827 [2208.03827]. +[33] S. May and V. Springel, The halo mass function and filaments in full cosmological simulations +with fuzzy dark matter, arXiv e-prints (2022) arXiv:2209.14886 [2209.14886]. +[34] M. Nori, A. V. Macciò and M. Baldi, Fuzzy Aquarius: evolution of a Milky-way like system in +the Fuzzy Dark Matter scenario, arXiv e-prints (2022) arXiv:2210.08022 [2210.08022]. +[35] S. M. L. Vogt, D. J. E. Marsh and A. Laguë, Improved Mixed Dark Matter Halo Model for +Ultralight Axions, arXiv e-prints (2022) arXiv:2209.13445 [2209.13445]. +[36] R. Hložek, D. J. E. Marsh and D. Grin, Using the full power of the cosmic microwave +background to probe axion dark matter, MNRAS 476 (2018) 3063 [1708.05681]. +[37] G. S. Farren, D. Grin, A. H. Jaffe, R. Hložek and D. J. Marsh, Ultralight axions and the +kinetic sunyaev-zel’dovich effect, Physical Review D 105 (2022) . +[38] A. Laguë, J. R. Bond, R. Hložek, K. K. Rogers, D. J. E. Marsh and D. Grin, Constraining +ultralight axions with galaxy surveys, J. Cosmology Astropart. Phys. 2022 (2022) 049 +[2104.07802]. +[39] M. Dentler, D. J. E. Marsh, R. Hložek, A. Laguë, K. K. Rogers and D. Grin, Fuzzy dark +matter and the Dark Energy Survey Year 1 data, MNRAS 515 (2022) 5646 [2111.01199]. +[40] A. Kunkel, T. Chiueh and B. M. Schäfer, A weak lensing perspective on nonlinear structure +formation with fuzzy dark matter, arXiv e-prints (2022) arXiv:2211.01523 [2211.01523]. +[41] V. Iršič, M. Viel, M. G. Haehnelt, J. S. Bolton and G. D. Becker, First constraints on fuzzy +dark matter from Lyman-α forest data and hydrodynamical simulations, ArXiv e-prints (2017) +[1703.04683]. +[42] T. Kobayashi, R. Murgia, A. De Simone, V. Iršič and M. Viel, Lyman-alpha Constraints on +Ultralight Scalar Dark Matter: Implications for the Early and Late Universe, ArXiv e-prints +(2017) [1708.00015]. +– 45 – + +[43] E. Armengaud, N. Palanque-Delabrouille, C. Yèche, D. J. E. Marsh and J. Baur, +Constraining the mass of light bosonic dark matter using SDSS Lyman-α forest, ArXiv +e-prints (2017) [1703.09126]. +[44] J. Baur, N. Palanque-Delabrouille, C. Yèche, C. Magneville and M. Viel, Lyman-alpha forests +cool warm dark matter, J. Cosmology Astropart. Phys. 8 (2016) 012 [1512.01981]. +[45] N. Dalal and A. Kravtsov, Excluding fuzzy dark matter with sizes and stellar kinematics of +ultrafaint dwarf galaxies, Phys. Rev. D 106 (2022) 063517. +[46] I. S. Goldstein, S. M. Koushiappas and M. G. Walker, Viability of ultralight bosonic dark +matter in dwarf galaxies, Phys. Rev. D 106 (2022) 063010 [2206.05244]. +[47] S. C. Hotinli, D. J. E. Marsh and M. Kamionkowski, Probing ultralight axions with the 21-cm +signal during cosmic dawn, Phys. Rev. D 106 (2022) 043529 [2112.06943]. +[48] J. B. Bauer, D. J. E. Marsh, R. Hložek, H. Padmanabhan and A. Laguë, Intensity mapping as +a probe of axion dark matter, MNRAS 500 (2021) 3162 [2003.09655]. +[49] J. Flitter and E. D. Kovetz, Closing the window on fuzzy dark matter with the 21-cm signal, +Phys. Rev. D 106 (2022) 063504 [2207.05083]. +[50] D. Antypas, A. Banerjee, C. Bartram, M. Baryakhtar, J. Betz, J. J. Bollinger et al., New +Horizons: Scalar and Vector Ultralight Dark Matter, arXiv e-prints (2022) arXiv:2203.14915 +[2203.14915]. +[51] C. B. Adams et al., Axion Dark Matter, in 2022 Snowmass Summer Study, 3, 2022, +2203.14923. +[52] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont et al., +Planck 2015 results. XIII. Cosmological parameters, A&A 594 (2016) A13 [1502.01589]. +[53] S. Aiola, E. Calabrese, L. Maurin, S. Naess, B. L. Schmitt, M. H. Abitbol et al., The Atacama +Cosmology Telescope: DR4 maps and cosmological parameters, J. Cosmology Astropart. Phys. +2020 (2020) 047 [2007.07288]. +[54] D. Dutcher, L. Balkenhol, P. A. R. Ade, Z. Ahmed, E. Anderes, A. J. Anderson et al., +Measurements of the E -mode polarization and temperature-E -mode correlation of the CMB +from SPT-3G 2018 data, Phys. Rev. D 104 (2021) 022003 [2101.01684]. +[55] O. H. E. Philcox and M. M. Ivanov, BOSS DR12 full-shape cosmology: Λ CDM constraints +from the large-scale galaxy power spectrum and bispectrum monopole, Phys. Rev. D 105 +(2022) 043517 [2112.04515]. +[56] BOSS collaboration, K. S. Dawson et al., The Baryon Oscillation Spectroscopic Survey of +SDSS-III, Astron. J. 145 (2013) 10 [1208.0022]. +[57] D. Grin, D. J. E. Marsh and R. Hlozek, “axionCAMB: Modification of the CAMB Boltzmann +code.” Astrophysics Source Code Library, record ascl:2203.026, Mar., 2022. +[58] D. Blas, J. Lesgourgues and T. Tram, The Cosmic Linear Anisotropy Solving System +(CLASS). Part II: Approximation schemes, J. Cosmology Astropart. Phys. 7 (2011) 34 +[1104.2933]. +[59] T. Cookmeyer, J. Cookmeyer, D. Grin and T. L. Smith, How sound are our ultralight axion +approximations?, Phys. Rev. D 101 (2020) 023501 [1909.11094]. +[60] S. Passaglia and W. Hu, Accurate effective fluid approximation for ultralight axions, Phys. +Rev. D 105 (2022) 123529 [2201.10238]. +[61] D. Baumann, A. Nicolis, L. Senatore and M. Zaldarriaga, Cosmological non-linearities as an +effective fluid, J. Cosmology Astropart. Phys. 2012 (2012) 051 [1004.2488]. +– 46 – + +[62] M. Simonović, T. Baldauf, M. Zaldarriaga, J. J. Carrasco and J. A. Kollmeier, Cosmological +perturbation theory using the FFTLog: formalism and connection to QFT loop integrals, J. +Cosmology Astropart. Phys. 2018 (2018) 030 [1708.08130]. +[63] G. Cabass, M. M. Ivanov, M. Lewandowski, M. Mirbabayi and M. Simonović, Snowmass +White Paper: Effective Field Theories in Cosmology, in 2022 Snowmass Summer Study, 3, +2022, 2203.08232. +[64] G. D’Amico, L. Senatore and P. Zhang, Limits on wCDM from the EFTofLSS with the +PyBird code, JCAP 01 (2021) 006 [2003.07956]. +[65] M. M. Ivanov, M. Simonović and M. Zaldarriaga, Cosmological Parameters from the BOSS +Galaxy Power Spectrum, JCAP 05 (2020) 042 [1909.05277]. +[66] M. M. Ivanov, M. Simonović and M. Zaldarriaga, Cosmological Parameters and Neutrino +Masses from the Final Planck and Full-Shape BOSS Data, Phys. Rev. D 101 (2020) 083504 +[1912.08208]. +[67] A. Chudaykin, M. M. Ivanov, O. H. E. Philcox and M. Simonović, Nonlinear perturbation +theory extension of the Boltzmann code CLASS, Phys. Rev. D 102 (2020) 063533 +[2004.10607]. +[68] M. M. Ivanov, Effective Field Theory for Large Scale Structure, 2212.08488. +[69] L. Senatore and M. Zaldarriaga, The Effective Field Theory of Large-Scale Structure in the +presence of Massive Neutrinos, arXiv e-prints (2017) arXiv:1707.04698 [1707.04698]. +[70] D. J. E. Marsh, WarmAndFuzzy: the halo model beyond CDM, ArXiv e-prints (2016) +[1605.05973]. +[71] S. M. L. Vogt, D. J. E. Marsh and A. Laguë, Improved Mixed Dark Matter Halo Model for +Ultralight Axions, arXiv e-prints (2022) arXiv:2209.13445 [2209.13445]. +[72] K. Heitmann, D. Higdon, M. White, S. Habib, B. J. Williams, E. Lawrence et al., The Coyote +Universe. II. Cosmological Models and Precision Emulation of the Nonlinear Matter Power +Spectrum, ApJ 705 (2009) 156 [0902.0429]. +[73] K. Heitmann, E. Lawrence, J. Kwan, S. Habib and D. Higdon, The Coyote Universe Extended: +Precision Emulation of the Matter Power Spectrum, ApJ 780 (2014) 111 [1304.7849]. +[74] E. Lawrence, K. Heitmann, J. Kwan, A. Upadhye, D. Bingham, S. Habib et al., The +Mira-Titan Universe. II. Matter Power Spectrum Emulation, ApJ 847 (2017) 50 +[1705.03388]. +[75] Z. Zhai, J. L. Tinker, M. R. Becker, J. DeRose, Y.-Y. Mao, T. McClintock et al., The Aemulus +Project. III. Emulation of the Galaxy Correlation Function, ApJ 874 (2019) 95 [1804.05867]. +[76] Euclid Collaboration, M. Knabenhans, J. Stadel, S. Marelli, D. Potter, R. Teyssier et al., +Euclid preparation: II. The EUCLIDEMULATOR - a tool to compute the cosmology +dependence of the nonlinear matter power spectrum, MNRAS 484 (2019) 5509 [1809.04695]. +[77] S. Bird, K. K. Rogers, H. V. Peiris, L. Verde, A. Font-Ribera and A. Pontzen, An emulator +for the Lyman-α forest, J. Cosmology Astropart. Phys. 2019 (2019) 050 [1812.04654]. +[78] K. K. Rogers, H. V. Peiris, A. Pontzen, S. Bird, L. Verde and A. Font-Ribera, Bayesian +emulator optimisation for cosmology: application to the Lyman-alpha forest, J. Cosmology +Astropart. Phys. 2019 (2019) 031 [1812.04631]. +[79] C. Pedersen, A. Font-Ribera, K. K. Rogers, P. McDonald, H. V. Peiris, A. Pontzen et al., An +emulator for the Lyman-α forest in beyond-ΛCDM cosmologies, J. Cosmology Astropart. +Phys. 2021 (2021) 033 [2011.15127]. +[80] K. K. Rogers, C. Dvorkin and H. V. Peiris, Limits on the Light Dark Matter-Proton Cross +Section from Cosmic Large-Scale Structure, Phys. Rev. Lett. 128 (2022) 171301 [2111.10386]. +– 47 – + +[81] M. Nori and M. Baldi, AX-GADGET: a new code for cosmological simulations of Fuzzy Dark +Matter and Axion models, MNRAS 478 (2018) 3935 [1801.08144]. +[82] H.-Y. Schive, T. Chiueh and T. Broadhurst, Cosmic structure as the quantum interference of +a coherent dark wave, Nature Physics 10 (2014) 496 [1406.6586]. +[83] P. Mocz, A. Fialkov, M. Vogelsberger, F. Becerra, M. A. Amin, S. Bose et al., First +Star-Forming Structures in Fuzzy Cosmic Filaments, Phys. Rev. Lett. 123 (2019) 141301 +[1910.01653]. +[84] X. Li, L. Hui and G. L. Bryan, Numerical and perturbative computations of the fuzzy dark +matter model, Phys. Rev. D 99 (2019) 063509 [1810.01915]. +[85] B. Schwabe, M. Gosenca, C. Behrens, J. C. Niemeyer and R. Easther, Simulating mixed fuzzy +and cold dark matter, Phys. Rev. D 102 (2020) 083518 [2007.08256]. +[86] M. Kulkarni, E. Visbal, G. L. Bryan and X. Li, If dark matter is fuzzy, the first stars form in +massive pancakes, arXiv e-prints (2022) arXiv:2210.11515 [2210.11515]. +[87] E. Abdalla, G. F. Abellán, A. Aboubrahim, A. Agnello, Ö. Akarsu, Y. Akrami et al., +Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology +associated with the cosmological tensions and anomalies, Journal of High Energy Astrophysics +34 (2022) 49 [2203.06142]. +[88] M. Lucca, Dark energy-dark matter interactions as a solution to the S8 tension, Physics of the +Dark Universe 34 (2021) 100899 [2105.09249]. +[89] V. Poulin, J. L. Bernal, E. Kovetz and M. Kamionkowski, The Sigma-8 Tension is a Drag, +arXiv e-prints (2022) arXiv:2209.06217 [2209.06217]. +[90] D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann and C. M. Wells, Atomic dark matter, J. +Cosmology Astropart. Phys. 2010 (2010) 021 [0909.0753]. +[91] F.-Y. Cyr-Racine and K. Sigurdson, Cosmology of atomic dark matter, Phys. Rev. D 87 +(2013) 103515 [1209.5752]. +[92] S. Bansal, J. Barron, D. Curtin and Y. Tsai, Precision Cosmological Constraints on Atomic +Dark Matter, arXiv e-prints (2022) arXiv:2212.02487 [2212.02487]. +[93] K. Enqvist, S. Nadathur, T. Sekiguchi and T. Takahashi, Decaying dark matter and the +tension in σ8, J. Cosmology Astropart. Phys. 2015 (2015) 067 [1505.05511]. +[94] K. L. Pandey, T. Karwal and S. Das, Alleviating the H0 and σ8 anomalies with a decaying +dark matter model, J. Cosmology Astropart. Phys. 2020 (2020) 026 [1902.10636]. +[95] T. Driskell, E. O. Nadler, J. Mirocha, A. Benson, K. K. Boddy, T. D. Morton et al., Structure +formation and the global 21-cm signal in the presence of Coulomb-like dark matter-baryon +interactions, Phys. Rev. D 106 (2022) 103525 [2209.04499]. +[96] A. Amon and G. Efstathiou, A non-linear solution to the S8 tension?, MNRAS 516 (2022) +5355 [2206.11794]. +[97] J. C. Hill, E. McDonough, M. W. Toomey and S. Alexander, Early dark energy does not +restore cosmological concordance, Phys. Rev. D 102 (2020) 043507 [2003.07355]. +[98] G. Ye, J. Zhang and Y.-S. Piao, Resolving both H0 and S8 tensions with AdS early dark +energy and ultralight axion, arXiv e-prints (2021) arXiv:2107.13391 [2107.13391]. +[99] S. Alexander, H. Bernardo and M. W. Toomey, Addressing the Hubble and S8 Tensions with a +Kinetically Mixed Dark Sector, arXiv e-prints (2022) arXiv:2207.13086 [2207.13086]. +[100] I. J. Allali, M. P. Hertzberg and F. Rompineve, Dark sector to restore cosmological +concordance, Phys. Rev. D 104 (2021) L081303 [2104.12798]. +– 48 – + +[101] U.-H. Zhang and T. Chiueh, Cosmological Perturbations of Extreme Axion in the Radiation +Era, Phys. Rev. D 96 (2017) 063522 [1705.01439]. +[102] F. X. L. Cedeño, A. X. González-Morales and L. A. Ureña López, Cosmological signatures of +ultralight dark matter with an axionlike potential, Phys. Rev. D 96 (2017) 061301 +[1703.10180]. +[103] K.-H. Leong, H.-Y. Schive, U.-H. Zhang and T. Chiueh, Testing extreme-axion wave-like dark +matter using the BOSS Lyman-alpha forest data, Mon. Not. Roy. Astron. Soc. 484 (2019) +4273 [1810.05930]. +[104] A. Arvanitaki, S. Dimopoulos, M. Galanis, L. Lehner, J. O. Thompson and K. Van Tilburg, +Large-misalignment mechanism for the formation of compact axion structures: Signatures +from the QCD axion to fuzzy dark matter, Phys. Rev. D 101 (2020) 083014 [1909.11665]. +[105] M. Pospelov, A. Ritz, C. Skordis, A. Ritz and C. Skordis, Pseudoscalar perturbations and +polarization of the cosmic microwave background, Phys. Rev. Lett. 103 (2009) 051302 +[0808.0673]. +[106] G. Sigl and P. Trivedi, Axion-like Dark Matter Constraints from CMB Birefringence, +1811.07873. +[107] M. A. Fedderke, P. W. Graham and S. Rajendran, Axion Dark Matter Detection with CMB +Polarization, Phys. Rev. D 100 (2019) 015040 [1903.02666]. +[108] I. Obata, Implications of the cosmic birefringence measurement for the axion dark matter +search, JCAP 09 (2022) 062 [2108.02150]. +[109] D. G. Levkov, A. G. Panin and I. I. Tkachev, Radio-emission of axion stars, Phys. Rev. D +102 (2020) 023501 [2004.05179]. +[110] D. Baumann, D. Green and B. Wallisch, New Target for Cosmic Axion Searches, Phys. Rev. +Lett. 117 (2016) 171301 [1604.08614]. +[111] F. D’Eramo, F. Hajkarim and S. Yun, Thermal Axion Production at Low Temperatures: A +Smooth Treatment of the QCD Phase Transition, Phys. Rev. Lett. 128 (2022) 152001 +[2108.04259]. +[112] W. Hu, Structure formation with generalized dark matter, Astrophys. J. 506 (1998) 485 +[astro-ph/9801234]. +[113] L. Amendola and R. Barbieri, Dark matter from an ultra-light pseudo-Goldsone-boson, Phys. +Lett. B 642 (2006) 192 [hep-ph/0509257]. +[114] J.-c. Hwang and H. Noh, Axion as a Cold Dark Matter candidate, Phys. Lett. B 680 (2009) 1 +[0902.4738]. +[115] D. J. E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1 [1510.07633]. +[116] E. Madelung, Eine anschauliche Deutung der Gleichung von Schrödinger, +Naturwissenschaften 14 (1926) 1004. +[117] M. Khlopov, B. A. Malomed and I. B. Zeldovich, Gravitational instability of scalar fields and +formation of primordial black holes, Mon. Not. Roy. Astron. Soc. 215 (1985) 575. +[118] D. J. E. Marsh and P. G. Ferreira, Ultra-Light Scalar Fields and the Growth of Structure in +the Universe, Phys. Rev. D 82 (2010) 103528 [1009.3501]. +[119] D. J. E. Marsh, D. Grin, R. Hložek and P. G. Ferreira, Axiverse cosmology and the energy +scale of inflation, Phys. Rev. D 87 (2013) 121701 [1303.3008]. +[120] R. Hložek, D. J. E. Marsh, D. Grin, R. Allison, J. Dunkley and E. Calabrese, Future CMB +tests of dark matter: Ultralight axions and massive neutrinos, Phys. Rev. D 95 (2017) 123511 +[1607.08208]. +– 49 – + +[121] J. C. Jackson, Fingers of God: A critique of Rees’ theory of primoridal gravitational radiation, +Mon. Not. Roy. Astron. Soc. 156 (1972) 1P [0810.3908]. +[122] M. M. Ivanov, O. H. E. Philcox, M. Simonović, M. Zaldarriaga, T. Nischimichi and +M. Takada, Cosmological constraints without nonlinear redshift-space distortions, +Phys. Rev. D 105 (2022) 043531 [2110.00006]. +[123] G. D’Amico, L. Senatore, P. Zhang and T. Nishimichi, Taming redshift-space distortion effects +in the EFTofLSS and its application to data, 2110.00016. +[124] N. Kaiser, Clustering in real space and in redshift space, MNRAS 227 (1987) 1. +[125] L. Senatore and M. Zaldarriaga, The IR-resummed Effective Field Theory of Large Scale +Structures, JCAP 02 (2015) 013 [1404.5954]. +[126] D. Blas, M. Garny, M. M. Ivanov and S. Sibiryakov, Time-Sliced Perturbation Theory II: +Baryon Acoustic Oscillations and Infrared Resummation, JCAP 07 (2016) 028 [1605.02149]. +[127] M. M. Ivanov and S. Sibiryakov, Infrared Resummation for Biased Tracers in Redshift Space, +JCAP 07 (2018) 053 [1804.05080]. +[128] C. Alcock and B. Paczynski, An evolution free test for non-zero cosmological constant, Nature +281 (1979) 358. +[129] M. M. Ivanov, O. H. E. Philcox, T. Nishimichi, M. Simonović, M. Takada and M. Zaldarriaga, +Precision analysis of the redshift-space galaxy bispectrum, Phys. Rev. D 105 (2022) 063512 +[2110.10161]. +[130] O. H. E. Philcox, M. M. Ivanov, G. Cabass, M. Simonović, M. Zaldarriaga and T. Nishimichi, +Cosmology with the redshift-space galaxy bispectrum monopole at one-loop order, Phys. Rev. D +106 (2022) 043530 [2206.02800]. +[131] G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore and P. Zhang, The BOSS bispectrum +analysis at one loop from the Effective Field Theory of Large-Scale Structure, 2206.08327. +[132] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell et al., The 6dF +Galaxy Survey: baryon acoustic oscillations and the local Hubble constant, MNRAS 416 +(2011) 3017 [1106.3366]. +[133] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, The +clustering of the SDSS DR7 main Galaxy sample - I. A 4 per cent distance measure at z = +0.15, MNRAS 449 (2015) 835 [1409.3242]. +[134] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek et al., The clustering of +galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological +analysis of the DR12 galaxy sample, MNRAS 470 (2017) 2617 [1607.03155]. +[135] M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas et al., Improved cosmological +constraints from a joint analysis of the SDSS-II and SNLS supernova samples, A&A 568 +(2014) A22 [1401.4064]. +[136] L. Balkenhol, D. Dutcher, A. Spurio Mancini, A. Doussot, K. Benabed, S. Galli et al., A +Measurement of the CMB Temperature Power Spectrum and Constraints on Cosmology from +the SPT-3G 2018 TT/TE/EE Data Set, arXiv e-prints (2022) arXiv:2212.05642 +[2212.05642]. +[137] J. Zuntz, M. Paterno, E. Jennings, D. Rudd, A. Manzotti, S. Dodelson et al., CosmoSIS: +Modular cosmological parameter estimation, Astronomy and Computing 12 (2015) 45 +[1409.3409]. +[138] J. Torrado and A. Lewis, Cobaya: code for Bayesian analysis of hierarchical physical models, +J. Cosmology Astropart. Phys. 2021 (2021) 057 [2005.05290]. +– 50 – + +[139] A. La Posta, T. Louis, X. Garrido and J. C. Hill, Constraints on prerecombination early dark +energy from spt-3g public data, Phys. Rev. D 105 (2022) 083519. +[140] O. H. E. Philcox, M. M. Ivanov, M. Simonović and M. Zaldarriaga, Combining Full-Shape and +BAO Analyses of Galaxy Power Spectra: A 1.6\% CMB-independent constraint on H0, JCAP +05 (2020) 032 [2002.04035]. +[141] SDSS collaboration, D. J. Eisenstein et al., SDSS-III: Massive Spectroscopic Surveys of the +Distant Universe, the Milky Way Galaxy, and Extra-Solar Planetary Systems, Astron. J. 142 +(2011) 72 [1101.1529]. +[142] O. H. E. Philcox, Cosmology without window functions: Quadratic estimators for the galaxy +power spectrum, Phys. Rev. D 103 (2021) 103504 [2012.09389]. +[143] O. H. E. Philcox, Cosmology without window functions. II. Cubic estimators for the galaxy +bispectrum, Phys. Rev. D 104 (2021) 123529 [2107.06287]. +[144] B. Kalus, W. J. Percival, D. J. Bacon, E. M. Mueller, L. Samushia, L. Verde et al., A +map-based method for eliminating systematic modes from galaxy clustering power spectra with +application to BOSS, Mon. Not. Roy. Astron. Soc. 482 (2019) 453 [1806.02789]. +[145] A. Chudaykin, K. Dolgikh and M. M. Ivanov, Constraints on the curvature of the Universe +and dynamical dark energy from the Full-shape and BAO data, Phys. Rev. D 103 (2021) +023507 [2009.10106]. +[146] F.-S. Kitaura et al., The clustering of galaxies in the SDSS-III Baryon Oscillation +Spectroscopic Survey: mock galaxy catalogues for the BOSS Final Data Release, Mon. Not. +Roy. Astron. Soc. 456 (2016) 4156 [1509.06400]. +[147] S. A. Rodríguez-Torres et al., The clustering of galaxies in the SDSS-III Baryon Oscillation +Spectroscopic Survey: modelling the clustering and halo occupation distribution of BOSS +CMASS galaxies in the Final Data Release, Mon. Not. Roy. Astron. Soc. 460 (2016) 1173 +[1509.06404]. +[148] D. J. Marsh, E. Macaulay, M. Trebitsch and P. G. Ferreira, Ultra-light Axions: Degeneracies +with Massive Neutrinos and Forecasts for Future Cosmological Observations, Phys. Rev. D 85 +(2012) 103514 [1110.0502]. +[149] O. Pisanti, A. Cirillo, S. Esposito, F. Iocco, G. Mangano, G. Miele et al., PArthENoPE: +Public algorithm evaluating the nucleosynthesis of primordial elements, Computer Physics +Communications 178 (2008) 956 [0705.0290]. +[150] T. Nishimichi, G. D’Amico, M. M. Ivanov, L. Senatore, M. Simonović, M. Takada et al., +Blinded challenge for precision cosmology with large-scale structure: results from effective field +theory for the redshift-space galaxy power spectrum, Phys. Rev. D 102 (2020) 123541 +[2003.08277]. +[151] T. Simon, P. Zhang, V. Poulin and T. L. Smith, On the consistency of effective field theory +analyses of BOSS power spectrum, 2208.05929. +[152] F. Feroz, M. P. Hobson and M. Bridges, MULTINEST: an efficient and robust Bayesian +inference tool for cosmology and particle physics, MNRAS 398 (2009) 1601 [0809.3437]. +[153] F. Feroz, M. P. Hobson, E. Cameron and A. N. Pettitt, Importance Nested Sampling and the +MultiNest Algorithm, The Open Journal of Astrophysics 2 (2019) 10 [1306.2144]. +[154] L. Amendola and R. Barbieri, Dark matter from an ultra-light pseudo-Goldsone-boson, +Physics Letters B 642 (2006) 192 [hep-ph/0509257]. +[155] V. Poulin, T. L. Smith, T. Karwal and M. Kamionkowski, Early Dark Energy can Resolve the +Hubble Tension, Phys. Rev. Lett. 122 (2019) 221301 [1811.04083]. +– 51 – + +[156] R. E. Kass and A. E. Raftery, Bayes factors, Journal of the American Statistical Association +90 (1995) 773 +[https://www.tandfonline.com/doi/pdf/10.1080/01621459.1995.10476572]. +[157] A. Heavens, Y. Fantaye, E. Sellentin, H. Eggers, Z. Hosenie, S. Kroon et al., No Evidence for +Extensions to the Standard Cosmological Model, Phys. Rev. Lett. 119 (2017) 101301 +[1704.03467]. +[158] T. M. C. Abbott, M. Aguena, A. Alarcon, S. Allam, O. Alves, A. Amon et al., Dark Energy +Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing, +Phys. Rev. D 105 (2022) 023520 [2105.13549]. +[159] C. Heymans, T. Tröster, M. Asgari, C. Blake, H. Hildebrandt, B. Joachimi et al., KiDS-1000 +Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering +constraints, A&A 646 (2021) A140 [2007.15632]. +[160] A. G. Sánchez, R. Scoccimarro, M. Crocce, J. N. Grieb, S. Salazar-Albornoz, C. Dalla Vecchia +et al., The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic +Survey: Cosmological implications of the configuration-space clustering wedges, MNRAS 464 +(2017) 1640 [1607.03147]. +[161] C. Blake, A. Amon, M. Childress, T. Erben, K. Glazebrook, J. Harnois-Deraps et al., The +2-degree Field Lensing Survey: design and clustering measurements, Monthly Notices of the +Royal Astronomical Society 462 (2016) 4240 +[https://academic.oup.com/mnras/article-pdf/462/4/4240/18517346/stw1990.pdf]. +[162] M. Raveri and C. Doux, Non-Gaussian estimates of tensions in cosmological parameters, +Phys. Rev. D 104 (2021) 043504 [2105.03324]. +[163] P. Lemos, M. Raveri, A. Campos, Y. Park, C. Chang, N. Weaverdyck et al., Assessing tension +metrics with dark energy survey and Planck data, MNRAS 505 (2021) 6179 [2012.09554]. +[164] N. Schöneberg, J. Lesgourgues and D. C. Hooper, The bao+bbn take on the hubble tension, +Journal of Cosmology and Astroparticle Physics 2019 (2019) 029. +[165] DESI Collaboration, A. Aghamousa, J. Aguilar, S. Ahlen, S. Alam, L. E. Allen et al., The +DESI Experiment Part I: Science,Targeting, and Survey Design, arXiv e-prints (2016) +arXiv:1611.00036 [1611.00036]. +[166] LSST Dark Energy Science Collaboration, Large Synoptic Survey Telescope: Dark Energy +Science Collaboration, arXiv e-prints (2012) arXiv:1211.0310 [1211.0310]. +[167] K. Bechtol, S. Birrer, F.-Y. Cyr-Racine, K. Schutz, S. Adhikari, A. Banerjee et al., +Snowmass2021 Cosmic Frontier White Paper: Dark Matter Physics from Halo Measurements, +arXiv e-prints (2022) arXiv:2203.07354 [2203.07354]. +[168] D. Grin, M. A. Amin, V. Gluscevic, R. Hlozek, D. J. E. Marsh, V. Poulin et al., Gravitational +probes of ultra-light axions, BAAS 51 (2019) 567 [1904.09003]. +[169] C. Dvorkin et al., Dark Matter Physics from the CMB-S4 Experiment, in 2022 Snowmass +Summer Study, 3, 2022, 2203.07064. +[170] M. M. Ivanov, E. McDonough, J. C. Hill, M. Simonović, M. W. Toomey, S. Alexander et al., +Constraining Early Dark Energy with Large-Scale Structure, Phys. Rev. D 102 (2020) 103502 +[2006.11235]. +– 52 – + diff --git a/k9E_T4oBgHgl3EQf6BwH/content/tmp_files/load_file.txt b/k9E_T4oBgHgl3EQf6BwH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b2bfed4b217f959e1707e7f1915fc95898d5156 --- /dev/null +++ b/k9E_T4oBgHgl3EQf6BwH/content/tmp_files/load_file.txt @@ -0,0 +1,3398 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf,len=3397 +page_content='Prepared for submission to JCAP Ultra-light axions and the S8 tension: joint constraints from the cosmic microwave background and galaxy clustering Keir K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers,a,1 Renée Hložek,a,b Alex Laguë,c,d,b,a Mikhail M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov,e,f Oliver H.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' London WC2R 2LS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' UK 1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08361v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='CO] 19 Jan 2023 E-mail: keir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='rogers@utoronto.' metadata={'source': 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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='uk, gcabass@ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='edu, kakitsu@ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='edu, david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='marsh@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='uk Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We search for ultra-light axions as dark matter (DM) and dark energy particle candidates, for axion masses 10−32 eV ≤ ma ≤ 10−24 eV, by a joint analysis of cosmic mi- crowave background (CMB) and galaxy clustering data – and consider if axions can resolve the tension in inferred values of the matter clustering parameter S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We give legacy con- straints from Planck 2018 CMB data, improving 2015 limits on the axion density Ωah2 by up to a factor of three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' CMB data from the Atacama Cosmology Telescope and the South Pole Telescope marginally weaken Planck bounds at ma = 10−25 eV, owing to lower (and theoretically-consistent) gravitational lensing signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We jointly infer, from Planck CMB and full-shape galaxy power spectrum and bispectrum data from the Baryon Oscillation Spectroscopic Survey (BOSS), that axions are, today, < 10% of the DM for ma ≤ 10−26 eV and < 1% for 10−30 eV ≤ ma ≤ 10−28 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data strengthen limits, in particular at higher ma by probing high-wavenumber modes (k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4h Mpc−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS alone finds a pref- erence for axions at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ, for ma = 10−26 eV, but Planck disfavours this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, axions in a window 10−28 eV ≤ ma ≤ 10−25 eV can improve consistency between CMB and galaxy clustering data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', reducing the S8 discrepancy from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6σ, since these axions suppress structure growth at the 8h−1 Mpc scales to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We expect im- proved constraints with upcoming high-resolution CMB and galaxy lensing and future galaxy clustering data, where we will further assess if axions can restore cosmic concordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Axion structure formation model 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Linear theory 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Axion cosmology 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 S8 and the linear matter power spectrum 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Cosmic microwave background 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Galaxy clustering and the effective field theory of large-scale structure 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Galaxy power spectrum and bispectrum multipoles 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Introduction to the effective field theory of large-scale structure 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Axions and the effective field theory of large-scale structure 10 3 Data 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Cosmic microwave background 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Planck 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Atacama Cosmology Telescope 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 South Pole Telescope 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Baryon acoustic oscillations & supernovae 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispectrum 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Parameter inference 15 4 Results 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Cosmic microwave background, baryon acoustic oscillations & supernovae 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Planck 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 All CMB, BAO & supernovae 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispectrum 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 ΛCDM 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Parameter tension metrics 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 BOSS-only axion constraints 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Joint Planck and BOSS axion constraints 31 5 Discussion 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Comparison to previous work 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 ma = 10−26 eV 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Comparison to other axion probes and future prospects 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Axions as a resolution to the S8 parameter tension 41 6 Conclusions 42 1 Introduction While evidence for dark matter (DM) exists observationally [1–6], the fundamental nature of dark matter remains one of the greatest unsolved problems in science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Axions are a well- motivated particle candidate for DM [7–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Axions were proposed to solve the strong CP – 1 – problem [15–17] and can arise in a string theory “axiverse” where many axions of different masses are produced, suggesting that no single axion, but rather a mixture, can dominate the dark sector [15, 17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Depending on their particle mass, axions behave either as DM or as a scalar field dark energy (DE) component [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Axions are proposed to resolve the so-called “small-scale crisis” in the clustering of matter [23–25], as they suppress the growth of small-scale (sub-Mpc) structure depending on the mass of the axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Improvements in our ability to model astrophysical effects in the formation of Galactic and sub-Galactic structure [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 26] and a more complete census of the Milky Way (MW) satellite galaxy population [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 27] have demonstrated the viability of astrophysical solutions to the “small-scale crisis.” Further, complementary constraints from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the Lyman-alpha forest [28] and the MW sub-halo mass function [29] have ruled out the axion mass ∼ 10−22 eV as being all the DM that is invoked to address small-scale issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, advances in our understanding of axion structure formation including astrophysical effects [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 30–35] now allows us to test empirically using cosmological data the existence of ultra-light axions, as motivated from fundamental theory, across the full mass range where their wavelength is astrophysically large (10−32 eV ≤ ma ≤ 10−18 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Observational constraints on the axion typically limit a combination of the axion mass and the axion energy density (or the fraction that axions make up of the total DM energy density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Multiple tracers have been used to constrain the allowed mass and density of axions including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the cosmic microwave background [CMB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 22, 36, 37], galaxy clustering [38], galaxy weak lensing [39, 40], the Lyman-alpha forest [28, 31, 41–44], dwarf galaxies [29, 45, 46], 21 cm observations [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A single axion species as the only DM candidate is ruled out by the Lyman-alpha forest for masses less than 2 × 10−20 eV (at 95% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=') [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, axions as a component of the DM or DE (either as a mixture of axions of different masses or in combination with other dark sector species) is still viable across the ultra-light mass range (10−32 eV ≤ ma ≤ 10−18 eV) and above (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [50, 51] for reviews of searches for axion-like particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we search for axions through their gravitational imprint in a compendium of CMB and large-scale structure data for ma ≤ 10−25 eV and allowing for sub-dominant axion energy densities Ωah2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We present legacy constraints from Planck 2018 CMB data [52], the first study of high-resolution CMB data from the Atacama Cosmology Telescope [ACT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 53] and the South Pole Telescope [SPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 54], and a full joint analysis of Planck CMB and full-shape galaxy power spectrum and bispectrum data from the Baryon Oscillation Spectroscopic Survey [BOSS, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We include high-resolution CMB data and we model axions in galaxy power spectrum data to smaller scales than previously considered (wavenumbers k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 h Mpc−1) as probing smaller scales gains sensitivity to the scale-dependent suppression of heavier axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The challenge is that exploiting smaller scales typically requires robust modelling of axion structure formation into the non-linear regime, moving beyond the well-established linear-order theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', axionCAMB [22, 57], AxiCLASS [58]) that is sufficient for Planck analyses1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] modelled axions into the mildly non-linear regime using the effective field theory of large- scale structure [EFT of LSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 61–68], finding that, similarly to massive neutrinos [66, 69], the galaxy bias and counterterm parameters capture non-linear effects after modifying the input linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Axion-induced wave effects are suppressed as they manifest on scales suppressed in the linear power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we find that current high-resolution CMB data from ACT-DR4 and SPT- 1Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [59, 60] discuss the robustness of the fluid approximations that are typically required to make linear-order axion perturbation calculations computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 2 – 3G can be modelled by linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, upcoming and proposed future high-resolution CMB lensing data from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', ACT, SPT, Simons Observatory, CMB-S4, and galaxy weak lensing data from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the Dark Energy Survey (DES), Rubin Observatory, Euclid will gain sensitivity to heavier axions (ma > 10−25 eV) but will probe fully non-linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [39] searched for axions as the only DM species in DES-Y1 galaxy shear data using an axion halo model that analytically captures the effect of axions on the formation and clustering of DM halos [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [71] extended this model to the case of mixed axion and cold DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A complementary approach is to capture non-linear modes using machine learning models called emulators which are trained on the outputs of cosmological simulations [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 72–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Emulators have been used successfully to set DM constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', with the Lyman- alpha forest [28, 31, 80], where astrophysical effects can be captured in training simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Accurate emulator predictions rely on accurate input simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There is much progress in our ability to simulate axion structure formation using fluid approximations [81] and by solving the full axion field equations [32–34, 82–86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There are discrepancies between CMB, galaxy clustering and galaxy shear inferences on the amplitude of matter density fluctuations [see 87, for a recent review].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is typically characterised by the matter clustering parameter σ8, the amplitude at redshift z = 0 when averaged over 8 h−1 Mpc scales, or by the degenerate combination S8 ≡ � Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 σ8 (where Ωm is the matter energy density), which is well constrained by large-scale structure experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The statistical significance of the so-called S8 tension ranges from 2 to 3 σ depending on the data considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' galaxy shear, in particular, drives the largest discrepancies with CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Notwithstanding undetected systematic errors in the data, the S8 tension has proposed solutions based on physics beyond ΛCDM typically by introducing either a time-dependence or a scale-dependence in the DM dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This can be achieved by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', coupling DM to DE [88, 89], a complex dark sector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', atomic DM [90–92]), decaying DM [93, 94], or baryon-DM interactions [95];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [96] more generally considers modifications to non-linear clustering including the effects of baryonic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ultra-light axions form a component of the dark sector with a scale-dependent growth factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore hypothesise that axions could alleviate the S8 tension, by behaving like standard cold DM at the scales probed by current CMB surveys, while suppressing the growth of structure at the smaller scales to which galaxy surveys are sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We investigate this hypothesis by jointly analysing CMB and galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The inclusion of galaxy shear measurements is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Another discrepancy in the ΛCDM model is the H0 tension, the ∼ 5σ difference in the Hubble expansion rate today H0 as inferred from different direct and indirect distance ladders [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Many proposed solutions to the H0 tension based on new physics, however, exacerbate the discrepancy in S8 [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Models of ultra-light axions, with ma ∼ (10−27 − 10−26) eV, combined with modifications to the dynamics of the DE component [98–100] are invoked to alleviate simultaneously both parameter tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we stress the importance of assessing tension in the full parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In testing the extent to which ultra-light axions can improve consistency between CMB and large-scale structure data, we therefore use metrics of tension that account for the full non-Gaussian posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In § 2, we introduce our model for axion structure formation: the linear theory in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 and the EFT of LSS that we use as our non-linear theory in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We discuss our data in § 3: CMB in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1, baryon acoustic oscillations (BAO) and supernovae in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, full-shape BOSS galaxy clustering in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 and our parameter inference methods in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We present results from the CMB, BAO and supernovae in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 and from BOSS galaxy clustering in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In – 3 – § 5, we discuss these results and draw conclusions in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2 Axion structure formation model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Linear theory 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Axion cosmology In order to model the effect of ultra-light axions (ULAs) on the cosmic microwave back- ground (CMB), we calculate linear-order perturbations using the Einstein-Boltzmann solver axionCAMB2 [22, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The fundamental equation governing the axion field φ is the Klein- Gordon equation: □φ − m2 aφ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) where □ is the d’Alembert operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We consider a temperature-independent axion mass, which is appropriate for string theory axions, where the mass switches on at a high energy scale (typically the geometric mean of the supersymmetry scale and the Planck scale [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We ignore self-interactions of the axion (valid for initial field misalignment angles that are not tuned close to π [101–104]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The axion-photon coupling can affect CMB polarisation if it is large (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [105–108]), but does not back-react significantly on the axion DM density (although see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [109]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmologically, all other axion couplings can lead only to a small thermal population of axions, which is negligible for couplings consistent with astrophysical limits, current constraints on the effective number of relativistic species Neff, and in the mass range that we consider (for related discussion, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [110, 111]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thus, we set the axion couplings to zero and consider only gravitational effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The gravitational couplings of the axion are contained in the metric dependence of □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' At early times, defined as when the Hubble expansion rate H ≫ ma, axionCAMB solves fluid equations, equivalent to the full Klein-Gordon equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1)) at linear order in spatial fluctuations of φ and metric perturbations, in the synchronous gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The homoge- neous axion field begins to oscillate when H ≈ ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' After this time, axionCAMB uses the WKB approximation to adopt an effective fluid description [112–115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The fluid model is equivalent to the Madelung formulation [116] and is accurate up to shell crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For further discussion of the accuracy of the adopted approximations, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [22, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The main physical features that distinguish ULAs from standard ΛCDM components, as pertains to cosmological observables, are two-fold [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' First, the slow roll of the axion field when H ≫ ma leads to a distinctive background evolution equivalent to a fraction of the matter component behaving like an early form of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This leads to differences in the diffusion damping and Sachs-Wolfe contributions to the CMB, changes the sound horizon, and changes the distance to the surface of last scattering (if axions begin their oscillation after matter-radiation equality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', for ma ≤ 10−28 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Second, the gradient terms in the Klein- Gordon equation appear as an effective pressure, opposing gravitational collapse, leading to a Jeans scale for the ULAs and, consequently, a suppression in the amplitude of density perturbations on small scales [21, 113, 117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 S8 and the linear matter power spectrum Figure 1 illustrates this Jeans scale (for wavenumbers k above the Jeans wavenumber, the linear matter power spectrum P linear(k) is suppressed relative to the ΛCDM limit) and how 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/dgrin1/axionCAMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 4 – −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 log � kP linear(k) � h−2 Mpc2�� ΛCDM, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='834 ma = 10−28 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='001, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='811 ma = 10−27 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='002, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='798 ma = 10−26 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='007, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='773 ma = 10−25 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='033, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='793 ma = 10−24 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='109, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='828 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 log � k � h Mpc−1�� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 P linear ULA DM(k) P linear ΛCDM(k) S8 integral kernel kPlanck max kBOSS max Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of ultra-light axions on the linear matter power spectrum (top panel) and the ratio to the ΛCDM limit (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Power spectra are shown at the 95% upper limit on the axion energy density Ωah2 given Planck CMB and BOSS galaxy clustering data and thus reflect the tightening density constraint at lower axion mass ma (see Table 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' all parameters fixed apart from ma, Ωah2, the cold DM density Ωch2 and the dark energy fraction ΩΛ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In the bottom panel, the shaded area shows the Fourier-space filter k2W 2(k) (in arbitrary units) of kP linear(k) in the integral calculation of the matter clumping factor S8 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2)), where W(k) is the Fourier transform of a top-hat filter in real space with radius 8 h−1 Mpc and Ωm is the (fixed) total matter energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The shaded area thus indicates the wavenumbers to which S8 is most sensitive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for ma ≤ 10−25 eV, axions suppress power and thus lower S8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for ma ≥ 10−24 eV, the axion-induced power suppression is at too large a wavenumber to change S8 significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The dashed line indicates the maximum wavenumber which we probe in the Planck likelihood (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the dotted line indicates the maximum wavenumber which we model in the BOSS galaxy power spectrum (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 5 – it depends on axion mass ma: the lighter the axion, the larger the power suppression scale (the smaller the suppression wavenumber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 1, we show linear matter power spectra at the 95 % upper limits on the axion energy density Ωah2 given a combination of Planck CMB and BOSS galaxy clustering data (see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As will be expanded later, these data set stronger constraints on the amount of axions at lower mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 1 thereby illustrates how a lower Ωah2 reduces the strength of the power suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The amplitude of the linear matter power spectrum today (redshift z = 0) is often summarised by the cosmological parameter σ8 = � dlnk k3 2πW 2(k)P linear(k), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) where W(k) is the Fourier transform of a top-hat filter in real space with radius 8 h−1 Mpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' large-scale structure (LSS) data are then typically used to constrain the parameter combina- tion S8 = � Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 σ8, where Ωm is the total matter energy density3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S8 is therefore sensitive to a filtered integral of the linear matter power spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 1 shows this filter and indicates the scales to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It can be seen that, for ma ≥ 10−24 eV, the power suppression is on too small scales to lower significantly S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For ma ≤ 10−28 eV, the data constraint is too strong to leave an appreciable amount of axions that significantly lowers S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, there is a window for ma ∈ [10−27, 10−25] eV, where the presence of axions significantly lowers S8 and is allowed by the data we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore discuss the prospects of axions resolving discrepancies in the inferred values of S8 given CMB and LSS data in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Cosmic microwave background In modelling CMB anisotropies, we consider only adiabatic initial perturbations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we defer a search for isocurvature perturbations to future work [see 36, 119, for consequences on the energy scale of inflation].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2 illustrates how axions change the CMB temperature TT, polarisation EE, cross TE and lensing potential φφ angular power spectra Cℓ as a function of multipole ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For the multipoles probed by current data (Planck, ACT-DR4, SPT-3G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3), the impact of axions for ma ≥ 10−25 eV on these data is small compared to statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For axions that become a dark matter component before matter-radiation equality (ma ≥ 10−27 eV), much of the data constraint comes from the change in the relative heights of acoustic peaks arising from the change in the matter-to-radiation ratio (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For axions that still behave like dark energy after matter-radiation equality (ma ≤ 10−28 eV), much of the data constraint (once the angular size of the sound horizon is constrained) comes from the change in the integrated Sachs-Wolfe effect at the smallest multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The lensing power spectrum is sensitive to all axion masses through a scale (multipole)-dependent suppression arising from the matter power spectrum (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [22, 36, 120] for summaries of the effects of axions on the CMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Since we conservatively ignore small-scale CMB lensing anisotropies (we use only mul- tipoles 8 ≤ L ≤ 400 as recommended by the Planck Collaboration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1), we ignore non-linear effects in the lensing power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [48, 71, 120] indicated that this is a good approximation as, for axion mass ma < 10−25 eV, axions with observationally-allowed 3The parameter combination S8 was historically optimised to project away parameter degeneracies in galaxy weak lensing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We typically use this parameter combination in this work with galaxy clustering since it still does a good job of projecting away degeneracies and it simplifies comparisons to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 6 – 0 1000 2000 3000 4000 0 2500 5000 ℓ(ℓ+1) 2π CTT ℓ � µK2� Max post (CMB+BAO+SNe): ma = 10−25 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03 ma = 10−25 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 ma = 10−27 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 Planck ACT-DR4 SPT-3G 0 1000 2000 3000 4000 −100 0 100 ℓ(ℓ+1) 2π CTE ℓ � µK2� 0 1000 2000 3000 4000 0 50 ℓ(ℓ+1) 2π CEE ℓ � µK2� 0 100 200 300 ℓ 0 1 107[ℓ(ℓ+1)]2 2π Cφφ ℓ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of ultra-light axions on (from top to bottom) the cosmic microwave background (CMB) TT, TE, EE and φφ angular power spectra, compared to data from Planck (blue), the Atacama Cosmology Telescope (ACT-DR4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' red) and the South Pole Telescope (SPT-3G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show the maximum posterior model (solid) given Planck, ACT and SPT CMB, galaxy baryon acoustic oscillation (BAO) and supernovae (SNe) data for axion mass ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We compare this to the cases where the axion energy density Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 for ma = 10−25, 10−27 eV (all other parameters fixed to their maximum posterior values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' At the multipoles currently probed, axions for ma ≥ 10−25 eV are poorly constrained by these data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' upcoming high-resolution CMB lensing measurements will increase sensitivity to heavier axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Data are shown as points with 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' errorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' energy densities do not non-linearly cluster at observationally-relevant wavenumbers and red- – 7 – 200 400 600 800 1000 L −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 ∆Cφφ L /Cφφ L ma = 10−25 eV Halo model − linear Planck CMB − S4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fractional difference in the cosmic microwave background (CMB) lensing potential angular power spectrum Cφφ L as a function of multipole L between a non-linear axion halo model [71] and the linear theory prediction from axionCAMB [22, 57] (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show the difference in the theoretical prediction for an axion of mass ma = 10−25 eV which constitutes 50% of the total dark matter energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There is a negligible difference for L ≤ 400 compared to the error in the Planck data that we use (orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The inclusion of non-linearities will however be necessary for future CMB surveys such as CMB-S4 (purple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' forecasted data error from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For ma ≥ 10−25 eV, the non-linear effects are only significant for larger multipoles than we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A halo model was developed to capture the effects of ultra-light axions on non-linear scales in CMB and galaxy lensing [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Using this halo model, we reconsider the impact of ignoring non-linear effects in our Planck CMB lensing analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We conclude that this observable is well captured using linear theory for the L range considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Forthcoming measurements of small-scale lensing anisotropies from ground-based CMB (and future galaxy weak lensing) experiments will increase sensitivity to smaller axion suppression scales and, hence, larger ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate that non-linear modelling will become necessary in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Galaxy clustering and the effective field theory of large-scale structure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Galaxy power spectrum and bispectrum multipoles In order to capture the anisotropic clustering in the galaxy distribution arising from redshift- space effects, we model galaxy power spectrum multipoles [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 67]: Pℓ(k, z) ≡ 2ℓ + 1 2 � 1 −1 dµ Lℓ(µ)Pg(k, µ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3) Here, Pg(k, µ, z) is the full anisotropic galaxy power spectrum depending on wavenumber k, the cosine of the angle between the wavenumber and the line-of-sight µ, and redshift z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 8 – Lℓ(µ) are Legendre polynomials indexed by multipole ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Non-linear redshift-space distortions (“fingers of God” [121]) are non-trivial to model with accuracy on small scales, while the power suppression effect of axions is stronger as wavenumber increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Therefore, to increase the constraining power from galaxy data and following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [122], we estimate the reconstructed real-space4 galaxy power spectrum Q0(k, z) ≡ P0(k, z) − 1 2P2(k, z) + 3 8P4(k, z) [see also 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [122] demonstrates that this estimator effectively down-weights information in line-of- sight modes (µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3) that are heavily contaminated by redshift-space distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, we extract information on the post-reconstructed baryon acoustic oscillation feature using the Alcock-Paczynski (AP) parameters: α∥(z) ≡ Hfid(z)rfid s (zd) H(z)rs(zd) , α⊥(z) ≡ DA(z)rfid s (zd) Dfid A (z)rs(zd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4) Here, H(z) is the Hubble parameter, rs(zd) is the sound horizon at the redshift of decoupling, DA(z) is the angular diameter distance, and “fid” indicates a fiducial cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For the first time, in order to extract information beyond the two-point statistics described above, we model the effect of axions on the galaxy bispectrum (Fourier transform of the three-point correlation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we consider only the angle-averaged bispectrum monopole (ℓ = 0) B0(k1, k2, k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Introduction to the effective field theory of large-scale structure In order to model the effect of ULAs on the redshift-space galaxy power spectrum and bis- pectrum, we calculate mildly non-linear perturbations using the effective field theory of large-scale structure [EFT of LSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 61–63, 68] as implemented in the Einstein-Boltzmann solver CLASS-PT5 [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Following the effective field theory of large-scale structure, this is a schematic view of our power spectrum model [65, 67]: Pℓ(k, z) = P tree ℓ (k, z) + P one−loop ℓ (k, z) + P counterterms ℓ (k, z) + P stochastic ℓ (k, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5) Here, P tree ℓ (k, z) captures linear bias and redshift-space distortions (∝ P linear(k, z), which is the linear matter power spectrum) [124];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P one−loop ℓ (k, z) captures perturbative corrections up to one loop in order (∝ k2P linear(k, z) on large scales);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P counterterms ℓ (k, z) captures ultra- violet counterterms that consistently account for small-scale physics (∝ k2P linear(k, z));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and P stochastic ℓ (k, z) captures stochastic effects including shot noise and fingers of God (∝ constant, plus corrections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We also include infrared resummation to account for non-perturbative long-wavelength displacements [67, 125–127], and account for the so-called Alcock-Paczynski distortion [128] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the effects of an incorrect fiducial cosmology above) by wavevector rescalings [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 55, 104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The bispectrum model can be given schematically in 3D space [129]: B(k1, k2, z) = Btree(k1, k2, z) + Bcounterterms(k1, k2, z) + Bstochastic(k1, k2, z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6) for wavevectors k1, k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Here, Btree(k1, k2, z) captures tree-level perturbations (∝ P 2 linear(k, z));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bcounterterms(k1, k2, z) is a proxy for ultraviolet fingers-of-God counterterms (∝ k2 ∥P 2 linear(k, z), where k∥ is the wavenumber for modes parallel to the line of sight);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and Bstochastic(k1, k2, z) 4I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', without redshift-space distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/Michalychforever/CLASS-PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Background evolution and linear matter power spec- trum calculations are done in axionCAMB, which are passed to CLASS-PT for non-linear corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 9 – captures stochastic effects (∝ Plinear(k, z) + constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For the bispectrum, the tree-level model is sufficiently accurate for the small wavenumbers that we consider in our data (k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 h Mpc−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We again account for infrared resummation [127] and the Alcock- Paczynski distortion as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We then integrate over external angles to calculate the bis- pectrum monopole B0(k1, k2, k3), which can be compared to data without additional window convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In comparing to BOSS data, we multiply B0 by a discreteness weight vector to account for the finite resoltuion of the Fourier grid6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Axions and the effective field theory of large-scale structure The EFT of LSS was originally developed with the assumption of cold, collisionless dark matter (CDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] in noting that the effect of ultra-light axion dark matter (not to cluster below a characteristic scale) is qualitatively the same as for free-streaming neutrinos (although the physical reason is different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [69] found that, to first order, the additional counterterms needed to account for the effect of neutrinos have the same functional form as existing CDM counterterms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore assume that the additional axion-induced counterterms will also have the same functional form, although with different constants of proportionality which must be marginalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] demonstrated that linear-order axion-wave corrections are negligible since they manifest on scales that are already heavily suppressed in the linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] also demonstrated that axion and cosmological parameters can be inferred without bias from a simulated BOSS galaxy catalogue in the presence of axions, when marginalising over the EFT of LSS model presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It follows that phenomenology of axions can be captured by only modifying the background evolution and linear matter power spectrum (presented in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 and calculated using axionCAMB) as input to the EFT of LSS model presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We marginalise over a full set of EFT of LSS nuisance parameters: linear b1, quadratic b2, tidal bG2 and third-order bΓ3 galaxy biases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' monopole c0, quadrupole c2, hexadecapole c4, fingers-of-God ˜c and bispectrum c1 counterterms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' power spectrum Pshot and bispectrum Bshot shot noise parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and power spectrum scale-dependent stochastic parameters a0 and a2 [more details are given in 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Figure 4 shows the effect of ultra-light axions on the galaxy power spectrum, for ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The solid line shows the power spectrum for axion energy density Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03, while the dashed line shows the power spectrum for Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09 (all other parameters fixed to the same values as for the solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There are two main effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The first is a scale-dependent suppression in the power spectrum, which gets stronger on smaller scales (and so is most significantly seen in the Q0 statistic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is qualitatively similar to the effect in the linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect is physically caused by axions not clustering on scales below their Jeans wavelength at matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The second effect is a small-scale enhancement in the galaxy quadrupole (and, to a much-lesser extent, hexadecapole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This effect is caused by a reduction in the fingers of God effect owing to lower peculiar velocities at weaker matter over-densities, although this is degenerate with the EFT of LSS counterterm parameter that controls the fingers of God amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These effects are discussed in further detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The dotted lines lower the primordial power spectrum amplitude As with respect to the solid lines and thus suppress the galaxy power spectrum at all wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5 shows the effect of ultra-light axions on the galaxy bispectrum monopole, for ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As above, the dashed line shows the effect of increasing the axion density 6BOSS discreteness weight vectors can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/oliverphilcox/full_shape_ likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 10 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 k [h Mpc−1] −1000 0 1000 2000 kPℓ(k) � h−2 Mpc2� Monopole P0(k) Quadrupole P2(k) Hexadecapole P4(k) Real-space Q0(k) Max like (BOSS+Planck): Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03 Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09 As = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 × 10−9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of ultra-light axions (mass ma = 10−25 eV, axion energy density Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09, dashed lines) on the Baryon Oscillation Spectroscopic Survey (BOSS) galaxy power spectrum, com- pared to maximum-likelihood model parameters (with Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03, solid lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' all other parameters fixed to maximum-likelihood values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The solid line shows the maximum likelihood given BOSS galaxy power spectrum and Planck cosmic microwave background (CMB) data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' here, we maximise the likelihood with respect to all cosmological and EFT of LSS parameters, including those that are usually analytically marginalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We also show the case (dotted lines) where we lower the primor- dial power spectrum amplitude from its best-fit value given Planck + BOSS (As = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 × 10−9) to its best-fit value given only BOSS galaxy power spectra (As = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='53 × 10−9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This illustrates the lack of degeneracy with heavier axions (ma = 10−25 eV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' nonetheless, a good fit to BOSS data is maintained given the addition of Planck data by reducing the best-fit linear galaxy bias (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate degeneracy between high As and high Ωah2 for ma ≤ 10−28 eV since the large Jeans scale suppresses all BOSS wavenumbers in a similar way as reducing As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show the galaxy power spectrum monopole P0(k) (blue), quadrupole P2(k) (red), hexadecapole P4(k) (orange), and the reconstructed real-space galaxy power spectrum Q0(k) (green), as a function of wavenumber k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data are shown as points with 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' errorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' while keeping all other parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect is a small scale-dependent suppression in the bispectrum, which gets stronger for smaller-scale triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, at the current statistical precision of BOSS data and on the relatively large scales modelled here in the bispectrum (k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 h Mpc−1), the effect is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate that axions will impact the smaller-scale, one-loop bispectrum [130, 131] more strongly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we will investigate this in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 11 – 0 10 20 30 40 50 60 Triangle index −40 −20 0 20 40 60 80 100 10−4k1k2k3B0(k1, k2, k3) � h−3 Mpc3� BOSS bispectrum data Max post (BOSS+Planck): Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='003 Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of ultra-light axions (ma = 10−25 eV, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09, dashed line) on the BOSS galaxy bispectrum monopole B0(k1, k2, k3), compared to maximum-posterior model parameters (with negligible axion densities, solid line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' all other parameters fixed to their maximum posterior values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The solid line shows the maximum posterior given BOSS power spectrum and bispectrum and Planck CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show B0 as a function of k1, k2, k3 wavenumber triangles, where triangle index increases first with k1, then with k2, and then with k3, for [k1, k2, k3] ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08] h Mpc−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', wavenumber triangles with smaller sides on the left and larger sides on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data are shown as points with 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' errorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Data Description Nuisance pars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' § Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Planck 2018 CTT,TE,EE,φφ ℓ : ℓ ≤ 2508, L ≤ 400 aPlanck 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 [2] ACT-DR4 CTT,TE,EE ℓ : 326 ≤ ℓ ≤ 4325 yp 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 [53] SPT-3G CTE,EE ℓ : 300 ≤ ℓ ≤ 2999 Fixed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 [54] BAO + SNe 6dFGS, MGS, BOSS DR12, JLA α, β, δM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 [132–135] BOSS full-shape P0,2,4(k, z), Q0(k, z), B0(k, z), AP EFT of LSS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 [55] Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A summary of the data used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3 Data In Table 1, we give a summary of the data used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3, we give more details about the data and, in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4, we give details on our parameter inference method including the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 12 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Cosmic microwave background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Planck We consider baseline Planck 2018 CMB temperature, polarisation and lensing angular power spectra [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use: the low-multipole (2 ≤ ℓ ≤ 29) temperature TT auto- spectrum likelihood commander_dx12_v3_2_29;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the low-multipole (2 ≤ ℓ ≤ 29) polari- sation EE auto-spectrum likelihood simall_100x143_offlike5_EE_Aplanck_B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the high- multipole, nuisance-marginalised, TT (30 ≤ ℓ ≤ 2508), TE and EE (30 ≤ ℓ ≤ 1996) likelihood plik_lite_v22_TTTEEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and the lensing φφ auto-spectrum likelihood (8 ≤ L ≤ 400) smicadx12_Dec5_ftl_mv2_ndlcpp_p_teb_consext8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As we use compressed, nuisance- marginalised likelihoods, we have remaining a single nuisance calibration parameter aPlanck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [22] demonstrated that there is no statistically-significant effect on axion parameter infer- ence if re-marginalising nuisance foreground parameters in an axion model with Planck data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The use of Planck 2018 data to constrain Ωah2 is an update from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36, 38], which used Planck 2015 data [52], and from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [39], which used Planck 2018 data to constrain only ma in the case where axions are all the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The main differences from 2015 data are a new low-ℓ polarisation likelihood and a larger-scale cut in the lensing likelihood, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Lmin goes from 40 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate improved bounds on the lightest axions from this additional large-scale information (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 7 for a breakdown of how 2018 data improves axion limits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Atacama Cosmology Telescope We consider Atacama Cosmology Telescope (ACT) data release 4 (DR4) temperature and polarisation angular power spectra [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use the baseline nuisance-marginalised (“CMB- only”) likelihood actpollite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This includes TT power spectra for 576 ≤ ℓ ≤ 4325 and TE and EE power spectra for 326 ≤ ℓ ≤ 4325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The foreground marginalisation leaves a single nuisance parameter yp, which is an overall polarisation efficiency that re-scales the TE and EE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Our baseline analysis combines ACT and Planck (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, the cross-covariance between these data has not yet been released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Therefore, to reduce the amount of cross-covariance that we ignore, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [53] in setting the minimum multipole in the ACT TT spectrum ℓmin = 1800, with no cut on the TE and EE spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [53] found that these approximations are sufficient to keep the underestimation of parameter uncertainties to less than 5%, including for one-parameter extensions of the standard cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 South Pole Telescope We consider South Pole Telescope (SPT-3G) TE and EE angular power spectra for 300 ≤ ℓ ≤ 29997 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use the baseline spt3g_2020 likelihood that has twenty nuisance foreground and calibration parameters8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In order to reduce the dimensionality of the parameter space, we fix the nuisance parameters to fiducial values9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We confirm that fixing these parameters makes no difference to the inferred cosmological posterior distribution by comparing to the case where all nuisance parameters are marginalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Our baseline analyses combine SPT with Planck 7In the latter stages of manuscript preparation, SPT-3G TT angular power spectra were released [136];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we will include these data in a future analysis, although we do not anticipate a significant change to our results since the multipoles contained in this release are already covered by the ACT data that we use (§ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 8We write a CosmoSIS [137] wrapper to the Cobaya [138] likelihood available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/ xgarrido/spt_likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 9We take fiducial values to be the maximum prior probability values from the fiducial SPT-3G analysis: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/xgarrido/spt_likelihoods/blob/master/spt3g_2020/TEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='yaml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 13 – (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) and ACT (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [54] in ignoring the cross-covariance between SPT and Planck since the survey sky overlap is small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we ignore cross-covariance between SPT and ACT for the same reason [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 139, for similar assumptions].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Baryon acoustic oscillations & supernovae We consider a compendium of galaxy baryon acoustic oscillation (BAO) data from: the 6dF Galaxy Survey (6dFGS) at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='106 [132];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the Sloan Digital Sky Survey data release 7 Main Galaxy Sample (SDSS DR7 MGS) at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 [133];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and the Baryon Oscillation Spectroscopic Survey data release 12 (BOSS DR12) at z = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='51, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61] [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These galaxy samples are largely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3, we will consider instead the full-shape galaxy power spectrum and bispectrum as measured from the BOSS DR12 galaxy sample, which captures the BOSS BAO information plus additional information in the full power spectrum and bispectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In the data in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3, the BAO information is extracted in the Alcock-Paczynski parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4) from the reconstructed power spectrum, taking into account their covariance with the full-shape information [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We never combine the full-shape BOSS likelihood in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 with the “standard” BOSS BAO likelihood (used in this section) as they contain identical BAO information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We consider a compendium of type Ia supernovae (SNe) data from the Joint Light-curve Analysis (JLA) [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We marginalise over the shape parameter α, the colour parameter β and the magnitude parameter δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispec- trum We use the twelfth data release of the Baryon Oscillation Spectrosopic Survey (BOSS DR12) [56, 134], which is part of the Sloan Digital Sky Survey (SDSS) [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This data release contains ∼ 8 × 105 galaxies across two redshift slices (LOWZ sample: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' CMASS sample: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75) and across both the north and south Galactic cap (NGC/SGC) sky cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use the window-free galaxy power spectrum and bispectrum measurements described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [55], which are measured respectively with the approaches of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [142] and [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We bin in k with ∆k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='005 h Mpc−1 for power spectra and ∆k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01 h Mpc−1 for bispectra, with minimum wavenumber kmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01 h Mpc−1 to avoid large-scale systematics [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As discussed in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, we fix the maximum wavenumber kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 h Mpc−1 for the power spectrum (using the ℓ = 0, 2, 4 multipoles) Pℓ(k, z) [55, 145], kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 h Mpc−1 for the bispectrum monopole [55, 129] B0(k, z), and we include the Q0(k, z) statistic for k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4] h Mpc−1 following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We extract from the BOSS galaxy power spectrum information on the post-reconstructed BAO feature using the AP parameters (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use power spectrum, bispectrum and AP measurements for each of the four redshift/sky cuts (NGC and SGC, both in redshift samples with central redshifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='38 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The BOSS power spectrum and bispectrum data for the North Galactic cap at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='38 are respectively shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These data are analysed with the EFT of LSS model presented in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we use the existing public BOSS likelihood10, with the covariance estimated from 2048 MultiDark-Patchy simulations [146, 147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The EFT of LSS nuisance parameters (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) are allowed to differ independently for each of the four BOSS data cuts due to their different redshifts and calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Only b1, b2 and bG2 enter the model non-linearly: the others are analytically marginalized and only this partially-marginalised likelihood is numerically sampled (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 for details about our numerical sampling approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/oliverphilcox/full_shape_likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 14 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Parameter inference All the likelihoods presented in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 are implemented in the cosmological parameter estimation code CosmoSIS [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We infer the posterior distribution for an axion cosmolog- ical model with uniform prior distributions on: the Hubble parameter h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the baryon energy density Ωbh2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the cold dark matter energy density Ωbh2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the axion energy density Ωah2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the primordial power spectrum amplitude As;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the primordial power spectrum tilt ns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and the reionisation optical depth τ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We fix the neutrino energy density Ωνh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0006442 with one massive neutrino at its minimally-allowed mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [71, 120, 148] discuss degeneracies between axion and neutrino density parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We consistently calculate the helium abun- dance given the baryon density and number of neutrinos using the bbn_consistency module [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We often show the posterior for derived cosmological parameters: the total matter energy density today Ωm (to which axions always contribute);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and the matter clumping fac- tor S8 ≡ � Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 σ8, where σ8 is the amplitude of the linear matter power spectrum averaged over 8 h−1 Mpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We do not vary the axion mass ma, but rather follow Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36, 38] by inferring the posterior for fixed values of ma ∈ [10−32 eV, 10−24 eV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is because the full posterior projected in the ma - Ωah2 plane has a highly non-trivial degeneracy, which is difficult to sample numerically in a converged manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We defer solving this sampling prob- lem to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We consider neither lighter axions as these are indistinguishable from a cosmological constant, nor heavier axions as these are indistinguishable from cold dark matter on the scales probed by the data we use [see 39, for more discussion and tests on axion prior choices].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use a uniform prior on the ACT calibration parameter yp and the three supernovae standardisation parameters [α, β, δM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use a Gaussian prior on the Planck calibration parameter aPlanck ∼ N(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For the EFT of LSS nuisance parameters (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2), we use the following priors (from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [55]): b1 ∼ U(0, 4), b2 ∼ N(0, 12), bG2 ∼ N(0, 12), bΓ3 ∼ N �23 42(b1 − 1), 12 � c0 [h−1 Mpc]2 ∼ N(0, 302), c2 [h−1 Mpc]2 ∼ N(30, 302), c4 [h−1 Mpc]2 ∼ N(0, 302), ˜c [h−1 Mpc]4 ∼ N(500, 5002), c1 [h−1 Mpc]2 ∼ N(0, 52), Pshot ∼ N(0, ¯n−2), Bshot ∼ N(1, ¯n−2), a0 ∼ N(0, ¯n−2), a2 ∼ N(0, ¯n−2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) where the inverse galaxy number density ¯n−1 = 5000 [h−1 Mpc]3 for the high-z samples and ¯n−1 = 3500 [h−1 Mpc]3 for the low-z samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For discussion on these priors and a comparison to the choices made in the PyBird implementation of the EFT of LSS model (whose parameters are a linear combination of the above and which was used in a previous axion analysis [38]), see § 5 and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [150] and [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We numerically sample posterior distributions using the importance nested sampling algorithm MultiNest [152, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We use 480 live points and we stop chains when posterior weights reach a tolerance of 1% of their maximum, thus ensuring that we sample the bulk of the posterior weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We check that our chains are converged with respect to the number of live points and tolerance by running test chains with 3600 live points and a tolerance of 10−5 and determining no shift in inferred distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 11When considering BOSS data alone, we do not vary τ since large-scale structure data are insensitive to this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 15 – −32 −30 −28 −26 −24 log [Axion mass ma (eV)] −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 log � Axion energy density Ωah2� BOSS Planck 2015 Planck 2018 Planck 2018 + BOSS −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 log � Ωah2 ΩDMh2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 � Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 95% credible upper limits on axion energy density Ωah2, as a function of axion mass ma, as inferred: from BOSS galaxy clustering data (blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' from Planck 2015 CMB data (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' from Planck 2018 CMB data (orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and as jointly inferred from Planck 2018 CMB and BOSS galaxy clustering data (green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' On the right-hand side, we show the 95% upper limit on the ratio of the axion energy density to the best-fit dark matter (DM) energy density as inferred from Planck in the ΛCDM model ΩDMh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The black horizontal dashed and dotted lines respectively indicate the energy densities at which axions form 10% and 1% of the DM today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Cosmic microwave background, baryon acoustic oscillations & supernovae 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Planck We first search for ultra-light axions (ULAs) in baseline Planck 2018 CMB temperature, polarisation and lensing data (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 for a description of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We note that this is an update from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36, 38] which considered older Planck 2015 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We now have access to a more robust measurement of the large-scale polarisation signal, and large-scale lensing anisotropies not previously released (Lmin goes from 40 to 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate improved bounds on the lightest axions that we consider since their effect is strong on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 6 shows the 95% upper limit on the axion energy density allowed by Planck 2018 as a function of mass (it also compares to joint constraints from Planck CMB and BOSS galaxy clustering data and constraints from BOSS alone, which we discuss in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' these results are also shown in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We see the typical “u”-shaped constraints [154] where axions in the “belly” (10−30 eV ≤ ma ≤ 10−28 eV) are heavily constrained, but dark energy (DE)-like axions for ma < 10−30 eV and dark matter (DM)-like axions for ma ≥ 10−27 eV can still be a significant cosmological component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, Planck data lose sensitivity for ma ≥ 10−25 eV as – 16 – ma Ωah2 (Planck) S8 (Planck) Ωah2 (Planck+BOSS) S8 (Planck+BOSS) ΛCDM – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='834+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 10−24 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='831 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='826+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='012 10−25 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='811+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='025 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='039 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='818+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='017 10−26 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='819 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='020 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='804+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='020 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='024 10−27 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='822+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='020 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='819+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 10−28 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='831+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='012 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='824 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 10−29 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='826 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 10−30 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='837+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 10−31 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='838+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 10−32 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='843+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='019 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='016 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='829+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Constraints on axion energy density Ωah2 and the matter clumping factor S8, as a function of axion mass ma (top to bottom), as inferred from Planck CMB data (left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) and as jointly inferred from Planck CMB and BOSS galaxy clustering data (right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For Ωah2, we give the 95% upper c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for S8, we give the maximum marginalised posterior with the asymmetric 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the scale-dependent suppression is on scales smaller than those that Planck probes and the background evolution is the same as ΛCDM deep into the radiation epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Our Planck 2018 results are consistent with previous Planck 2015 limits [36] for ma ≥ 10−27 eV, but stronger for ma ≤ 10−28 eV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 7 investigates which parts of the updated 2018 data are most important in improving axion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In systematically replacing parts of the 2015 likelihood12 with 2018 updates, we find that it is the inclusion of the 2018 low-ℓ likelihood that accounts for the vast majority of the improvement in the axion energy density bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The main difference at low ℓ in 2018 data, arising from an analysis of high (electromagnetic) frequency polarisation modes, is a stronger and slightly lower constraint on the reionisation optical depth τ thanks to measurement of the large-scale reionisation bump in the EE power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This τ measurement breaks degeneracies with the primordial power spectrum amplitude As and the axion energy density Ωah2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show results for ma = 10−30 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These are indicative for all ma ≤ 10−28 eV, since the effect of the lightest DE-like axions in CMB data is restricted to the largest scales (through the integrated Sachs-Wolfe effect) that are degenerate with the primordial amplitude [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hence, the improved τ measurement does not improve axion constraints for heavier DM-like axions (ma ≥ 10−27 eV), whose effect is restricted to smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Figure 8 shows the Planck constraints on other cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' DE-like axions (ma < 10−27 eV) are consistent with lower values of h as they drive accelerated expansion after matter-radiation equality [22]13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As DE-like axions have all started oscillating (and so behave like DM) by today, they count towards the total matter energy density Ωm, but are not degenerate with cold DM in the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thus, for ma ≤ 10−27 eV, larger values of Ωm are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This also drives compatibility with larger values of the matter clumping factor 12We consider the same Planck 2015 CMB likelihood as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36]: the low-ℓ likelihood lowl_SMW_70_dx11d_2014_10_03_v5c_Ap for 2 ≤ ℓ ≤ 29, the high-ℓ likelihood plik_lite_v18_TTTEEE for 30 ≤ ℓ ≤ 2508 (TT power spectrum) and 30 ≤ ℓ ≤ 1996 (TE and EE power spectra), and the lensing likelihood smica_g30_ftl_full_pp for 40 ≤ L ≤ 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 13We are considering different axion models than those that are typically invoked to increase h (and so address the Hubble parameter tension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These so-called “early dark energy” axions are contrived to induce a burst of accelerated expansion before recombination and typically require non-trivial axion potentials [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 17 – Planck 2015 Planck 2015 low-ℓ & lensing + 2018 high-ℓ Planck 2015 lensing + 2018 low-ℓ & high-ℓ Planck 2018 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As [×10−9] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 Ωah2 [×10−3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As [×10−9] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 Ωah2 [×10−3] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of updating from Planck 2015 (blue) to Planck 2018 CMB data on axion constraints for ma = 10−30 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We systematically update parts of the 2015 data with 2018 results: first the high-multipole ℓ likelihood (red), then also the low-ℓ likelihood (orange), and then finally also the lensing likelihood (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that the vast majority of improvement in the axion energy density bound comes from 2018 low-ℓ information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the measurement of the large-scale reionisation bump breaks degeneracies between the reionisation optical depth τ, the primordial power spectrum amplitude As and the axion energy density Ωah2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' since S8 ∝ √Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Conversely, DM-like axions (ma > 10−27 eV) are degenerate with cold DM (CDM) in the CMB and so, in the high-mass limit where axions are poorly constrained, the CDM density is also poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, DM-like axions suppress the matter power spectrum on scales below their de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thus, when DM-like axions can – 18 – Planck (ΛCDM) h Ωbh2 Ωch2 As Planck (ma = 10−24 eV) Planck (ma = 10−25 eV) Planck (ma = 10−26 eV) Planck (ma = 10−27 eV) Planck (ma = 10−28 eV) Planck (ma = 10−29 eV) Planck (ma = 10−30 eV) Planck (ma = 10−31 eV) Planck (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 ×10−9 Planck (ΛCDM) ns τ Ωm S8 aPlanck Planck (ma = 10−24 eV) Planck (ma = 10−25 eV) Planck (ma = 10−26 eV) Planck (ma = 10−27 eV) Planck (ma = 10−28 eV) Planck (ma = 10−29 eV) Planck (ma = 10−30 eV) Planck (ma = 10−31 eV) Planck (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0025 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of axion mass ma on cosmological parameter constraints from Planck CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We see how dark energy-like axions (ma < 10−27 eV) have degeneracy with lower values of the Hubble parameter h, while dark matter (DM)-like axions (ma ≥ 10−27 eV) have degeneracy with the cold DM density Ωch2 and lower values of the matter clumping factor S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' These degeneracies are explored further in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Each point indicates the maximum marginalised posterior, while the errorbar indicates the marginalised 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' comprise a significant fraction (≳ 2%) of the total DM budget (10−27 eV ≤ ma ≤ 10−25 eV), Planck data are compatible with lower values of S8 than in the ΛCDM model, since S8 integrates over lower-amplitude modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For ma > 10−25 eV, the power spectrum suppression is on scales smaller than those to which the S8 parameter is most sensitive and so ΛCDM values of S8 are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This suggests that axions with ma ∈ [10−27, 10−25] eV could help to resolve the so-called S8 tension by bringing CMB data into compatibility with the lower S8 values inferred from large-scale structure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 9 explicitly illustrates that it is degeneracy with the axion energy density that allows lower values of h and higher values of Ωm for DE-like axions (ma ∼ 10−30 eV), and lower values of S8 for DM-like axions (ma ∼ [10−26−10−25] eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although axions with ma = 10−25 eV can comprise the dark matter according to Planck data, there is no preference for such a model compared to ΛCDM according to the Bayesian evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The log-ratio of model evidences (or Bayes factor) given Planck data is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 in favour of ΛCDM (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This amounts to “positive” evidence in favour of ΛCDM according to the Jeffreys scale as given by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This lack of preference for extended cosmological models is consistent with previous studies [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 157], and there is no improvement in the maximum likelihood (or minimum chi-squared).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 19 – Planck CMB (ΛCDM) Planck CMB (ma = 10−24 eV) Planck CMB (ma = 10−25 eV) Planck CMB (ma = 10−26 eV) Planck CMB (ma = 10−30 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='675 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='88 S8 10−3 10−1 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='675 h 10−3 10−1 Ωah2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of ultra-light axions on the matter clumping factor S8, matter energy density Ωm and Hubble parameter h, inferred from Planck, as a function of axion mass ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dark matter (DM)- like axions for ma ∈ [10−26, 10−25] eV give lower S8 values by a scale-dependent power spectrum suppression, while dark energy-like axions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', ma = 10−30 eV) give lower h values by causing accelerated expansion after matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' DM-like axions with ma = 10−24 eV have a negligible effect on S8 as the power spectrum suppression is on scales smaller than those to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 All CMB, BAO & supernovae For the first time in a ULA search, we consider the addition of higher-resolution CMB data from the ACT and SPT experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We defer a systematic search of the axion mass param- eter space to future work, in anticipation of upcoming high-resolution lensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this – 20 – Data ma Bayes factor relative to ΛCDM Planck 10−25 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 Planck + ACT-DR4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 Planck + SPT-3G 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 All CMB + BAO + SNe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 10−24 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 10−25 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 10−26 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 10−27 eV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 Planck + BOSS 10−28 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 10−29 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 10−30 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 10−31 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 10−32 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bayes factor (log-ratio of model evidences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' right column) for the indicated data (left column) given the indicated axion model (middle column) relative to the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For all the data combinations shown, the Bayesian evidence favours the ΛCDM model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' although we find that axions can improve consistency between datasets (see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2), there is no preference for an extension beyond ΛCDM given these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Data S8 (ΛCDM) Ωah2 (ma = 10−25 eV) S8 (ma = 10−25 eV) Planck 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='834+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='811+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='025 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='039 Planck + ACT-DR4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='835+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='012 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='789+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='041 Planck + SPT-3G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='828+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='799+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='046 All CMB + BAO + SNe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='010 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10610 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='774+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='037 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Constraints on axion energy density Ωah2 and the matter clumping factor S8, for different CMB, galaxy BAO and supernovae data combinations (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 for a description of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For Ωah2, we give the 95% upper c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for S8, we give the maximum marginalised posterior with the asymmetric 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' study, we focus on the impact of current ACT (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) and SPT (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3) data (and a compendium of low-z galaxy BAO and supernovae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) on DM-like axions that most significantly increase compatibility with low values of S8: ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 10 illustrates the effect on the S8 - Ωm - Ωah2 planes from adding these data to the Planck data considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The posterior shifts with respect to Planck alone are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There is a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5σ decrease in Ωm when adding BAO and SNe data (∼ 1σ decrease seen in the ΛCDM case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, the axion energy density bounds at ma = 10−25 eV are slightly weakened with the addition of these data (see also Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Correspondingly, there is a shift to even lower values of S8 driven by its parameter degeneracy with Ωah2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This weakening of constraints is consistent with previous searches for massive neutrinos in high-resolution CMB data [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Similarly to neutrinos, DM-like axions are constrained in primary CMB anisotropy power spectra through the lensing-induced smoothing of acoustic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Here, gravitational lensing by lower-redshift (mostly z < 2) large-scale structure dampens the amplitude of peaks in angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It follows that the amount of lensing-induced smoothing is sensitive to the presence of ultra-light axions or neutrinos which suppress the growth of structure and thus reduce the amount of smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The amount of lensing relative to the best-fit ΛCDM – 21 – Planck (ma = 10−25 eV) Planck + ACT-DR4 (ma = 10−25 eV) Planck + SPT-3G (ma = 10−25 eV) All CMB + BAO + SNe (ma = 10−25 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 Ωah2 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of current higher-resolution CMB data (Planck and ACT-DR4 in red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Planck and SPT-3G in orange), galaxy baryon acoustic oscillations (BAO) and supernovae (SNe) (all com- bined with Planck in green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Planck only in blue) on axion constraints for ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' From left to right, S8 is the matter clumping factor, Ωm is the matter energy density and Ωah2 is the physical axion energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' expectation is quantified by the multiplicative correction to the theoretical expectation AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, both ACT-DR4 (AL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11) [53] and SPT-3G (AL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12) [54] prefer lower values of AL compared to Planck (AL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='065) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This means that when adding ACT or SPT data to Planck, constraints on models that suppress structure and lower the lensing signal are weakened, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', massive neutrinos [53] or ultra-light axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 11 shows marginalised constraints on all other cosmological parameters, also comparing – 22 – Planck (ΛCDM) Ωah2 h Ωbh2 Ωch2 As Planck (ULADM) Planck+ACT-DR4 (ΛCDM) Planck+ACT-DR4 (ULADM) Planck+SPT-3G (ΛCDM) Planck+SPT-3G (ULADM) CMB+BAO+SNe (ΛCDM) CMB+BAO+SNe (ULADM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 ×10−9 Planck (ΛCDM) ns τ Ωm S8 aPlanck Planck (ULADM) Planck+ACT-DR4 (ΛCDM) Planck+ACT-DR4 (ULADM) Planck+SPT-3G (ΛCDM) Planck+SPT-3G (ULADM) CMB+BAO+SNe (ΛCDM) CMB+BAO+SNe (ULADM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='005 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of current higher-resolution CMB data (ACT-DR4, SPT-3G), galaxy BAO and supernovae (SNe) on ultra-light axion and cosmological constraints for ma = 10−25 eV (ULA DM), also comparing to ΛCDM and Planck-only constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Each point indicates the marginalised mean, while the errorbar indicates the marginalised 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' to the ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We see the typical degeneracy for DM-like axions with standard cold DM meaning that weakened constraints on Ωah2 lead to correspondingly-weakened constraints on Ωch2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We note a ∼ 1σ increase in the Planck calibration parameter aPlanck when adding ACT data which is seen in both ΛCDM and axion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Similarly as for Planck data, there is no preference given these combined datasets for an axion model compared to ΛCDM according to the Bayesian evidence (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The Bayes factors amount to evidence in favour of ΛCDM that ranges from “positive” to “not worth more than a bare mention” according to the Jeffreys scale [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We demonstrate above that, in an axion model with ma = 10−25 eV, the combination of Planck, ACT-DR4 and SPT-3G CMB, galaxy BAO and supernovae data are compatible with lower values of the matter clumping factor (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='774+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='037) than in ΛCDM (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 12 compares this result to fiducial ΛCDM constraints from combined galaxy weak lensing and clustering (3 × 2) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We consider ΛCDM constraints (with fixed neutrino energy density) from the combination of galaxy clustering, galaxy lensing shear and galaxy – galaxy lensing two-point correlation functions (3 × 2) as measured by the Dark Energy Survey (DES) [158]14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We also consider ΛCDM constraints (with fixed neutrino energy density) from the same combination of three × two-point correlation functions as measured by the Kilo-Degree Survey (KiDS) [159], which includes redshift-space galaxy clustering data from BOSS [160] and galaxy – galaxy lensing data from the survey overlap between KiDS, BOSS and the spectroscopic 2-degree Field Lensing Survey (2dFLenS) [161]15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We note that the KiDS 3 × 2 constraints are therefore not entirely independent of the CMB + BAO + SNe 14This is the publicly-released posterior chain chain_3x2pt_fixednu_lcdm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15This is the publicly-released posterior chain samples_multinest_blindC_EE_nE_w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 23 – All CMB + BAO + SNe (ΛCDM) All CMB + BAO + SNe (ma = 10−25 eV) DES-Y3 3 × 2 (ΛCDM) KiDS 3 × 2 (ΛCDM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='40 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='84 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='40 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 Ωah2 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of CMB (Planck, ACT-DR4, SPT-3G), galaxy BAO and supernovae (SNe) constraints with fiducial galaxy weak lensing and clustering (3 × 2) ΛCDM constraints from the Dark Energy Survey (DES) and the Kilo-Degree Survey (KiDS) (all with fixed neutrino mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In ΛCDM, CMB, BAO and SNe data prefer systematically higher values of the matter clumping factor S8 than is inferred from fiducial 3 × 2 analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' When axions of ma = 10−25 eV contribute to the energy budget with energy density Ωah2, CMB, BAO and SNe data are consistent with lower values of S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In order to assess consistency between data in an axion model, it is necessary to re-analyse the 3 × 2 data in the presence of axions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, we consider the first part with galaxy clustering from BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' From left to right, S8 is the matter clumping factor, Ωm is the matter energy density and Ωah2 is the physical axion energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' compendium we consider, since part of the BAO measurements we use is derived from the same BOSS data (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) as goes into the KiDS 3 × 2 measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, we note – 24 – that the addition of BAO and SNe data makes only a small difference to the S8 constraint from Planck + ACT and Planck + SPT (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate that, in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, we will consider the full-shape galaxy clustering power spectrum from BOSS, which will be much more significantly correlated with the KiDS 3×2 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Despite this proviso, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 12 illustrates how, in the ΛCDM model, 3 × 2 analyses from both DES (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='783 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='020) and KiDS (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='765±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='017) prefer systematically lower values of S8 than the compendium of CMB, BAO and SNe data (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is a manifestation of the so-called “S8 tension”, where many galaxy clustering, weak lensing and galaxy cluster observations prefer lower values of S8 than is inferred from CMB observations, with statistical significance ranging from 2 to 3 σ depending on the data comparison [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 87, for a recent review].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, when axions of ma = 10−25 eV contribute to the energy budget, the CMB, BAO and SNe compendium is compatible with the low S8 values preferred by DES and KiDS in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore hypothesise that axions could resolve the S8 tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In order to assess this, we must reanalyse the 3 × 2 data in the axion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we consider the first part of this in analysing full-shape galaxy clustering information from BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We present these results in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Baryon Oscillation Spectroscopic Survey galaxy power spectrum & bispec- trum We now consider the effect on ultra-light axion constraints from the galaxy power spectrum and bispectrum as measured from the Baryon Oscillation Spectroscopic Survey (BOSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 for a description of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 ΛCDM Before studying the combination of Planck CMB and BOSS galaxy clustering data, we assess constraints independently from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 13 shows ΛCDM cosmological constraints from the BOSS galaxy power spectrum only (P0, P2, P4, Q0 and the post-reconstructed BAO Alcock-Paczynski parameters), the BOSS galaxy power spectrum and bispectrum monopole (additionally B0), and Planck CMB data (previously shown in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, we consider BOSS constraints without a prior on the baryon energy density Ωbh2 or any other cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It is striking how much more constraining is Planck data on the full cosmological model than BOSS data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, as is typical of large-scale structure experiments, BOSS provides more competitive constraints when projected onto the plane of derived parameters Ωm and S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 Parameter tension metrics In order to assess consistency between datasets in their cosmological constraints, we consider three metrics of parameter tension (the difference in S8 only, the difference in the S8 - Ωm plane, and the difference in the full posterior distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We now describe these metrics in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The first metric, which is most widely quoted in the literature, is the discrepancy in the marginalised S8 constraint from two datasets (labelled 1 and 2), defined as ∆S8 σS8 = µ1 − µ2 � σ2 1 + σ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) Here, µi and σi are respectively the parameter posterior mean and standard deviation given experiment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This metric is given in the third column of Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We also consider a second – 25 – BOSS galaxy power spectrum (ΛCDM) BOSS galaxy power spectrum + bispectrum (ΛCDM) Planck cosmic microwave background (ΛCDM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03 Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='18 Ωch2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='90 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03 Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='18 Ωch2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='90 S8 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of BOSS galaxy clustering and Planck CMB constraints on ΛCDM cosmo- logical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data alone (in particular without an Ωbh2 prior) are much less constraining than Planck data on the standard cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the darker and lighter shaded contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' metric, which is an extension of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) to higher dimensions: χ2 = (µ1 − µ2)T(C1 + C2)−1(µ1 − µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) Here, µi is now the vector of parameter posterior means and Ci is the posterior covariance, both given experiment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We then calculate the probability p to exceed χ2 (for a χ2 distri- bution with degrees of freedom equal to the number of parameters) and convert this to a – 26 – number N of σ using the standard Gaussian interpretation16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) are good measures of parameter discrepancy in the limit of Gaussian posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore give, in the fourth column of Table 5, the metric defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2) evaluated for the marginalised posterior in the S8 − Ωm plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this plane, the BOSS data are most constraining and the distribution is reasonably Gaus- sian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It is important nonetheless also to consider consistency in the full set of parameters constrained by both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, the full BOSS posterior distribution appears highly non-Gaussian and so the metrics defined above will not be a good measure of consistency in the full parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore elect to calculate the full posterior distribution of the parameter difference ∆θ (marginalised over the parameters θ) [162]: P(∆θ) = � dθP1(θ)P2(θ − ∆θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3) Here, Pi(θ) is the posterior distribution given experiment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We can then calculate the significance of the inferred parameter shift (relative to none) by integrating P(∆θ) above the iso-probability contour that goes through ∆θ = 0 (this probability to exceed can be converted to a number of σ as above)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this way, this third tension metric accounts for non- Gaussianities in the parameter posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore give, in the final column of Table 5, the metric derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3) as evaluated in the volume of all parameters constrained by both Planck and BOSS [h, Ωbh2, Ωch2, As, ns, Ωm, S8 and Ωah2 when part of the model].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There are many proposed approaches to evaluating parameter consistency in high- dimensional and non-Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although these different approaches tend to agree in terms of trend (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', they typically agree with respect to an increasing or decreasing tension) [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 163], they typically disagree as to the particular value of tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We there- fore urge caution when interpreting Table 5 that it is most useful as a measure of relative tension given different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' All three metrics considered in Table 5 (and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 13) illus- trate that the addition of BOSS bispectrum data B0 increases the discrepancy with respect to Planck mostly by preferring slightly lower values of the primordial power spectrum amplitude As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This in turn pushes S8 to slightly lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 BOSS-only axion constraints Figure 14 shows the same set of posterior contours as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 13 but for an axion model with ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although BOSS is less constraining than Planck on ΛCDM parameters, it is significantly more constraining on the axion energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is driven by the addition of smaller-scale data in the reconstructed real-space galaxy power spectrum Q0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 19 and discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, since Planck alone is unconstraining on the axion energy density at this mass (see also § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1), it is more consistent with the lower values of S8 that BOSS (and indeed other large-scale structure experiments) prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This means that there is more posterior overlap in the S8 - Ωm plane and this is reflected in the improved tension metrics in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Notably, the tension in S8 when comparing Planck to full BOSS data is reduced from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='70 σ (ΛCDM) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63 σ (for ma = 10−25 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, there is no degeneracy between Ωah2 and As at ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is because Ωah2 largely affects the small-scale power spectrum, while As is constrained by the overall normalisation at all wavenumbers (see 16N = √ 2 erf−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 17We numerically evaluate this integral using the tensiometer package: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/mraveri/ tensiometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 27 – BOSS galaxy power spectrum (ma = 10−25 eV) BOSS galaxy power spectrum + bispectrum (ma = 10−25 eV) Planck cosmic microwave background (ma = 10−25 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='030 Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='16 Ωch2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='030 Ωbh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='16 Ωch2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of BOSS galaxy clustering and Planck CMB constraints on axion and cosmological parameters, for axion mass ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data alone are more constraining than Planck data on axion energy density Ωah2 since BOSS probes smaller scales (k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 h Mpc−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' in the extended axion model, there is more posterior overlap in the S8 - Ωm plane than in ΛCDM (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the darker and lighter shaded contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This means there is no improvement in the As discrepancy between Planck and BOSS even in the presence of axions at ma = 10−25 eV and this is reflected in the full parameter space tension metric given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Indeed, when not including bispectrum data, the full parameter tension increases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 6, we show the 95 % upper limits on Ωah2 derived from BOSS data across the full mass range that we consider (see also Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For ma < 10−25 eV, the BOSS-only – 28 – Data Model S8 (σ) S8 − Ωm (σ) All parameters (σ) Planck, BOSS [no B0] ΛCDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='77 ma = 10−25 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14 Planck, BOSS ΛCDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='36 ma = 10−25 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='70 ma = 10−26 eV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='38 ma = 10−27 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63 ma = 10−28 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='31 ma = 10−29 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='19 ma = 10−30 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11 ma = 10−31 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='95 ma = 10−32 eV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='19 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Discrepancy in parameters (given in the top row) as inferred from the two datasets given in the first column, for the model given in the second column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The third column is the discrepancy in the marginalised S8 constraint, and the fourth column is the discrepancy in the marginalised S8 −Ωm plane, both with the reasonable approximation of a Gaussian posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The final column is the discrepancy in the marginalised constraint on all cosmological (and axion) parameters, where we account for non-Gaussianity by calculating the full parameter difference posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The full details of the tension metrics that we use are given in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' ma Ωah2 (BOSS) S8 (BOSS) ΛCDM – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='723+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='041 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='037 10−24 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='718+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='038 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='039 10−25 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='709+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='043 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='037 10−26 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='653 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 10−27 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='719+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='038 10−28 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='742+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 10−29 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='759+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='044 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='043 10−30 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='745+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='041 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 10−31 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='744+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='042 10−32 eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='737+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='040 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='038 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Constraints on axion energy density Ωah2 and the matter clumping factor S8, as a function of axion mass ma (top to bottom), as inferred from BOSS galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For Ωah2, we give the 95% upper c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for S8, we give the maximum marginalised posterior with the asymmetric 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For consistency with other masses, at ma = 10−26 eV, we give the upper limit on the axion density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' nonetheless, Ωah2 = 0 is disfavoured at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', the maximum marginalised posterior Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0100+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0048 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' constraints are weaker than Planck, although the BOSS data are crucial in strengthening the combined CMB and galaxy clustering limit at nearly all masses (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, we see the typical “u”-shaped constraints (that we see with CMB data) also given BOSS alone: at higher mass, BOSS loses sensitivity since the scale-dependent suppression manifests at larger wavenumbers than those we model in BOSS (crucially, BOSS probes smaller scales than Planck and so we have improved sensitivity for ma = 10−25 eV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' at lower mass, BOSS loses sensitivity owing to degeneracy with As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The degeneracy with As for ma ≤ 10−28 eV is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15, where we show how axions impact BOSS constraints on all cosmological – 29 – BOSS (ΛCDM) Ωah2 h Ωbh2 Ωch2 As BOSS (ma = 10−24 eV) BOSS (ma = 10−25 eV) BOSS (ma = 10−26 eV) BOSS (ma = 10−27 eV) BOSS (ma = 10−28 eV) BOSS (ma = 10−29 eV) BOSS (ma = 10−30 eV) BOSS (ma = 10−31 eV) BOSS (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ×10−9 BOSS (ΛCDM) ns Ωm S8 b1 BOSS (ma = 10−24 eV) BOSS (ma = 10−25 eV) BOSS (ma = 10−26 eV) BOSS (ma = 10−27 eV) BOSS (ma = 10−28 eV) BOSS (ma = 10−29 eV) BOSS (ma = 10−30 eV) BOSS (ma = 10−31 eV) BOSS (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of axion mass ma on cosmological parameter constraints from BOSS galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We stress that even the ΛCDM constraints differ from those reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [55] as, in this work, we do not use a Big Bang nucleosynthesis (BBN) prior on the baryon energy density Ωbh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Each point indicates the marginalised mean, while the errorbar indicates the marginalised 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' b1 is the linear galaxy bias at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61 in the north Galactic cap (NGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' similar values are found in all four redshift/sky samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This degeneracy arises at low mass since the axion Jeans wavenumber is then smaller than the smallest wavenumber that we model in BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This means that axions suppress all BOSS wavenumbers, which is degenerate with lowering As and so lowering the overall power amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS data are therefore compatible with higher values of S8 (driven by higher As and also higher Ωm) than for ΛCDM for ma ≤ 10−28 eV (the effects of higher As and Ωah2 do not cancel perfectly at the scales to which S8 is sensitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This drives an increase in compatibility between BOSS and Planck around ma ∼ 10−29 eV, including (unlike with heavier axions) with regards to the As discrepancy (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Beyond degeneracy with As, we also see the typical degeneracy with Ωch2 for (heavier) DM-like axions and degeneracy with Ωm for (lighter) DE-like axions since they additionally count as matter by today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15 also reveals that at ma = 10−26 eV, rather than an upper limit on the axion density, BOSS data alone disfavour no axions at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ significance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we discuss this in more detail below (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 18 and surrounding discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 30 – BOSS galaxy power spectrum (ma = 10−25 eV) Planck (ma = 10−25 eV) Planck + BOSS galaxy power spectrum (ma = 10−25 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 Ωah2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 b1(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' NGC) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='35 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 b1(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' NGC) Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of BOSS galaxy power spectrum (blue), Planck CMB (orange) and joint (black) constraints on axion and cosmological parameters, for axion mass ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The strongest bound on the axion energy density Ωah2 comes from combining the datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' in order to maintain a good fit to the galaxy data in the joint constraint, lower (though still physically plausible) values of the linear galaxy bias b1 are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the darker and lighter shaded contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' From left to right, Ωah2 is the physical axion energy density, Ωm is the matter energy density, S8 is the matter clumping factor and b1(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' NGC) is the linear galaxy bias at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61 in the north Galactic cap (NGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' similar values are found in all four redshift/sky samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Joint Planck and BOSS axion constraints In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 16, we show the joint constraint from Planck and the BOSS galaxy power spectrum on axions for ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The strongest limit on the axion energy density comes from – 31 – Planck (ΛCDM) h Ωbh2 Ωch2 As Planck+BOSS (ΛCDM) Planck+BOSS (ma = 10−24 eV) Planck+BOSS (ma = 10−25 eV) Planck+BOSS (ma = 10−26 eV) Planck+BOSS (ma = 10−27 eV) Planck+BOSS (ma = 10−28 eV) Planck+BOSS (ma = 10−29 eV) Planck+BOSS (ma = 10−30 eV) Planck+BOSS (ma = 10−31 eV) Planck+BOSS (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='075 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='150 ×10−9 Planck (ΛCDM) ns τ Ωm S8 b1 Planck+BOSS (ΛCDM) Planck+BOSS (ma = 10−24 eV) Planck+BOSS (ma = 10−25 eV) Planck+BOSS (ma = 10−26 eV) Planck+BOSS (ma = 10−27 eV) Planck+BOSS (ma = 10−28 eV) Planck+BOSS (ma = 10−29 eV) Planck+BOSS (ma = 10−30 eV) Planck+BOSS (ma = 10−31 eV) Planck+BOSS (ma = 10−32 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of axion mass ma on cosmological parameter constraints from the joint inference of Planck CMB and BOSS galaxy clustering data, and a comparison to the Planck ΛCDM inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Since Planck is much more constraining than BOSS alone on ΛCDM cosmological parameters, the joint constraints on these parameters are broadly consistent with the Planck ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Each point indicates the marginalised mean, while the errorbar indicates the marginalised 68% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As is in units of 10−9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' b1 is the linear galaxy bias at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='61 in the north Galactic cap (NGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' similar values are found in all four redshift/sky samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The Ωch2 constraint at ma = 10−24 eV extends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we zoom-in for clarity at other masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' combining the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Since Planck is significantly more constraining than BOSS alone on ΛCDM parameters, the joint constraint on those parameters is largely driven by Planck (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS (and galaxy clustering data in general) are constraining on a degenerate combination b1S8 of the power spectrum amplitude S8 and the linear galaxy bias b1, since this combination scales the large-scale galaxy power spectrum (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' although this degeneracy is partly broken by the quadrupole’s sensitivity to fσ8, where f is the growth rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It follows that, in the joint constraint, since Planck drives higher values of the power spectrum amplitude (even in the presence of axions) that a good fit to BOSS data is maintained by preferring a lower value of b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 16, where the joint constraint on b1 is lower than for BOSS alone (moving along the b1S8 degeneracy), but still has a value b1 ∼ 2 that is consistent with previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This behaviour is observed at other axion masses and in the ΛCDM case (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Figure 6 shows the joint limit from Planck and BOSS on the axion energy density across the full axion mass range to which we are sensitive (10−32 eV ≤ ma ≤ 10−24 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 32 – BOSS (ma = 10−26 eV) Planck (ma = 10−26 eV) Planck + BOSS (ma = 10−26 eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='75 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 ns Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of BOSS galaxy clustering (all data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' blue), Planck CMB (orange) and joint (black) constraints on axion and cosmological parameters, for axion mass ma = 10−26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although BOSS data give a hint of a significant axion energy density at this mass, Planck data disfavour this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The consequence is that the joint axion constraint is weaker than for Planck data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, we note that, unlike axions at other masses, axions with ma = 10−26 eV increase the discrepancy between Planck and BOSS data with respect to ΛCDM (see Table 5) and so the joint constraint should be considered with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the darker and lighter shaded contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' From left to right, Ωah2 is the physical axion energy density, Ωm is the matter energy density, S8 is the matter clumping factor and ns is the primordial power spectrum tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see also Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' At nearly all masses, the strongest bound comes from combining the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 17 shows the joint constraints on the other cosmological parameters and the – 33 – linear galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As discussed above (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 16), since Planck is much more constraining on ΛCDM parameters, the joint Planck + BOSS constraints on these parameters is largely driven by Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, we note the typical degeneracy of Ωah2 with, for DE-like axions, lower values of h and higher values of Ωm, and for DM-like axions, with lower values of Ωch2 (see also Planck data in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' BOSS, in general, strengthens the limit on the amount of axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, in the DM-like mass range to which we are sensitive (10−27 eV ≤ ma ≤ 10−25 eV), the joint bound leaves enough axions still to drive consistency with lower values of S8 (see also Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Below, we consider which parts of the BOSS data are most responsible for improving constraints (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 19) and discuss further the implications of these results for the S8 tension (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 20 and 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Similarly as for the CMB data considered in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1, with the addition of BOSS data, there remains no preference for axion models according to the Bayesian evidence (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The Bayes factors amount to evidence in favour of ΛCDM ranging from “positive” to “strong” [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It is striking that the addition of BOSS data strengthens axion bounds at all masses apart from ma = 10−26 eV, where in fact the bound is weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 18 breaks down the constraint at this mass into its constituent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' While Planck alone sets a 95% credible upper limit Ωah2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00615, BOSS alone actually favours a contribution of axions Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0100+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0048 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0037, which excludes no axions at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ (the best-fit model with respect to BOSS data has a chi-squared reduced by ∆χ2 = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This discrepancy in the axion constraint, however, increases the tension between all parameters inferred from Planck and BOSS, as seen in all three tension metrics shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, the preference in BOSS data for axions of ma = 10−26 eV increases the discrepancy in S8 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='70 σ in the ΛCDM case to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is because the power suppression of axions combines with the already-low value of As (Ωah2 and As are constrained from different parts of the galaxy power spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4) to lower further the power spectrum amplitude S8 that is inferred from BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For completeness, we show the joint constraint although we caution that it derives from two datasets that are in more discrepancy than in the ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As before, Planck dominates the constraint on ΛCDM parameters, while the joint limit on Ωah2 is slightly weaker than for Planck alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] in their analysis of previous BOSS data do not report a preference for axions at this mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There are a number of differences with respect to this study (summarised at the start of § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, in particular, previously, the primordial power spectrum tilt was fixed: ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='9611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 18 illustrates that fixing ns at this value will break degeneracy with Ωah2 such that the preference for a non-zero contribution is removed (this degeneracy with ns in this mass range is also seen in Planck data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 8)18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This preference for axions is not seen at any other mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' At all other masses, BOSS data strengthen the axion limit and also increase consistency between Planck and BOSS datasets (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The fixed axion masses which we consider are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thus, this result means that there is a preference in BOSS data alone for a contribution of axions with a mass in a window ma ∈ [10−27, 10−25] eV, which motivates future work where we additionally sample ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Notwithstanding ma = 10−26 eV, BOSS data otherwise always improve axion limits with respect to Planck alone, and axions improve consistency between the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 19 illustrates which parts of the BOSS data are most constraining at ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We 18Using a Big Bang nucleosynthesis (BBN) prior on the baryon energy density Ωbh2 ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02268, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00038) [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' ] reduces the significance for an axion component at ma = 10−26 eV to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' further adding a Planck- motivated prior ns ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='9649, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0042) [2] reduces the significance to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weakening the prior on the EFT of LSS bias and counterterm parameters (see § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4) (by doubling the standard deviation in Gaussian prior distributions and doubling the width in uniform prior distributions) increases the significance to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 34 – Planck Planck + BOSS [BAO, P0, P2, P4] Planck + BOSS [BAO, P0, P2, P4, Q0] Planck + BOSS [BAO, P0, P2, P4, Q0, B0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 ≠m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 ≠ah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 ≠m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 ≠ah2 Planck Planck + BOSS [BAO, P0, P2, P4] Planck + BOSS [BAO, P0, P2, P4, Q0] Planck + BOSS [BAO, P0, P2, P4, Q0, B0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 ≠m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 ≠ah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='34 ≠m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10 ≠ah2 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The effect of adding different parts of the BOSS galaxy clustering data on axion energy density Ωah2 constraints for ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We systematically add to the Planck CMB likelihood (blue) different parts of the BOSS likelihood: first, BAO and power spectrum multipoles [P0, P2, P4] up to maximum wavenumber kmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 h Mpc−1 (red);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' then, also the reconstructed real-space power spectrum Q0 for k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4] h Mpc−1 (orange);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and finally, also the bispectrum monopole B0 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that the vast majority of the improvement in the bound comes from the addition of smaller- scale information in the Q0 likelihood, since the suppression effect of axions is stronger on smaller scales (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each data cut, we show the 1D marginalised posterior for Ωah2, where the 68% credible region is shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' systematically add different parts of the BOSS data to a joint constraint with Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that it is the addition of the small-scale reconstructed real-space galaxy power spectrum Q0 for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 h Mpc−1 < k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 h Mpc−1 which drives the vast majority of the improvement in the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This arises because the power suppression effect of axions is always stronger on smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Figure 20 illustrates the degeneracy between Ωah2 and S8 within the 95% credible upper limits on Ωah2 that are allowed by the joint analysis of Planck and BOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As we saw above, there is no such degeneracy for DE-like axions (ma < 10−28 eV) or for DM-like axions where the power suppression scale is too small (ma ≥ 10−24 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In the mass window (10−28 eV ≤ ma ≤ 10−25 eV) however, the joint constraint still allows enough axions to drive consistency with lower values of S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although more axions are allowed at higher masses (in the DM- like regime), since the suppression scale is smaller at higher mass, there is less total power suppression at the wavenumbers to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The lowest values of S8 are in fact found at ma = 10−26 eV (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='804+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='020 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='024;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see also Table 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However we caution that this constraint arises from two datasets that are in stronger tension than the ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, at ma = 10−25 eV (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='818+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='017) and ma = 10−27 eV (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='819+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='013 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='014), the parameter discrepancy between Planck and BOSS is reduced and the joint constraint on S8 is shifted to lower values than the ΛCDM case (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 21 updates Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 12 with the joint Planck + BOSS constraints (see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 for details about the DES and KiDS ΛCDM contours that we show).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In comparison to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 12, we note how the addition of BOSS data more strongly constrains the axion energy density and in turn reduces the extent to which low values of S8 are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, there remains a tail in the posterior to lower values of S8 in the presence of axions with ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fully assessing the consistency with galaxy weak lensing experiments like DES and KiDS requires re-analysing these data in – 35 – Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 95% credible upper limits on axion energy density Ωah2, as a function of axion mass ma, as jointly inferred from Planck CMB and BOSS galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We illustrate the degeneracy with the matter clumping factor S8 by colouring (unweighted) posterior samples according to their S8 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The lowest values of S8 are allowed for dark matter-like axions with ma ∈ [10−27, 10−25] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' On the right-hand side, we show the 95% upper limit on the ratio of the axion energy density to the best-fit dark matter (DM) energy density as inferred from Planck ΩDMh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the axion models we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We discuss the prospects for this in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5 Discussion In § 4, we present several new results in searching for ultra-light axions in a compendium of CMB and large-scale structure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1, we present legacy constraints on the axion energy density from Planck 2018 CMB temperature, polarisation and lensing anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that, compared to previous Planck 2015 results [36], a new measurement of the optical depth to reionisation (through large-scale polarisation) breaks parameter degeneracies and improves energy density bounds for DE-like axions (ma ≤ 10−28 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, we search for axions in a compendium of higher-resolution CMB data (ACT-DR4, SPT-3G), galaxy BAO and supernovae data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that the addition of these data marginally weakens the axion energy density bound for ma = 10−25 eV (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 36 – S: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0 32 30 28 26 24 log [Axion mass ma (eV)]Planck + BOSS (ΛCDM) Planck + BOSS (ma = 10−25 eV) DES-Y3 3 × 2 (ΛCDM) KiDS 3 × 2 (ΛCDM) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='40 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='84 S8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='050 Ωah2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='40 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='050 Ωah2 Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Comparison of joint Planck CMB and BOSS galaxy clustering constraints (in both axion and ΛCDM models) with fiducial galaxy weak lensing and clustering (3 × 2) ΛCDM constraints from the Dark Energy Survey (DES) and the Kilo-Degree Survey (KiDS) (all with fixed neutrino mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Planck and BOSS data are consistent with lower values of S8 in the presence of axions with mass ma = 10−25 eV compared to the ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In order to assess consistency between all data in an axion model, it is necessary to re-analyse the 3×2 data in the presence of axions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we discuss the future analysis of cosmic shear data in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We note caution in assessing parameter tension by eye, especially as the Planck + BOSS and KiDS datasets are not independent, since KiDS uses BOSS clustering information in their 3 × 2 measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For each set, the inner and outer contours respectively indicate the 68% and 95% credible regions of the 2D marginalised posterior distribution, with the 1D marginalised posteriors on the diagonal, where 68% credible regions are shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' From left to right, S8 is the matter clumping factor, Ωm is the matter energy density and Ωah2 is the physical axion energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 37 – In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, we present axion constraints from BOSS galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that the addition of BOSS to Planck improves axion energy density bounds at nearly all axion masses that we consider (10−32 eV ≤ ma ≤ 10−25 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Crucially, we find that the inclusion of new small-scale modes (Q0 for k ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4] h Mpc−1) strengthens the constraint at ma = 10−25 eV with respect to Planck only (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is driven by gaining sensitivity to larger wavenumbers where the power suppression of heavier axions manifests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Gains in sensitivity from BOSS data to lighter, DE-like axions (ma ≤ 10−28 eV) are limited by degeneracy between Ωah2 and As at those masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This arises since the axion-induced power suppression occurs at wavenumbers smaller than we model in BOSS data and so the axion effect is degenerate with an overall re-scaling of the galaxy power spectrum amplitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This suggests that robustly modelling larger-volume galaxy surveys can improve sensitivity to DE-like axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Robustly modelling smaller-scale correlations in galaxy positions will be extremely challenging owing to the non-trivial way that galaxies trace dark matter on small scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', non-linear galaxy bias).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore suggest alternative probes like galaxy and CMB weak lensing (that are insensitive to galaxy bias) to increase sensitivity at ma ≥ 10−24 eV (see above and below for more discussion about probes complementary to galaxy correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, our results demonstrate the power in combining CMB and large-scale structure data when constraining dark matter models beyond standard CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1 Comparison to previous work There are a number of differences between this study and a previous BOSS analysis presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' First, as discussed above, we model more of the BOSS data, in particular, addi- tionally, the galaxy power spectrum hexadecapole P4, the small-scale real-space galaxy power spectrum Q0 (where we conservatively project away hard-to-model non-linear redshift-space distortions) and the galaxy bispectrum monopole B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, we choose less informative priors on cosmological parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', we do not use BBN information to place a prior on the baryon energy density Ωbh2 and, importantly, we do not fix the primordial power spec- trum tilt ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The latter is important as we do in general observe degeneracy between Ωah2 and ns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15) and this degeneracy will be broken by fixing ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We thus find that our bounds from BOSS alone are weaker than those reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] combined Planck and BOSS through a Planck-motivated prior on cosmological and axion parameters (except ns which remained fixed) combined with the BOSS likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Instead, in this work, for the first time, we jointly sample the Planck and BOSS likelihoods in a full axion and cosmological model in setting axion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find in general that our combined con- straints are stronger than those reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We attribute a large degree of this to the information gained by updating to Planck 2018 data (Planck 2015 data was previously considered) for low masses (see above) and using the small-scale Q0 statistic for higher masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] are affected by an error in the BOSS data weights, which has since been corrected and does not affect the results presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There are further pipeline differences between the two analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In particular, beyond the different and more complete compression of the BOSS data discussed above, we use different implementations of the BOSS likelihood and EFT of LSS theory calculations (namely, CLASS-PT/full_shape_likelihoods and, previously, PyBird) and, correspondingly, different EFT of LSS parameter priors (namely, so-called “East Coast” and, previously, “West Coast” priors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In general, the different prior choices will lead to differences in parameter inference given the same set of BOSS data (in ΛCDM, the cosmological constraints are consistent within ∼ 1σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [151]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Importantly, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [150] and [151] demonstrated that, with – 38 – external CMB information from Planck, the prior sensitivity is significantly reduced, while future larger-volume galaxy surveys will have sufficient constraining power also to lose prior sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We defer to future work a detailed study of the effect of EFT of LSS priors on BOSS axion constraints since, in this work, it is non-trivial to disentangle the other analysis differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 ma = 10−26 eV A striking difference between this work and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] is the axion constraint at ma = 10−26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Unlike at other axion masses that we consider, at ma = 10−26 eV, rather than setting an upper limit on the axion energy density, we find, given BOSS data only, Ωah2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0100+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0048 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0037 that excludes no axions at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' However, such a large contribution of axions at this mass is disfavoured by Planck (Ωah2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00615) and so the tension in parameter inference between these datasets is increased at this axion mass with respect to ΛCDM (at all other masses, the tension is reduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see more discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For completeness, we consider the joint constraint (which is thus weaker than Planck alone) although we caution that this derives from two datasets in greater discrepancy than in the standard CDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that the preference for axions at this mass opens up degeneracy with other cosmological parameters, in particular ns and b1 (although in an opposite sense as at other masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 15 and 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This explains why this preference was not observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] where ns was fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' indeed, when giving a BBN prior on Ωbh2 and a Planck prior on ns, the significance of the axion preference is reduced to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Further, if an inflation-motivated prior that excluded low values of ns was imposed, we anticipate that the significance would also be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The anomalous (with respect to other masses) degeneracy between higher Ωah2 and lower ns and b1 suggests an effect from marginalisation over other EFT of LSS parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' indeed, weakening the prior on EFT of LSS bias and counterterm parameters increases the axion pref- erence to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As discussed in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3, in order to reduce the dimensions of the sampling task, we analytically marginalise over a number of bias and counterterm parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore leave for future work a detailed study of the effect of nuisance parameter marginalisation by numerically sampling the full joint cosmological and EFT of LSS posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We stress, nonetheless, owing to the way that we consider axion masses only at a number of fixed values, that there is an element of the look-elsewhere effect where it is not surprising to find one of the nine axion masses has a mild preference unlike the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Future galaxy data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', from the Dark Energy Spectroscopic Instrument [165] or the Rubin Observatory [166]) will be crucial in determining if this preference is only a statistical anomaly or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Reconcili- ation with the Planck bound may be connected to the AL anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' As discussed above, we find that CMB datasets with lower (and theoretically-consistent) amounts of lensing weaken axion bounds and so we hypothesise that the AL anomaly in Planck is strengthening the bound and increasing the discrepancy with BOSS at ma = 10−26 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We will investigate this hypothesis in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3 Comparison to other axion probes and future prospects Notwithstanding the mild preference for axions at ma = 10−26 eV given BOSS data only, our Planck and joint Planck and BOSS analyses set strong limits on the axion energy density for ma ≤ 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Axions are well-motivated in a range of particle masses and can be produced in a mixture with other axions (the so-called “axiverse” [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 21]) and/or with other DM and DE particle candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This motivates a search for axions across a wide range of masses and – 39 – 10−32 10−30 10−28 10−26 10−24 10−22 10−20 10−18 Axion mass ma (eV) 10−4 10−3 10−2 10−1 100 Axion energy density Ωa CMB-S4 GUT-scale fa Lyα BOSS+Hi-Res Lyα DESI+Hi-Res +MW-Rubin PTA +WL-DES +BOSS CMB-Planck 2018 +MW-DES IM-SKA CMB-HD Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 95% c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' axion energy density Ωa bounds presented in this work from Planck 2018 CMB data (top left) and from a joint analysis of Planck CMB and BOSS galaxy clustering data (+BOSS) compared to other cosmological bounds (shaded solid) and projected bounds (thick lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Our results are complementary to existing bounds at higher masses derived from probes of smaller- scale structure: a joint analysis of Planck CMB and galaxy weak lensing data from the Dark Energy Survey (+WL-DES) [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the Milky Way sub-halo mass function from DES (+MW-DES) [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the strongest lower limit for axions being all the DM (ma > 2 × 10−20 eV) comes from high-resolution Lyman-alpha forest data (Lyα Hi-Res) [28, 31], while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [42] considered a sub-dominant axion contribution (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [43] for a BOSS Lyman-alpha forest analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We show projected bounds for: the CMB-S4 experiment [120];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the CMB-HD experiment using the Ostriker-Vishniac signal [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' the Rubin Observatory using the MW sub-halo mass function [167];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' future Lyman-alpha forest data from the Dark Energy Spectroscopic Instrument (DESI) and high-resolution quasar spectra (Hi-Res) [168];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' intensity mapping from the Square Kilometre Array (IM-SKA) [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and pulsar timing array (PTA) residuals [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We indicate (between the black dotted lines) the parameter space where the axion decay constant fa is at the Grand Unified Theory (GUT) scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Where bounds exclude axions being all the DM, we additionally exclude higher energy densities (up to Ωa = 1) by enforcing that the Universe is not over-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The projections rely on different assumptions and have varying degrees of rigour and so are only indicative of future progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' for sub-dominant energy densities so that an axion of a particular mass is not prematurely excluded by assuming that it constitutes the entirety of the DM or DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2219 compares our new bounds for ma ≤ 10−25 eV to other cosmological bounds across the mass range where the gravitational effect of axions is distinguishable in the large-scale structure from standard cold DM (10−32 eV ≤ ma ≤ 10−18 eV)20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The details of each experiment and current and projected bounds are given in the caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We stress that, to probe across the parameter space, it is necessary to use complementary probes of large- and small-scale structure to search for, respectively, lighter and heavier axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We anticipate progress in this regard from ongoing, upcoming and proposed CMB (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', ACT, SPT, Simons Observatory, CMB-S4 [169], CMB-HD), large-scale structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Rubin, DESI), intensity mapping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', SKA) and pulsar timing array observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 19An up-to-date version of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 22 is maintained at https://keirkwame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='io/DM_limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 20There are many ongoing and proposed experimental efforts to probe axions at and above this mass range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [50, 51] for recent reviews;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 22 shows only cosmological probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 40 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4 Axions as a resolution to the S8 parameter tension A key aim of this study is not only to search for axions as a DM and DE candidate, but also to consider the extent to which axions can improve consistency between CMB and large-scale structure datasets in their parameter inference, in particular with respect to the so-called S8 tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The cosmological parameter S8, through its dependence on the matter power spectrum amplitude σ8, is a measure of the clustering of matter at z = 0 when averaged over 8 h−1 Mpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' CMB experiments prefer higher values of S8 than various large-scale structure analyses with statistical significance ranging from 2 to 3 σ depending on the data comparison [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 87, for a recent review].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Since CMB experiments generally probe struc- ture at higher redshift (even CMB lensing is more sensitive to structure at earlier times than current galaxy surveys), most concrete model solutions to the S8 tension invoke a redshift- dependent suppression in the growth of structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', decaying dark matter [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this way, it is argued that this explains why probes of later-time structure have lower ampli- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, we investigate the hypothesis that the S8 tension is a discrepancy between probes of larger- and smaller-scale structure, with axions as a concrete model, and with no need to invoke a late-time decay in the nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 1 illustrates our hypothesis by showing how Planck CMB data lose sensitivity to small-scale modes to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It follows that it is possible to invoke a scale- dependent suppression in the matter power spectrum that is consistent with current CMB data on large scales, while lowering the value of S8 to improve compatibility with galaxy surveys (in particular, galaxy weak lensing) as a more direct probe of the scales to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Indeed, we find (in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1) axions with ma ∈ [10−28, 10−25] eV as a good candidate where they are compatible with a compendium of “large-scale” probes (CMB, galaxy BAO and supernovae) and the lower values of S8 that are inferred from fiducial galaxy clustering and weak lensing (3 × 2) analyses (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lighter axions are largely incompatible with Planck data (except as a highly sub-dominant contribution that does little to S8), while heavier axions are unconstrained by these data but suppress wavenumbers larger than those to which S8 is sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In order to assess whether axions can resolve parameter tensions between CMB and large-scale structure data, it is necessary also to analyse large-scale structure data in an axion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2, we analyse BOSS galaxy clustering data and model the effect of axions in the mildly non-linear regime using the effective field theory of large-scale structure (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find two regimes in which the S8 discrepancy between Planck and BOSS is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The first is for ma ∼ 10−25 eV, where, as discussed above, large axion contributions are unconstrained by CMB data and so bring CMB data into compatibility with lower values of S8: the S8 discrepancy is reduced from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='70σ in ΛCDM to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='63σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The second regime is for ma ∼ 10−28 eV, where, instead, the effect of axions in BOSS data is partly degenerate with the overall amplitude of the galaxy power spectrum since all BOSS wavenumbers are suppressed by axions of this mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This weakens BOSS constraints on the lightest axions, while allowing higher values of the primordial power spectrum amplitude As and also S8 (the effects of higher As and higher Ωah2 not cancelling exactly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thus, the S8 discrepancy is reduced to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='78σ, but, importantly, values of As (which are low in this BOSS analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [151] for a discussion on the effect of EFT of LSS priors on cosmological parameter inference) are brought into greater compatibility with the higher values inferred from Planck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Indeed, although S8 is a reasonably good compression of the information contained in large-scale structure data (though not necessarily optimal for all experiments), it is necessary – 41 – to assess tension in the full posterior, in particular accounting for non-Gaussianity in the distribution (which one-parameter tension metrics do not capture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In this work, in order better to capture tension in the full parameter space, we estimate the posterior distribution of the parameter difference [162] inferred given the two experiments (Planck and BOSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' If the two experiments are in perfect agreement, the parameter difference posterior will peak at zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' we assign the significance of the discrepancy between experiments to the amount of shift from perfect agreement (see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We find that this measure of tension improves from ΛCDM at nearly all axion masses apart from ma = 10−26 eV (as discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' There is no single tension metric on which the community has converged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' different metrics tend to disagree in terms of absolute value though they agree with regards to increasing or decreasing tension [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The metrics we use in this work (and quite generally) depend on the parameterisation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Since S8 and σ8 may not be optimal measures of the matter clustering information directly probed by CMB and even many large-scale structure experiments, we defer to future work studies of the agreement between datasets directly in the linear matter power spectrum using Bayesian metrics like the posterior predictive distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nonetheless, our results suggest that axions with masses in a window [10−28, 10−25] eV can be a promising candidate to improve consistency between CMB and large-scale structure observations, in particular by bringing Planck CMB and BOSS galaxy clustering data into consistency with lower values of S8 that are preferred by galaxy weak lensing data (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 20 and 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We stress, though, that this is achieved through only upper limits on the axion energy density and there is no preference for model extensions beyond ΛCDM given these data according to the Bayesian evidence in any of our analysis (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A more stringent test of the ability for axions to address cosmological parameter tension is the inclusion of galaxy weak lensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' It is common in the literature to include the effect of weak lensing through a prior on S8 derived from ΛCDM S8 constraints, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [97, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is a good measure of the information content in the ΛCDM model, but we caution that this may not be the case in extended models like axions which affect in a non-trivial way the non-linear modes probed by galaxy shear (indeed, we see with BOSS how the S8 constraint changes with axions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We therefore leave for future work an analysis of galaxy shear and 3×2 clustering and shear data using a fully non-linear halo model of axion structure formation [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This will build on initial studies of DES cosmic shear in the limited case that axions comprise the entirety of the DM [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 6 Conclusions We present a comprehensive search for ultra-light axions as a well-motivated dark matter and dark energy particle candidate using a compendium of CMB and large-scale structure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We set the strongest bounds to date on the axion energy density for axion masses ma ∈ [10−32, 10−25] eV through a joint analysis of Planck 2018 CMB and BOSS full-shape galaxy power spectrum and bispectrum data, modelling the effect of axions in the mildly non-linear regime using the effective field theory of large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We exclude axions being more than 10% of the DM today for ma ≤ 10−26 eV and more than 1% for ma ∈ [10−30, 10−28] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We give legacy constraints from Planck 2018 CMB data and find that measurements of the optical depth to reionisation break parameter degeneracies and improve bounds for DE- like axions (ma ≤ 10−28 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' For the first time, we consider high-resolution CMB data from the Atacama Cosmology Telescope and the South Pole Telescope (in combination with – 42 – galaxy BAO and supernovae data), which we find to weaken marginally axion bounds at ma = 10−25 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Similarly to the effect of massive neutrinos, we attribute this weakening to the lower (and theoretically-consistent) amounts of lensing observed in ACT and SPT angular power spectra, which allow more structure suppression arising from axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In the first full joint analysis of Planck 2018 and BOSS full-shape data, we find that galaxy clustering information strengthens axion energy density limits at nearly all masses that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The exception is at ma = 10−26 eV, where BOSS data alone have a mild preference for a non-zero axion contribution, excluding no axions at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The significance is reduced to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7σ when including Big Bang nucleosynthesis constraints on the baryon energy density Ωbh2 and Planck constraints on the primordial power spectrum tilt ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Such an axion contribution is, further, disfavoured by Planck and we caution that the look-elsewhere effect applies owing to the large number of axion masses that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Future galaxy data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', DESI, Rubin) will be crucial in assessing the significance of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We propose axions as a candidate to address the so-called “S8 tension”, where CMB experiments infer systematically higher values of S8 (which is sensitive to the matter power spectrum amplitude at z = 0) than various large-scale structure datasets, with significance ranging from 2 to 3 σ [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We hypothesise that the scale-dependent power spectrum suppression (relative to standard cold DM) arising from axion DM can reconcile current CMB data (which probe larger scales and prefer higher amplitude) with the more direct probes of smaller-scale structure in galaxy clustering and weak lensing that prefer lower amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' We indeed find that a compendium of “large-scale” data (CMB, galaxy BAO and supernovae) are compatible with lower values of S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='774+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='032 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='037 for ma = 10−25 eV than in ΛCDM (S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='827 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' This is achieved since this data combination is not sensitive to the small-scale suppression arising from axions of this mass, while the axion suppression still occurs at wavenumbers to which S8 is sensitive, thus lowering its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Although BOSS full-shape data, in general, strengthen axion density bounds (apart from at ma = 10−26 eV), we find that axions can improve inferred parameter consistency between Planck and BOSS and that the joint Planck and BOSS constraint is still consistent with lower values of S8 than ΛCDM in a window of masses [10−28, 10−25] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' In future work, we will assess consistency with upcoming CMB and galaxy weak lensing data using a fully non-linear (halo) model of axion structure formation [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 39, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Acknowledgments The authors thank Daniel Grin for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' RH is a CIFAR Azrieli Global Scholar (Gravity & the Extreme Universe Program 2019) and a 2020 Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sloan Research Fellow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' and is supported by the Natural Sciences and En- gineering Research Council of Canada Discovery Grant Program and the Connaught Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' MMI is supported by the National Aeronautics and Space Administration (NASA) through the NASA Hubble Fellowship grant #HST-HF2-51483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astron- omy, Incorporated, under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' OHEP is a Junior Fellow of the Simons Society of Fellows and thanks the Institute for Advanced Study for their hospitality and abundance of baked goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' KA is supported by Japan Society for the Promotion of Science (JSPS) Overseas Research Fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' DJEM is supported by an Ernest Rutherford Fellowship from the Science and Technologies Facilities Council in the United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 43 – References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Brout, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Scolnic, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Popovic, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Carr, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zuntz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Pantheon+ Analysis: Cosmological Constraints, ApJ 938 (2022) 110 [2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04077].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [2] Planck Collaboration, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aghanim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Akrami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ashdown, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aumont, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baccigalupi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmological parameters, A&A 641 (2020) A6 [1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bennett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Larson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weiland, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Jarosik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hinshaw, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Odegard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results, ApJS 208 (2013) 20 [1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [4] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Reichardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Story, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Follin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Keisler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Constraints on Cosmology from the Cosmic Microwave Background Power Spectrum of the 2500 deg2 SPT-SZ Survey, ApJ 782 (2014) 74 [1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Madhavacheril, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sehgal and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Slatyer, Current dark matter annihilation constraints from CMB and low-redshift data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 89 (2014) 103508 [1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3815].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sievers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hlozek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nolta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Acquaviva, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Addison, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Atacama Cosmology Telescope: cosmological parameters from three seasons of data, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2013 (2013) 060 [1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0824].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wise, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Georgi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Glashow, SU(5) and the Invisible Axion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 47 (1981) 402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dine, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fischler and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Srednicki, A simple solution to the strong CP problem with a harmless axion, Physics Letters B 104 (1981) 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dine, Axions: Visible and Invisible, in Novel Results in Particle Physics - 1982: Fifth International Conference on Particle Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 93 of American Institute of Physics Conference Series, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 66–76, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 1982, DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Abbott, Axion Cosmology, in Relativity, Cosmology, Topological Mass and Supergravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' SILARG IV, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 100, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Preskill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wise and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wilczek, Cosmology of the invisible axion, Physics Letters B 120 (1983) 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Steinhardt and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Turner, Saving the invisible axion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B 129 (1983) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kim, Light pseudoscalars, particle physics and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 150 (1987) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [14] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Berezhiani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sakharov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Khlopov, Primordial background of cosmological axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Soviet Journal of Nuclear Physics 55 (1992) 1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peccei and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Quinn, CP Conservation in the Presence of Instantons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 38 (1977) 1440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weinberg, A New Light Boson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 40 (1978) 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [17] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wilczek, Problem of Strong p and t Invariance in the Presence of Instantons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 40 (1978) 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Witten, Some properties of O(32) superstrings, Physics Letters B 149 (1984) 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Svrcek and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Witten, Axions In String Theory, JHEP 06 (2006) 051 [hep-th/0605206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weinberg, A new light boson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 40 (1978) 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Arvanitaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dimopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dubovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kaloper and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' March-Russell, String Axiverse, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 81 (2010) 123530 [0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4720].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hlozek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ferreira, A search for ultralight axions using precision cosmological data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 91 (2015) 103512 [1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2896].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 44 – [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bullock and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Boylan-Kolchin, Small-Scale Challenges to the ΛCDM Paradigm, ARA&A 55 (2017) 343 [1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weinberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bullock, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Governato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kuzio de Naray and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peter, Cold dark matter: Controversies on small scales, Proceedings of the National Academy of Science 112 (2015) 12249 [1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0913].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ostriker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tremaine and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Witten, Ultralight scalars as cosmological dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 95 (2017) 043541 [1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08297].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pontzen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Governato, Cold dark matter heats up, Nature 506 (2014) 171 [1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1764].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Drlica-Wagner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bechtol, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' McNanna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nadler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Milky Way Satellite Census.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The Observational Selection Function for Milky Way Satellites in DES Y3 and Pan-STARRS DR1, ApJ 893 (2020) 47 [1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, Strong Bound on Canonical Ultralight Axion Dark Matter from the Lyman-Alpha Forest, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 126 (2021) 071302 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12705].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [29] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nadler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Drlica-Wagner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bechtol, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wechsler, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Gluscevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Constraints on Dark Matter Properties from Observations of Milky Way Satellite Galaxies, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 126 (2021) 091101 [2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bond, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Söding, Evolving ultralight scalars into non-linearity with Lagrangian perturbation theory, MNRAS 504 (2021) 2391 [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08482].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, General framework for cosmological dark matter bounds using N -body simulations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 103 (2021) 043526 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13751].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dome, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fialkov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mocz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schäfer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Boylan-Kolchin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Vogelsberger, On the Cosmic Web Elongation in Fuzzy Dark Matter Cosmologies, arXiv e-prints (2022) arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03827 [2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03827].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' May and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Springel, The halo mass function and filaments in full cosmological simulations with fuzzy dark matter, arXiv e-prints (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14886 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14886].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Macciò and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baldi, Fuzzy Aquarius: evolution of a Milky-way like system in the Fuzzy Dark Matter scenario, arXiv e-prints (2022) arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08022 [2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Vogt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, Improved Mixed Dark Matter Halo Model for Ultralight Axions, arXiv e-prints (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13445 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, Using the full power of the cosmic microwave background to probe axion dark matter, MNRAS 476 (2018) 3063 [1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05681].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [37] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Farren, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Jaffe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, Ultralight axions and the kinetic sunyaev-zel’dovich effect, Physical Review D 105 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bond, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, Constraining ultralight axions with galaxy surveys, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2022 (2022) 049 [2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07802].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dentler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, Fuzzy dark matter and the Dark Energy Survey Year 1 data, MNRAS 515 (2022) 5646 [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kunkel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chiueh and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schäfer, A weak lensing perspective on nonlinear structure formation with fuzzy dark matter, arXiv e-prints (2022) arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01523 [2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01523].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [41] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Iršič, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Viel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Haehnelt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bolton and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Becker, First constraints on fuzzy dark matter from Lyman-α forest data and hydrodynamical simulations, ArXiv e-prints (2017) [1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04683].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [42] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kobayashi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Murgia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' De Simone, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Iršič and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Viel, Lyman-alpha Constraints on Ultralight Scalar Dark Matter: Implications for the Early and Late Universe, ArXiv e-prints (2017) [1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 45 – [43] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Armengaud, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Palanque-Delabrouille, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Yèche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baur, Constraining the mass of light bosonic dark matter using SDSS Lyman-α forest, ArXiv e-prints (2017) [1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baur, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Palanque-Delabrouille, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Yèche, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Magneville and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Viel, Lyman-alpha forests cool warm dark matter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 8 (2016) 012 [1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01981].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [45] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dalal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kravtsov, Excluding fuzzy dark matter with sizes and stellar kinematics of ultrafaint dwarf galaxies, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 063517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [46] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Goldstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Koushiappas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Walker, Viability of ultralight bosonic dark matter in dwarf galaxies, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 063010 [2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hotinli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kamionkowski, Probing ultralight axions with the 21-cm signal during cosmic dawn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 043529 [2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06943].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bauer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Padmanabhan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, Intensity mapping as a probe of axion dark matter, MNRAS 500 (2021) 3162 [2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09655].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Flitter and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kovetz, Closing the window on fuzzy dark matter with the 21-cm signal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 063504 [2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05083].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [50] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Antypas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Banerjee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bartram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baryakhtar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Betz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bollinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', New Horizons: Scalar and Vector Ultralight Dark Matter, arXiv e-prints (2022) arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14915 [2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14915].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Axion Dark Matter, in 2022 Snowmass Summer Study, 3, 2022, 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='14923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [52] Planck Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ade, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aghanim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Arnaud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ashdown, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aumont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Planck 2015 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmological parameters, A&A 594 (2016) A13 [1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01589].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [53] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aiola, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Calabrese, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Maurin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Naess, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Abitbol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Atacama Cosmology Telescope: DR4 maps and cosmological parameters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2020 (2020) 047 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07288].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [54] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dutcher, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Balkenhol, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ade, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ahmed, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Anderes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Measurements of the E -mode polarization and temperature-E -mode correlation of the CMB from SPT-3G 2018 data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 104 (2021) 022003 [2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01684].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [55] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, BOSS DR12 full-shape cosmology: Λ CDM constraints from the large-scale galaxy power spectrum and bispectrum monopole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 043517 [2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04515].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [56] BOSS collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Baryon Oscillation Spectroscopic Survey of SDSS-III, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 145 (2013) 10 [1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hlozek, “axionCAMB: Modification of the CAMB Boltzmann code.” Astrophysics Source Code Library, record ascl:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='026, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [58] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lesgourgues and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tram, The Cosmic Linear Anisotropy Solving System (CLASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Part II: Approximation schemes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 7 (2011) 34 [1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2933].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [59] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cookmeyer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cookmeyer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Smith, How sound are our ultralight axion approximations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 101 (2020) 023501 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11094].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Passaglia and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hu, Accurate effective fluid approximation for ultralight axions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 123529 [2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10238].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [61] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baumann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nicolis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, Cosmological non-linearities as an effective fluid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2012 (2012) 051 [1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2488].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 46 – [62] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baldauf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Carrasco and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kollmeier, Cosmological perturbation theory using the FFTLog: formalism and connection to QFT loop integrals, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2018 (2018) 030 [1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cabass, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lewandowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mirbabayi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, Snowmass White Paper: Effective Field Theories in Cosmology, in 2022 Snowmass Summer Study, 3, 2022, 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [64] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D’Amico, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang, Limits on wCDM from the EFTofLSS with the PyBird code, JCAP 01 (2021) 006 [2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07956].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [65] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, Cosmological Parameters from the BOSS Galaxy Power Spectrum, JCAP 05 (2020) 042 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [66] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, Cosmological Parameters and Neutrino Masses from the Final Planck and Full-Shape BOSS Data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 101 (2020) 083504 [1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chudaykin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, Nonlinear perturbation theory extension of the Boltzmann code CLASS, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 063533 [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10607].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [68] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, Effective Field Theory for Large Scale Structure, 2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [69] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, The Effective Field Theory of Large-Scale Structure in the presence of Massive Neutrinos, arXiv e-prints (2017) arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04698 [1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04698].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [70] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, WarmAndFuzzy: the halo model beyond CDM, ArXiv e-prints (2016) [1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05973].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [71] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Vogt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Laguë, Improved Mixed Dark Matter Halo Model for Ultralight Axions, arXiv e-prints (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13445 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13445].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [72] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Heitmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Higdon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Habib, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Williams, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lawrence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Coyote Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmological Models and Precision Emulation of the Nonlinear Matter Power Spectrum, ApJ 705 (2009) 156 [0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0429].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [73] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Heitmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lawrence, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kwan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Habib and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Higdon, The Coyote Universe Extended: Precision Emulation of the Matter Power Spectrum, ApJ 780 (2014) 111 [1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='7849].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [74] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lawrence, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Heitmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kwan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Upadhye, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bingham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Habib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Mira-Titan Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Matter Power Spectrum Emulation, ApJ 847 (2017) 50 [1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03388].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [75] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tinker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Becker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' DeRose, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' McClintock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The Aemulus Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Emulation of the Galaxy Correlation Function, ApJ 874 (2019) 95 [1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05867].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [76] Euclid Collaboration, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Knabenhans, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Stadel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marelli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Potter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Teyssier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Euclid preparation: II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' The EUCLIDEMULATOR - a tool to compute the cosmology dependence of the nonlinear matter power spectrum, MNRAS 484 (2019) 5509 [1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04695].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [77] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bird, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Verde, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Font-Ribera and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pontzen, An emulator for the Lyman-α forest, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2019 (2019) 050 [1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04654].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [78] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pontzen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bird, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Verde and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Font-Ribera, Bayesian emulator optimisation for cosmology: application to the Lyman-alpha forest, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2019 (2019) 031 [1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04631].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [79] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pedersen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Font-Ribera, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' McDonald, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pontzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', An emulator for the Lyman-α forest in beyond-ΛCDM cosmologies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2021 (2021) 033 [2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [80] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rogers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dvorkin and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Peiris, Limits on the Light Dark Matter-Proton Cross Section from Cosmic Large-Scale Structure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 128 (2022) 171301 [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10386].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 47 – [81] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nori and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baldi, AX-GADGET: a new code for cosmological simulations of Fuzzy Dark Matter and Axion models, MNRAS 478 (2018) 3935 [1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [82] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schive, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chiueh and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nature Physics 10 (2014) 496 [1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6586].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [83] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mocz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fialkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Vogelsberger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Becerra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', First Star-Forming Structures in Fuzzy Cosmic Filaments, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 123 (2019) 141301 [1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01653].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [84] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hui and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bryan, Numerical and perturbative computations of the fuzzy dark matter model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 99 (2019) 063509 [1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01915].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [85] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schwabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Gosenca, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Behrens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Niemeyer and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Easther, Simulating mixed fuzzy and cold dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 083518 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [86] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kulkarni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Visbal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bryan and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Li, If dark matter is fuzzy, the first stars form in massive pancakes, arXiv e-prints (2022) arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11515 [2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11515].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [87] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Abdalla, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Abellán, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aboubrahim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Agnello, Ö.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Akarsu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Akrami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies, Journal of High Energy Astrophysics 34 (2022) 49 [2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [88] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lucca, Dark energy-dark matter interactions as a solution to the S8 tension, Physics of the Dark Universe 34 (2021) 100899 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [89] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Poulin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bernal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kovetz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kamionkowski, The Sigma-8 Tension is a Drag, arXiv e-prints (2022) arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06217 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [90] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kaplan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Krnjaic, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rehermann and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wells, Atomic dark matter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2010 (2010) 021 [0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0753].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [91] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cyr-Racine and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sigurdson, Cosmology of atomic dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 87 (2013) 103515 [1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5752].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [92] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bansal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Barron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Curtin and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tsai, Precision Cosmological Constraints on Atomic Dark Matter, arXiv e-prints (2022) arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02487 [2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02487].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [93] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Enqvist, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nadathur, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sekiguchi and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Takahashi, Decaying dark matter and the tension in σ8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2015 (2015) 067 [1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05511].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [94] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pandey, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Karwal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Das, Alleviating the H0 and σ8 anomalies with a decaying dark matter model, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2020 (2020) 026 [1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10636].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [95] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Driskell, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nadler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mirocha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Benson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Boddy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Structure formation and the global 21-cm signal in the presence of Coulomb-like dark matter-baryon interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 103525 [2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04499].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [96] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amon and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Efstathiou, A non-linear solution to the S8 tension?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', MNRAS 516 (2022) 5355 [2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11794].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [97] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hill, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' McDonough, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Toomey and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alexander, Early dark energy does not restore cosmological concordance, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 043507 [2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07355].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [98] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Piao, Resolving both H0 and S8 tensions with AdS early dark energy and ultralight axion, arXiv e-prints (2021) arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13391 [2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13391].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [99] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alexander, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bernardo and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Toomey, Addressing the Hubble and S8 Tensions with a Kinetically Mixed Dark Sector, arXiv e-prints (2022) arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13086 [2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13086].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [100] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Allali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hertzberg and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rompineve, Dark sector to restore cosmological concordance, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 104 (2021) L081303 [2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='12798].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 48 – [101] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chiueh, Cosmological Perturbations of Extreme Axion in the Radiation Era, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 96 (2017) 063522 [1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='01439].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [102] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cedeño, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' González-Morales and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ureña López, Cosmological signatures of ultralight dark matter with an axionlike potential, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 96 (2017) 061301 [1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [103] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Leong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schive, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chiueh, Testing extreme-axion wave-like dark matter using the BOSS Lyman-alpha forest data, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 484 (2019) 4273 [1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05930].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [104] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Arvanitaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dimopoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Galanis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lehner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Thompson and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Van Tilburg, Large-misalignment mechanism for the formation of compact axion structures: Signatures from the QCD axion to fuzzy dark matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 101 (2020) 083014 [1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11665].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [105] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pospelov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ritz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Skordis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ritz and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Skordis, Pseudoscalar perturbations and polarization of the cosmic microwave background, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 103 (2009) 051302 [0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0673].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [106] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sigl and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Trivedi, Axion-like Dark Matter Constraints from CMB Birefringence, 1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07873.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [107] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fedderke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Graham and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rajendran, Axion Dark Matter Detection with CMB Polarization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 100 (2019) 015040 [1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02666].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [108] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Obata, Implications of the cosmic birefringence measurement for the axion dark matter search, JCAP 09 (2022) 062 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [109] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Levkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Panin and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tkachev, Radio-emission of axion stars, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 023501 [2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [110] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Baumann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Green and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Wallisch, New Target for Cosmic Axion Searches, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 117 (2016) 171301 [1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08614].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [111] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D’Eramo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hajkarim and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Yun, Thermal Axion Production at Low Temperatures: A Smooth Treatment of the QCD Phase Transition, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 128 (2022) 152001 [2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [112] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hu, Structure formation with generalized dark matter, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 506 (1998) 485 [astro-ph/9801234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [113] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amendola and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Barbieri, Dark matter from an ultra-light pseudo-Goldsone-boson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B 642 (2006) 192 [hep-ph/0509257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [114] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hwang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Noh, Axion as a Cold Dark Matter candidate, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B 680 (2009) 1 [0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4738].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [115] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, Axion Cosmology, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 643 (2016) 1 [1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07633].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [116] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Madelung, Eine anschauliche Deutung der Gleichung von Schrödinger, Naturwissenschaften 14 (1926) 1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [117] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Khlopov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Malomed and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zeldovich, Gravitational instability of scalar fields and formation of primordial black holes, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 215 (1985) 575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [118] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ferreira, Ultra-Light Scalar Fields and the Growth of Structure in the Universe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 82 (2010) 103528 [1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3501].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [119] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ferreira, Axiverse cosmology and the energy scale of inflation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 87 (2013) 121701 [1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [120] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hložek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Allison, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dunkley and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Calabrese, Future CMB tests of dark matter: Ultralight axions and massive neutrinos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 95 (2017) 123511 [1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 49 – [121] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Jackson, Fingers of God: A critique of Rees’ theory of primoridal gravitational radiation, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 156 (1972) 1P [0810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3908].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [122] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nischimichi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Takada, Cosmological constraints without nonlinear redshift-space distortions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 043531 [2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [123] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D’Amico, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nishimichi, Taming redshift-space distortion effects in the EFTofLSS and its application to data, 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [124] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kaiser, Clustering in real space and in redshift space, MNRAS 227 (1987) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [125] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, The IR-resummed Effective Field Theory of Large Scale Structures, JCAP 02 (2015) 013 [1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='5954].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [126] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Garny, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sibiryakov, Time-Sliced Perturbation Theory II: Baryon Acoustic Oscillations and Infrared Resummation, JCAP 07 (2016) 028 [1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [127] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sibiryakov, Infrared Resummation for Biased Tracers in Redshift Space, JCAP 07 (2018) 053 [1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05080].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [128] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alcock and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Paczynski, An evolution free test for non-zero cosmological constant, Nature 281 (1979) 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [129] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nishimichi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Takada and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, Precision analysis of the redshift-space galaxy bispectrum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 063512 [2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [130] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cabass, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nishimichi, Cosmology with the redshift-space galaxy bispectrum monopole at one-loop order, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 106 (2022) 043530 [2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02800].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [131] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D’Amico, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Donath, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lewandowski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang, The BOSS bispectrum analysis at one loop from the Effective Field Theory of Large-Scale Structure, 2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [132] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Beutler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blake, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Colless, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Staveley-Smith, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant, MNRAS 416 (2011) 3017 [1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3366].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [133] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ross, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Samushia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Howlett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Percival, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Burden and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Manera, The clustering of the SDSS DR7 main Galaxy sample - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A 4 per cent distance measure at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15, MNRAS 449 (2015) 835 [1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3242].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [134] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bailey, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Beutler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bizyaev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blazek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, MNRAS 470 (2017) 2617 [1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [135] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Betoule, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kessler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Guy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mosher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hardin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples, A&A 568 (2014) A22 [1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='4064].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [136] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Balkenhol, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dutcher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Spurio Mancini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Doussot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Benabed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Galli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', A Measurement of the CMB Temperature Power Spectrum and Constraints on Cosmology from the SPT-3G 2018 TT/TE/EE Data Set, arXiv e-prints (2022) arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05642 [2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05642].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [137] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zuntz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Paterno, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Jennings, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rudd, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Manzotti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dodelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', CosmoSIS: Modular cosmological parameter estimation, Astronomy and Computing 12 (2015) 45 [1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3409].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [138] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Torrado and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lewis, Cobaya: code for Bayesian analysis of hierarchical physical models, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cosmology Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 2021 (2021) 057 [2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05290].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 50 – [139] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' La Posta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Louis, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Garrido and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hill, Constraints on prerecombination early dark energy from spt-3g public data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 083519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [140] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zaldarriaga, Combining Full-Shape and BAO Analyses of Galaxy Power Spectra: A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='6\\% CMB-independent constraint on H0, JCAP 05 (2020) 032 [2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04035].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [141] SDSS collaboration, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', SDSS-III: Massive Spectroscopic Surveys of the Distant Universe, the Milky Way Galaxy, and Extra-Solar Planetary Systems, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 142 (2011) 72 [1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1529].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [142] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, Cosmology without window functions: Quadratic estimators for the galaxy power spectrum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 103 (2021) 103504 [2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09389].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [143] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Philcox, Cosmology without window functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cubic estimators for the galaxy bispectrum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 104 (2021) 123529 [2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06287].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [144] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kalus, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Percival, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bacon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mueller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Samushia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Verde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', A map-based method for eliminating systematic modes from galaxy clustering power spectra with application to BOSS, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 482 (2019) 453 [1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='02789].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [145] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chudaykin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dolgikh and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, Constraints on the curvature of the Universe and dynamical dark energy from the Full-shape and BAO data, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 103 (2021) 023507 [2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [146] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kitaura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: mock galaxy catalogues for the BOSS Final Data Release, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 456 (2016) 4156 [1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06400].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [147] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rodríguez-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: modelling the clustering and halo occupation distribution of BOSS CMASS galaxies in the Final Data Release, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 460 (2016) 1173 [1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='06404].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [148] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Macaulay, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Trebitsch and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ferreira, Ultra-light Axions: Degeneracies with Massive Neutrinos and Forecasts for Future Cosmological Observations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 85 (2012) 103514 [1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0502].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [149] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pisanti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cirillo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Esposito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Iocco, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Mangano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Miele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', PArthENoPE: Public algorithm evaluating the nucleosynthesis of primordial elements, Computer Physics Communications 178 (2008) 956 [0705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0290].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [150] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Nishimichi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D’Amico, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Senatore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Takada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Blinded challenge for precision cosmology with large-scale structure: results from effective field theory for the redshift-space galaxy power spectrum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 123541 [2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='08277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [151] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Zhang, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Poulin and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Smith, On the consistency of effective field theory analyses of BOSS power spectrum, 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='05929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [152] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Feroz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hobson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bridges, MULTINEST: an efficient and robust Bayesian inference tool for cosmology and particle physics, MNRAS 398 (2009) 1601 [0809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='3437].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [153] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Feroz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hobson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cameron and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Pettitt, Importance Nested Sampling and the MultiNest Algorithm, The Open Journal of Astrophysics 2 (2019) 10 [1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='2144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [154] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amendola and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Barbieri, Dark matter from an ultra-light pseudo-Goldsone-boson, Physics Letters B 642 (2006) 192 [hep-ph/0509257].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [155] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Poulin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Smith, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Karwal and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kamionkowski, Early Dark Energy can Resolve the Hubble Tension, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 122 (2019) 221301 [1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='04083].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 51 – [156] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kass and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Raftery, Bayes factors, Journal of the American Statistical Association 90 (1995) 773 [https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='tandfonline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1080/01621459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='10476572].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [157] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Heavens, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Fantaye, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sellentin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Eggers, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hosenie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Kroon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', No Evidence for Extensions to the Standard Cosmological Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' 119 (2017) 101301 [1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03467].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [158] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Abbott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aguena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alarcon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Allam, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alves, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 105 (2022) 023520 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='13549].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [159] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Heymans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Tröster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Asgari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blake, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hildebrandt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Joachimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints, A&A 646 (2021) A140 [2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='15632].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [160] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Sánchez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Scoccimarro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Crocce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grieb, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Salazar-Albornoz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dalla Vecchia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Cosmological implications of the configuration-space clustering wedges, MNRAS 464 (2017) 1640 [1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [161] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Blake, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Childress, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Erben, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Glazebrook, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Harnois-Deraps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The 2-degree Field Lensing Survey: design and clustering measurements, Monthly Notices of the Royal Astronomical Society 462 (2016) 4240 [https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='com/mnras/article-pdf/462/4/4240/18517346/stw1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='pdf].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [162] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Raveri and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Doux, Non-Gaussian estimates of tensions in cosmological parameters, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 104 (2021) 043504 [2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='03324].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [163] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lemos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Raveri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Campos, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Chang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Weaverdyck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Assessing tension metrics with dark energy survey and Planck data, MNRAS 505 (2021) 6179 [2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09554].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [164] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schöneberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Lesgourgues and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hooper, The bao+bbn take on the hubble tension, Journal of Cosmology and Astroparticle Physics 2019 (2019) 029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [165] DESI Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aghamousa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Aguilar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ahlen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', The DESI Experiment Part I: Science,Targeting, and Survey Design, arXiv e-prints (2016) arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00036 [1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='00036].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [166] LSST Dark Energy Science Collaboration, Large Synoptic Survey Telescope: Dark Energy Science Collaboration, arXiv e-prints (2012) arXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0310 [1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='0310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [167] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Bechtol, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Birrer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Cyr-Racine, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Schutz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Adhikari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Snowmass2021 Cosmic Frontier White Paper: Dark Matter Physics from Halo Measurements, arXiv e-prints (2022) arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07354 [2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [168] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Grin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Amin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Gluscevic, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hlozek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Marsh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Poulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Gravitational probes of ultra-light axions, BAAS 51 (2019) 567 [1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='09003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [169] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Dvorkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Dark Matter Physics from the CMB-S4 Experiment, in 2022 Snowmass Summer Study, 3, 2022, 2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='07064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' [170] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Ivanov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' McDonough, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Hill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Simonović, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Toomey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=', Constraining Early Dark Energy with Large-Scale Structure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' D 102 (2020) 103502 [2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content='11235].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} +page_content=' – 52 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E_T4oBgHgl3EQf6BwH/content/2301.08361v1.pdf'} diff --git a/kb_test/content/tmp_files/2301.00001v1.pdf.txt b/kb_test/content/tmp_files/2301.00001v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3e12a9b0225dc645f46513352fd9fdf1cd8f8a6 --- /dev/null +++ b/kb_test/content/tmp_files/2301.00001v1.pdf.txt @@ -0,0 +1,629 @@ +NFTrig: Using Blockchain Technologies for Math Education +JORDAN THOMPSON, Augustana College, USA +RYAN BENAC, Augustana College, USA +KIDUS OLANA, Augustana College, USA +TALHA HASSAN, Augustana College, USA +ANDREW SWARD, Augustana College, USA +TAUHEED KHAN MOHD, Augustana College, USA +NFTrig is a web-based application created for use as an educational tool to teach trigonometry and block +chain technology. Creation of the application includes front and back end development as well as integration +with other outside sources including MetaMask and OpenSea. The primary development languages include +HTML, CSS (Bootstrap 5), and JavaScript as well as Solidity for smart contract creation. The application itself +is hosted on Moralis utilizing their Web3 API. This technical report describes how the application was created, +what the application requires, and smart contract design with security considerations in mind. The NFTrig +application has underwent significant testing and validation prior to and after deployment. Future suggestions +and recommendations for further development, maintenance, and use in other fields for education are also +described. +CCS Concepts: • Computer systems organization → Redundancy; Robotics; • Networks → Network +reliability. +Additional Key Words and Phrases: Matic, Metamask, polygon, bootstrap5, Solidity +1 +INTRODUCTION +The purpose of this report is to describe the technical details involved in the development of the +NFTrig application. This includes both the front end website design, the back end smart contract, +and NFT creation. It will mainly focus on the technical details of the project outlining software +requirements, design through programming languages, client and server side interactions, and +validation testing. This allows the reader to undertake further development, fixes, or maintenance +of the software, as this forms part of the documentation for the software. +The NFTrig project is based around the creation of a web-based game application that allows +interaction of NFTs (non-fungible token) with trigonometric function designs. NFts are digital +assets, for example a picture, that has a unique identification and can generally be freely traded +with cryptocurrency [33]. Through this application, users are able to purchase digital artwork of +many different trigonometric functions and combine them using mathematical operations. Current +supported operations include multiplication and division of the trigonometry functions, and the +output of each operation is a new NFT card that would be the result of an operation. The old cards +will then be removed from the user’s possession and burned using the smart contact. For example, +if a user combined the two cards Sin(x) and Cos(x) using multiplication, they would lose their two +old cards and receive the new card Tan(x). Further, the NFT cards are assigned one of the following +rarity levels: common, uncommon, rare, and legendary. The probability of each of these levels is +defined later in this report. +The application also allows a user to connect to MetaMask, a digital wallet capable of storing a +user’s cryptocurrency and NFTs as well as a way to connect to block chain. The NFTrig application +Authors’ addresses: Jordan Thompson, jordanthompson18@augustana.edu, Augustana College, Rock Island, USA; Ryan +Benac, ryanbenac18@augustana.edu, Augustana College, Rock Island, USA; Kidus Olana, kidusolana18@augustana.edu, +Augustana College, Rock Island, USA; Talha Hassan, talhahassan18@augustana.edu, Augustana College, Rock Island, +USA; Andrew Sward, andrewsward@augustana.edu, Augustana College, Rock Island, USA; Tauheed Khan Mohd, +tauheedkhanmohd@augustana.edu, Augustana College, Rock Island, USA. +arXiv:2301.00001v1 [cs.HC] 21 Dec 2022 + +2 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +can also display the NFTs owned by the user and allow them to connect to OpenSea to sell the +NFTrig cards on a public marketplace. The application is hosted on Moralis employing their Web3 +API. Technical languages used in this project, which will be discussed in detail throughout this +paper, include front end web development languages HTML, CSS (specifically Bootstrap5), and +JavaScript as well as the back end smart contract development language Solidity. +In order to attract users, this application also allows a user to answer trivia questions and gain +experience points. These points can then be used to unlock new sets of NFT cards or upgrade existing +cards in a user’s wallet. This game-like design should appeal to a younger audience and encourage +them to answer trigonometry or math based questions. This will have an incredible educational +benefit for the user because they will be both learning and playing a game simultaneously. +2 +MOTIVATION +The purpose of this application is as an educational tool for students who are attempting to +understand the ways that trigonometric functions interact with each other. As opposed to just +graphing these functions by hand, students will be able to generate new NFTs by combining +whatever trigonometric functions they already own. In fact, using technology is shown to influence +and better educational processes by increasing interaction between those in the classroom [9]. +Technology is becoming increasingly prevalent in every sphere of daily life, so the use of technology +in a classroom setting is not only logical, but it increases the educational benefit of students [29]. +However, as the technology continues to evolve, "the gap between traditional course material +taught to students in B.S./M.S. programs at universities and the cutting edge of technology used in +industry is widening at an unprecedented rate" [30]. By creating this project, it will give students +the opportunity to gain experience with block chain, and hopefully be a starting place for narrowing +that ever growing gap. After much research, it is likely that this proposed application is the first of +its kind that utilizes NFTs to teach mathematical concepts. +Aside from user benefit of this application, there is also an intellectual merit in the block chain +and education fields. Best described by Carmen Holotescu, "As education becomes more open, +diversified, democratised, and decentralised, the block chain technology is taken in consideration +by researchers, teachers and institutions, to maintain reputation, trust in certification, and proof of +learning" [17]. Further, development of this project continues research on NFT and block chain +technologies. This application can also serve as the boilerplate basis for other NFT-based educational +tools and resources. Research for this project provides opportunities for training computer science +students on how to use NFTs in general, but more specifically in educational contexts. +NFTrig was developed by computer science students as a final senior inquiry project at Augustana +College. In conjunction and with funding by the Department of Mathematics and Computer Science, +this project employs a variety of software development skills and techniques that further the +research and understanding of the block chain and web development field. +3 +RELATED WORK +Block chain technology has enabled the formation of decentralized distributed records of digital +data which does not require any third party to moderate any transactions [34]. The decentralized +nature of block chain also renders it easy for use in a ranging variety of applications in several fields +such as healthcare [16], internet of things [7], gaming [2], banking [6], and education (explored in +greater detail in subsection 3.1). Non Fungible token (NFTs) are a relatively new phenomena within +the field of block chain based technologies, but its application in aforementioned fields are already +being studied. Specifically within the healthcare context, NFT’s are solving long term issues such as +storing patients’ private data more safely as well as maintaining better records while giving better +autonomy and privacy to both patients and healthcare providers [22]. The application of NFTs in + +NFTrig: Using Blockchain Technologies for Math Education +3 +education is still an understudied area. These next related work sections explore the broader use of +block chain based technologies for educational purposes, gamification, and overall collaborative +learning. +3.1 +Block chain Based Technologies for Educational Purposes +There has been extensive work concerning how block chain based technologies are enabling better +ownership and sharing of personal records for students and supporting collaborative learning +environments. Yumna et al. conducted a systematic literature review of the use of block chain +technologies in educational sector [35]. They also propose several uses of existing block chain +based technologies in educational sector that leverage the decentralized and traceable consensus +making mechanisms of block chain. Researchers have examined the use of block chain to allow +students to maintain educational records such as transcripts, credentials, diplomas, and learning +activities [5, 14, 31]. Similarly, research has also explored learning management systems design +based on block chain based technology. The technology can potentially verify a students records as +well as enable the design of an automatic decentralized enrollment system which does not require +moderation from school staff [31]. +Another elegant use of block chain in the field of education is the ability to support life-long +learning applications. The educational sector is becoming more diverse with a variety of different +types of classrooms and learning modalities. E-learning has also allowed students to acquire +licences and accreditation online. Therefore, it is imperative to maintain the learning journeys of +students over time to understand the different types of learning that they have been engaging in +and improving on over time. The traceable nature of block chain based technologies (defined as +one of the salient features in the aforementioned systematic review by [35]) enables all of these +applications. +The decentralized nature of block chains coupled with the consensus making algorithms also +makes it suitable for collaborative environments. Prior research has looked at how block chain +based technologies can enable better developmental experiences in the realm of business [11] but +there is very minimal work on its application within the field of education application[3]. +3.2 +Applications in Education Application and Collaborative Learning +Although preliminary in nature, limited prior work has explored the utilization of NFTs for design- +ing various different independent learning environments for students. There are some proposed +commercial systems that have analogous functioning to some of the systems described in the prior +section. For example, commercial systems are looking at leveraging NFTs to award “Pass" status +to students for different courses 1. NFTs enjoy a key advantage over conventional block chain +technologies as they are typically designed using the more secure Ethereum block chain enabling an +even more secure record and identity management. Researchers have shown that there is promise +in using NFTs as academic tokens to represent student transcripts and other records as well that +can be more easily verified [9]. However, there is still a dearth of academic literature in this field. +Student incentivization is heavily advocated in pedagogical literature [12]. NFTs make it easier +to tie incentivization to learning outcomes as they can be automatically acquired by students at +any time upon completion of learning outcomes. This gives NFTs based certifications an advantage +over the more traditional learning settings where students have to strongly adhere to semester +timelines. Elmessiry et al. has looked at designing an incentive mechanism that can be used by +teachers and students to achieve better learning outcomes in an effective and cost-efficient manner +1A teacher at Pepperdine University using NFTs to award course completion certifications to students: https://upcea.edu/tech- +trends-in-higher-ed-metaverse-nft-and-dao/ + +4 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +[9]. They also concluded there was better engagement outcomes for students. On several metrics +of usability, the students reported more than 80% preference for buying, using, and collecting +NFTs. Such independent learning methods were particularly more useful during the COVID-19 +pandemic to accommodate the need of remote independent learning options. Architecturally, this +project takes inspiration from [9], and applies it to a more narrower, focused domain of learning +mathematical operations in this study. Further, these NFTs are also easier to share on social media +[20]. Therefore, it also allows students to more readily share their accomplishments. +3.3 +Gamification to Support Mathematical Learning +Since the proposed application teaches mathematical and trigonometric formulas to students, the +literature on use of gamification to support mathematical learning should be better described. +Gamification, in combination with incentivization explained in the previous section, will allow +for the success of this application. Gaming settings have traditionally been used to teach simple +mathematical operations to students. More recently, researchers have also proposed systems that +teach advanced concepts to students including College Algebra [10]. These learning environments +make it easier for students to relate the learning concepts with more daily life phenomena. While +gamification itself cannot guarantee better learning outcomes, it can improve students’ interest +and performance by encouraging them to engage with the content for a longer duration of time +[18]. The simpler, more systematic, and operational nature of mathematics as a subject also makes +it easier for incorporation in gaming environments because final answers are usually short and +numerical as opposed to long and descriptive answer that might be found in social or natural +sciences. Trigonometry especially can easily be broken down into a series of operations and steps +which simulates a similar environment found in other online games where users play to find +different “rewards" and “collectables". Despite all these benefits there are some limitations of +gamification as well. For example, it is hard to know how a student arrived a solution and give +feedback [4]. Not being able to solve trigonometric equations can also lead to frustration and +impeded learning experience. Foresight into the project’s future looks to mitigate these concerns +by fostering better communication between different game players and providing links to useful +learning resources in the application. Prior research has extensively explored the use of gamification +in different mathematical fields. This application is likely the first to extend the use of NFTs and +block chain to aid in teaching trigonometric equations. +Research shows that technology, specifically games are shown to be excellent educational tools. +In fact, "one of the most successful positive reinforcement mechanisms [in education] known is +gamification" [9]. This includes taking a topic transforming it into a game with positive reinforce- +ment. This leverages educational benefits in students and encourages them to continue playing the +game to learn. Nftrig has future plans to add a game function which will allow the user to answer +trigonometry trivia and math questions. This will aid in both their learning and the continued use +of the NFTrig application. Further, the ability to combine owned NFTs with math functions also +aids in the education of trigonometry for the student. +4 +EXPERIMENTAL SETUP +4.1 +Software Development Requirements +The NFTrig application employs a variety of software development requirements that cover the +range of the project. From front end web development to back end smart contract creation and +NFT storage, this section describes the requirements and software used to complete the project. +4.1.1 +Compiling IDE. The smart contracts created for NFTrig are hosted on Remix. Remix is an +an open source online compiler IDE that can be used to test and deploy smart contracts [1]. The + +NFTrig: Using Blockchain Technologies for Math Education +5 +platform can be accessed by any browser, and it allows the developer to write and deploy smart +contracts on an actual or test server simultaneously. The current deployment is on a test server. In +order to test and debug the smart contract, Visual Studio Code is used. Visual Studio was found +to be the best code editor because a developer can easily upload most file types, and edit them +[19]. For NFTrig, it was used to develop front end HTML and CSS files, as well as back end solidity +contract editing. The required installed plugins for Visual Studio (VS) include Solidity and Block +chain development. [21] These allowed for simple, straightforward development of code. +4.1.2 +Moralis. Moralis SDK is the primary back end platform for the project. The platform allows +connection of the front end web application to the smart contract. [8] The Moralis platform uses +a combination of server management and a JavaScript SDK to allow for maximum interaction +and simplicity. A developer can do many tasks through this including authentication of users, +getting necessary user data, and connecting with MetaMask in a non-complicated and simply coded +process. The only expectation is that a developer will need to have programming knowledge in +JavaScript as well as a familiarity with Moralis and MetaMask, experience querying a database, and +some knowledge of Web3 development to ensure maximum results and efficiency. Moralis also has +the ability to easily connect to MetaMask. +4.1.3 +MetaMask. MetaMask is the digital wallet required for participation in the NFTrig game +application. It allows the collection of purchases from the user, and it can be installed as an extension +on a browser for increased ease of use [28]. MetaMask stores all NFTs owned by the user, and in +connection with the NFTrig application, can view and upgrade or modify existing NFTs at a users +discretion. Connection to the browser extension is required for the application to access anything +owned by a user [24]. Because MetaMask is easily integrated into Moralis, and thus NFTrig, there +is little a user needs to do to create a connection aside from installing the MetaMask extension, and +clicking connect. +4.1.4 +Front End Design. Front end design was accomplished primarily through Visual Studio. The +Live Server extension was installed which allows each developer to "host" their developed website +using a native web application. Doing so allowed simplified testing and front end development. +Instead of creating CSS files from scratch, the NFTrig interface heavily employs Bootstrap5, which +simplifies the process of modifying the content layout and design of buttons and other content +[25]. Moralis and Bootstrap5 each have extensive documentation to understand and support front +end web development. These tools have been utilized to a near maximum extent. +4.1.5 +Web Hosting Platform. The initial testing of NFTrig, as previously explained, was hosted on +a local live server through Visual Studio. After initial development, the project was moved to a web +server hosted by Augustana College so that initial testing could begin. It is currently unclear how +the site will ultimately be hosted. One option for hosting the web application is directly through +Google [32]. This would allow the website to be named something easily searchable and accessible. +A second option would be to host directly through Moralis, but a limitation of this would be a +more diluted website naming convention along with a more confusion process of uploading and +modifying website content. Currently, the NFTrig application will remain on the local Augustana +College Server. +5 +SOFTWARE DESIGN +This section covers all of the decisions necessary to understand the development of NFTrig, as well +as the technical implementation of each technology used in the design process. + +6 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +5.1 +Software Architecture +The architecture of this project follows the model-server design architecture [27]. Using this model, +the clients send transactions and requests to a proxy smart contract stored on the block chain +which then makes the appropriate calls to the logic smart contract which is also stored on the block +chain. This style of architecture is required for this project because the smart contracts must be +stored on the server-side chain in order to be functional. The use of proxy contracts also allows our +smart contracts to be fully upgradeable with any future updates that may need to be implemented. +5.2 +Choice of Programming Language +This section examines and explains the benefit of each chosen language employed in NFTrig. Front +end languages include HTML and CSS and the back end includes Solidity and JavaScript. Each has +been chosen because they were found to be the best option for development. +5.2.1 +Solidity. Solidity is the programming language of choice when it comes to coding smart +contracts. Solidity is "similar to JavaScript and yet has some features of object-oriented languages +such as Java and C++" [26]. This is a leading language for the development of smart contracts and +use on block chain technologies. This project utilizes the solidity library openzeppelin in order to +create a solid foundation for the smart contracts. Hardhat and Node JS are then used for the testing +and deployment of the smart contracts to the Polygon blockchain. +5.2.2 +JavaScript. In the NFTrig application, JavaScript (JS) is primarily used in the front end +application. The primary purpose of this language is generally to create dynamic and interactive +web content [15]. For the client, JS was used in the navigation bar to allow for clickable links and +resizing of the navigation bar in smaller screens. This language was also used to give buttons +functionality ranging from logging in to MetaMask to purchasing NFTrig cards. Further, JS was used +to test the logic of the front-end combination page until the smart contract was applied. Aside from +augmenting HTML and CSS application pages, JavaScript is also used in this project to connect the +back end smart contract with the from end web application. This application was also developed +using Next JS and deployed via an application known as vercel. +5.2.3 +HTML and CSS. Web development of the user interface was primarily completed using +HTML and CSS (Bootstrap5). These languages are equally popular and necessary to develop the +web pages [13]. Instead of creating all CSS requirements from scratch, Bootstrap5 was utilized to +allow for cleaner design across web pages and better alignment of web page elements. Bootstrap5 +also simplifies the need to explicitly code buttons and other interactive items. +5.3 +Security Considerations +Throughout this project, there have been several security considerations discovered that threatened +the safety and use of the application. One such discovered issue was initially, there was no code +written to block a user from looking at another users token. Further, before minting a new NFT +card, the smart contracts check to ensure that the card does not already exist, the cards used for +combining are owned by the user, and that the newly minted card follows the correct probabilities +of outcomes shows in 2. These probabilities are coded into the smart contract. +5.4 +Smart Contract Design +The smart contract for this project is broken up into two separate contracts. The first of which is +the NFTrig logic contract which contains the logic for purchasing packs of cards as well as the logic +for how cards will interact with each other. The second contract is the marketplace contract which +will allow users to trade their own NFTs with other users through the website. Within the NFTrig + +NFTrig: Using Blockchain Technologies for Math Education +7 +contract, there are functions for multiplying and dividing cards, purchasing randomized packs of +cards, and tracking the details of each individual token as transactions are made. The marketplace +contract contains information about sale history as well as the functionality to post new sales and +purchase items for sale. Both of these contracts were deployed as upgradeable contracts so they +can have updates implemented in the future. +5.5 +NFT Storage and Naming Conventions +All NFT images are stored on the server with the HTML, CSS, and JS files. The naming convention +for each image references what image it is in four numbers. The first number is the power of sin, the +second is the power of cos, the third is the rarity or color of the card (0-3 is green, blue, purple, and +red respectively), and the final number is the text variant (0-3). These files were named accordingly +to better determine the output if cards were combined using a mathematical function. For example, +a sin card might have the naming convention: 1023.jpg. 10 defines it is a sin card, 2 defines it is +rarity purple, and 3 defines it is text variant 3. The purpose of naming the files in this way is so +that the front end can easily determine which image corresponds to a particular NFT by simply +looking at the four features of each token which match the four numbers in the file name. +5.6 +Client Design +The NFTrig application interface was designed using HTML and CSS. The primary use of CSS was +often replaced by Bootstrap5. Bootstrap 5, a library for CSS, allows for easier scaling and alignment +of objects in the HTML file, and thus the computer screen [23]. Documentation on the Bootstrap5 +has utilized to a full extent. Each section examines the layout and use of each application page. +5.6.1 +NFTrig Home. The interface is designed to allow a user to access the marketplace, their +individual current collections, and their profile. The navigational bar contains links to the client-side +facing pages: NFTrigHome, MyCards, CombineCards, Marketplace, and Game. We used a total of +three colors to enable good contrast and make it easier for our users to view complex graphs and +formula without a cluttered background 2. The JavaScript elements declared are reusable across +multiple screens. They support functions and interactions such as a user hovering over a cell or +clicking a cell and providing both feedback and error handling to the user. The navigation bar is +also, the top bar changes color to indicate the tab that the user is on. +5.6.2 +Combination. The main purpose of the combination page is for users to choose cards that +they currently own, and see options for combining them using either multiplication or division. +Figure 1 displays the layout of the screen where user selected cards are shown on the left, and +potential results are shown on the right. +The page utilizes Bootstrap5 capabilities to format effectively to different screen sizes and +resolutions. It connects with a back end script to the smart contract. This provides functionality to +the buttons and easy generation of possible NFT results. Below shows the probabilities of generated +NFT outcomes based on the selected input cards. +5.6.3 +Marketplace and MyCards. Marketplace and MyCards are similar pages, as they connect to +a data source and display NFTs. The Marketplace tab shows all NFT cards available for purchase +both from other users who own NFTs and cards owned by the NFTrig project. MyCards however +specifically shows all cards owned by a user. The layout for each generates all necessary NFT +images and information about the rarity. The rarity is signified by the color and the text option of +the card. Figure 3 shows the actual layout displayed on the page. +2Background-color:#333, Color: #f2f2f2, + +8 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +Fig. 1. Interface where users will combine NFTrigs +Fig. 2. Probabilities of outcomes depending on rarity of selected cards +5.6.4 +Quality attributes of client-side interface and code. In order to have an application of quality, +consistency, and accuracy, the project followed the following guidelines: +(1) The code is written in a manner that components and layouts can be rearranged to support +any structural changes in the front end. +(2) The code has consistent style and format, such as the padding used in individual NFTrig +elements and the purchase page’s color. +(3) The code contains comments and is well indented for easy maintenance and understanding. +(4) Consistent colors and feedback systems are provided so the system is easy to learn for users. +(5) Page-level styling was avoided when possible to keep design consistent. +(6) Thorough testing was completed for basic accessibility features. +5.6.5 +Testing the Client Design. Basic unit testing of different elements was initially conducted +to ensure easy navigation between front end pages. In order to ensure that testing would cover + +Combination +Choosetwo cards and a mathfunction and hit combine! +PossibleResults +SIN(x) +TAN(x) +Sin(x)*Tan(X) +SN()TAN(a) +SIN(TAN) +Sin(x)*Tan(X) +Rarity: Green +Rarity Blue +Font: 0 +Fonto +Liklihood: 75% +Liklihood: 20% +sin(r)jtao(s) +sin(z+y) +d +=sin(z)cos(y)+sin(y)cos(z) +tan(r)=sec() +dar +SIN(o)-TAN() +Sin(x)*Tan(X) +SIN(a)-TAN(a) +Sin(x)*Tan(X) +Rarity: Blue +Rarity: Pink +Borcler +Font:o +Sin(X) Green +Tan(x) Bue +Font:0 +Liklihood: 20% +Multiply +Divide +Likliho0d: 20% +(r+V +Combine +Copyright@2022-AllRightsRes +rved-Augustana CollegeNFTrigCOMBINATIONS +COMMON +UNCOMMON +RARE +LEGENDARY +Common+Common +20% +60% +15% +5% +Common+Legendary +10% +25% +35% +30% +Common+Rare +10% +50% +25% +15% +Common+Uncommon +10% +60% +20% +10% +Rare+Rare +5% +15% +30% +50% +Uncommon+Uncommon +5% +20% +60% +15% +Uncommon+Rare +5% +10% +60% +25% +Uncommon+Legendary +5% +10% +55% +30% +Rare +Legendary +0% +10% +30% +60% +Legendary+Legendary +0% +5% +20% +75%NFTrig: Using Blockchain Technologies for Math Education +9 +Fig. 3. Interface displaying NFTrig Marketplace +most application uses, three user cases were devised: a user browsing NFTrigs, a user making a +purchase, and a user combining NFTrigs. All assumptions and expected actions expected from +the system were listed and analyzed through testing. Further, testing through some edge cases +were also pursued. Currently, the application works as intended, however future plans involve +rigorous testing with JavaScript code and external APIs (if any are devised). This will ensure a fully +functional, secure, and usable application that can also be used as a boiler plate project for other +educational blockchain technologies. +5.6.6 +Future Work: Game. Future work for this project will include the ability for users to play +a trivia and trigonometric equation game. This allows a user to gain experience points that they +can then use to purchase new NFTs. This eliminates the need to always need cryptocurrency to +purchase individual or group NFT cards. Although there is not currently an interface for this page +written in HTML, functionality exists for the trivia game itself. The files are currently stored on +the server, but they are disabled and there is no navigable way to get there through the application. +6 +METHODS +Most methods for completing this project have been thoroughly explained in the sections above. +However, the final intended version of this project will be hosted in a different location than it +resides currently. The initial portion of this project had the front end website hosted on a local +Augustana College server and the back end smart contract hosted on the Polygon test net. This +allowed initial testing and validation that the smart contract operated as expected, as well as give +time and opportunity to discover security vulnerabilities. The future of this project will be hosted +on a decentralized web application online so that users can access it and begin to interact with the +smart contract. Further, a redesign of the website user interface is likely. This will require transition +from BootStrap5 to NextJS which allows cards to be generated, displayed, and interactable through +a version of JavaScript. +7 +RESULTS +This project successfully allowed the exploration and creation of applying NFT and block chain +technology to math education. Although preliminary in use and nature, this project allows for +initial project creation as a boiler plate project. The smart contract is currently deployed on the + +Marketplace +Collections +Profile +About +Metamask +Search.. +Q +NFTrig +SEC(aCSC(a) +SEC(a)CSC() +SEC(a)-CSC(a) +SEC()CSa) +SEC()CGCR +U +2 csc(2x) +to be added +sec2(x)+csc2(x) +cot(x) + tan(x) +2 csc(2x) +to be added +SEC(-CSCa) +SEC(u)CSC(a) +SEC(-CSC() +sec2(x)+csc2(x) +cot(x) + tan(x) +2 csc(2x) +to be added +sec2(x)+csc2(x) +cot(x) + tan(x)10 +Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd +Polygon testnet and can be interacted with using test Matic. Each web page has functionality to +display the user’s owned NFTs as well as the NFTs they have put for sale on the marketplace. Using +NextJS will also allow the Combination page to have functionality and smart contract use. It is also +worth noting that the created web page is not required to interact with the NFTrig smart contracts. +8 +RECOMMENDATIONS FOR FUTURE WORK +The goal for this project was a working Beta demo that shows application functionality, and correct +smart contract execution. There are many other features planned for the continued work of this +project. The first, as earlier explained, is a game option which challenges the user with trigonometry +trivia and math problems. Answering these questions successfully will increase the experience +points of a user. The user can then use these experience points to purchase individual or packs of +NFTrig cards, or they can be used to combine cards. +REFERENCES +[1] Rana M Amir Latif, Khalid Hussain, NZ Jhanjhi, Anand Nayyar, and Osama Rizwan. 2020. A remix IDE: smart +contract-based framework for the healthcare sector by using Blockchain technology. Multimedia Tools and Applications +(2020), 1–24. +[2] Mohsen Attaran and Angappa Gunasekaran. 2019. Blockchain for Gaming. In Applications of Blockchain Technology in +Business. Springer, 85–88. +[3] Rocsana Bucea-Manea-T, oniş, Oliva Martins, Radu Bucea-Manea-T, oniş, Cătălin Gheorghit,ă, Valentin Kuleto, Milena P +Ilić, and Violeta-Elena Simion. 2021. Blockchain Technology Enhances Sustainable Higher Education. Sustainability +13, 22 (2021), 12347. +[4] Juan José Bullón, Ascensión Hernández Encinas, M. Jesús Santos Sánchez, and Víctor Gayoso Martínez. 2018. Analysis +of student feedback when using gamification tools in math subjects. In 2018 IEEE Global Engineering Education +Conference (EDUCON). 1818–1823. https://doi.org/10.1109/EDUCON.2018.8363455 +[5] Guang Chen, Bing Xu, Manli Lu, and Nian-Shing Chen. 2018. Exploring blockchain technology and its potential +applications for education. Smart Learning Environments 5, 1 (2018), 1–10. +[6] Luisanna Cocco, Andrea Pinna, and Michele Marchesi. 2017. Banking on blockchain: Costs savings thanks to the +blockchain technology. Future internet 9, 3 (2017), 25. +[7] Marco Conoscenti, Antonio Vetro, and Juan Carlos De Martin. 2016. Blockchain for the Internet of Things: A systematic +literature review. In 2016 IEEE/ACS 13th International Conference of Computer Systems and Applications (AICCSA). IEEE, +1–6. +[8] Oscar Delgado-Mohatar, Ruben Tolosana, Julian Fierrez, and Aythami Morales. 2020. Blockchain in the Internet of +Things: Architectures and Implementation. In 2020 IEEE 44th Annual Computers, Software, and Applications Conference +(COMPSAC). 1072–1077. https://doi.org/10.1109/COMPSAC48688.2020.0-131 +[9] A Elmessiry, M Elmessiry, and L Bridgesmith. 2021. NFT STUDENT TEACHER INCENTIVE SYSTEM (NFT-STIS). In +Proceedings of EDULEARN21 Conference, Vol. 5. 6th. +[10] Usef Faghihi, Albert Brautigam, Kris Jorgenson, David Martin, Angela Brown, Elizabeth Measures, and Sioui Maldonado- +Bouchard. 2014. How Gamification Applies for Educational Purpose Specially with College Algebra. Procedia Computer +Science 41 (2014), 182–187. +https://doi.org/10.1016/j.procs.2014.11.102 5th Annual International Conference on +Biologically Inspired Cognitive Architectures, 2014 BICA. +[11] Julian Alberto Garcia-Garcia, Nicolás Sánchez-Gómez, David Lizcano, María José Escalona, and Tomás Wojdyński. +2020. Using blockchain to improve collaborative business process management: Systematic literature review. IEEE +Access 8 (2020), 142312–142336. +[12] Susan Gass, Koen Van Gorp, and Paula Winke. 2019. Using different carrots: How incentivization affects proficiency +testing outcomes. Foreign Language Annals 52, 2 (2019), 216–236. +[13] Ammar Yanuar Ghulam. 2021. Konseptual Desain Website Aplikasi Penyedia Jasa Kursus Mengemudi Mobil Di +Purwokerto Menggunakan Framework Bootstrap 5. (2021). +[14] Alexander Grech and Anthony F Camilleri. 2017. Blockchain in education. Luxembourg: Publications Office of the +European Union. +[15] Marijn Haverbeke. 2018. Eloquent javascript: A modern introduction to programming. No Starch Press. +[16] Marko Hölbl, Marko Kompara, Aida Kamišalić, and Lili Nemec Zlatolas. 2018. A systematic review of the use of +blockchain in healthcare. Symmetry 10, 10 (2018), 470. + +NFTrig: Using Blockchain Technologies for Math Education +11 +[17] Carmen Holotescu et al. 2018. Understanding blockchain opportunities and challenges. In Conference proceedings of» +eLearning and Software for Education «(eLSE), Vol. 4. ” Carol I” National Defence University Publishing House, 275–283. +[18] Tomislav Jagušt, Ivica Botički, and Hyo-Jeong So. 2018. Examining competitive, collaborative and adaptive gamification +in young learners’ math learning. Computers Education 125 (2018), 444–457. https://doi.org/10.1016/j.compedu.2018. +06.022 +[19] Bruce Johnson. 2012. Professional visual studio 2012. John Wiley & Sons. +[20] Arnav Kapoor, Dipanwita Guhathakurta, Mehul Mathur, Rupanshu Yadav, Manish Gupta, and Ponnurungam Ku- +maraguru. 2022. TweetBoost: Influence of Social Media on NFT Valuation. arXiv preprint arXiv:2201.08373 (2022). +[21] Parth Khandelwal, Rahul Johari, Varnika Gaur, and Dharm Vashisth. 2021. BlockChain Technology based Smart +Contract Agreement on REMIX IDE. In 2021 8th International Conference on Signal Processing and Integrated Networks +(SPIN). 938–942. https://doi.org/10.1109/SPIN52536.2021.9565983 +[22] Kristin Kostick-Quenet, Kenneth D. Mandl, Timo Minssen, I. Glenn Cohen, Urs Gasser, Isaac Kohane, and Amy L. +McGuire. 2022. How NFTs could transform health information exchange. Science 375, 6580 (2022), 500–502. https: +//doi.org/10.1126/science.abm2004 arXiv:https://www.science.org/doi/pdf/10.1126/science.abm2004 +[23] Jörg Krause. 2020. Introduction to Bootstrap. In Introducing Bootstrap 4. Springer, 1–17. +[24] Wei-Meng Lee. 2019. Using the metamask chrome extension. In Beginning Ethereum Smart Contracts Programming. +Springer, 93–126. +[25] Raoul LePage and Lynne Billard. 1992. Exploring the limits of bootstrap. Vol. 270. John Wiley & Sons. +[26] Debajani Mohanty. 2018. Basic solidity programming. In Ethereum for Architects and Developers. Springer, 55–103. +[27] Haroon Shakirat Oluwatosin. 2014. Client-server model. IOSRJ Comput. Eng 16, 1 (2014), 2278–8727. +[28] Deni Pramulia and Bayu Anggorojati. 2020. Implementation and evaluation of blockchain based e-voting system with +Ethereum and Metamask. In 2020 International Conference on Informatics, Multimedia, Cyber and Information System +(ICIMCIS). 18–23. https://doi.org/10.1109/ICIMCIS51567.2020.9354310 +[29] R Raja and PC Nagasubramani. 2018. Impact of modern technology in education. Journal of Applied and Advanced +Research 3, 1 (2018), 33–35. +[30] A Ravishankar Rao and Riddhi Dave. 2019. Developing hands-on laboratory exercises for teaching STEM students the +internet-of-things, cloud computing and blockchain applications. In 2019 IEEE Integrated STEM Education Conference +(ISEC). IEEE, 191–198. +[31] Diane J Skiba et al. 2017. The potential of blockchain in education and health care. Nursing education perspectives 38, 4 +(2017), 220–221. +[32] Craig Standing. 2002. Methodologies for developing Web applications. Information and Software Technology 44, 3 +(2002), 151–159. https://doi.org/10.1016/S0950-5849(02)00002-2 +[33] Qin Wang, Rujia Li, Qi Wang, and Shiping Chen. 2021. Non-fungible token (NFT): Overview, evaluation, opportunities +and challenges. arXiv preprint arXiv:2105.07447 (2021). +[34] Hafiza Yumna, Muhammad Murad Khan, Maria Ikram, and Sabahat Ilyas. 2019. Use of Blockchain in Education: A +Systematic Literature Review. In Intelligent Information and Database Systems, Ngoc Thanh Nguyen, Ford Lumban +Gaol, Tzung-Pei Hong, and Bogdan Trawiński (Eds.). Springer International Publishing, Cham, 191–202. +[35] Hafiza Yumna, Muhammad Murad Khan, Maria Ikram, and Sabahat Ilyas. 2019. Use of blockchain in education: a +systematic literature review. In Asian Conference on Intelligent Information and Database Systems. Springer, 191–202. + diff --git a/kb_test/content/tmp_files/98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments.pdf.txt b/kb_test/content/tmp_files/98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78aaa183d25fe30879e2b6e9df02ed8ea45f0520 --- /dev/null +++ b/kb_test/content/tmp_files/98e2f027-c8ee-45d3-9b9f-9bfd2d232293-01-2018-A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments.pdf.txt @@ -0,0 +1,1769 @@ +RESEARCH +Open Access +A metagenomics roadmap to the +uncultured genome diversity in hypersaline +soda lake sediments +Charlotte D. Vavourakis1 +, Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y. Sorokin3,4 +and Gerard Muyzer1* +Abstract +Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity. Despite the +high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but +the microbiome of soda lake sediments received much less attention of microbiologists. Here, we performed metagenomic +sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, +extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages. +Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and +a salt content between 70 and 400 g L−1. The recovered 16S rRNA gene sequences were mostly from Bacteria, even in +the salt-saturated lakes. Most OTUs were assigned to uncultured families. We reconstructed 871 metagenome-assembled +genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla +Radiation (CPR). Five new species of CPR were among the most dominant community members. Novel dominant +lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen +cycling. Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla +never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the +Actinobacteria. +Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important +advances. First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR +and several hundred other novel prokaryote lineages. The soda lake CPR is a functionally diverse group, but the most +abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation. Second, +we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those +encompassing known homo-acetogens, sulfate-reducers, and methanogens. Since only few environmental metagenomics +studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant +not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine +and freshwater sediments. +Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl +pathway +* Correspondence: G.Muijzer@uva.nl +†Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this +work. +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands +Full list of author information is available at the end of the article +© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 +International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and +reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to +the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver +(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. +Vavourakis et al. Microbiome (2018) 6:168 +https://doi.org/10.1186/s40168-018-0548-7 + +MicrobiomeBackground +Soda lakes are evaporative, athallasic salt lakes with low cal- +cium and magnesium concentrations and a high-alkaline +pH up to 11 buffered by dissolved (bi-) carbonate ions [1]. +They are constrained to arid regions across the globe, +mainly the tropical East African Rift Valley [2], the Libyan +Desert [3], the deserts in California and Nevada [4], and the +dry steppe belt of Central Asia that spans to southern Si- +beria, north-eastern Mongolia, and Inner Mongolia in +China [1]. On top of the extreme salinity and alkaline pH, +the Eurasian soda lakes experience extreme seasonal +temperature differences, causing highly unstable water re- +gimes and fluctuating salinities [5]. Yet, soda lakes harbor +diverse communities of haloalkaliphilic microbes, mostly +prokaryotes that are well adapted to survive and grow in +these extreme environments and consist of similar func- +tional groups in soda lakes around the world [1, 2, 6]. The +relative abundance of different groups is typically governed +by the salinity of the brine [1, 7, 8], and microbial-mediated +nutrient +cycles +become +partially +hampered +only +at +salt-saturating conditions [1]. +So far, all characterized prokaryotic lineages cultured +from soda lakes comprise over 70 different species within +more than 30 genera [1, 6, 9, 10]. From these, only a lim- +ited number of genomes have been sequenced today, +mostly from chemolithoautotrophic sulfur-oxidizing bac- +teria belonging to the genus Thioalkalivibrio (class Gam- +maproteobacteria) [1, 11, 12]. It is well established that +metagenomics enables the recovery of genomes and the +identification of novel genetic diversity where culturing ef- +forts fail [13, 14]. In recent years, next-generation sequen- +cing has recovered a massive number of genomes from +previously unknown groups of prokaryotes [15, 16], +including a strikingly large and diverse group called +“Candidate Phyla Radiation” (CPR), only distantly related +to other cultured bacterial lineages [17]. Previously, we +conducted a metagenomics study on soda lakes and re- +constructed novel genomes from uncultured Bacteroidetes +and “Candidatus Nanohaloarchaeaota” living in hypersa- +line Siberian soda brines [7]. Here, we turned our atten- +tion to the far more complex prokaryotic communities +living in the sediments of the hypersaline soda lakes from +the same region. We give a broad overview of all the +taxonomic groups sequenced and focus on the metabolic +diversity found in the reconstructed genomes of the most +abundant, uncultured organisms. +Results +Overall prokaryote community structure +The salinities from the studied soda lakes ranged from +moderately hypersaline (between 70 and 110 g L−1) to +salt-saturated (400 g L−1 salt). The soluble carbonate al- +kalinity was in the molar range, and the pH in all lakes +was around ten (see Additional file 1: Table S1). To give +an overview of the overall prokaryotic community com- +position in each of the samples, we looked at the taxo- +nomic classification of 16S rRNA genes recovered both +by amplicon sequencing and direct metagenomics se- +quencing (Fig. 1, see also Additional file 2: Figure S1; +Additional file 3). The prokaryotic communities of all +five sediment samples were highly diverse and consisted +mostly of uncultured taxonomic groups. Bacteria were +more abundant than Archaea, regardless of the salinity +of the overlaying brine [7] (Fig. 1). Euryarchaeota were +the second and third largest group in the sediments of +the two salt-saturated lakes comprising ~ 10 and ~ 20% +of the 16S rRNA genes in the metagenomes. Most +Euryarchaeota-related OTUs detected by amplicon se- +quencing belonged either to the uncultured Thermoplas- +mata group KTK 4A (SILVA classification) or the genera +Halohasta and Halorubrum (class Halobacteria). In ac- +cordance with cultivation-dependent studies [6], most +OTUs assigned to methanogens were from the class +Methanomicrobia, +especially +the +lithotrophic +genus +Methanocalculus (up to ~ 3%) and the methylotrophic +genus Methanosalsum (Additional file 3). +The varying ratio of the three dominant bacterial groups, +Firmicutes, Bacteroidetes (including the newly proposed +phyla +Rhodothermaeota +and +Balneolaeota +[18]), +and +Gammaproteobacteria, showed no clear trend in relation to +the salinity in the lakes, but when Firmicutes were domin- +ant, Bacteroidetes were less abundant and vice versa. Most +Firmicutes belonged to the order Clostridales. Uncultured +members from the family Syntrophomonadaceae had a +relative abundance of more than 5% in all five metagen- +omes and comprised in two lakes even ~ 11–20% of the +recovered amplicon sequences. The second most abundant +Firmicutes order was Halanaerobiales, particularly the +genus Halanaerobium (family Halanaerobiaceae) and un- +cultured members of the Halobacteroidaceae. The majority +of Bacteroidetes-related OTUs could not be assigned down +to the genus level. The uncultured ML635J-40 aquatic +group (order Bacteroidales) comprised at least 5% of all five +prokaryotic communities. This group has been previously +found to be abundant in Mono Lake [4] (a soda lake) and +in an anoxic bioreactor degrading cyanobacterial biomass +under haloalkaline conditions [19]. Two other highly abun- +dant (up to ~ 8%) uncultured groups from the class Balneo- +lia (proposed new phylum Balneolaeota [18]) were also +detected in other soda lakes before [3, 4]. Within the Gam- +maproteobacteria, the genus Thioalkalivibrio was abundant +(~ 3% of the total community), but also uncultured +members of HOC36 were prevailing at moderate salinities. +Members of the Deltaproteobacteria, Alphaproteobacteria, +and Chloroflexi comprised up to ~ 10% of the detected 16S +rRNA gene in some of the metagenomes. The GIF9 family +of the class Dehalococcoidia was among the top three most +abundant OTUs in two lakes. The extremely salt-tolerant +Vavourakis et al. Microbiome (2018) 6:168 +Page 2 of 18 + +and alkaliphilic genera Desulfonatronobacter (order Desulfo- +bacterales) and Desulfonatronospira (order Desulfovibrio- +nales) +were +the +dominant +Deltaproteobacteria. +Highly +abundant OTUs, within the Actinobacteria belonged to the +class Nitriliruptoria and within the Alphaproteobacteria to +the family Rhodobacteraceae and the genus Roseibaca. The +important nitrifying genus Nitrobacter (Alphaproteobacteria) +was present in only one of the lakes with moderate salinity +(Additional file 3). +Some bacterial top-level taxa appeared less dominant +(< 5%) from the 16S rRNA genes recovered from the +metagenomes but were represented mainly by a single +highly abundant OTU in the amplicon sequences, in- +cluding the haloalkaliphilic genus Truepera within the +phylum Deinococcus-Thermus, the genus Spirochaeata +within the phylum Spirochaetes, the family BSN166 +within the phylum Ignavibacteriae, the BD2–11 terres- +trial group within the Gemmatimonadetes, and the +WCHB1–41 +order +within +the +Verrucomicrobia. +All +OTUs +within +the +Thermotogae +and +Lentisphaerae +belonged to uncultured genera from the family Kosmoto- +gaceae and Oligosphaeraceae, respectively. All Tenericu- +tes-related OTUs belonged to the class Mollicutes, and +especially the order NB1-n was dominant. In contrast, +the phylum Planctomycetes was relatively diverse, with +at least 11 different genus-level OTUs spread over four +class-level groups. +High-throughput genome recovery +We obtained 717 medium-quality (≥ 50% complete, +< 10% contamination) and 154 near-complete (≥ 90% +complete, < 5% contamination) metagenome-assembled +genomes (MAGs) across three major prokaryote groups: +Archaea, Bacteria, and CPR (see Additional file 4 and +Additional file 2: Figure S2). Figures 2 and 3 show the +top-level phylogeny of all MAGs based on 16 ribosomal +proteins. The reference database used contains a repre- +sentative for each major prokaryote lineage [17]. We +a +b +Fig. 1 Abundant prokaryotic groups in five hypersaline soda lake sediments. a Relative abundance of the top-level taxa (those with ≥ 1% abundance +in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets. b Relative abundance of the 16S rRNA OTUs (those with sum +of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level. Three of the assessed soda +lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) +Vavourakis et al. Microbiome (2018) 6:168 +Page 3 of 18 + +colored the different phyla from which we obtained a +MAG +in +alternate +blue +and +orange +colors, +and +highlighted the MAGs obtained here in a darker shade. +Many MAGs belonged to uncultured groups commonly +detected in soda lake 16S rRNA gene surveys, over 100 +MAGs still belonged to candidate prokaryote phyla and +divisions that to our knowledge were never detected be- +fore in soda lakes, including CPR. Although only few +MAGs had near-complete 16S rRNA genes, in most +cases we were able to link available taxonomic gene an- +notations and ribosomal protein phylogeny to the SILVA +taxonomy of the OTUs assigned to the amplicon se- +quences, while cross-checking the abundance profiles of +both MAGs (Additional file 5) and OTUs. +The soda lake CPR recovered from the metagenomes was +restricted to a few distinct phyla within the Parcubacteria +group, mostly affiliating with “Candidatus Nealsonbacteria” +and “Ca. Zambryskibacteria” [15] (Fig. 2). The first group of +MAGs encompassed four different branches in our riboso- +mal protein tree, suggesting a high-phylogenetic diversity, +with 33 putative new species sampled here (ANI and con- +DNA matrices given in Additional file 6). The “Ca. Zambrys- +kibacteria-”related MAGs consisted of at least five new +species. Few MAGs were recovered from CPR groups also +detected by amplicon sequencing (see Additional file 2: +Figure S1), namely the “Ca. Dojkabacteria” (former WS6), +“Ca. Saccharibacteria” (former TM7), CPR2, and “Ca. +Katanobacteria” (former WWE3). +Fig. 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins. The archaeal tree is unrooted. The CPR tree is rooted +to the Wirthbacteria. Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of +this study are highlighted by darker shades (labeled as “MAG present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this +study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show +confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 4 of 18 + +Most archaeal MAGs belonged to the phylum Euryarch- +aeota and the abundant classes Halobacteria, Methanomi- +crobia, and Thermoplasmata (related to OTU KTK 4A) +within. In addition, three Thermoplasmata-related MAGs +that encoded for the key enzyme for methanogenesis +(methyl-coenzyme M reductase, mcr) affiliated with refer- +ence genomes from Methanomassilicoccales, the seventh +order of methanogens have been recovered [20, 21]. +Another MCR-encoding MAG was closely related to the +latest +discovered +group +of +poly-extremophilic, +methyl-reducing methanogens from hypersaline lakes +from the class Methanonatronarchaeia [9] (related to +OTU ST-12K10A). We recovered also one MAG from the +class Methanobacteria and a high-quality MAG from the +WCHA1–57 +group +(“Candidatus +Methanofastidiosa” +[22]) in the candidate division WSA2 (Arc I). Several +MAGs were recovered from the DPANN archaeal +groups “Ca. Diapherotrites,” “Ca. Aenigmarchaeota,” +(see Additional file 2: Figure S3) and “Ca. Woesearch- +aeota” (former Deep Sea Hydrothermal Vent Group 6, +DHVEG-6). Although we did not reconstruct any +reasonable-sized MAGs from the TACK superphylum, +we found several 16S rRNA genes on the assembled +contigs that affiliated to the Thaumarchaeota (see +Additional file 1: Table S2). +Nearly every known bacterial phylum had an extremo- +philic lineage sampled from our hypersaline soda lake +sediments (Fig. 3). In most cases, the soda lake lineages +clearly formed separate branches appearing as sister +groups to known reference lineages. The highest genome +recovery was from the same top-level taxonomic groups +that were also abundant in our 16S rRNA gene analysis. +From the Verrucomicrobia, most MAGs belonged to the +order WCHB1-41 (16S rRNA gene identity 92–100%). +However, in our ribosomal protein tree, they branched +within the phylum Lentisphaerae. Sixteen Tenericutes +MAGs from at least 12 different species (Additional file 6) +were closely related to the NB1-n group of Mollicutes. +Based on the recovered genome size and encoded meta- +bolic potential, these organisms are free-living anaerobic +fermenters of simple sugars, similar to what has recently +been +proposed +for +“Candidatus +Izimaplasma” +[23]. +Fig. 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins. Alternate orange and blue colors show phyla/ +classes from which we obtained MAGs (labeled as “Phyla present”). Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG +present”). Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not +present”), and the numerical labels are represented at the bottom. Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) +Vavourakis et al. Microbiome (2018) 6:168 +Page 5 of 18 + +Several MAGs belonged to the candidate phyla “Ca. +Omnitrophica,” “Ca. Atribacteria,” and “Ca. Acetother- +mia” (former OP1), which were moderately abundant +also in some sediment (see Additional file 2: Figure S1). +For the latter phylum, we suspect that four MAGs were +more closely related to ca. div. WS1 and “Ca. Lindow- +bacteria” for which only few reference genomes are +currently available in NCBI (see Additional file 2: +Figure S4). Due to a high-sequencing coverage, we also +managed to reconstruct several MAGs from rare Bacteria +(< 100 amplicon sequences detected, see Additional file 2: +Figure S1), including the phyla “Ca. Hydrogenedentes,” +“Ca. Cloacimonetes,” ca. div. BRC1, Elusimicrobia, Caldi- +serica, and “Ca. Latescibacteria.” The MAGs from the +latter phylum were more closely related to the recently +proposed phylum “Ca. Handelsmanbacteria” [15]. Two +additional MAGs with 16S rRNA gene fragments with +94–95% identity to the class MD2898-B26 (Nitrospinae) +were more likely members of ca. div. KSB3 (proposed +“Ca. Moduliflexus” [24], see Additional file 2: Figure S5). +Draft genomes of haloalkaliphilic CPR +Strikingly, members of the CPR related to “Ca. Nealson- +bacteria” and “Ca. Vogelbacteria” were among the top +5% of abundant organisms in the surface sediments of +the soda lakes, especially those with moderate salinity +(Fig. 4). Like most members of the CPR, the MAGs of +the four most abundant “Ca. Nealsonbacteria” seem to +be anaerobic fermenters [25]. They lacked a complete +TCA cycle and most complexes from the oxidative elec- +tron transfer chain, except for the subunit F of a +NADH-quinone oxidoreductase (complex I, nuoF, nuoG, +nuoA) and coxB genes (complex II). All CPR MAGs had +a near-complete glycolysis pathway (Embden-Meyerhof- +Parnas) encoded, but pentose phosphate pathways were +severely truncated. The commonly encoded F- and +V-type ATPase can establish a membrane potential for +symporter-antiporters by utilizing the ATP formed by +substrate-level phosphorylation during fermentation. All +CPR have V-type ATPases that can translocate Na+ in +addition to H+ (see Additional file 2: Figure S6), while +only two members of the “Ca. Falkowbacteria” had puta- +tive Na+-coupled F-type ATPases (see Additional file 2: +Figure S7). The coupling of ATP hydrolysis to sodium +translocation is advantageous to maintain pH homeosta- +sis in alkaline environments. Interestingly, with only two +exceptions [26, 27], all CPR genomes recovered from +other environments with neutral pH were reported to +encode only F-type ATPases [28–32]. One low-abundant +MAG affiliated to “Ca. Peregrinibacteria” contained both +the +large +subunit +of +a +RuBisCO +(type +II/III, +see +Additional file 2: Figure S8) and a putative phosphoribu- +lokinase (PRK, K00855) encoded in the same contig. +This is remarkable because PRK homologs were not +previously identified among CPR, and RuBisCo form II/ +III was inferred to function in a nucleoside salvage path- +way [33]. One “Ca. Saccharibacteria” MAG encoded for +a putative channelrhodopsin (see Additional file 2: +Figure S9). This is the first rhodopsin found among the +CPR and suggests that this enigmatic group of organ- +isms may have acquired evolutionary adaptations to a +life in sunlit surface environments. +A previous study showed that most CPR has coccoid +cell morphotypes with a monoderm cell envelope resem- +bling those from Gram-positives and Archaea but with a +distinct S-layer [34]. Thick peptidoglycans coated with +acidic surface polymers such as teichoic acids help pro- +tect the cells of Gram-positives against reactive hydroxyl +ions in highly alkaline environments [35] (Fig. 5a). All +soda lake CPR had indeed the capability for peptidogly- +can biosynthesis, but we found proteins typical for +Gram-negatives for the biosynthesis of lipopolysaccha- +rides (see Additional file 1: Table S3), homologous to the +inner membrane proteins of type II secretion systems +and +to +several +proteins +associated +to +the +outer +membrane and peptidoglycan, including OmpA. +It remains to be determined whether the soda lake +CPR also lacks an outer membrane and perhaps anchor +lipopolysaccharides, S-layer proteins, and lipoproteins to +the inner cell membrane or peptidoglycan. We also +found gene encoding cardiolipin and squalene synthases. +Increased levels of cardiolipin and the presence of squa- +lene make the cytoplasmic membrane less leaky for +protons [36]. In addition, cation/proton exchangers are +known to play a crucial role for pH homeostasis in alka- +liphilic prokaryotes as they help acidify the cytoplasm +during the extrusion of cations [35]. Putative Na+/H+ +exchangers of the Nha-type and multi-subunit Mnh-type +were found only within a few soda lake CPR. Secondary +active transport of K+ might be mediated in most soda +lake CPR by KefB (COG0475)/kch Kef-type, glutathione- +dependent K+ transport systems, with or without H+ +antiport (67,68). +Various soda lake CPR had an acidic proteome, with +pI curves resembling those found in extremely halophilic +Bacteria. Intracellular proteins enriched in acidic amino +acids might be an adaptation to a “salt-in” strategy, i.e., +maintaining high intracellular potassium (K+) concentra- +tions to keep osmotic balance [7, 37] (Fig. 5b; see +Additional file 2: Figure S10). Such a strategy is energet- +ically favorable over de novo synthesis or import of +osmolytes such as ectoine and glycine betaine. We did +not find genes for the synthesis of organic osmolytes and +homologs of ABC-type transporters for primary active +uptake of proline/glycine betaine which were encoded +only in one MAG (Fig. 5a). For the “Ca. Nealsonbac- +teria” and “Ca. Vogelbacteria,” the salt-in strategy might +be a unique feature for the soda lake species explaining +Vavourakis et al. Microbiome (2018) 6:168 +Page 6 of 18 + +their high abundance in the hypersaline soda lake sedi- +ments, as we did not found an acidic proteome pre- +dicted from genomes obtained from other non-saline +environments (See Additional file 2: Figure S11). The +uptake of K+ ions remains enigmatic for most soda lake +CPR. Low-affinity Trk-type K+ uptake transporters (gen- +erally with symport of H+) (67,68) were encoded only by +a limited number of MAGs. We found three MAGs +Fig. 4 Relative abundance and metabolic potential of the dominant species. Abundance values, expressed as reads per kilobase of MAG per gigabase +of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, +Additional file 6). Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets +(cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom. The metabolic potential derived from functional marker +genes (Additional file 7) is depicted by the colored symbols. CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, +fix. = fixation, red. = reduction, ox. = oxidation, dis. = disproportionation +Vavourakis et al. Microbiome (2018) 6:168 +Page 7 of 18 + +a +b +Fig. 5 (See legend on next page.) +Vavourakis et al. Microbiome (2018) 6:168 +Page 8 of 18 + +encoding for Kdp-type sensor kinases (kdpD) but no +corresponding genes for the response regulator (kdpE) +or for Kdp-ATPases that function as the inducible, high- +affinity K+ transporters in other Bacteria (67,68). Finally, +mechanosensitive ion channels (mscS, mscL) and ABC- +type multidrug transport systems (AcrAB, ccmA, EmrA, +MdlB, NorM) and sodium efflux permeases (NatB) were +encoded in almost every MAG. The first might rapidly +restore the turgor pressure under fluctuating salinity +conditions by releasing cytoplasmic ions [38]. +Novel abundant groups involved in sulfur, nitrogen, and +carbon cycles +A new species of Thioalkalivibrio (family Ectothiorhodospir- +aceae) was by far the most abundant in the sediments of +the two salt-saturated lakes (Fig. 4). In the sediment of +Bitter-1, also a purple sulfur bacterium from the same fam- +ily was highly abundant. It was closely related to Halorho- +dospira, a genus also frequently cultured from hypersaline +soda lakes [1]. None of the abundant Ectothiorhodospira- +ceae spp. had already a species-representative genome +sequenced (Additional file 6). The potential of the Thioalk- +alivibrio spp. for chemolithotrophic sulfur oxidation was +evident (Additional file 7; see Additional file 8: Information +S1). Interestingly, the encoded nitrogen metabolisms were +quite versatile. While Thioalkalivibrio sp. 1 had the poten- +tial for nitrate reduction to nitrite, Thioalkalivibrio sp. 2 +might perform dissimilatory nitrite reduction to ammonia +(DNRA; see Additional file 2: Figure S12). +Two +deltaproteobacterial +lineages +of +dissimilatory +sulfate-reducing bacteria (SRB) were highly abundant in +the soda lake sediment of Bitter-1. One MAG from the +family Desulfobacteraceae (order Desulfobacterales) is +the first genome from the genus Desulfonatronobacter. It +encodes the genes for complete sulfate reduction to sul- +fide using various electron donors, as well as for the +complete oxidation of volatile fatty acids and alcohols, a +unique +feature +for +the +genus +Desulfonatronobacter +among haloalkaliphilic SRB [10] (see Additional file 8: +Information S2). Fumarate and nitrite (DNRA, NrfAH) +could be used as alternative electron acceptors. The sec- +ond dominant lineage was a new species from the genus +Desulfonatronospira (family Desulfohalobiaceae, order +Desulfovibrionales). Like other members of this genus, it +had the potential to reduce or disproportionate partially +reduced sulfur compounds. In addition, it could also use +nitrite as an alternative electron acceptor (NrfAH) [6]. +A novel lineage of gammaproteobacterial SOB was +highly abundant in the sediments of the moderately hy- +persaline Cock Soda Lake. It appeared as a sister group of +the family Xanthomonadaceae in the ribosomal protein +tree. This heterotrophic organism could conserve energy +through aerobic respiration. It might detoxify sulfide by +oxidizing it to elemental sulfur (sqr) with subsequent re- +duction or disproportionation of the polysulfides (psrA) +chemically formed from the sulfur. It also encoded the po- +tential for DNRA (nrfA and napC). Genes likely involved +in sulfide detoxification (sqr and psrA) were found also in +several other abundant MAGs of heterotrophs, including +one new abundant species from the family of Nitrilirup- +toraceae (class Nitriliruptoria, phylum Actinobacteria). +We found a wide variety of carbohydrate-active enzymes +in these MAGs, such as cellulases (GH1 family) in +addition to genes for glycolysis and TCA cycle and a +chlorophyll/bacteriochlorophyll a/b synthase (bchG gene). +The latter was also found in other Actinobacteria from the +genus Rubrobacter [39]. No evidence was found for +nitrile-degrading potential. +A second novel, uncultured lineage of Gammaproteo- +bacteria that was highly abundant at moderate salinities +branched in our ribosomal protein tree as a sister group +to the family Halothiobacillaceae. The MAGs encoded +for a versatile metabolism typical for purple non-sulfur +bacteria. The MAGs contained puf genes, bch genes, +genes for carotenoid biosynthesis (not shown), and a +Calvin cycle for photoautotrophic growth. Alternatively, +energy may be conserved through aerobic respiration, +while acetate and proprionate could be taken up via an +acetate permease (actP) and further used for acetyl-CoA +biosynthesis and carbon assimilation. Since the sqr gene +was present, but no dsr or sox genes, the organism +might oxidize sulfide only to elemental sulfur. One bin +contained also nifDKH genes suggesting putative diazo- +trophy, as well as a coenzyme F420 hydrogenase suggest- +ing photoproduction of hydrogen [40]. +The abundant Euryarchaeota organism showed a clear +preference for higher salinities. We obtained one highly +abundant MAG from the class Thermoplasmata that +encoded a full-length 16S rRNA gene only distantly re- +lated (91,2% identity, e value 0) to that of a member of +the KTK 4A group found in a hypersaline endoevaporitic +microbial mat [8]. The abundant soda lake organism is +likely a new genus and species. All KTK 4A-related +MAGs found here encoded for similar heterotrophic, +fermentative +metabolisms, +with +the +potential +for +(See figure on previous page.) +Fig. 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla. a Membrane transporters, +channels, and lipids. Peptidoglycan is depicted in gray and S-layer proteins in cyan. b Predicted isoelectric points (bin width 0.2) for the coding +sequences of MAGs. A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also +Additional file 2: Figure S11) +Vavourakis et al. Microbiome (2018) 6:168 +Page 9 of 18 + +anaerobic formate and CO oxidation. The KTK 4A +might be also primary degraders since they encoded pu- +tative cellulases (CAZY-families GH1, GH5) and chiti- +nases (GH18). Interestingly, half of the MAGs encoded a +putative +chlorophyll/bacteriochlorophyll +a/b +synthase +(bchG), which is highly unusual for Archaea. Although +little can be inferred from the presence of only one +marker gene, a functional bchG was previously also +found in Crenarchaeota [41]. The remaining two highly +abundant Euryarchaeota-related MAGs belonged to a +new species of Halorubrum (Additional file 6). +Key genes of the Wood-Ljungdahl pathway found in +novel phylogenetic groups +More than 50 MAGs were related to the family Syntro- +phomonadaceae (class Clostridia, phylum Firmicutes) +based on ribosomal protein phylogeny. All 16S rRNA +gene sequences found in the MAGS had 86–95% iden- +tity to sequences obtained from uncultured organisms +related to the genus Dethiobacter. While an isolated +strain of Dethiobacter alkaliphilus is a facultative auto- +troph +that +respires +thiosulfate, +elemental +sulfur +or +polysulfides with hydrogen as an electron donor [42] or +disproportionates +sulfur +[43], +other +haloalkaliphilic +members +of +the +Syntrophomonadaceae +are +reverse +acetogens, oxidizing acetate in syntrophy with a hydro- +genotrophic partner [44]. Two populations (different +species, Additional file 6) were especially abundant in +Cock Soda Lake (Fig. 4). They encoded for a full +CODH/ACS complex, the key enzyme for the reductive +acetyl-CoA or Wood-Ljungdahl pathway (WL) and a +complete +Eastern +branch +for +CO2 +conversion +to +5-methyl-tetrahydrofolate (Additional file 9) [45, 46]. +Acetogens use the WL to reduce CO2 to acetyl-CoA, +which can be fixed into the cell or used to conserve en- +ergy via acetogenesis. Syntrophic acetate oxidizers, some +sulfate reducing bacteria and aceticlastic methanogens +run the WL in reverse. Syntrophomonadaceae sp. 2 +encoded for a putative thiosulfate/polysulfide reductase +as well as a phosphotransacetylase (pta) and an acetate +kinase (ack) for the ATP-dependent conversion of acet- +ate to acetyl-CoA. Although alternative pathways for the +latter interconversion can exist, this second species has +the complete potential for (reversed) acetogenesis. +Highly remarkable was the presence of a bacterial-type +CODH/ACS +complex +and +a +near-complete +eastern +branch of the WL in a highly abundant species in Cock +Soda Lake from the family Coriobacteriaceae (phylum +Actinobacteria). This prompted us to scan all 871 MAGs +for the presence of acsB encoding for the beta-subunit +of the oxido-reductase module of CODH/ACS. We con- +firmed an encoded +(near)-complete +WL in several +additional organisms belonging to phylogenetic groups +not +previously +associated +with +this +pathway +[46] +(Additional file 9). We removed the Coriobacteriaceae +acsB genes from the final dataset to construct a phylo- +genetic tree since they were < 500 aa (Fig. 6) but found +seven MAGs from the OPB41 class within the Actino- +bacteria (16S rRNA gene fragment identity 94–96%). +The eastern branch of WL can function independently +in folate-dependent C1 metabolism [45], but the pres- +ence of the Western-branch in a phylum that comprises +mostly aerobic isolates is very surprising. The WL in +combination with the potential for acetate to acetyl-CoA +interconversion (pta/ack) and a glycolysis pathway were +also present in the soda lake MAGs from the phyla “Ca. +Handelsmanbacteria,” “Ca. Atribacteria” (latter branched +within the “Ca. Acetothermia”), and the class LD1-PA32 +(Chlamydiae), suggesting all these uncultured organisms +might be heterotrophic acetogens. However, it should be +noted that a PFOR typically connecting glycolysis to the +WL was only encoded in the LD1-PA32 MAGs. More- +over, from the genetic make-up alone, it cannot be +excluded that acetate is activated, and the WL run in +reverse for syntrophic acetate oxidation. Finally, the +novel acsB genes from soda lake Halanaerobiaceae, +Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) +and from Brocadiaceae and Planctomycetaceae (Plancto- +mycetes) disrupt the previously proposed dichotomy +between Terrabacteria and Gracilicutes bacterial groups +unifying 16S rRNA and acsB gene phylogenies [46] and +suggest a far more complex evolutionary history of the +WL pathway than previously anticipated. +Discussion +Extensive +classical +microbiology +efforts +have +been +already undertaken to explore the unique extremophilic +microbial communities inhabiting soda lakes. These un- +covered the presence of most of the functional groups +participating in carbon, nitrogen, sulfur, and minor +element cycling at haloalkaline conditions. The main re- +sults of this work are summarized in several recent re- +views [1, 6, 47, 48]. Since most microbes, including +those living in soda lakes, still evade all cultivation ef- +forts, a very effective way to discover new microbes and +assess their physiology based on their genetic repertoire +is either through single cell genomics or by directly se- +quenced environmental DNA. This exploratory metage- +nomics +study +performed +on +soda +lake +sediments +effectively overcame the existing cultivation bottleneck. +First, we expanded the known diversity of CPR consider- +ably with the first genomes of poly-extremophiles sam- +pled from soda lake sediments. Although the presence of +16S rRNA genes from CPR in marine sediments and hy- +persaline microbial mats was previously shown [34], +until now, CPR MAGs were mainly obtained from deep, +subsurface environments [15, 26, 29, 32, 49–52], and hu- +man microbiota [30]. Despite being highly abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 10 of 18 + +100 % +90-100 % +70-90 % +50-70 % +some MAGs +all MAGs +Bootstraps +Genes present +Glycolysis (EMP) +PFOR +WL-Eastern branch +H4MPT +TH4 +WL-Western branch +CODH/ACS +Acetogenesis/ +acetate activation +(pta/ack) +0.4 +PVC group (Chlamydiae LD1-PA32) +Syntrophorhabdus aromaticivorans +PVC group bacterium CSSed11_184 +Aerophobetes bacterium SCGC_AAA255-F10 +Ca. Acetothermia +Ca. Handelsmanbacteria +Planctomycetaceae +Anaerolineae +Firmicutes +Brocadiaceae +Planctomycetes +Methanomassiliicoccales +Halobacteroidaceae +Natranaerobiaceae +Methanomicrobiales +Desulfonatronospira +Firmicutes +Dehalococcoidia +Armatimonadetes bacterium CSP1-3 +Deltaproteobacteria +Thermodesulfobacteria +Desulfobulbaceae +Halanaerobiaceae +Nitrospirae +Actinobacteria (OPB41) +Fig. 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs. Only sequences ≥ 500 aa +were included. Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence +of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see +also Additional file 9: Dataset S6). Additional lineages found in this study are marked in blue. The three was rooted according to [46]. +Circles at the nodes show confidence percentage of the bootstraps analysis (100×). EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin +oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = +tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase. PVC group bacterium CSSed11_184 is likely a member +of the WCHB1-41 class within the Verrucomicrobia +Vavourakis et al. Microbiome (2018) 6:168 +Page 11 of 18 + +here, CPR went unnoticed in previous amplicon sequen- +cing studies. This might be due to the fact that many +CPR representatives have random inserts of various +length in their 16S rRNA genes or due to primer mis- +matches [29, 34]. This illustrates also that direct metage- +nomics should not only be preferred over amplicon +sequencing to infer functional potential, but the former +is far more effective for the discovery of novel organ- +isms. Second, we obtained many more genomes from +“traditional” bacterial phyla such as the Planctomycetes +and Chloroflexi, as well as candidate phyla, for which no +soda lake isolates, hence no genomes were previously +obtained. Third, even within the sulfur cycle, the most +active and frequently studied element cycle in soda lakes +[1], we found considerable metabolic novelty. Finally, we +found the Wood-Ljungdahl pathway in several novel +phyla, not closely related to any known acetogens, +methanogens, or sulfate-reducing bacteria [46]. The lat- +ter shows that our sequencing recovery effort has also +significantly contributed to the discovery of metabolic +novelty within various prokaryote phylogenetic groups. +Salinity is often considered to be the major factor +shaping prokaryote community composition in diverse +habitats [53, 54]. Extreme halophilic Euryarchaeota +seem to be always the dominant group in salt-saturated +hypersaline brines, both those with neutral or alkaline +pH [1, 7, 37]. Here, we found that although these +haloarchaea are still relatively more abundant in the sed- +iments exposed to brines with salt-saturating conditions, +the clear majority of microbes in all investigated hyper- +saline soda lake sediments are Bacteria. It could be +hypothesized that the sediment is a hide-out for the +extreme alkalinity and salinity governing the water +column, and that sediment stratification, especially in +the anoxic part, offers plenty of opportunities for niche +diversification. On the other hand, it should no longer +be a surprise that soda lakes are such productive and +biodiverse +systems. +First, +it +has +been +previously +elaborated that soda lake organisms are exposed to +approximately half the osmotic pressure in sodium +carbonate-dominated +brines +compared +to +sodium +chloride-dominated brines with the same Na+ molarity +[47]. Second, nitrogen limitation in the community can +be overcome when many members contribute to the +fixation of atmospheric N2, and various forms of organic +nitrogen are efficiently recycled. The soda lakes exam- +ined in this study were also eutrophic, and sulfur com- +pounds were abundant. Sulfide is also far less toxic at +high pH as it mostly occurs in the form of bisulfide +(HS−). Besides the evident high metabolic and taxo- +nomic diversity of dissimilatory sulfur-cycling bacteria, a +diverse heterotrophic community can be sustained com- +prising both generalist and very specialized carbon de- +graders. Less eutrophic soda lakes might not suffer from +carbon +limitation +either, +due +to +a +presence +of +high-bicarbonate concentrations. These effectively elim- +inate the inorganic carbon limitation for primary pro- +ducers who are highly active in soda lakes, especially +Cyanobacteria [55, 56]. Third, light that penetrates the +surface of the sediment seems to stimulate oxygenic and +anoxygenic phototrophic growth. Moreover, various het- +erotrophs, such as the rhodopsin-containing haloarchaea +and Bacteroidetes, have the option to tap into this un- +limited energy source for example to help sustain the +costly maintenance of osmotic balance. Unexpectedly, +we even found the first rhodopsin encoded by a member +of the CPR. Fourth, tight syntrophic relations, as pro- +posed for CPR members and Syntrophomonadaceae +spp., might be the solution to successful growth in an +energetically challenging environment. +Since our metagenomes are snapshots in time and space, +the failure to reconstruct specific MAGs gives no conclu- +sive evidence for the absence of certain microbial-mediated +element transformation in hypersaline soda lake sediments. +Additionally, technical limitations of the assembly and bin- +ning of highly micro-diverse genome populations might +hamper genome recovery [57]. More importantly, the +abundance of a specific microbe is not necessarily corre- +lated to the importance of its performance in an ecosystem. +Many metabolic capacities are redundant, and often key +transformations are reserved for a few rare organisms that +might proliferate for a short time-span when specific condi- +tions allow for it. For example, although no MAGs were re- +covered from chemolithoautotrophic nitrifiers [58], we did +detect a Nitrobacter-related OTU by amplicon sequencing +and assembled 16S rRNA genes from Thaumarchaeota, +suggesting bacterial and archaeal nitrifiers are present in +the surface sediments of soda lakes at very low abundance. +Finally, the method of DNA isolation might impact the +community composition apparent in the final metagenome +sequenced. Environmental samples contain complex mix- +tures of different organisms, and it is impossible to find a +protocol where the DNA from every single organism is ex- +tracted as efficiently without compromising the final quality +of the extracted DNA. However, since we find all the im- +portant taxonomic and functional groups known from pre- +vious cultivation-dependent studies back in either our +amplicon sequencing datasets or our directly sequenced +metagenomes, we are confident that the community com- +position and the MAGs presented here are representative +for the microbiomes of the soda lake sediments in the +Kulunda Steppe. +Conclusion +Years of intensive microbiological research on soda lakes +seem to have paid off, since many of the described gen- +era we could detect here have a cultured representative +from soda lakes. However, as many of the abundant +Vavourakis et al. Microbiome (2018) 6:168 +Page 12 of 18 + +lineages and groups found in soda lake sediments are +still uncultured, metagenomics proved to be a helpful +tool to gain primary insights in the potential physiology +and ecology of these poly-extremophilic prokaryotes. +We reconstructed the first genomes for many of such +organisms and proposed new functional roles for the +most abundant ones. Future studies should provide +more in depth analyses of these genomes, especially +from the less abundant organisms that might perform +key ecological processes, such as methanogens and nitri- +fiers. In addition, they should focus on gaining physio- +logical culture-based evidence or proof for in situ +activity for the abundant organisms described here. The +key metabolic insights provided by this metagenomics +study can lead to the design of new cultivation strategies. +In general, sediment communities are far more complex +than those found in the corresponding water column +[53, 59] and are therefore often considered too complex +for efficient metagenomic analysis. Many of the novel +lineages found here may therefore have related neutro- +philic lineages in marine and freshwater sediments that +await discovery. We demonstrate here that, by providing +sufficient sequencing depth, the “state of the art metage- +nomics toolbox” can effectively be used on sediments as +well. +Methods +Site description and sample collection +The top 10 cm sediments from four hypersaline, eutrophic +soda lakes located in the Kulunda Steppe (south-western +Siberia, Altai, Russia) were sampled in July of 2010 and +2011. General features and exact location of the sampled +soda lakes are summarized in Additional file 1: Table S1; a +map of the area was published previously [5]. Cock Soda +Lake (a stand-alone lake, sampled both in 2010 and 2011) +and Tanatar-3 (Tanatar system) were moderately hypersa- +line (~ 100 g L−1) with sandy sediment, while Tanatar-1 +and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) +with sulfide-rich sapropel sediments underlined by rock +trona deposits [7, 60]. Especially, Bitter-1 harbors a very +active microbial community, probably due to its high- +organic and -mineral content. Surface sediments were col- +lected by a plastic corer into sterile glass containers and +transported to the laboratory in a cooler. +DNA isolation, 16S rRNA amplicon, and metagenomic +sequencing +The colloidal fraction of each sediment sample (~ 10% +of 50 g) was separated from the course sandy fraction by +several short (30–60 s) low-speed (1–2,000 rpm in +50 mL Falcon tubes) centrifugation steps and washed +with 1–2 M NaCl solution. The pelleted colloidal sedi- +ment fraction was first subjected to 3 cycles of freezing +in liquid nitrogen/thawing, then re-suspended in 0.1 M +Tris (pH 8)/10 mM EDTA, and then subjected to harsh +bead beating treatment. Next, the samples were incu- +bated with lysozyme (15 mg/mL) for 2 h at 37 °C +followed by a SDS (10% w/v) and proteinase K (10 μg/ +mL) treatment for 30 min. at 45 °C. High molecular +weight DNA was isolated using phenol/chloroform ex- +traction, quality-checked, and sequenced as described +previously [7]. Direct high-throughput sequencing of the +DNA was performed on an Illumina HiSeq 2000 plat- +form to generate 150 b paired-end reads. Amplification +of the V4-V6 region of prokaryote 16S rRNA genes +using barcoded 926F-1392R primers, amplicon purifica- +tion, quantification, and Roche (454)-sequencing was +performed together in a batch with brine samples from +the same sampling campaigns. Barcodes and adapter se- +quences were removed from de-multiplexed amplicon +sequence reads and analyzed with the automated NGS +analysis pipeline of the SILVA rRNA gene database pro- +ject [61] (SILVAngs 1.3, database release version 128) +using default parameters. The OTUs (97% identity) +assigned down to the genus level were only considered +when they had a relative abundance ≥ 0.1% in at least +one of the five datasets. +Processing metagenomics reads, assembly, binning, and +post-binning +Metagenomic raw reads were quality trimmed using +Sickle [62] (version 1.33), and only reads ≥ 21 b were +retained. The prokaryotic community structure at taxo- +nomic top levels was extrapolated from ten million ran- +domly sampled singletons from each dataset. Candidate +16S rRNA fragments > 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. Extremophiles. +2014;18:791–809. +2. +Oduor SO, Kotut K. Soda lakes of the East African Rift System: the past, the +present and the future. In: Schagerl M, editor. Soda lakes of East Africa. +Berlin: Springer; 2016. p. 365–74. +3. +Mesbah NM, Abou-El-Ela SH, Wiegel J. Novel and unexpected prokaryotic +diversity in water and sediments of the alkaline, hypersaline lakes of the +Wadi An Natrun, Egypt. Microb Ecol. 2007;54:598–617. +4. +Humayoun SB, Bano N, James T, Hollibaugh JT. Depth distribution of +microbial diversity in Mono Lake, a meromictic soda lake in California. Appl +Environ Microbiol. 2003;69:1030–42. +5. +Foti MJ, Sorokin DY, Zacharova EE, Pimenov NV, Kuenen JG, Muyzer G. +Bacterial diversity and activity along a salinity gradient in soda lakes of the +Kulunda Steppe (Altai, Russia). Extremophiles. 2008;12:133–45. +6. +Sorokin DY. Anaerobic haloalkaliphiles. eLS. 2017; https://doi.org/10.1002/ +9780470015902.a0027654. +7. +Vavourakis CD, Ghai R, Rodriguez-Valera F, Sorokin DY, Tringe SG, +Hugenholtz P, et al. Metagenomic insights into the uncultured diversity and +physiology of microbes in four hypersaline soda lake brines. Front Microbiol. +2016;7:211. +8. +Sørensen KB, Canfield DE, Oren A. Salinity responses of benthic microbial +communities in a solar saltern (Eilat, Israel). Appl Environ Microbiol. 2004;70: +1608–16. +9. +Sorokin DY, Makarova KS, Abbas B, Ferrer M, Golyshin PN, Galinski EA, et al. +Discovery of extremely halophilic, methyl-reducing euryarchaea provides +insights into the evolutionary origin of methanogenesis. Nat Microbiol. +2017;2:17081. +10. +Sorokin DY, Chernyh NA, Poroshina MN. Desulfonatronobacter acetoxydans +sp. nov.,: a first acetate-oxidizing, extremely salt-tolerant alkaliphilic SRB from +a hypersaline soda lake. Extremophiles. 2015;19:899–907. +11. +Ahn A-C, Meier-Kolthoff JP, Overmars L, Richter M, Woyke T, Sorokin DY, +et al. Genomic diversity within the haloalkaliphilic genus Thioalkalivibrio. +PLoS One. 2017;12:e0173517. +12. +Sorokin DY, Kuenen JG. Haloalkaliphilic sulfur-oxidizing bacteria in soda +lakes. FEMS Microbiol Rev. 2005;29:685–702. +13. +Albertsen M, Hugenholtz P, Skarshewski A, Nielsen KL, Tyson GW, Nielsen +PH. Genome sequences of rare, uncultured bacteria obtained by differential +coverage binning of multiple metagenomes. Nat Biotechnol. 2013;31:533–8. +14. +Qin J, Li R, Raes J, Arumugam M, Burgdorf KS, Manichanh C, et al. A human +gut microbial gene catalogue established by metagenomic sequencing. +Nature. 2010;464:59–65. +15. +Anantharaman K, Brown CT, Hug LA, Sharon I, Castelle CJ, Probst AJ, et al. +Thousands of microbial genomes shed light on interconnected +biogeochemical processes in an aquifer system. Nat Commun. 2016;7:13219. +16. +Parks DH, Rinke C, Chuvochina M, Chaumeil P-A, Woodcroft BJ, Evans PN, +et al. Recovery of nearly 8,000 metagenome-assembled genomes +substantially expands the tree of life. Nat Microbiol. 2017;2:1533–42. +17. +Hug LA, Baker BJ, Anantharaman K, Brown CT, Probst AJ, Castelle CJ, et al. A +new view of the tree of life. Nat Microbiol. 2016;1:16048. +18. +Hahnke RL, Meier-Kolthoff JP, García-López M, Mukherjee S, Huntemann M, +Ivanova NN, et al. Genome-based taxonomic classification of Bacteroidetes. +Front Microbiol. 2016;7:2003. +19. +Nolla-Ardevol V, Strous M, Tegetmeyer HE. Anaerobic digestion of the +microalga Spirulina at extreme alkaline conditions: biogas production, +metagenome and metatranscriptome. Front Microbiol. 2015;6:597. +20. +Borrel G, Parisot N, Harris HM, Peyretaillade E, Gaci N, Tottey W, et al. +Comparative genomics highlights the unique biology of +Methanomassiliicoccales, a Thermoplasmatales-related seventh order +of methanogenic archaea that encodes pyrrolysine. BMC Genomics. +2014;15:679. +21. +Sorokin DY, Abbas B, Geleijnse M, Pimenov NV, Sukhacheva MV, van +Loosdrecht MCM. Methanogenesis at extremely haloalkaline conditions in the +soda lakes of Kulunda Steppe (Altai, Russia). FEMS Microbiol Ecol. 2015;91:4. +22. +Nobu MK, Narihiro T, Kuroda K, Mei R, Liu WT. Chasing the elusive +Euryarchaeota class WSA2: genomes reveal a uniquely fastidious methyl- +reducing methanogen. ISME J. 2016;10:2478–87. +23. +Skennerton CT, Haroon MF, Briegel A, Shi J, Jensen GJ, Tyson GW, et al. +Phylogenomic analysis of Candidatus “Izimaplasma” species: free-living +representatives from a Tenericutes clade found in methane seeps. ISME J. +2016;10:2679–92. +24. +Sekiguchi Y, Ohashi A, Parks DH, Yamauchi T, Tyson GW, Hugenholtz P. First +genomic insights into members of a candidate bacterial phylum +responsible for wastewater bulking. PeerJ. 2015;3:e740. +25. +Wrighton KC, Thomas BC, Sharon I, Miller CS, Castelle CJ, VerBerkmoes NC, +et al. Fermentation, hydrogen, and sulfur metabolism in multiple +uncultivated bacterial phyla. Science. 2012;337:1661–5. +26. +León-Zayas R, Peoples L, Biddle JF, Podell S, Novotny M, Cameron J, et al. +The metabolic potential of the single cell genomes obtained from the +Challenger Deep, Mariana Trench within the candidate superphylum +Parcubacteria (OD1). Environ Microbiol. 2017;19:2769–84. +27. +Castelle CJ, Brown CT, Thomas BC, Williams KH, Banfield JF. Unusual +respiratory capacity and nitrogen metabolism in a Parcubacterium (OD1) of +the Candidate Phyla Radiation. Sci Rep. 2017;7:40101. +28. +Anantharaman K, Brown CT, Burstein D, Castelle CJ, Probst AJ, Thomas BC, +et al. Analysis of five complete genome sequences for members of the class +Peribacteria in the recently recognized Peregrinibacteria bacterial phylum. +PeerJ. 2016;4:e1607. +29. +Brown CT, Hug LA, Thomas BC, Sharon I, Castelle CJ, Singh A, et al. Unusual +biology across a group comprising more than 15% of domain Bacteria. +Nature. 2015;523:208–11. +30. +Campbell JH, O ‘donoghue P, Campbell AG, Schwientek P, Sczyrba A, +Woyke T, et al. UGA is an additional glycine codon in uncultured SR1 +bacteria from the human microbiota. 2013; doi:https://doi.org/10.1073/pnas. +1303090110. +31. +Hanke A, Hamann E, Sharma R, Geelhoed JS, Hargesheimer T, Kraft B, et al. +Recoding of the stop codon UGA to glycine by a BD1-5/SN-2 bacterium +and niche partitioning between Alpha- and Gammaproteobacteria in a tidal +sediment microbial community naturally selected in a laboratory chemostat. +Front Microbiol. 2014;5:231. +32. +Kantor RS, Wrighton KC, Handley KM, Sharon I, Hug LA, Castelle CJ, et al. +Small genomes and sparse metabolisms of sediment-associated bacteria +from four candidate phyla. MBio. 2013;4:1–11. +33. +Wrighton KC, Castelle CJ, Varaljay VA, Satagopan S, Brown CT, Wilkins MJ, +et al. RubisCO of a nucleoside pathway known from Archaea is found in +diverse uncultivated phyla in bacteria. ISME J. 2016;10:2702–14. +34. +Luef B, Frischkorn KR, Wrighton KC, Holman HYN, Birarda G, Thomas BC, +et al. Diverse uncultivated ultra-small bacterial cells in groundwater. Nat +Commun. 2015;6:1–8. +35. +Krulwich TA, Sachs G, Padan E. Molecular aspects of bacterial pH sensing +and homeostasis. Nat Rev Microbiol. 2011;9:330–43. +36. +Hauß T, Dante S, Dencher NA, Haines TH. Squalane is in the midplane of +the lipid bilayer: implications for its function as a proton permeability +barrier. Biochim Biophys Acta Bioenerg. 2002;1556:149–54. +37. +Oren A. Life at high salt concentrations, intracellular KCl concentrations, and +acidic proteomes. Front Microbiol. 2013;4:315. +Vavourakis et al. Microbiome (2018) 6:168 +Page 16 of 18 + +Published online: 19 September 201838. +Levina N. Protection of Escherichia coli cells against extreme turgor by +activation of MscS and MscL mechanosensitive channels: identification of +genes required for MscS activity. EMBO J. 1999;18:1730–7. +39. +Gupta RS, Khadka B. Evidence for the presence of key chlorophyll- +biosynthesis-related proteins in the genus Rubrobacter (phylum +Actinobacteria) and its implications for the evolution and origin of +photosynthesis. Photosynth Res. 2016;127:201–18. +40. +Basak N, Das D. The prospect of purple non-sulfur (PNS) photosynthetic +bacteria for hydrogen production:the present state of the art. World J +Microbiol Biotechnol. 2007;23:31–42. +41. +Meng J, Wang F, Wang F, Zheng Y, Peng X, Zhou H, et al. An uncultivated +crenarchaeota contains functional bacteriochlorophyll a synthase. ISME J. +2009;3:106–16. +42. +Sorokin DY, Tourova TP, Mußmann M, Muyzer G. Dethiobacter alkaliphilus +gen. nov. sp. nov., and Desulfurivibrio alkaliphilus gen. nov. sp. nov.: two novel +representatives of reductive sulfur cycle from soda lakes. Extremophiles. +2008;12:431–9. +43. +Poser A, Lohmayer R, Vogt C. Extremophiles KK-, 2013 U. Disproportionation +of elemental sulfur by haloalkaliphilic bacteria from soda lakes. +Extremophiles. 2013;17:1003–12. +44. +Sorokin DY, Abbas B, Tourova TP, Bumazhkin BK, Kolganova TV, Muyzer G. +Sulfate-dependent acetate oxidation under extremely natron-alkaline +conditions by syntrophic associations from hypersaline soda lakes. +Microbiology. 2014;160(Pt_4):723–32. +45. +Ragsdale SW. Enzymology of the Wood-Ljungdahl pathway of acetogenesis. +Ann N Y Acad Sci. 2008;1125:129–36. +46. +Adam PS, Borrel G, Gribaldo S. Evolutionary history of carbon monoxide +dehydrogenase/acetyl-CoA synthase, one of the oldest enzymatic +complexes. Proc Natl Acad Sci. 2018;115:E1166–73. +47. +Sorokin DY, Banciu HL, Muyzer G. Functional microbiology of soda lakes. +Curr Opin Microbiol. 2015;25:88–96. +48. +Grant WD, Jones BE. Bacteria, Archaea and viruses of soda lakes. In: Schagerl +M, editor. Soda lakes of East Africa. Berlin: Springer; 2016. p. 97–147. +49. +Bruno A, Sandionigi A, Rizzi E, Bernasconi M, Vicario S, Galimberti A, et al. +Exploring the under-investigated “microbial dark matter” of drinking water +treatment plants. Sci Rep. 2017;7:1–7. +50. +Danczak RE, Johnston MD, Kenah C, Slattery M, Wrighton KC, Wilkins MJ. +Members of the Candidate Phyla Radiation are functionally differentiated by +carbon- and nitrogen-cycling capabilities. Microbiome. 2017;5:112. +51. +Hu P, Tom L, Singh A, Thomas BC, Baker BJ, Piceno YM, et al. Genome- +resolved metagenomic analysis reveals roles for candidate phyla and other +microbial community members in biogeochemical transformations in oil +reservoirs. MBio. 2016;7:e01669–15. +52. +Probst AJ, Castelle CJ, Singh A, Brown CT, Anantharaman K, Sharon I, et al. +Genomic resolution of a cold subsurface aquifer community provides +metabolic insights for novel microbes adapted to high CO2 concentrations. +Environ Microbiol. 2017;19:459–74. +53. +Lozupone CA, Knight R. Global patterns in bacterial diversity. Proc Natl Acad +Sci. 2007;104:11436–40. +54. +Thompson LR, Sanders JG, McDonald D, Amir A, Ladau J, Locey KJ, et al. A +communal catalogue reveals Earth’s multiscale microbial diversity. Nature. +2017;551:457. +55. +Samylina OS, Sapozhnikov FV, Gainanova OY, Ryabova AV, Nikitin MA, +Sorokin DY. Algo-bacterial communities of the Kulunda steppe (Altai region, +Russia) Soda Lakes. Microbiology. 2014;83:849–60. +56. +Krienitz L, Schagerl M. Tiny and tough: microphytes of east African soda lakes. In: +Schagerl M, editor. Soda lakes of East Africa. Berlin: Springer; 2016. p. 149–77. +57. +Nelson WC, Maezato Y, Wu Y-W, Romine MF, Lindemann SR. Identification +and resolution of microdiversity through metagenomic sequencing of +parallel consortia. Appl Environ Microbiol. 2015;82:255–67. +58. +Hansel C. Small but mighty: how minor components drive major +biogeochemical cycles. Environ Microbiol Rep. 2017;9:8–10. +59. +Zinger L, Amaral-Zettler LA, Fuhrman JA, Horner-Devine MC, Huse SM, +Welch DBM, et al. Global patterns of bacterial beta-diversity in seafloor and +seawater ecosystems. PLoS One. 2011;6:e24570. +60. +Isachenko BL. Chloride sulfate and soda lakes of Kulunda steppe and its +biogenic processes. In: Selected works, vol. 2. Leningrad: Academy of +Sciences USSR; 1951. p. 143–62. +61. +Quast C, Pruesse E, Yilmaz P, Gerken J, Schweer T, Yarza P, et al. The SILVA +ribosomal RNA gene database project: improved data processing and web- +based tools. Nucleic Acids Res. 2012;41:D590–6. +62. +Joshi NA, Fass JN. Sickle: a sliding-window, adaptive, quality-based trimming +tool for FastQ files (Version 1.33). 2011. +63. +Ghai R, Pašić L, Fernández AB, Martin-Cuadrado A-B, Mizuno CM, McMahon +KD, et al. New abundant microbial groups in aquatic hypersaline +environments. Sci Rep. 2011;1:135. +64. +Li D, Liu CM, Luo R, Sadakane K, Lam TW. MEGAHIT: An ultra-fast single- +node solution for large and complex metagenomics assembly via succinct +de Bruijn graph. Bioinformatics. 2015;31:1674–6. +65. +Hyatt D, Chen G-L, LoCascio PF, Land ML, Larimer FW, Hauser LJ. Prodigal: +prokaryotic gene recognition and translation initiation site identification. +BMC Bioinformatics. 2010;11:119. +66. +Huang Y, Li W, Finn PW, Perkins DL. Ribosomal RNA identification in +metagenomic and metatranscriptomic datasets. In: De Bruijn FJ, +editor. Handbook of Molecular Microbial Ecology I. Hoboken: Wiley; +2011. p. 387–91. +67. +Lowe TM, Eddy SR. tRNAscan-SE: a program for improved detection of +transfer RNA genes in genomic sequence. Nucleic Acids Res. 1997;25: +955–64. +68. +Kanehisa M, Sato Y, Morishima K. BlastKOALA and GhostKOALA: KEGG tools +for functional characterization of genome and metagenome sequences. J +Mol Biol. 2016;428:726–31. +69. +Lauro FM, Demaere MZ, Yau S, Brown MV, Ng C, Wilkins D, et al. An +integrative study of a meromictic lake ecosystem in Antarctica. ISME J. 2010; +5:879–95. +70. +Hernsdorf AW, Amano Y, Miyakawa K, Ise K, Suzuki Y, Anantharaman K, et al. +Potential for microbial H2 and metal transformations associated with novel +bacteria and archaea in deep terrestrial subsurface sediments. Nat Publ Gr. +2017;11:1915–29. +71. +Llorens-Marès T, Yooseph S, Goll J, Hoffman J, Vila-Costa M, Borrego CM, +et al. Connecting biodiversity and potential functional role in modern +euxinic environments by microbial metagenomics. ISME J. 2015;9:1648–61. +72. +Kang DD, Froula J, Egan R, Wang Z. MetaBAT, an efficient tool for accurately +reconstructing single genomes from complex microbial communities. PeerJ. +2015;3:e1165. +73. +Langmead B, Salzberg SL. Fast gapped-read alignment with bowtie 2. Nat +Methods. 2012;9:357–9. +74. +Parks DH, Imelfort M, Skennerton CT, Hugenholtz P, Tyson GW. CheckM: +assessing the quality of microbial genomes recovered from isolates, single +cells, and metagenomes. Genome Res. 2015;25:1043–55. +75. +Eren AM, Esen ÖC, Quince C, Vineis JH, Morrison HG, Sogin ML, et al. Anvi’o: +an advanced analysis and visualization platform for ‘omics data. PeerJ. 2015; +3:e1319. +76. +Eren AM, Delmot TO. Predicting CPR genomes in metagenomic bins. http:// +merenlab.org/2016/04/17/predicting-CPR-Genomes/. +77. +Edgar RC. Search and clustering orders of magnitude faster than BLAST. +Bioinformatics. 2010;26:2460–1. +78. +Rinke C, Schwientek P, Sczyrba A, Ivanova NN, Anderson IJ, Cheng JF, et al. +Insights into the phylogeny and coding potential of microbial dark matter. +Nature. 2013;499:431–7. +79. +Bushnell B. BBMap short read aligner. 2016. +80. +Bowers RM, Kyrpides NC, Stepanauskas R, Harmon-Smith M, Doud D, Reddy +TBK, et al. Minimum information about a single amplified genome (MISAG) +and a metagenome-assembled genome (MIMAG) of bacteria and archaea. +Nat Biotechnol. 2017;35:725–31. +81. +Price MN, Dehal PS, Arkin AP. FastTree 2--approximately maximum- +likelihood trees for large alignments. PLoS One. 2010;5:e9490. +82. +Katoh K, Standley DM. MAFFT multiple sequence alignment software +version 7: improvements in performance and usability. Mol Biol Evol. 2013; +30:772–80. +83. +Letunic I, Bork P. Interactive tree of life (iTOL) v3: an online tool for the +display and annotation of phylogenetic and other trees. Nucleic Acids Res. +2016;44:W242–5. +84. +FigTree. http://tree.bio.ed.ac.uk/software/figtree/. +85. +Seemann T. Prokka: rapid prokaryotic genome annotation. Bioinformatics. +2014;30:2068–9. +86. +Jones P, Binns D, Chang H-Y, Fraser M, Li W, McAnulla C, et al. InterProScan +5: genome-scale protein function classification. Bioinformatics. 2014;30: +1236–40. +87. +Yin Y, Mao X, Yang J, Chen X, Mao F, Xu Y. dbCAN: a web resource for +automated carbohydrate-active enzyme annotation. Nucleic Acids Res. 2012; +40:W445–51. +Vavourakis et al. Microbiome (2018) 6:168 +Page 17 of 18 + +88. +Cantarel BL, Coutinho PM, Rancurel C, Bernard T, Lombard V, Henrissat B. +The Carbohydrate-Active EnZymes database (CAZy): an expert resource for +glycogenomics. Nucleic Acids Res. 2009;37:D233–8. +89. +Nayfach S, Pollard KS. Average genome size estimation improves +comparative metagenomics and sheds light on the functional ecology of +the human microbiome. Genome Biol. 2015;16:1–18. +90. +Goris J, Konstantinidis KT, Klappenbach JA, Coenye T, Vandamme P, Tiedje +JM. DNA-DNA hybridization values and their relationship to whole-genome +sequence similarities. Int J Syst Evol Microbiol. 2007;57:81–91. +91. +Huerta-Cepas J, Szklarczyk D, Forslund K, Cook H, Heller D, Walter MC, et al. +eggNOG 4.5: a hierarchical orthology framework with improved functional +annotations for eukaryotic, prokaryotic and viral sequences. Nucleic Acids +Res. 2016;44:D286–93. +92. +Tikhonova TV, Slutsky A, Antipov AN, Boyko KM, Polyakov KM, Sorokin DY, +et al. Molecular and catalytic properties of a novel cytochrome c nitrite +reductase from nitrate-reducing haloalkaliphilic sulfur-oxidizing bacterium +Thioalkalivibrio nitratireducens. Biochim Biophys Acta - Proteins Proteomics. +2006;1764:715–23. +93. +Tikhonova T, Tikhonov A, Trofimov A, Polyakov K, Boyko K, Cherkashin E, +et al. Comparative structural and functional analysis of two octaheme nitrite +reductases from closely related Thioalkalivibrio species. FEBS J. 2012;279: +4052–61. +94. +Tabita FR, Hanson TE, Li H, Satagopan S, Singh J, Chan S. Function, structure, +and evolution of the RuBisCO-like proteins and their RuBisCO homologs. +Microbiol Mol Biol Rev. 2007;71:576–99. +95. +Eddy SR. Accelerated profile HMM searches. PLoS Comput Biol. 2011;7: +e1002195. +Vavourakis et al. Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_test/content/tmp_files/load_file.txt b/kb_test/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1918aae016eb16a67cb725488828f2130172a01 --- /dev/null +++ b/kb_test/content/tmp_files/load_file.txt @@ -0,0 +1,1583 @@ +filepath=D:\projects\langchain-ChatGLM-master\knowledge_base\kb_test\content\test.pdf,len=972 +page_content='Bandit approach to conflict-free multi-agent Q-learning in view of photonic implementation Hiroaki Shinkawa 1, *, Nicolas Chauvet 1, Andr´e R¨ohm 1, Takatomo Mihana 1, Ryoichi Horisaki 1, Guillaume Bachelier 2, and Makoto Naruse 1 1Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2Univ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Grenoble Alpes, CNRS, Institut N´eel, 38000 Grenoble, France.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' *Corresponding author.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Email: gokukyukyoku555@gmail.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='com Abstract Recently, extensive studies on photonic reinforcement learning to accelerate the process of calculation by exploiting the physical nature of light have been conducted.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Previous studies utilized quantum interference of photons to achieve collective decision-making without choice conflicts when solving the competitive multi-armed bandit problem, a fundamental example of reinforcement learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, the bandit problem deals with a static environment where the agent’s action does not influence the reward probabilities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study aims to extend the conventional approach to a more general multi-agent reinforcement learning targeting the grid world problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Unlike the conventional approach, the proposed scheme deals with a dynamic environment where the reward changes because of agents’ actions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A successful photonic re- inforcement learning scheme requires both a photonic system that contributes to the quality of learning and a suitable algorithm.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study proposes a novel learning algorithm, discon- tinuous bandit Q-learning, in view of a potential photonic implementation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Here, state-action pairs in the environment are regarded as slot machines in the context of the bandit problem and an updated amount of Q-value is regarded as the reward of the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We perform numerical simulations to validate the effectiveness of the bandit algorithm.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In addition, we propose a multi-agent architecture in which agents are indirectly connected through quantum interference of light and quantum principles ensure the conflict-free property of state-action pair selections among agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We demonstrate that multi-agent reinforcement learning can be accelerated owing to conflict avoidance among multiple agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1 Introduction Reinforcement learning is a machine learning technique that enables an agent to perform the desired task through repeated trials and errors in a particular environment .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Methods implemented in previous studies have yielded remarkable results, including victories over professional human players in board games, such as Go .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Recently, photonic approaches to reinforcement learning to outsource the computational costs and exploit the physical nature of light have been proposed [4–8].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Previous studies solved the bandit problem, a fundamental reinforcement learning model, using the quantum nature of photons [9–12].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The bandit problem is a frequently used model of human decision-making .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Multiple slot machines probabilistically generate a reward and an agent at- tempts to maximize the cumulative reward from the machines under the constraint that they can only play one machine at a time .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Because the agent lacks prior knowledge of the reward 1 arXiv:2212.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='09926v1 [cs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='AI] 20 Dec 2022 probabilities of the machines, they must play with various machines, including bad machines at that time, in the early stages of the game to accurately estimate the reward probabilities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This results from the stochastic nature of reward generation; that is, a machine should not be considered as having a low reward probability just because it has not been generating many rewards at that time.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, the agent would suffer a loss if they played bad machines excessively; therefore, they must concentrate on the machines that have the highest reward probabilities in the latter stages of the game.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The former aspect is called exploration, whereas the latter is called exploitation; balancing these two conflicting demands is the key to solving this problem .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The softmax rule is a model that balances exploration and exploitation through probabilistic decision-making and is considered as the model that best fits human decision-making .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The quantum nature of photons can be applied to solve the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In particular, by mapping the selection of a machine to the observation of a photon’s state, probabilistic decision- making can be implemented because the state observed is determined probabilistically .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Further- more, the role of photons in decision-making becomes critical owing to entanglement and quantum interference, which are inherent properties of quantum physics [10–12].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, consider a situation in which two agents solve the bandit problem simultaneously but the selection of the same machine reduces the total reward.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is analogous to a real-world situation when multiple peo- ple or devices simultaneously connect to the same wireless channel, resulting in the degradation of the individual communication speed [14, 16–18].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' By observing the states of a two-photon pair whose polarizations are entangled, the two agents can ensure that their choices always differ in such circumstances.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' That is, entanglement avoids selection conflicts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' theoretically and experimentally showed that the competitive multi-armed bandit problem, which deals with the aforementioned situation, can be resolved with no conflict of choices by two agents faced with two machines using photon pairs whose polarizations are entangled .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Their system is remarkable in that the agents can avoid selection conflicts without directly communicating with each other about the machine to be selected because of quantum entanglement.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Furthermore, Amakasu et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' theoretically showed that the system could be extended to handle three or more machines using quantum interference of orbital angular momentum of light .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Accordingly, they developed a photonic system that ensured conflict-free selections by two agents with an arbitrary number of machines.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In addition, Shinkawa et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' formulated a problem in which people individually have a probabilistic preference over options, derived the optimal joint decision-making in terms of satisfaction , and demonstrated that a system based on quantum interference of photons can provide a heuristic solution to this problem .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is another example of the coordination of multi-individual decision-making by a photonic system.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study aims to demonstrate the potential of a photonic reinforcement learning scheme, which requires the combination of a suitable algorithm and a photonic system that leverages the unique physical nature of photons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Based on previous studies, a photonic system can be used to solve the bandit problem, a simple reinforcement learning task.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, to tackle challenging problems, the photonic system must be extended such that it can handle three or more agents, and the algorithm must be modified accordingly.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The environment in the bandit problem is static, whereas that in a general reinforcement learning problem is generally dynamic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In particular, the environment (reward 2 probabilities) is independent of the agent’s action in the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Conversely, in a general reinforcement learning problem, the state of the environment changes because of the action, which must be considered in the learning process.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study presents a modified algorithm that can solve a broader class of reinforcement learning problems.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' While the extension of the photonic system with more than two agents remains open and must be addressed in future studies, this study lays the foundation for a photonic reinforcement learning scheme that can be implemented once the photonic system is developed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We consider the grid world problem as a dynamic environment .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' It is a collection of cells in which an agent can either implement an up, down, left, or right action.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Depending on the combination of cells and actions, the agent receives different rewards from the environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Because different cells have different reward environments, the grid world is a dynamic environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' While Q-learning is generally used as an algorithm for reinforcement learning [22–24], this study proposed a combination of Q-learning with the bandit algorithm, called discontinuous bandit Q- learning (DBQL).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Although Q-learning aims to learn the optimal paths, this study aims to learn the value of each state-action pair in the entire environment with high accuracy.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Thus, suppose the agent deviates from the optimal paths.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In that case, it will accurately return to the optimal paths from any location in the environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In the proposed DBQL method, each agent selects a state-action pair in the environment at each time step and updates the corresponding Q-value (a detailed definition is given in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Decisions on the state-action pair to be selected have a similar structure to the bandit problem because the agent must balance two demands; the first is the demand for exploitation, which is to update state-action pairs that are likely to have a large value of ∆Q (the change in the Q-value) for the moment, to accelerate learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The second is the demand for exploration to accurately estimate the expected value of ∆Q for other state-action pairs that have not yet been visited frequently.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Thus, by considering the state-action pairs as machines and ∆Q as a reward, the accurate estimation of Q-values for the entire environment can be viewed as a bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Furthermore, we consider a case in which multiple agents participate in the learning and follow DBQL simultaneously.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We demonstrate the learning can be accelerated by avoiding the selection of the same state-action pair simultaneously; that is, by forcing the agents to make conflict-free decisions.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' As earlier mentioned, we have not conceived a photonic system that enables conflict-free selections among more than two agents without direct communication yet.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Accordingly, this study algorithmically realized conflict-free selections, which essentially means forcing the agents to disclose their selections.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Once a photonic system with more than two agents is developed in the future, our scheme will be implemented by the mixture of the photonic system and our proposed algorithm, thus eliminating the necessity for the agents to share their selections.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1 describes the experimental envi- ronment and the grid world.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Section 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2 explains Q-learning followed by a detailed description of the proposed method DBQL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Section 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3 provides the environment’s response when multiple agents explore simultaneously, and Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='4 illustrates a selection conflict avoidance system using quantum interference of photons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Section 3 demonstrates the result of performing an actual search in the grid world using DBQL to quantify the impact of the bandit algorithm on learning and the impact of 3 avoiding selection conflicts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Finally, Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4 discusses the results and future perspectives.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2 Materials and Methods 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1 Experimental Design The schematic of the grid world, which is often used as a model in previous studies on reinforcement learning is shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' An agent exists in the grid world and moves around in the environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A A’ B B’ +10 +5 Figure 1: 5 × 5 grid world.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The agent implements one of the four actions at each time step and receives a reward and the next state.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In special cells A and B, the reward is large and the agent jumps to another cell.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In this example, the world is represented by a 5 × 5 cell grid, where each cell is called a “state.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' At each time step, the agent selects an “action” either up, down, left, or right.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In the grid world, when an agent is in a state st at a time step t, the chosen action at determines the reward rt and the next state st+1, which is provided by the environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In this study, we assumed the environment is Markovian, meaning the next state st+1 is determined only by the current state st and action taken at.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, if an agent is in the top left corner cell and selects the action “right,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' the agent earns a specific reward and moves to cell A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Herein, the rule that determines the action to be chosen by the agent in each state is called a “policy.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In this study, we confine the policy to be deterministic.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Then, the “action-value function” Qπ(s, a) is determined for each state-action pair (s, a) when the agent follows a particular policy π.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This function represents the total future rewards when the agent is currently in a state s and takes an action a followed by a series of actions that π instructs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Qπ(s, a) = ∞ � t=0 γtrt, (1) where γ is the time discount.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The time discount is applied to reflect that distant-future rewards matter less than near-future rewards and to ensure the convergence of the function.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Note that a 4 larger γ means the agent is more concerned about a long-term benefit.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' If the reward is determined stochastically, the expected value of the reward E[rt] is used instead of rt.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Suppose the grid world problem is fully known, meaning all the possible states, actions, and rewards are known in advance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In that case, we can use dynamic programming algorithms, such as value iteration or policy iteration, which solve the Bellman equation to derive the optimal action- value function ˜Q(s, a) and policy ˜π(s), where ˜π(s) = argmax a ˜Q(s, a) (2) is satisfied .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study aims to ensure that the agent without the knowledge of the envi- ronment accurately learns the optimal action-value function ˜Q(s, a) for all state-action pairs (s, a) using the information they obtain from the environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The initial values are Q(s, a) = 0, and the value of Q(s, a) during the learning is called “Q-value.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2 Discontinuous Bandit Q-Learning Q-learning is generally used to solve the grid world.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Algorithm 1 provides an overview of Q-learning .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Algorithm 1 Q-learning 1: Pick an initial state s0 2: while t ≤ T do 3: if rand() < ϵ then 4: at ← random 5: else 6: at ← argmax a Q(st, a) 7: end if 8: Implement an action at and obtain a reward rt and the next state st+1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Then, update Q(st, at): 9: Q(st, at) ← Q(st, at) + α · � rt + γ max a′ Q(st+1, a′) − Q(st, at) � 10: end while First, the agent randomly chooses the initial state s0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The policy π is the ϵ-greedy method.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' That is, the agent usually chooses an action with the largest Q(st, at); however, the action is chosen uniformly at random with a probability ϵ.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Notably, the possible actions are confined to the four actions in the current cell, unlike in discontinuous Q-learning proposed later.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Thus, the agent receives a reward rt and the next state st+1 from the environment after executing the action at.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Finally, it updates Q(st, at) according to the following rule: Q(st, at) ← Q(st, at) + α · � rt + γ max a′ Q(st+1, a′) − Q(st, at) � , (3) where α is the learning rate and γ is the time discount.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Note that if α = 0, nothing is learned; 5 however, if α = 1, all the previous experiences are forgotten and only the last experience is considered.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This process can be repeated to obtain Q-values as good approximations of the optimal action-value functions ˜Q(s, a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In this study, we first propose discontinuous Q-learning to interpret the original Q-learning as a decision-making problem about what state-action pair (s, a) the agent selects at each time step.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Algorithm 2 provides an overview of discontinuous Q-learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Unlike basic Q-learning, in discontinuous Q-learning, the new state s′ obtained from the environment because of the action at is ignored and a new state-action pair (st+1, at+1) is chosen from all the possible pairs in the environment, which are not limited to pairs whose states are s′.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Although the algorithm implies that the agent “jumps” at every time step, this assumption is realistic in reinforcement learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The initial position is already often randomly chosen in the existing algorithms at the start of every iteration (thus, the position “jumps” from the final position in the last iteration) in famous problems such as the cart-pole problem or the maze-solving problem .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Algorithm 2 Discontinuous Q-learning 1: while t ≤ T do 2: Select one state-action pair (st, at) based on specific criteria 3: Implement an action at and obtain a reward rt and next state s′.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Then, update Q(st, at): 4: Q(st, at) ← Q(st, at) + α · � rt + γ max a′ Q(s′, a′) − Q(st, at) � 5: end while Algorithm 2 shows that the state-action pair (st, at) updated by the agent at every time step is determined based on “specific criteria.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study demonstrates that the bandit algorithm can function effectively as the selection criterion.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Now, ∆Q(s, a) is defined as the absolute value of the updated amount of Q(s, a) at each time step: ∆Q(s, a) := ���α · � rt + γ max a′ Q(s′, a′) − Q(st, at) ���� .' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (4) Larger ∆Q(s, a) means faster learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Therefore, the agent should choose a state-action pair (s, a) with a high expected value of ∆Q(s, a) to make the learning process more efficient.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, the potential update ∆Q(s, a) for other state-action pairs (s, a) could be higher and will also vary as the update proceeds.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Thus, the agent cannot rely on just selecting the same state-action pair (s, a) over and over.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The agent also needs to explore other pairs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This structure is similar to that of the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Therefore, by regarding each state-action pair (s, a) as a slot machine and the change in Q(s, a) as the reward in the context of bandit problems for the discontinuous Q-learning algorithm, we can associate the agent’s attempt to select the state-action pair (s, a) with large ∆Q as “exploitation” and the investigation of ∆Q for other state-action pairs (s, a) as “exploration.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We thus define discontinuous bandit Q-learning (DBQL) as an algorithm that follows discontinuous Q-learning in which the bandit algorithm functions as the selection criterion.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In DBQL, the agent follows the softmax algorithm, a widely used algorithm to successfully solve the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The agent records ∆Q for each state-action pair (s, a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Let µt(s, a) be the 6 empirical mean of ∆Q(s, a) by time step t.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The probability of the agent selecting the state-action pair (si, aj) at the next time step t + 1 is calculated as follows: pt+1(si, aj) = eβ·µt(si,aj) � (s,a) eβ·µt(s,a) , (5) where β controls the degree of exploration and exploitation.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3 Multi-agent Learning In this study, multiple agents participate in simultaneously updating the global lookup table of Q(s, a) based on DBQL to accelerate the learning process as shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A situation is considered in which the agents share the global lookup table of Q(s, a) while individually recording a separate table of ∆Q(s, a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' At time step t, each agent refers to the ∆Q table it has recorded and determines the state-action pair (st, at) to update based on the softmax algorithm in Eq.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Next, it retrieves the value of Q(st, at) from the global lookup table, calculates the updated value of Q(st, at) according to Eq.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (3), and sends it back to the global table.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' An important rule is that when two or more agents attempt to update the same state-action pair (s, a) at the same time step, only one of the updates is randomly reflected to the global lookup table.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This depends on the problem settings; however, real-world examples exist in which the same investigation of state-action pairs by multiple agents is detrimental.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, consider an exploratory scenario where sonic waves are used to reveal the underlying stratigraphy of the seafloor.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Multiple agents simultaneously conducting the same location exploration would result in interference and yield poor results.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Moreover, even if we were to allow simultaneous updates by multiple agents and calculate the sum of ∆Q to reflect to the global table, this could disturb the convergence of Q-learning, because taking the sum essentially means changing the learning rate α locally for this particular time step.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='4 Cooperative Decision-Making through Quantum Interference This section explains how the quantum interference of photons can be leveraged such that mul- tiple agents can avoid selecting the same state-action pair (s, a) at the same time step without any direct knowledge of the selections of the other agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' As already mentioned, we have been unable to extend the conventional cooperative decision-making system with two agents, and thus the numerical demonstrations shown in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3 uses an algorithmic way to avoid selection conflicts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Therefore, we will only cover the core concepts in this section and outline how in principle a photonic implementation may function.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Amakasu et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' proposed a conflict-free collective decision-making system with two agents using the orbital angular momentum (OAM) of light.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A photon can carry theoretically an infinite 7 Global lookup table of Each agent records table individually Look up Send back updated Agent 1 Agent 2 Agent Agent Selections of are coordinated through quantum interference of photons Figure 2: Structure of the DBQL by multiple agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Each agent looks up a Q-value from the global lookup table, updates it using the generated reward and the next state provided by the environment, and sends it back to the global table.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In our scheme, agents are not directly connected and cannot communicate with one another; yet, their state-action selections are coordinated owing to the quantum interference of photons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, we coordinated their selections in this study using an algorithm because we could not extend the photonic system with two agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 8 number of OAMs, and the state of the photon is described as a superposition of different OAMs |k⟩: |Φ⟩ = 1 √ K K � k=1 eiφk| + k⟩.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (6) Because of the quantum property of photons, the detection probability of each OAM is calculated using the modulus square of the probability amplitude.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In addition, the usage of attenuators enables us to control the probability amplitudes, thus changing the observation probabilities.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In their pro- posed system, Amakasu et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' set K equal to the number of options (in our case, this is the number of state-action pairs) and designed a protocol in which the agent selects the option whose index is the same as the detected OAM number.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, if an OAM of | + 1⟩ is detected by the first agent, the agent selects the first option.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This protocol enables probabilistic decision-making because the control of the probability amplitudes by the attenuators results in the control of the selection probabilities of the options.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The use of quantum physics makes a difference when two agents simultaneously make decisions based on probability following the aforementioned protocol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A quantum effect called the Hong- Ou-Mandel interference exists, whereby different OAMs are always observed when a photon-pair connected by this effect is observed by two detectors.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Based on the protocol, the two agents always select different options without informing each other of their selections; that is, conflict-free selections are possible.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The implementation of the Hong-Ou-Mandel interference is quite simple and can be accomplished with only very basic optical instruments, such as spatial light modulators and beam splitters, as shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' SLM SLM BS BS SLM : Spatial light modulator BS : Beam splitter Figure 3: Two-photon Hong-Ou-Mandel interference.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' |Φ⟩ and |Ψ⟩ represent the states of photons that are controlled by spatial light modulators.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Although a detailed design is yet to be devised, it is likely that conflict-free probabilistic decision- making can be realized with three or more agents by cascading multiple spatial light modulators and beam splitters as well as appropriately configuring the input OAM states as an extension of the previous setup.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, the schematic of a photonic configuration with three photons is shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This system eliminates selections completely in which all the agents select the same option; however, selections in which only two select the same option still remain.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Numerous studies 9 on quantum interference among multiple photons have been conducted, including Refs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' [29–31], thus successfully integrating these methods with the usage of OAMs has a guiding significance in developing a photonic system that completely eliminates selection conflicts in the future.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' BS BS : Beam splitter BS Figure 4: Photonic configuration with three photons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' |Φ⟩, |Ψ⟩, |Ξ⟩ represent the states of photons.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In our scheme, we assumed that the multi-photon conflict-free system can be realized and uti- lized to coordinate the probabilistic decision-making of N agents through quantum interference of photons regarding the choice of the state-action pairs (s, a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This enables the agents to prevent selection conflicts without communicating with each other about the selection of pairs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Not only is the learning accelerated because unnecessary updates are avoided, but also resources required to exchange information about state-action pair selections can be reduced.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Note that, simultaneous updates by different agents will be a waste except for one of them as explained in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study realized conflict-free selections using an algorithm by numerically computing the joint selection probabilities on a computer instead of using a photonic system.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3 Results A 5 × 5 grid world shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1 was considered in this study, and we analyzed the situation in which multiple agents updates a state-action pair (s, a) at every time step according to discontinuous Q-learning (Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1 Rules in the Grid World The state-action combination in the grid world determines the rewards received by an agent and the cell to which it moves.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The following settings are used in this study.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 10 When any action is taken from cell A, the chances of the cell generating a reward of +10 is 50% and the agent jumps to cell A’.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' If no reward is generated, it remains in cell A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' When any action is taken from cell B, the chances of the cell generating a reward of +5 is 50% and the agent jumps to cell B’.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' If no reward is generated, it remains in cell B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In any other cell, no reward is generated and the destination follows the action except when the agent hits a wall.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In such cases, a reward of −1 is generated and the agent remains on the current cell.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2 Objectives Each agent selects a state-action pair (s, a) at each time step and updates Q(s, a) according to Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In this study, we considered 10–100 agents with 100 state-action pairs (25 cells and four actions in each cell).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' To quantify the learning accuracy, we defined the loss Lt as the average absolute error between the true action-value function ˜Q(s, a) and Q-values learned at time step t, which we refer to as Qt(s, a), over all the state-action pairs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Lt = 1 100 � (s,a) | ˜Q(s, a) − Qt(s, a)|.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (7) One hundred trials were performed and the average loss Lt over the trials was calculated as a metric to measure the gap between the optimal action-value functions and the Q-values.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The smaller the average loss, the more successful the learning process.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Based on the rules in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1, the ground truth values of the action-value function ˜Q(s, a) are summarized in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='59 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='59 9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='43 9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='43 12.' metadata={'source': 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='84 7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='45 9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='39 9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='39 7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='45 8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='39 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='43 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} 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'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='88 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='43 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='22 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='22 13.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='57 11.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='59 Figure 5: Ground truth values of the action-value function ˜Q(s, a).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Two major points were tested in this study: First, we tested whether the bandit algorithm outperforms random selections in Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' That is, we compared the loss trajectories between DBQL and the case in which the agents make decisions uniformly at random instead of using the softmax algorithm as the selection criteria in Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 11 Second, we considered the effect of conflict avoidance in the state-action pair selection on learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' As noted in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3, if multiple agents simultaneously select the same state-action pair (s, a), only one of their updates will be reflected in the global Q-table.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hence, the learning process will always accelerate by avoiding selection conflicts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This study aims to quantify this effect.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Furthermore, we demonstrate the significance of conflict avoidance especially when using DBQL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The parameters are as follows: the number of iterations T is 20000, learning rate α in Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2 is initially set to 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='035 that decays linearly to α = 0 at t = 20000, and the time discount γ is 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='9.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' β in Eq.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, which controls the degree of exploration and exploitation of the softmax algorithm used in DBQL, is initially set to 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0 and grows linearly to β = 5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0 at t = 20000 because more exploitation is necessary in the later stage of learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3 Performance Comparison The average loss Lt, which quantifies the gap between the optimal action-value functions and the Q-values during learning when the number of agents is 10, 50, or 90, are shown in Figs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6 (a), (b), and (c), respectively.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Therein, the blue and orange curves represent the cases in which different agents were allowed to simultaneously select the same state-action pair, denoted by “conflict.”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The procedure for state-action pair selections differs for the blue and orange curves.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The blue curve denoted by the legend “uniform random/conflict” is based on random selections, whereas the orange curve denoted by “bandit/conflict” is based on the softmax algorithm in Eq.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (5).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Similarly, the green and red lines represent the cases in which the state-action pair selections are conducted in a “conflict-free” manner, as marked by the latter half of the legend.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Agents were not allowed to select the same state-action pair at the same time step.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In addition, actions were selected in a uniformly random manner in the green curve, whereas the red curve was based on a bandit- based approach.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hence, the green and red curves were denoted as “uniform random/conflict-free” and “bandit/conflict-free,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' respectively.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' First, we compared the random-based lines with the bandit-based lines to examine the effect of the bandit algorithm.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' By comparing the blue and orange lines or the green and red lines, we observed that the learning is faster when the agents follow the bandit algorithm.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This validates the effectiveness of DBQL, which considers the change in the Q-value (∆Q) as the reward of the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' As the number of agents approaches a hundred, the difference in performances between uniform random/conflict-free and bandit/conflict-free narrows.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is because in situations where decision overlaps are not allowed, the significance of each agent’s sensible selection is reduced if the number of agents is sufficiently large.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In particular, when the number of agents is one hundred, the two performances are exactly the same, irrespective of the choice made by the agents because only one agent is always assigned to each state-action pair.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Moreover, we analyzed the impact of conflict avoidance in state-action pair selections on learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' While the results are rather obvious considering the environmental rules, learning is faster when the selections are conflict-free.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We defined Sunder as the area under the learning curve to quantify the 12 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='5000 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='10000 ' metadata={'source': 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+page_content='agents ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='Figure ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='6: ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='Comparison ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='of ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='the ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='four ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='learning ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='methods.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The average loss, representing the gap between optimal action-value functions and the Q-values during learning, is shown for the four learning methods.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The learning methods are divided along two axes: whether the bandit algorithm is used for selection criteria and whether selection conflicts are allowed among different agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 13 learning efficiency across the entire learning process.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sunder = T � t=1 Lt.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' (8) Table 1 lists the Sunder ratios of uniform random/conflict by uniform random/conflict-free and bandit/conflict by bandit/conflict-free to quantify the benefit of the conflict avoidance in state-action pair selections.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Table 1: Benefit of conflict avoidance in the selection of state-action pairs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Conflict avoidance is crucial when the number of agents increases.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Number of agents 10 20 30 40 50 60 70 80 90 100 Uniform random 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='06 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='12 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='19 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='26 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='32 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='39 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='46 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='53 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='59 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='67 Bandit 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='13 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='26 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='34 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='4 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='43 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='47 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='49 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='52 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='55 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='56 As the number of agents increases, the ratio also increases, indicating that conflict avoidance provides a more significant benefit.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is because the probability of multiple agents selecting the same state-action pair (s, a) increases and more selections are discarded when selection conflicts are allowed, as only one of them is valid.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Thus, we defined Rvalid as the proportion of valid choices to quantitatively evaluate such effects.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The change in Rvalid for bandit/conflict as the number of agents changes is shown in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Rvalid decreased as the number of agents increased, indicating that conflict avoidance is more significant for an increased number of agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' When the number of agents is one hundred, approximately 60% of the updates are wasted if the agents’ selections are not coordinated.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 Number of agents 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='4 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='6 0.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='8 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0 Rvalid Figure 7: Valid selection rate Rvalid.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The larger the number of agents, the more frequently the selections of the agents overlap, validating the significance of conflict avoidance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Furthermore, comparing the convergence values in Fig.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6, bandit/conflict has a larger value than random/conflict, indicating that uniform random/conflict outperforms bandit/conflict if only the final accuracy is compared.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is because, in bandit/conflict, most agents decide to update cell A as they proceed with learning; therefore, other cells are not updated, thus resulting in residual action-value function errors for those cells.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Figure 8 shows an example of the number of agents that choose each state-action pair (s, a) in the final time step when the number of agents is a hundred for 14 the bandit/conflict case.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Approximately 90% of the agents update cell A and most of the remaining agents update cell B, which has the second highest reward.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Therefore, ensuring the conflict-free property is crucial to avoiding most agents getting “stuck” when using a bandit-derived algorithm.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Figure 8: Number of agents that choose each state-action pair (s, a) at the final time step for the bandit-based algorithm when conflicts are allowed.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Approximately 90% of the agents update cell A, which generated the highest reward, while the other cells remain largely unexplored.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4 Discussion This study proposed a photonic reinforcement learning scheme, which required both a novel algo- rithm and photonic system, and demonstrated its performance.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We employed the grid world problem, a frequently used model in reinforcement learning, with the aim to learn the optimal action-value functions for all state-action pairs (s, a) with high precision, involving multiple agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The details presented in the study are summarized as follows.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' First, we proposed DBQL, a learning algorithm in which each agent selected one of all the state- action pairs (s, a) in the environment in each time step t and updated Q(s, a) based on the same formula as in the original Q-learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The decision-making problem of selecting a state-action pair (s, a) was similar to the bandit problem.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This is because if we define the amount of Q(s, a)-update at each time step as ∆Q, the agent must strike a balance between the demand to select state-action pairs (s, a) with large ∆Q to accelerate the learning process (“exploitation” in the context of the bandit problem) and the demand to investigate the values of ∆Q for other state-action pairs (s, a) (“exploration” in the bandit problem).' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We compared the case in which the bandit algorithm was used as the decision-making criteria for Alg.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2 with that in which a uniform random selection was used, to validate the effectiveness of DBQL.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The former resulted in a faster learning process as described in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Second, we proposed a multi-agent architecture where multiple agents make conflict-free decisions in the learning, and then quantitatively evaluated the impact of the conflict avoidance on learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' As demonstrated in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, the learning was indeed accelerated by avoiding selection conflicts 15 particularly when the number of agents increased.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Moreover, cooperative decision-making is essential when multiple agents follow DBQL to avoid getting stuck in the later stage of learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Without coordinating the selection, most agents would be highly likely to select the cell with largest reward in the latter parts of the learning.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Although the particular configuration of the system is not yet established, a photonic system with cascaded spatial light modulators and beam splitters is expected to enable cooperative decision-making by three or more agents for avoiding selection conflicts.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Once this system is conceived in the future, it can be incorporated into our proposed scheme and obviate the necessity for the agents to communicate with each other to coordinate their selections.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' While Amakasu et al.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' provided the concept of the base idea that addressed the competitive bandit problem, this study addresses a general reinforcement learning problem with the grid world problem as an example.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' The two problems differ in that, in the bandit problem, the machines’ reward probabilities are invariant regardless of the agent’s action; however, in the grid world problem, state transitions, which correspond to the changes in the reward probabilities in the context of the bandit problem, occur because of the agent’s action.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Our proposed scheme applies to such challenging problems in a dynamic environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Next, we discuss some of the limitations of this study and how they may be addressed in the future.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' First, in DBQL, the agent’s actions are discontinuous.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This can be resolved by restricting the possible state-action pairs (s, a) that can be chosen by the agent at each time step to those in the current cell.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' However, with this method, if more than four agents end up in a particular cell, at least two will have to choose the same state-action pair (s, a) in the next time step.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' This requires rule making for exception handling.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Second, when the number of state-action pairs is sufficiently larger than the number of agents, conflicts of choice occur less frequently, and the advantage of conflict avoidance by quantum interference may be reduced.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Regarding this concern, as indicated in the birthday paradox, the probability that the choices of two agents overlap is greater than our intuition even if the number of agents is such smaller than that of pairs.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' For example, suppose 100 state-action pairs exist and 10 agents are to make uniform choices at random, the probability that at least two agents make the same choice is over 37%.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Furthermore, as mentioned in Sec.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, conflict avoidance is essential in DBQL because the probability of multiple agents intending to make the same choice increases significantly as learning proceeds.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' In the future, our first priority is to design a system that allows conflict-free decision-making by three or more agents.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Additionally, we would like to develop algorithms that allow agents to take continuous actions and apply DBQL to other reinforcement learning problems that are more complex than the grid world.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' To the best of our knowledge, this study is the first to connect the notion of photonic cooperative decision-making with Q-learning and apply it to a dynamic environment.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' We believe this study makes a valuable contribution to the field of decision-making using physical processes.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Data Availability Data used in this study are available from the corresponding author upon request.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 16 Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Acknowledgments This study was supported in part by the CREST project (JPMJCR17N2) funded by the Japan Science and Technology Agency, Grants-in-Aid for Scientific Research (JP20H00233) and Trans- formative Research Areas (A) (JP22H05197) funded by the Japan Society for the Promotion of Science.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' AR was funded by the Japan Society for the Promotion of Science as an International Research Fellow.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' References R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sutton and A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Barto, Reinforcement learning: An introduction.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' MIT press, 2018.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1109/TNN.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1998.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='712192.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Silver, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Huang, C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Maddison, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Guez, L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sifre, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Van Den Driessche, J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Schrittwieser, I.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Antonoglou, V.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Panneershelvam, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Lanctot, et al., “Mastering the game of go with deep neural networks and tree search,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Nature, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 529, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 7587, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 484–489, 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/ nature16961.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Silver, T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hubert, J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Schrittwieser, I.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Antonoglou, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Lai, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Guez, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Lanctot, L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sifre, D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kumaran, T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Graepel, et al., “A general reinforcement learning algorithm that masters chess, shogi, and go through self-play,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Science, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 362, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6419, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1140–1144, 2018.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1126/science.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='aar6404.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' F.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Flamini, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hamann, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Jerbi, L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Trenkwalder, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Nautrup, and H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Briegel, “Photonic architecture for reinforcement learning,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' New Journal of Physics, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 22, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 045 002, 2020.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1088/1367-2630/ab783c.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Steinbrecher, J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Olson, D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Englund, and J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Carolan, “Quantum optical neural net- works,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' npj Quantum Information, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 60, 2019.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/s41534-019- 0174-7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' V.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Saggio, B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' E.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Asenbeck, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hamann, T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Str¨omberg, P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Schiansky, V.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Dunjko, N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Friis, N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Harris, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hochberg, D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Englund, et al., “Experimental quantum speed-up in reinforcement learning agents,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Nature, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 591, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 7849, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 229–233, 2021.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/s41586-021- 03242-7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bukov, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Day, D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sels, P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Weinberg, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Polkovnikov, and P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Mehta, “Reinforcement learning in different phases of quantum control,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Physical Review X, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 8, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 031 086, 2018.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1103/PhysRevX.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='031086.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bueno, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Maktoobi, L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Froehly, I.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Fischer, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Jacquot, L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Larger, and D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Brunner, “Rein- forcement learning in a large-scale photonic recurrent neural network,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Optica, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 756–760, 2018.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1364/OPTICA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='000756.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 17 M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Berthel, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Drezet, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Huant, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Aono, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hori, and S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='-J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kim, “Single-photon decision maker,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Scientific Reports, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1–9, 2015.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/srep13253.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet, D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Jegouso, B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Boulanger, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Saigo, K.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Okamura, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hori, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Drezet, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Huant, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bachelier, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, “Entangled-photon decision maker,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Scientific Reports, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 9, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4832, 2019.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/s41598-019-48647-7.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bachelier, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Huant, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Saigo, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hori, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, “Entangled n-photon states for fair and optimal social decision making,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Scientific Reports, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 10, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 20 420, 2020.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/s41598-020-77340-3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Amakasu, N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bachelier, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Huant, R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Horisaki, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, “Conflict- free collective stochastic decision making by orbital angular momentum of photons through quantum interference,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Scientific Reports, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 11, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 21 117, 2021.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/s41598- 021-00493-2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' D.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Daw, J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' O’doherty, P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Dayan, B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Seymour, and R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Dolan, “Cortical substrates for exploratory decisions in humans,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Nature, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 441, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 7095, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 876–879, 2006.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1038/nature04766.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Maghsudi and E.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Hossain, “Multi-armed bandits with application to 5g small cells,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' IEEE Wireless Communications, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 23, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 64–73, 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1109/MWC.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='7498076.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' March, “Exploration and exploitation in organizational learning,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Organization Science, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 71–87, 1991.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1287/orsc.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='71.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Lai, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' El Gamal, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Jiang, and H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' V.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Poor, “Cognitive medium access: Exploration, ex- ploitation, and competition,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' IEEE Transactions on Mobile Computing, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 10, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 239– 253, 2010.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1109/TMC.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2010.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='65.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='-J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kim, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Aono, “Harnessing the computational power of fluids for op- timization of collective decision making,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Philosophies, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 245–260, 2016.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3390/philosophies1030245.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Besson and E.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kaufmann, “Multi-player bandits revisited,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' in Algorithmic Learning The- ory, PMLR, 2018, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 56–92.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' [Online].' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Available: https://proceedings.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='mlr.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='press/v83/ besson18a.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='html.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Shinkawa, N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bachelier, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R¨ohm, R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Horisaki, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, “Optimal preference satisfaction for conflict-free joint decisions,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='00799, 2022.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='48550/arXiv.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2205.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='00799.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Shinkawa, N.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Chauvet, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R¨oohm, T.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Mihana, R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Horisaki, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bachelier, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Naruse, “Conflict-free joint sampling for preference satisfaction through quantum interference,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Physi- cal Review Applied, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 18, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 064 018, 2022.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1103/PhysRevApplied.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='18.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='064018.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sutton, “Integrated architectures for learning, planning, and reacting based on ap- proximating dynamic programming,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' in Machine Learning Proceedings 1990, Elsevier, 1990, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 216–224.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1016/B978-1-55860-141-3.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='50030-4.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Watkins, “Learning from delayed rewards,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1989.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 18 C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Watkins and P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Dayan, “Q-learning,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Machine learning, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 8, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 279–292, 1992.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1007/BF00992698.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Jang, M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kim, G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Harerimana, and J.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' W.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Kim, “Q-learning algorithms: A comprehensive classification and applications,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' IEEE Access, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 7, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 133 653–133 667, 2019.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1109/ ACCESS.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2019.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2941229.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bellman, “Dynamic programming,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Science, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 153, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 3731, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 34–37, 1966.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1126/science.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='153.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='3731.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='34.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' ——, “The theory of dynamic programming,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Bulletin of the American Mathematical Society, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 60, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 6, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 503–515, 1954.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1090/S0002-9904-1954-09848-8.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' G.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Barto, R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sutton, and C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' W.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Anderson, “Neuronlike adaptive elements that can solve difficult learning control problems,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' IEEE Transactions on Systems, Man, and Cybernetics, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 834–846, 1983.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1109/TSMC.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1983.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='6313077.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' L.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Samuel, “Some studies in machine learning using the game of checkers,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' IBM Journal of Research and Development, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 44, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2, pp.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 206–226, 2000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1147/rd.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='441.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='0206.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' ˙Zukowski, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Zeilinger, and M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Horne, “Realizable higher-dimensional two-particle en- tanglements via multiport beam splitters,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Physical Review A, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 55, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 2564, 1997.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1103/PhysRevA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='55.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='2564.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Campos, “Three-photon hong-ou-mandel interference at a multiport mixer,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Physical Review A, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 62, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 1, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 013 809, 2000.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1103/PhysRevA.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='62.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='013809.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' M.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Tillmann, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='-H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Tan, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' E.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Stoeckl, B.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' C.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Sanders, H.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' De Guise, R.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Heilmann, S.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Nolte, A.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Szameit, and P.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Walther, “Generalized multiphoton quantum interference,”' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' Physical Review X, vol.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 5, no.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 4, p.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 041 015, 2015.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' doi: 10.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='1103/PhysRevX.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='5.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content='041015.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +page_content=' 19' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\kb_test\\content\\test.pdf'} +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf,len=609 +page_content='NFTrig: Using Blockchain Technologies for Math Education JORDAN THOMPSON, Augustana College, USA RYAN BENAC, Augustana College, USA KIDUS OLANA, Augustana College, USA TALHA HASSAN, Augustana College, USA ANDREW SWARD, Augustana College, USA TAUHEED KHAN MOHD, Augustana College, USA NFTrig is a web-based application created for use as an educational tool to teach trigonometry and block chain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Creation of the application includes front and back end development as well as integration with other outside sources including MetaMask and OpenSea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The primary development languages include HTML, CSS (Bootstrap 5), and JavaScript as well as Solidity for smart contract creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The application itself is hosted on Moralis utilizing their Web3 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This technical report describes how the application was created, what the application requires, and smart contract design with security considerations in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The NFTrig application has underwent significant testing and validation prior to and after deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Future suggestions and recommendations for further development, maintenance, and use in other fields for education are also described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' CCS Concepts: • Computer systems organization → Redundancy; Robotics; • Networks → Network reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Additional Key Words and Phrases: Matic, Metamask, polygon, bootstrap5, Solidity 1 INTRODUCTION The purpose of this report is to describe the technical details involved in the development of the NFTrig application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This includes both the front end website design, the back end smart contract, and NFT creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' It will mainly focus on the technical details of the project outlining software requirements, design through programming languages, client and server side interactions, and validation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This allows the reader to undertake further development, fixes, or maintenance of the software, as this forms part of the documentation for the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The NFTrig project is based around the creation of a web-based game application that allows interaction of NFTs (non-fungible token) with trigonometric function designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFts are digital assets, for example a picture, that has a unique identification and can generally be freely traded with cryptocurrency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Through this application, users are able to purchase digital artwork of many different trigonometric functions and combine them using mathematical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Current supported operations include multiplication and division of the trigonometry functions, and the output of each operation is a new NFT card that would be the result of an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The old cards will then be removed from the user’s possession and burned using the smart contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For example, if a user combined the two cards Sin(x) and Cos(x) using multiplication, they would lose their two old cards and receive the new card Tan(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, the NFT cards are assigned one of the following rarity levels: common, uncommon, rare, and legendary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The probability of each of these levels is defined later in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The application also allows a user to connect to MetaMask, a digital wallet capable of storing a user’s cryptocurrency and NFTs as well as a way to connect to block chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The NFTrig application Authors’ addresses: Jordan Thompson, jordanthompson18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Ryan Benac, ryanbenac18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Kidus Olana, kidusolana18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Talha Hassan, talhahassan18@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Andrew Sward, andrewsward@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA; Tauheed Khan Mohd, tauheedkhanmohd@augustana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu, Augustana College, Rock Island, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='00001v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='HC] 21 Dec 2022 2 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd can also display the NFTs owned by the user and allow them to connect to OpenSea to sell the NFTrig cards on a public marketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The application is hosted on Moralis employing their Web3 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Technical languages used in this project, which will be discussed in detail throughout this paper, include front end web development languages HTML, CSS (specifically Bootstrap5), and JavaScript as well as the back end smart contract development language Solidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In order to attract users, this application also allows a user to answer trivia questions and gain experience points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These points can then be used to unlock new sets of NFT cards or upgrade existing cards in a user’s wallet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This game-like design should appeal to a younger audience and encourage them to answer trigonometry or math based questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This will have an incredible educational benefit for the user because they will be both learning and playing a game simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2 MOTIVATION The purpose of this application is as an educational tool for students who are attempting to understand the ways that trigonometric functions interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' As opposed to just graphing these functions by hand, students will be able to generate new NFTs by combining whatever trigonometric functions they already own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In fact, using technology is shown to influence and better educational processes by increasing interaction between those in the classroom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Technology is becoming increasingly prevalent in every sphere of daily life, so the use of technology in a classroom setting is not only logical, but it increases the educational benefit of students .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' However, as the technology continues to evolve, "the gap between traditional course material taught to students in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' programs at universities and the cutting edge of technology used in industry is widening at an unprecedented rate" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' By creating this project, it will give students the opportunity to gain experience with block chain, and hopefully be a starting place for narrowing that ever growing gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' After much research, it is likely that this proposed application is the first of its kind that utilizes NFTs to teach mathematical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Aside from user benefit of this application, there is also an intellectual merit in the block chain and education fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Best described by Carmen Holotescu, "As education becomes more open, diversified, democratised, and decentralised, the block chain technology is taken in consideration by researchers, teachers and institutions, to maintain reputation, trust in certification, and proof of learning" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, development of this project continues research on NFT and block chain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This application can also serve as the boilerplate basis for other NFT-based educational tools and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Research for this project provides opportunities for training computer science students on how to use NFTs in general, but more specifically in educational contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFTrig was developed by computer science students as a final senior inquiry project at Augustana College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In conjunction and with funding by the Department of Mathematics and Computer Science, this project employs a variety of software development skills and techniques that further the research and understanding of the block chain and web development field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 3 RELATED WORK Block chain technology has enabled the formation of decentralized distributed records of digital data which does not require any third party to moderate any transactions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The decentralized nature of block chain also renders it easy for use in a ranging variety of applications in several fields such as healthcare , internet of things , gaming , banking , and education (explored in greater detail in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Non Fungible token (NFTs) are a relatively new phenomena within the field of block chain based technologies, but its application in aforementioned fields are already being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Specifically within the healthcare context, NFT’s are solving long term issues such as storing patients’ private data more safely as well as maintaining better records while giving better autonomy and privacy to both patients and healthcare providers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The application of NFTs in NFTrig: Using Blockchain Technologies for Math Education 3 education is still an understudied area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These next related work sections explore the broader use of block chain based technologies for educational purposes, gamification, and overall collaborative learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 Block chain Based Technologies for Educational Purposes There has been extensive work concerning how block chain based technologies are enabling better ownership and sharing of personal records for students and supporting collaborative learning environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Yumna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' conducted a systematic literature review of the use of block chain technologies in educational sector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' They also propose several uses of existing block chain based technologies in educational sector that leverage the decentralized and traceable consensus making mechanisms of block chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Researchers have examined the use of block chain to allow students to maintain educational records such as transcripts, credentials, diplomas, and learning activities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Similarly, research has also explored learning management systems design based on block chain based technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The technology can potentially verify a students records as well as enable the design of an automatic decentralized enrollment system which does not require moderation from school staff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Another elegant use of block chain in the field of education is the ability to support life-long learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The educational sector is becoming more diverse with a variety of different types of classrooms and learning modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' E-learning has also allowed students to acquire licences and accreditation online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Therefore, it is imperative to maintain the learning journeys of students over time to understand the different types of learning that they have been engaging in and improving on over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The traceable nature of block chain based technologies (defined as one of the salient features in the aforementioned systematic review by ) enables all of these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The decentralized nature of block chains coupled with the consensus making algorithms also makes it suitable for collaborative environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Prior research has looked at how block chain based technologies can enable better developmental experiences in the realm of business but there is very minimal work on its application within the field of education application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2 Applications in Education Application and Collaborative Learning Although preliminary in nature, limited prior work has explored the utilization of NFTs for design- ing various different independent learning environments for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' There are some proposed commercial systems that have analogous functioning to some of the systems described in the prior section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For example, commercial systems are looking at leveraging NFTs to award “Pass" status to students for different courses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFTs enjoy a key advantage over conventional block chain technologies as they are typically designed using the more secure Ethereum block chain enabling an even more secure record and identity management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Researchers have shown that there is promise in using NFTs as academic tokens to represent student transcripts and other records as well that can be more easily verified .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' However, there is still a dearth of academic literature in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Student incentivization is heavily advocated in pedagogical literature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFTs make it easier to tie incentivization to learning outcomes as they can be automatically acquired by students at any time upon completion of learning outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This gives NFTs based certifications an advantage over the more traditional learning settings where students have to strongly adhere to semester timelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Elmessiry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' has looked at designing an incentive mechanism that can be used by teachers and students to achieve better learning outcomes in an effective and cost-efficient manner 1A teacher at Pepperdine University using NFTs to award course completion certifications to students: https://upcea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='edu/tech- trends-in-higher-ed-metaverse-nft-and-dao/ 4 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' They also concluded there was better engagement outcomes for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' On several metrics of usability, the students reported more than 80% preference for buying, using, and collecting NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Such independent learning methods were particularly more useful during the COVID-19 pandemic to accommodate the need of remote independent learning options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Architecturally, this project takes inspiration from , and applies it to a more narrower, focused domain of learning mathematical operations in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, these NFTs are also easier to share on social media .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Therefore, it also allows students to more readily share their accomplishments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='3 Gamification to Support Mathematical Learning Since the proposed application teaches mathematical and trigonometric formulas to students, the literature on use of gamification to support mathematical learning should be better described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Gamification, in combination with incentivization explained in the previous section, will allow for the success of this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Gaming settings have traditionally been used to teach simple mathematical operations to students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' More recently, researchers have also proposed systems that teach advanced concepts to students including College Algebra .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These learning environments make it easier for students to relate the learning concepts with more daily life phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' While gamification itself cannot guarantee better learning outcomes, it can improve students’ interest and performance by encouraging them to engage with the content for a longer duration of time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The simpler, more systematic, and operational nature of mathematics as a subject also makes it easier for incorporation in gaming environments because final answers are usually short and numerical as opposed to long and descriptive answer that might be found in social or natural sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Trigonometry especially can easily be broken down into a series of operations and steps which simulates a similar environment found in other online games where users play to find different “rewards" and “collectables".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Despite all these benefits there are some limitations of gamification as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For example, it is hard to know how a student arrived a solution and give feedback .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Not being able to solve trigonometric equations can also lead to frustration and impeded learning experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Foresight into the project’s future looks to mitigate these concerns by fostering better communication between different game players and providing links to useful learning resources in the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Prior research has extensively explored the use of gamification in different mathematical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This application is likely the first to extend the use of NFTs and block chain to aid in teaching trigonometric equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Research shows that technology, specifically games are shown to be excellent educational tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In fact, "one of the most successful positive reinforcement mechanisms [in education] known is gamification" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This includes taking a topic transforming it into a game with positive reinforce- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This leverages educational benefits in students and encourages them to continue playing the game to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Nftrig has future plans to add a game function which will allow the user to answer trigonometry trivia and math questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This will aid in both their learning and the continued use of the NFTrig application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, the ability to combine owned NFTs with math functions also aids in the education of trigonometry for the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4 EXPERIMENTAL SETUP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 Software Development Requirements The NFTrig application employs a variety of software development requirements that cover the range of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' From front end web development to back end smart contract creation and NFT storage, this section describes the requirements and software used to complete the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 Compiling IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The smart contracts created for NFTrig are hosted on Remix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Remix is an an open source online compiler IDE that can be used to test and deploy smart contracts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The NFTrig: Using Blockchain Technologies for Math Education 5 platform can be accessed by any browser, and it allows the developer to write and deploy smart contracts on an actual or test server simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The current deployment is on a test server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In order to test and debug the smart contract, Visual Studio Code is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Visual Studio was found to be the best code editor because a developer can easily upload most file types, and edit them .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For NFTrig, it was used to develop front end HTML and CSS files, as well as back end solidity contract editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The required installed plugins for Visual Studio (VS) include Solidity and Block chain development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These allowed for simple, straightforward development of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2 Moralis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Moralis SDK is the primary back end platform for the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The platform allows connection of the front end web application to the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The Moralis platform uses a combination of server management and a JavaScript SDK to allow for maximum interaction and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' A developer can do many tasks through this including authentication of users, getting necessary user data, and connecting with MetaMask in a non-complicated and simply coded process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The only expectation is that a developer will need to have programming knowledge in JavaScript as well as a familiarity with Moralis and MetaMask, experience querying a database, and some knowledge of Web3 development to ensure maximum results and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Moralis also has the ability to easily connect to MetaMask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='3 MetaMask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' MetaMask is the digital wallet required for participation in the NFTrig game application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' It allows the collection of purchases from the user, and it can be installed as an extension on a browser for increased ease of use .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' MetaMask stores all NFTs owned by the user, and in connection with the NFTrig application, can view and upgrade or modify existing NFTs at a users discretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Connection to the browser extension is required for the application to access anything owned by a user .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Because MetaMask is easily integrated into Moralis, and thus NFTrig, there is little a user needs to do to create a connection aside from installing the MetaMask extension, and clicking connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='4 Front End Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Front end design was accomplished primarily through Visual Studio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The Live Server extension was installed which allows each developer to "host" their developed website using a native web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Doing so allowed simplified testing and front end development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Instead of creating CSS files from scratch, the NFTrig interface heavily employs Bootstrap5, which simplifies the process of modifying the content layout and design of buttons and other content .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Moralis and Bootstrap5 each have extensive documentation to understand and support front end web development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These tools have been utilized to a near maximum extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5 Web Hosting Platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The initial testing of NFTrig, as previously explained, was hosted on a local live server through Visual Studio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' After initial development, the project was moved to a web server hosted by Augustana College so that initial testing could begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' It is currently unclear how the site will ultimately be hosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' One option for hosting the web application is directly through Google .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This would allow the website to be named something easily searchable and accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' A second option would be to host directly through Moralis, but a limitation of this would be a more diluted website naming convention along with a more confusion process of uploading and modifying website content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Currently, the NFTrig application will remain on the local Augustana College Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5 SOFTWARE DESIGN This section covers all of the decisions necessary to understand the development of NFTrig, as well as the technical implementation of each technology used in the design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 6 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 Software Architecture The architecture of this project follows the model-server design architecture .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Using this model, the clients send transactions and requests to a proxy smart contract stored on the block chain which then makes the appropriate calls to the logic smart contract which is also stored on the block chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This style of architecture is required for this project because the smart contracts must be stored on the server-side chain in order to be functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The use of proxy contracts also allows our smart contracts to be fully upgradeable with any future updates that may need to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2 Choice of Programming Language This section examines and explains the benefit of each chosen language employed in NFTrig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Front end languages include HTML and CSS and the back end includes Solidity and JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Each has been chosen because they were found to be the best option for development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 Solidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Solidity is the programming language of choice when it comes to coding smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Solidity is "similar to JavaScript and yet has some features of object-oriented languages such as Java and C++" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This is a leading language for the development of smart contracts and use on block chain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This project utilizes the solidity library openzeppelin in order to create a solid foundation for the smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Hardhat and Node JS are then used for the testing and deployment of the smart contracts to the Polygon blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2 JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In the NFTrig application, JavaScript (JS) is primarily used in the front end application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The primary purpose of this language is generally to create dynamic and interactive web content .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For the client, JS was used in the navigation bar to allow for clickable links and resizing of the navigation bar in smaller screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This language was also used to give buttons functionality ranging from logging in to MetaMask to purchasing NFTrig cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, JS was used to test the logic of the front-end combination page until the smart contract was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Aside from augmenting HTML and CSS application pages, JavaScript is also used in this project to connect the back end smart contract with the from end web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This application was also developed using Next JS and deployed via an application known as vercel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='3 HTML and CSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Web development of the user interface was primarily completed using HTML and CSS (Bootstrap5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These languages are equally popular and necessary to develop the web pages .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Instead of creating all CSS requirements from scratch, Bootstrap5 was utilized to allow for cleaner design across web pages and better alignment of web page elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Bootstrap5 also simplifies the need to explicitly code buttons and other interactive items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='3 Security Considerations Throughout this project, there have been several security considerations discovered that threatened the safety and use of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' One such discovered issue was initially, there was no code written to block a user from looking at another users token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, before minting a new NFT card, the smart contracts check to ensure that the card does not already exist, the cards used for combining are owned by the user, and that the newly minted card follows the correct probabilities of outcomes shows in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These probabilities are coded into the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='4 Smart Contract Design The smart contract for this project is broken up into two separate contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The first of which is the NFTrig logic contract which contains the logic for purchasing packs of cards as well as the logic for how cards will interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The second contract is the marketplace contract which will allow users to trade their own NFTs with other users through the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Within the NFTrig NFTrig: Using Blockchain Technologies for Math Education 7 contract, there are functions for multiplying and dividing cards, purchasing randomized packs of cards, and tracking the details of each individual token as transactions are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The marketplace contract contains information about sale history as well as the functionality to post new sales and purchase items for sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Both of these contracts were deployed as upgradeable contracts so they can have updates implemented in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5 NFT Storage and Naming Conventions All NFT images are stored on the server with the HTML, CSS, and JS files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The naming convention for each image references what image it is in four numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The first number is the power of sin, the second is the power of cos, the third is the rarity or color of the card (0-3 is green, blue, purple, and red respectively), and the final number is the text variant (0-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' These files were named accordingly to better determine the output if cards were combined using a mathematical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' For example, a sin card might have the naming convention: 1023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='jpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 10 defines it is a sin card, 2 defines it is rarity purple, and 3 defines it is text variant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The purpose of naming the files in this way is so that the front end can easily determine which image corresponds to a particular NFT by simply looking at the four features of each token which match the four numbers in the file name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6 Client Design The NFTrig application interface was designed using HTML and CSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The primary use of CSS was often replaced by Bootstrap5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Bootstrap 5, a library for CSS, allows for easier scaling and alignment of objects in the HTML file, and thus the computer screen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Documentation on the Bootstrap5 has utilized to a full extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Each section examines the layout and use of each application page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1 NFTrig Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The interface is designed to allow a user to access the marketplace, their individual current collections, and their profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The navigational bar contains links to the client-side facing pages: NFTrigHome, MyCards, CombineCards, Marketplace, and Game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' We used a total of three colors to enable good contrast and make it easier for our users to view complex graphs and formula without a cluttered background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The JavaScript elements declared are reusable across multiple screens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' They support functions and interactions such as a user hovering over a cell or clicking a cell and providing both feedback and error handling to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The navigation bar is also, the top bar changes color to indicate the tab that the user is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2 Combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The main purpose of the combination page is for users to choose cards that they currently own, and see options for combining them using either multiplication or division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Figure 1 displays the layout of the screen where user selected cards are shown on the left, and potential results are shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The page utilizes Bootstrap5 capabilities to format effectively to different screen sizes and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' It connects with a back end script to the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This provides functionality to the buttons and easy generation of possible NFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Below shows the probabilities of generated NFT outcomes based on the selected input cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='3 Marketplace and MyCards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Marketplace and MyCards are similar pages, as they connect to a data source and display NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The Marketplace tab shows all NFT cards available for purchase both from other users who own NFTs and cards owned by the NFTrig project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' MyCards however specifically shows all cards owned by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The layout for each generates all necessary NFT images and information about the rarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The rarity is signified by the color and the text option of the card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Figure 3 shows the actual layout displayed on the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2Background-color:#333, Color: #f2f2f2, 8 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Interface where users will combine NFTrigs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Probabilities of outcomes depending on rarity of selected cards 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='4 Quality attributes of client-side interface and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In order to have an application of quality, consistency, and accuracy, the project followed the following guidelines: (1) The code is written in a manner that components and layouts can be rearranged to support any structural changes in the front end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (2) The code has consistent style and format, such as the padding used in individual NFTrig elements and the purchase page’s color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (3) The code contains comments and is well indented for easy maintenance and understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (4) Consistent colors and feedback systems are provided so the system is easy to learn for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (5) Page-level styling was avoided when possible to keep design consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (6) Thorough testing was completed for basic accessibility features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5 Testing the Client Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Basic unit testing of different elements was initially conducted to ensure easy navigation between front end pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In order to ensure that testing would cover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Combination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Choosetwo cards and a mathfunction and hit combine! ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='PossibleResults ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='SIN(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='TAN(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='SN()TAN(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='SIN(TAN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rarity: Green ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rarity Blue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Font: 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Fonto ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Liklihood: 75% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Liklihood: 20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='sin(r)jtao(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='sin(z+y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='=sin(z)cos(y)+sin(y)cos(z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='tan(r)=sec() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='dar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='SIN(o)-TAN() ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='SIN(a)-TAN(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Sin(x)*Tan(X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rarity: Blue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rarity: Pink ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Borcler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Font:o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Sin(X) Green ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Tan(x) Bue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Font:0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Liklihood: 20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Multiply ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Divide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Likliho0d: 20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='(r+V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Combine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Copyright@2022-AllRightsRes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='rved-Augustana CollegeNFTrigCOMBINATIONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='COMMON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='UNCOMMON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='RARE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='LEGENDARY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Common+Common ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='15% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Common+Legendary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='25% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='35% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Common+Rare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='25% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='15% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Common+Uncommon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rare+Rare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='15% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Uncommon+Uncommon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='15% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Uncommon+Rare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='25% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Uncommon+Legendary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='55% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Rare +Legendary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Legendary+Legendary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='75%NFTrig: Using Blockchain Technologies for Math Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Interface displaying NFTrig Marketplace most application uses, three user cases were devised: a user browsing NFTrigs, a user making a purchase, and a user combining NFTrigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' All assumptions and expected actions expected from the system were listed and analyzed through testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, testing through some edge cases were also pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Currently, the application works as intended, however future plans involve rigorous testing with JavaScript code and external APIs (if any are devised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This will ensure a fully functional, secure, and usable application that can also be used as a boiler plate project for other educational blockchain technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='6 Future Work: Game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Future work for this project will include the ability for users to play a trivia and trigonometric equation game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This allows a user to gain experience points that they can then use to purchase new NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This eliminates the need to always need cryptocurrency to purchase individual or group NFT cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Although there is not currently an interface for this page written in HTML, functionality exists for the trivia game itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The files are currently stored on the server, but they are disabled and there is no navigable way to get there through the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 6 METHODS Most methods for completing this project have been thoroughly explained in the sections above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' However, the final intended version of this project will be hosted in a different location than it resides currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The initial portion of this project had the front end website hosted on a local Augustana College server and the back end smart contract hosted on the Polygon test net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This allowed initial testing and validation that the smart contract operated as expected, as well as give time and opportunity to discover security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The future of this project will be hosted on a decentralized web application online so that users can access it and begin to interact with the smart contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Further, a redesign of the website user interface is likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' This will require transition from BootStrap5 to NextJS which allows cards to be generated, displayed, and interactable through a version of JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 7 RESULTS This project successfully allowed the exploration and creation of applying NFT and block chain technology to math education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Although preliminary in use and nature, this project allows for initial project creation as a boiler plate project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The smart contract is currently deployed on the Marketplace Collections Profile About Metamask Search..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Q NFTrig SEC(aCSC(a) SEC(a)CSC() SEC(a)-CSC(a) SEC()CSa) SEC()CGCR U 2 csc(2x) to be added sec2(x)+csc2(x) cot(x) + tan(x) 2 csc(2x) to be added SEC(-CSCa) SEC(u)CSC(a) SEC(-CSC() sec2(x)+csc2(x) cot(x) + tan(x) 2 csc(2x) to be added sec2(x)+csc2(x) cot(x) + tan(x)10 Jordan Thompson, Ryan Benac, Kidus Olana, Talha Hassan, Andrew Sward, and Tauheed Khan Mohd Polygon testnet and can be interacted with using test Matic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Each web page has functionality to display the user’s owned NFTs as well as the NFTs they have put for sale on the marketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Using NextJS will also allow the Combination page to have functionality and smart contract use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' It is also worth noting that the created web page is not required to interact with the NFTrig smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 8 RECOMMENDATIONS FOR FUTURE WORK The goal for this project was a working Beta demo that shows application functionality, and correct smart contract execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' There are many other features planned for the continued work of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The first, as earlier explained, is a game option which challenges the user with trigonometry trivia and math problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Answering these questions successfully will increase the experience points of a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The user can then use these experience points to purchase individual or packs of NFTrig cards, or they can be used to combine cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' REFERENCES Rana M Amir Latif, Khalid Hussain, NZ Jhanjhi, Anand Nayyar, and Osama Rizwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' A remix IDE: smart contract-based framework for the healthcare sector by using Blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Multimedia Tools and Applications (2020), 1–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Mohsen Attaran and Angappa Gunasekaran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Blockchain for Gaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Applications of Blockchain Technology in Business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer, 85–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Rocsana Bucea-Manea-T, oniş, Oliva Martins, Radu Bucea-Manea-T, oniş, Cătălin Gheorghit,ă, Valentin Kuleto, Milena P Ilić, and Violeta-Elena Simion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Blockchain Technology Enhances Sustainable Higher Education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Sustainability 13, 22 (2021), 12347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Juan José Bullón, Ascensión Hernández Encinas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Jesús Santos Sánchez, and Víctor Gayoso Martínez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Analysis of student feedback when using gamification tools in math subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2018 IEEE Global Engineering Education Conference (EDUCON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 1818–1823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1109/EDUCON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='8363455 Guang Chen, Bing Xu, Manli Lu, and Nian-Shing Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Exploring blockchain technology and its potential applications for education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Smart Learning Environments 5, 1 (2018), 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Luisanna Cocco, Andrea Pinna, and Michele Marchesi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Banking on blockchain: Costs savings thanks to the blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Future internet 9, 3 (2017), 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Marco Conoscenti, Antonio Vetro, and Juan Carlos De Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Blockchain for the Internet of Things: A systematic literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2016 IEEE/ACS 13th International Conference of Computer Systems and Applications (AICCSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Oscar Delgado-Mohatar, Ruben Tolosana, Julian Fierrez, and Aythami Morales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Blockchain in the Internet of Things: Architectures and Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2020 IEEE 44th Annual Computers, Software, and Applications Conference (COMPSAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 1072–1077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1109/COMPSAC48688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='0-131 A Elmessiry, M Elmessiry, and L Bridgesmith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFT STUDENT TEACHER INCENTIVE SYSTEM (NFT-STIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Proceedings of EDULEARN21 Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 6th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Usef Faghihi, Albert Brautigam, Kris Jorgenson, David Martin, Angela Brown, Elizabeth Measures, and Sioui Maldonado- Bouchard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' How Gamification Applies for Educational Purpose Specially with College Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Procedia Computer Science 41 (2014), 182–187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='procs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='102 5th Annual International Conference on Biologically Inspired Cognitive Architectures, 2014 BICA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Julian Alberto Garcia-Garcia, Nicolás Sánchez-Gómez, David Lizcano, María José Escalona, and Tomás Wojdyński.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Using blockchain to improve collaborative business process management: Systematic literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' IEEE Access 8 (2020), 142312–142336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Susan Gass, Koen Van Gorp, and Paula Winke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Using different carrots: How incentivization affects proficiency testing outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Foreign Language Annals 52, 2 (2019), 216–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Ammar Yanuar Ghulam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Konseptual Desain Website Aplikasi Penyedia Jasa Kursus Mengemudi Mobil Di Purwokerto Menggunakan Framework Bootstrap 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Alexander Grech and Anthony F Camilleri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Blockchain in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Luxembourg: Publications Office of the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Marijn Haverbeke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Eloquent javascript: A modern introduction to programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' No Starch Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Marko Hölbl, Marko Kompara, Aida Kamišalić, and Lili Nemec Zlatolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' A systematic review of the use of blockchain in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Symmetry 10, 10 (2018), 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' NFTrig: Using Blockchain Technologies for Math Education 11 Carmen Holotescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Understanding blockchain opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Conference proceedings of» eLearning and Software for Education «(eLSE), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' ” Carol I” National Defence University Publishing House, 275–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Tomislav Jagušt, Ivica Botički, and Hyo-Jeong So.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Examining competitive, collaborative and adaptive gamification in young learners’ math learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Computers Education 125 (2018), 444–457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='compedu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='022 Bruce Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Professional visual studio 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Arnav Kapoor, Dipanwita Guhathakurta, Mehul Mathur, Rupanshu Yadav, Manish Gupta, and Ponnurungam Ku- maraguru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' TweetBoost: Influence of Social Media on NFT Valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='08373 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Parth Khandelwal, Rahul Johari, Varnika Gaur, and Dharm Vashisth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' BlockChain Technology based Smart Contract Agreement on REMIX IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2021 8th International Conference on Signal Processing and Integrated Networks (SPIN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 938–942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1109/SPIN52536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='9565983 Kristin Kostick-Quenet, Kenneth D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Mandl, Timo Minssen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Glenn Cohen, Urs Gasser, Isaac Kohane, and Amy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' McGuire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' How NFTs could transform health information exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Science 375, 6580 (2022), 500–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='abm2004 arXiv:https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='abm2004 Jörg Krause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Introduction to Bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Introducing Bootstrap 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer, 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Wei-Meng Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Using the metamask chrome extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Beginning Ethereum Smart Contracts Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer, 93–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Raoul LePage and Lynne Billard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Exploring the limits of bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Debajani Mohanty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Basic solidity programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Ethereum for Architects and Developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer, 55–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Haroon Shakirat Oluwatosin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Client-server model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' IOSRJ Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Eng 16, 1 (2014), 2278–8727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Deni Pramulia and Bayu Anggorojati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Implementation and evaluation of blockchain based e-voting system with Ethereum and Metamask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2020 International Conference on Informatics, Multimedia, Cyber and Information System (ICIMCIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 18–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1109/ICIMCIS51567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='9354310 R Raja and PC Nagasubramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Impact of modern technology in education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Journal of Applied and Advanced Research 3, 1 (2018), 33–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' A Ravishankar Rao and Riddhi Dave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Developing hands-on laboratory exercises for teaching STEM students the internet-of-things, cloud computing and blockchain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In 2019 IEEE Integrated STEM Education Conference (ISEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' IEEE, 191–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Diane J Skiba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' The potential of blockchain in education and health care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Nursing education perspectives 38, 4 (2017), 220–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Craig Standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Methodologies for developing Web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Information and Software Technology 44, 3 (2002), 151–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='1016/S0950-5849(02)00002-2 Qin Wang, Rujia Li, Qi Wang, and Shiping Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Non-fungible token (NFT): Overview, evaluation, opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' arXiv preprint arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content='07447 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Hafiza Yumna, Muhammad Murad Khan, Maria Ikram, and Sabahat Ilyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Use of Blockchain in Education: A Systematic Literature Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Intelligent Information and Database Systems, Ngoc Thanh Nguyen, Ford Lumban Gaol, Tzung-Pei Hong, and Bogdan Trawiński (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer International Publishing, Cham, 191–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Hafiza Yumna, Muhammad Murad Khan, Maria Ikram, and Sabahat Ilyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Use of blockchain in education: a systematic literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' In Asian Conference on Intelligent Information and Database Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} +page_content=' Springer, 191–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_test/content/2301.00001v1.pdf'} diff --git a/kb_test/content/tmp_files/test.pdf.txt b/kb_test/content/tmp_files/test.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..527b0ed3d8061ceec1036582f67eaeb2420eea80 --- /dev/null +++ b/kb_test/content/tmp_files/test.pdf.txt @@ -0,0 +1,940 @@ +Bandit approach to conflict-free multi-agent Q-learning in view of +photonic implementation +Hiroaki Shinkawa +1, *, Nicolas Chauvet +1, Andr´e R¨ohm +1, Takatomo Mihana +1, Ryoichi +Horisaki +1, Guillaume Bachelier +2, and Makoto Naruse +1 +1Department of Information Physics and Computing, Graduate School of Information Science and +Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. +2Univ. Grenoble Alpes, CNRS, Institut N´eel, 38000 Grenoble, France. +*Corresponding author. Email: gokukyukyoku555@gmail.com +Abstract +Recently, extensive studies on photonic reinforcement learning to accelerate the process of +calculation by exploiting the physical nature of light have been conducted. Previous studies +utilized quantum interference of photons to achieve collective decision-making without choice +conflicts when solving the competitive multi-armed bandit problem, a fundamental example of +reinforcement learning. However, the bandit problem deals with a static environment where +the agent’s action does not influence the reward probabilities. This study aims to extend the +conventional approach to a more general multi-agent reinforcement learning targeting the grid +world problem. Unlike the conventional approach, the proposed scheme deals with a dynamic +environment where the reward changes because of agents’ actions. A successful photonic re- +inforcement learning scheme requires both a photonic system that contributes to the quality +of learning and a suitable algorithm. This study proposes a novel learning algorithm, discon- +tinuous bandit Q-learning, in view of a potential photonic implementation. Here, state-action +pairs in the environment are regarded as slot machines in the context of the bandit problem and +an updated amount of Q-value is regarded as the reward of the bandit problem. We perform +numerical simulations to validate the effectiveness of the bandit algorithm. +In addition, we +propose a multi-agent architecture in which agents are indirectly connected through quantum +interference of light and quantum principles ensure the conflict-free property of state-action +pair selections among agents. We demonstrate that multi-agent reinforcement learning can be +accelerated owing to conflict avoidance among multiple agents. +1 +Introduction +Reinforcement learning is a machine learning technique that enables an agent to perform the desired +task through repeated trials and errors in a particular environment [1]. Methods implemented in +previous studies have yielded remarkable results, including victories over professional human players +in board games, such as Go [2, 3]. +Recently, photonic approaches to reinforcement learning to +outsource the computational costs and exploit the physical nature of light have been proposed [4–8]. +Previous studies solved the bandit problem, a fundamental reinforcement learning model, using +the quantum nature of photons [9–12]. The bandit problem is a frequently used model of human +decision-making [13]. Multiple slot machines probabilistically generate a reward and an agent at- +tempts to maximize the cumulative reward from the machines under the constraint that they can +only play one machine at a time [1, 14]. Because the agent lacks prior knowledge of the reward +1 +arXiv:2212.09926v1 [cs.AI] 20 Dec 2022 + +probabilities of the machines, they must play with various machines, including bad machines at that +time, in the early stages of the game to accurately estimate the reward probabilities. This results +from the stochastic nature of reward generation; that is, a machine should not be considered as +having a low reward probability just because it has not been generating many rewards at that time. +However, the agent would suffer a loss if they played bad machines excessively; therefore, they must +concentrate on the machines that have the highest reward probabilities in the latter stages of the +game. The former aspect is called exploration, whereas the latter is called exploitation; balancing +these two conflicting demands is the key to solving this problem [15]. The softmax rule is a model +that balances exploration and exploitation through probabilistic decision-making and is considered +as the model that best fits human decision-making [13]. +The quantum nature of photons can be applied to solve the bandit problem. In particular, by +mapping the selection of a machine to the observation of a photon’s state, probabilistic decision- +making can be implemented because the state observed is determined probabilistically [9]. Further- +more, the role of photons in decision-making becomes critical owing to entanglement and quantum +interference, which are inherent properties of quantum physics [10–12]. For example, consider a +situation in which two agents solve the bandit problem simultaneously but the selection of the same +machine reduces the total reward. This is analogous to a real-world situation when multiple peo- +ple or devices simultaneously connect to the same wireless channel, resulting in the degradation +of the individual communication speed [14, 16–18]. By observing the states of a two-photon pair +whose polarizations are entangled, the two agents can ensure that their choices always differ in such +circumstances. That is, entanglement avoids selection conflicts. +Chauvet et al. theoretically and experimentally showed that the competitive multi-armed bandit +problem, which deals with the aforementioned situation, can be resolved with no conflict of choices +by two agents faced with two machines using photon pairs whose polarizations are entangled [10, +11]. Their system is remarkable in that the agents can avoid selection conflicts without directly +communicating with each other about the machine to be selected because of quantum entanglement. +Furthermore, Amakasu et al. theoretically showed that the system could be extended to handle three +or more machines using quantum interference of orbital angular momentum of light [12]. Accordingly, +they developed a photonic system that ensured conflict-free selections by two agents with an arbitrary +number of machines. In addition, Shinkawa et al. formulated a problem in which people individually +have a probabilistic preference over options, derived the optimal joint decision-making in terms of +satisfaction [19], and demonstrated that a system based on quantum interference of photons can +provide a heuristic solution to this problem [20]. This is another example of the coordination of +multi-individual decision-making by a photonic system. +This study aims to demonstrate the potential of a photonic reinforcement learning scheme, which +requires the combination of a suitable algorithm and a photonic system that leverages the unique +physical nature of photons. Based on previous studies, a photonic system can be used to solve the +bandit problem, a simple reinforcement learning task. However, to tackle challenging problems, the +photonic system must be extended such that it can handle three or more agents, and the algorithm +must be modified accordingly. The environment in the bandit problem is static, whereas that in a +general reinforcement learning problem is generally dynamic. In particular, the environment (reward +2 + +probabilities) is independent of the agent’s action in the bandit problem. Conversely, in a general +reinforcement learning problem, the state of the environment changes because of the action, which +must be considered in the learning process. This study presents a modified algorithm that can solve +a broader class of reinforcement learning problems. While the extension of the photonic system with +more than two agents remains open and must be addressed in future studies, this study lays the +foundation for a photonic reinforcement learning scheme that can be implemented once the photonic +system is developed. +We consider the grid world problem as a dynamic environment [21]. It is a collection of cells +in which an agent can either implement an up, down, left, or right action. +Depending on the +combination of cells and actions, the agent receives different rewards from the environment. Because +different cells have different reward environments, the grid world is a dynamic environment. +While Q-learning is generally used as an algorithm for reinforcement learning [22–24], this study +proposed a combination of Q-learning with the bandit algorithm, called discontinuous bandit Q- +learning (DBQL). Although Q-learning aims to learn the optimal paths, this study aims to learn +the value of each state-action pair in the entire environment with high accuracy. Thus, suppose the +agent deviates from the optimal paths. In that case, it will accurately return to the optimal paths +from any location in the environment. +In the proposed DBQL method, each agent selects a state-action pair in the environment at +each time step and updates the corresponding Q-value (a detailed definition is given in Sec. 2). +Decisions on the state-action pair to be selected have a similar structure to the bandit problem +because the agent must balance two demands; the first is the demand for exploitation, which is to +update state-action pairs that are likely to have a large value of ∆Q (the change in the Q-value) for +the moment, to accelerate learning. The second is the demand for exploration to accurately estimate +the expected value of ∆Q for other state-action pairs that have not yet been visited frequently. Thus, +by considering the state-action pairs as machines and ∆Q as a reward, the accurate estimation of +Q-values for the entire environment can be viewed as a bandit problem. +Furthermore, we consider a case in which multiple agents participate in the learning and follow +DBQL simultaneously. We demonstrate the learning can be accelerated by avoiding the selection +of the same state-action pair simultaneously; that is, by forcing the agents to make conflict-free +decisions. As earlier mentioned, we have not conceived a photonic system that enables conflict-free +selections among more than two agents without direct communication yet. Accordingly, this study +algorithmically realized conflict-free selections, which essentially means forcing the agents to disclose +their selections. Once a photonic system with more than two agents is developed in the future, our +scheme will be implemented by the mixture of the photonic system and our proposed algorithm, +thus eliminating the necessity for the agents to share their selections. +The remainder of this paper is organized as follows. Sec. 2.1 describes the experimental envi- +ronment and the grid world. Section 2.2 explains Q-learning followed by a detailed description of +the proposed method DBQL. Section 2.3 provides the environment’s response when multiple agents +explore simultaneously, and Sec. 2.4 illustrates a selection conflict avoidance system using quantum +interference of photons. Section 3 demonstrates the result of performing an actual search in the grid +world using DBQL to quantify the impact of the bandit algorithm on learning and the impact of +3 + +avoiding selection conflicts. Finally, Sec. 4 discusses the results and future perspectives. +2 +Materials and Methods +2.1 +Experimental Design +The schematic of the grid world, which is often used as a model in previous studies on reinforcement +learning [21] is shown in Fig. +1. +An agent exists in the grid world and moves around in the +environment. +A +A’ +B +B’ ++10 ++5 +Figure 1: 5 × 5 grid world. The agent implements one of the four actions at each time step and +receives a reward and the next state. In special cells A and B, the reward is large and the agent +jumps to another cell. +In this example, the world is represented by a 5 × 5 cell grid, where each cell is called a “state.” +At each time step, the agent selects an “action” either up, down, left, or right. In the grid world, +when an agent is in a state st at a time step t, the chosen action at determines the reward rt and the +next state st+1, which is provided by the environment. In this study, we assumed the environment +is Markovian, meaning the next state st+1 is determined only by the current state st and action +taken at. For example, if an agent is in the top left corner cell and selects the action “right,” the +agent earns a specific reward and moves to cell A. Herein, the rule that determines the action to +be chosen by the agent in each state is called a “policy.” In this study, we confine the policy to be +deterministic. +Then, the “action-value function” Qπ(s, a) is determined for each state-action pair (s, a) when +the agent follows a particular policy π. This function represents the total future rewards when the +agent is currently in a state s and takes an action a followed by a series of actions that π instructs. +Qπ(s, a) = +∞ +� +t=0 +γtrt, +(1) +where γ is the time discount. The time discount is applied to reflect that distant-future rewards +matter less than near-future rewards and to ensure the convergence of the function. Note that a +4 + +larger γ means the agent is more concerned about a long-term benefit. If the reward is determined +stochastically, the expected value of the reward E[rt] is used instead of rt. +Suppose the grid world problem is fully known, meaning all the possible states, actions, and +rewards are known in advance. In that case, we can use dynamic programming algorithms, such as +value iteration or policy iteration, which solve the Bellman equation to derive the optimal action- +value function ˜Q(s, a) and policy ˜π(s), where +˜π(s) = argmax +a +˜Q(s, a) +(2) +is satisfied [25, 26]. This study aims to ensure that the agent without the knowledge of the envi- +ronment accurately learns the optimal action-value function ˜Q(s, a) for all state-action pairs (s, a) +using the information they obtain from the environment. The initial values are Q(s, a) = 0, and the +value of Q(s, a) during the learning is called “Q-value.” +2.2 +Discontinuous Bandit Q-Learning +Q-learning is generally used to solve the grid world. Algorithm 1 provides an overview of Q-learning +[23]. +Algorithm 1 Q-learning +1: Pick an initial state s0 +2: while t ≤ T do +3: +if rand() < ϵ then +4: +at ← random +5: +else +6: +at ← argmax +a +Q(st, a) +7: +end if +8: +Implement an action at and obtain a reward rt and the next state st+1. Then, update Q(st, at): +9: +Q(st, at) ← Q(st, at) + α · +� +rt + γ max +a′ Q(st+1, a′) − Q(st, at) +� +10: end while +First, the agent randomly chooses the initial state s0. The policy π is the ϵ-greedy method. +That is, the agent usually chooses an action with the largest Q(st, at); however, the action is chosen +uniformly at random with a probability ϵ. Notably, the possible actions are confined to the four +actions in the current cell, unlike in discontinuous Q-learning proposed later. Thus, the agent receives +a reward rt and the next state st+1 from the environment after executing the action at. Finally, it +updates Q(st, at) according to the following rule: +Q(st, at) ← Q(st, at) + α · +� +rt + γ max +a′ Q(st+1, a′) − Q(st, at) +� +, +(3) +where α is the learning rate and γ is the time discount. Note that if α = 0, nothing is learned; +5 + +however, if α = 1, all the previous experiences are forgotten and only the last experience is considered. +This process can be repeated to obtain Q-values as good approximations of the optimal action-value +functions ˜Q(s, a). +In this study, we first propose discontinuous Q-learning to interpret the original Q-learning +as a decision-making problem about what state-action pair (s, a) the agent selects at each time +step. Algorithm 2 provides an overview of discontinuous Q-learning. Unlike basic Q-learning, in +discontinuous Q-learning, the new state s′ obtained from the environment because of the action +at is ignored and a new state-action pair (st+1, at+1) is chosen from all the possible pairs in the +environment, which are not limited to pairs whose states are s′. Although the algorithm implies +that the agent “jumps” at every time step, this assumption is realistic in reinforcement learning. +The initial position is already often randomly chosen in the existing algorithms at the start of every +iteration (thus, the position “jumps” from the final position in the last iteration) in famous problems +such as the cart-pole problem [27] or the maze-solving problem [28]. +Algorithm 2 Discontinuous Q-learning +1: while t ≤ T do +2: +Select one state-action pair (st, at) based on specific criteria +3: +Implement an action at and obtain a reward rt and next state s′. Then, update Q(st, at): +4: +Q(st, at) ← Q(st, at) + α · +� +rt + γ max +a′ Q(s′, a′) − Q(st, at) +� +5: end while +Algorithm 2 shows that the state-action pair (st, at) updated by the agent at every time step +is determined based on “specific criteria.” This study demonstrates that the bandit algorithm can +function effectively as the selection criterion. +Now, ∆Q(s, a) is defined as the absolute value of the updated amount of Q(s, a) at each time +step: +∆Q(s, a) := +���α · +� +rt + γ max +a′ Q(s′, a′) − Q(st, at) +���� . +(4) +Larger ∆Q(s, a) means faster learning. Therefore, the agent should choose a state-action pair (s, a) +with a high expected value of ∆Q(s, a) to make the learning process more efficient. However, the +potential update ∆Q(s, a) for other state-action pairs (s, a) could be higher and will also vary as +the update proceeds. Thus, the agent cannot rely on just selecting the same state-action pair (s, a) +over and over. The agent also needs to explore other pairs. This structure is similar to that of the +bandit problem. +Therefore, by regarding each state-action pair (s, a) as a slot machine and the change in Q(s, a) +as the reward in the context of bandit problems for the discontinuous Q-learning algorithm, we can +associate the agent’s attempt to select the state-action pair (s, a) with large ∆Q as “exploitation” +and the investigation of ∆Q for other state-action pairs (s, a) as “exploration.” We thus define +discontinuous bandit Q-learning (DBQL) as an algorithm that follows discontinuous Q-learning in +which the bandit algorithm functions as the selection criterion. +In DBQL, the agent follows the softmax algorithm, a widely used algorithm to successfully solve +the bandit problem. The agent records ∆Q for each state-action pair (s, a). Let µt(s, a) be the +6 + +empirical mean of ∆Q(s, a) by time step t. The probability of the agent selecting the state-action +pair (si, aj) at the next time step t + 1 is calculated as follows: +pt+1(si, aj) = +eβ·µt(si,aj) +� +(s,a) +eβ·µt(s,a) , +(5) +where β controls the degree of exploration and exploitation. +2.3 +Multi-agent Learning +In this study, multiple agents participate in simultaneously updating the global lookup table of +Q(s, a) based on DBQL to accelerate the learning process as shown in Fig. +2. +A situation is +considered in which the agents share the global lookup table of Q(s, a) while individually recording +a separate table of ∆Q(s, a). At time step t, each agent refers to the ∆Q table it has recorded +and determines the state-action pair (st, at) to update based on the softmax algorithm in Eq. (5). +Next, it retrieves the value of Q(st, at) from the global lookup table, calculates the updated value +of Q(st, at) according to Eq. (3), and sends it back to the global table. +An important rule is that when two or more agents attempt to update the same state-action +pair (s, a) at the same time step, only one of the updates is randomly reflected to the global lookup +table. +This depends on the problem settings; however, real-world examples exist in which the +same investigation of state-action pairs by multiple agents is detrimental. For example, consider +an exploratory scenario where sonic waves are used to reveal the underlying stratigraphy of the +seafloor. Multiple agents simultaneously conducting the same location exploration would result in +interference and yield poor results. +Moreover, even if we were to allow simultaneous updates by multiple agents and calculate the +sum of ∆Q to reflect to the global table, this could disturb the convergence of Q-learning, because +taking the sum essentially means changing the learning rate α locally for this particular time step. +2.4 +Cooperative Decision-Making through Quantum Interference +This section explains how the quantum interference of photons can be leveraged such that mul- +tiple agents can avoid selecting the same state-action pair (s, a) at the same time step without +any direct knowledge of the selections of the other agents. As already mentioned, we have been +unable to extend the conventional cooperative decision-making system with two agents, and thus +the numerical demonstrations shown in Sec. 3 uses an algorithmic way to avoid selection conflicts. +Therefore, we will only cover the core concepts in this section and outline how in principle a photonic +implementation may function. +Amakasu et al. [12] proposed a conflict-free collective decision-making system with two agents +using the orbital angular momentum (OAM) of light. A photon can carry theoretically an infinite +7 + +Global lookup table +of +Each agent records + table individually +Look up +Send back updated +Agent 1 +Agent 2 +Agent +Agent +Selections of + are coordinated +through quantum interference of photons +Figure 2: Structure of the DBQL by multiple agents. Each agent looks up a Q-value from the +global lookup table, updates it using the generated reward and the next state provided by the +environment, and sends it back to the global table. In our scheme, agents are not directly connected +and cannot communicate with one another; yet, their state-action selections are coordinated owing +to the quantum interference of photons. However, we coordinated their selections in this study using +an algorithm because we could not extend the photonic system with two agents. +8 + +number of OAMs, and the state of the photon is described as a superposition of different OAMs |k⟩: +|Φ⟩ = +1 +√ +K +K +� +k=1 +eiφk| + k⟩. +(6) +Because of the quantum property of photons, the detection probability of each OAM is calculated +using the modulus square of the probability amplitude. In addition, the usage of attenuators enables +us to control the probability amplitudes, thus changing the observation probabilities. In their pro- +posed system, Amakasu et al. set K equal to the number of options (in our case, this is the number +of state-action pairs) and designed a protocol in which the agent selects the option whose index is +the same as the detected OAM number. For example, if an OAM of | + 1⟩ is detected by the first +agent, the agent selects the first option. This protocol enables probabilistic decision-making because +the control of the probability amplitudes by the attenuators results in the control of the selection +probabilities of the options. +The use of quantum physics makes a difference when two agents simultaneously make decisions +based on probability following the aforementioned protocol. +A quantum effect called the Hong- +Ou-Mandel interference exists, whereby different OAMs are always observed when a photon-pair +connected by this effect is observed by two detectors. Based on the protocol, the two agents always +select different options without informing each other of their selections; that is, conflict-free selections +are possible. The implementation of the Hong-Ou-Mandel interference is quite simple and can be +accomplished with only very basic optical instruments, such as spatial light modulators and beam +splitters, as shown in Fig. 3. +SLM +SLM +BS +BS +SLM : Spatial light modulator +BS : Beam splitter +Figure 3: Two-photon Hong-Ou-Mandel interference. |Φ⟩ and |Ψ⟩ represent the states of photons +that are controlled by spatial light modulators. +Although a detailed design is yet to be devised, it is likely that conflict-free probabilistic decision- +making can be realized with three or more agents by cascading multiple spatial light modulators +and beam splitters as well as appropriately configuring the input OAM states as an extension of +the previous setup. For example, the schematic of a photonic configuration with three photons is +shown in Fig. 4. This system eliminates selections completely in which all the agents select the same +option; however, selections in which only two select the same option still remain. Numerous studies +9 + +on quantum interference among multiple photons have been conducted, including Refs. [29–31], +thus successfully integrating these methods with the usage of OAMs has a guiding significance in +developing a photonic system that completely eliminates selection conflicts in the future. +BS +BS : Beam splitter +BS +Figure 4: Photonic configuration with three photons. |Φ⟩, |Ψ⟩, |Ξ⟩ represent the states of photons. +In our scheme, we assumed that the multi-photon conflict-free system can be realized and uti- +lized to coordinate the probabilistic decision-making of N agents through quantum interference of +photons regarding the choice of the state-action pairs (s, a). This enables the agents to prevent +selection conflicts without communicating with each other about the selection of pairs. Not only +is the learning accelerated because unnecessary updates are avoided, but also resources required to +exchange information about state-action pair selections can be reduced. Note that, simultaneous +updates by different agents will be a waste except for one of them as explained in Sec. 2.3. This +study realized conflict-free selections using an algorithm by numerically computing the joint selection +probabilities on a computer instead of using a photonic system. +3 +Results +A 5 × 5 grid world shown in Fig. 1 was considered in this study, and we analyzed the situation in +which multiple agents updates a state-action pair (s, a) at every time step according to discontinuous +Q-learning (Alg. 2). +3.1 +Rules in the Grid World +The state-action combination in the grid world determines the rewards received by an agent and the +cell to which it moves. The following settings are used in this study. +10 + +• When any action is taken from cell A, the chances of the cell generating a reward of +10 is +50% and the agent jumps to cell A’. If no reward is generated, it remains in cell A. +• When any action is taken from cell B, the chances of the cell generating a reward of +5 is 50% +and the agent jumps to cell B’. If no reward is generated, it remains in cell B. +• In any other cell, no reward is generated and the destination follows the action except when +the agent hits a wall. In such cases, a reward of −1 is generated and the agent remains on the +current cell. +3.2 +Objectives +Each agent selects a state-action pair (s, a) at each time step and updates Q(s, a) according to Alg. +2. In this study, we considered 10–100 agents with 100 state-action pairs (25 cells and four actions +in each cell). To quantify the learning accuracy, we defined the loss Lt as the average absolute error +between the true action-value function ˜Q(s, a) and Q-values learned at time step t, which we refer +to as Qt(s, a), over all the state-action pairs. +Lt = +1 +100 +� +(s,a) +| ˜Q(s, a) − Qt(s, a)|. +(7) +One hundred trials were performed and the average loss Lt over the trials was calculated as a metric +to measure the gap between the optimal action-value functions and the Q-values. The smaller the +average loss, the more successful the learning process. Based on the rules in Sec. 3.1, the ground +truth values of the action-value function ˜Q(s, a) are summarized in Fig. 5. +11.59 +11.59 +9.43 +9.43 +12.88 +12.88 +10.59 +10.43 +14.31 +14.31 +11.88 +11.59 +15.9 +15.9 +13.31 +12.88 +17.66 +14.9 +14.9 +14.31 +10.43 +12.88 +10.43 +10.59 +11.59 +14.31 +11.59 +11.59 +12.88 +15.9 +12.88 +12.88 +14.31 +17.66 +14.31 +14.31 +20.61 +20.61 +20.61 +20.61 +9.39 +11.59 +11.59 +9.43 +10.43 +12.88 +12.88 +10.43 +11.59 +14.31 +14.31 +11.59 +12.88 +15.9 +15.9 +12.88 +13.57 +14.9 +17.66 +14.31 +8.45 +10.43 +10.43 +8.39 +9.39 +11.59 +11.59 +9.39 +10.43 +12.88 +12.88 +10.43 +11.59 +13.57 +14.31 +11.59 +15.84 +15.84 +15.84 +15.84 +7.45 +9.39 +9.39 +7.45 +8.39 +10.43 +10.43 +8.45 +9.43 +11.59 +11.59 +9.39 +10.59 +12.22 +12.88 +10.43 +11.22 +11.22 +13.57 +11.59 +Figure 5: Ground truth values of the action-value function ˜Q(s, a). +Two major points were tested in this study: First, we tested whether the bandit algorithm +outperforms random selections in Alg. 2. That is, we compared the loss trajectories between DBQL +and the case in which the agents make decisions uniformly at random instead of using the softmax +algorithm as the selection criteria in Alg. 2. +11 + +Second, we considered the effect of conflict avoidance in the state-action pair selection on learning. +As noted in Sec. 2.3, if multiple agents simultaneously select the same state-action pair (s, a), only +one of their updates will be reflected in the global Q-table. Hence, the learning process will always +accelerate by avoiding selection conflicts. This study aims to quantify this effect. Furthermore, we +demonstrate the significance of conflict avoidance especially when using DBQL. The parameters are +as follows: the number of iterations T is 20000, learning rate α in Alg. 2 is initially set to 0.035 that +decays linearly to α = 0 at t = 20000, and the time discount γ is 0.9. β in Eq. 5, which controls the +degree of exploration and exploitation of the softmax algorithm used in DBQL, is initially set to 1.0 +and grows linearly to β = 5.0 at t = 20000 because more exploitation is necessary in the later stage +of learning. +3.3 +Performance Comparison +The average loss Lt, which quantifies the gap between the optimal action-value functions and the +Q-values during learning when the number of agents is 10, 50, or 90, are shown in Figs. 6 (a), (b), +and (c), respectively. Therein, the blue and orange curves represent the cases in which different +agents were allowed to simultaneously select the same state-action pair, denoted by “conflict.” The +procedure for state-action pair selections differs for the blue and orange curves. The blue curve +denoted by the legend “uniform random/conflict” is based on random selections, whereas the orange +curve denoted by “bandit/conflict” is based on the softmax algorithm in Eq. (5). +Similarly, the green and red lines represent the cases in which the state-action pair selections are +conducted in a “conflict-free” manner, as marked by the latter half of the legend. Agents were not +allowed to select the same state-action pair at the same time step. In addition, actions were selected +in a uniformly random manner in the green curve, whereas the red curve was based on a bandit- +based approach. Hence, the green and red curves were denoted as “uniform random/conflict-free” +and “bandit/conflict-free,” respectively. +First, we compared the random-based lines with the bandit-based lines to examine the effect +of the bandit algorithm. By comparing the blue and orange lines or the green and red lines, we +observed that the learning is faster when the agents follow the bandit algorithm. This validates the +effectiveness of DBQL, which considers the change in the Q-value (∆Q) as the reward of the bandit +problem. +As the number of agents approaches a hundred, the difference in performances between uniform +random/conflict-free and bandit/conflict-free narrows. This is because in situations where decision +overlaps are not allowed, the significance of each agent’s sensible selection is reduced if the number +of agents is sufficiently large. In particular, when the number of agents is one hundred, the two +performances are exactly the same, irrespective of the choice made by the agents because only one +agent is always assigned to each state-action pair. +Moreover, we analyzed the impact of conflict avoidance in state-action pair selections on learning. +While the results are rather obvious considering the environmental rules, learning is faster when the +selections are conflict-free. We defined Sunder as the area under the learning curve to quantify the +12 + +0 +5000 +10000 +15000 +20000 +Time step t +0 +5 +10 +15 +Gap with the optimal Q +Uniform random / Conflict +Bandit / Conflict +Uniform random / Conflict-free +Bandit / Conflict-free +(a) 10 agents +0 +5000 +10000 +15000 +20000 +Time step t +0 +5 +10 +15 +Gap with the optimal Q +Uniform random / Conflict +Bandit / Conflict +Uniform random / Conflict-free +Bandit / Conflict-free +(b) 50 agents +0 +5000 +10000 +15000 +20000 +Time step t +0 +5 +10 +15 +Gap with the optimal Q +Uniform random / Conflict +Bandit / Conflict +Uniform random / Conflict-free +Bandit / Conflict-free +(c) 90 agents +Figure 6: Comparison of the four learning methods. The average loss, representing the gap between +optimal action-value functions and the Q-values during learning, is shown for the four learning +methods. The learning methods are divided along two axes: whether the bandit algorithm is used +for selection criteria and whether selection conflicts are allowed among different agents. +13 + +learning efficiency across the entire learning process. +Sunder = +T +� +t=1 +Lt. +(8) +Table 1 lists the Sunder ratios of uniform random/conflict by uniform random/conflict-free and +bandit/conflict by bandit/conflict-free to quantify the benefit of the conflict avoidance in state-action +pair selections. +Table 1: Benefit of conflict avoidance in the selection of state-action pairs. Conflict avoidance is +crucial when the number of agents increases. +Number of agents +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Uniform random +1.06 +1.12 +1.19 +1.26 +1.32 +1.39 +1.46 +1.53 +1.59 +1.67 +Bandit +1.13 +1.26 +1.34 +1.4 +1.43 +1.47 +1.49 +1.52 +1.55 +1.56 +As the number of agents increases, the ratio also increases, indicating that conflict avoidance +provides a more significant benefit. This is because the probability of multiple agents selecting the +same state-action pair (s, a) increases and more selections are discarded when selection conflicts are +allowed, as only one of them is valid. +Thus, we defined Rvalid as the proportion of valid choices to quantitatively evaluate such effects. +The change in Rvalid for bandit/conflict as the number of agents changes is shown in Fig. 7. Rvalid +decreased as the number of agents increased, indicating that conflict avoidance is more significant +for an increased number of agents. When the number of agents is one hundred, approximately 60% +of the updates are wasted if the agents’ selections are not coordinated. +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Number of agents +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rvalid +Figure 7: Valid selection rate Rvalid. The larger the number of agents, the more frequently the +selections of the agents overlap, validating the significance of conflict avoidance. +Furthermore, comparing the convergence values in Fig. 6, bandit/conflict has a larger value than +random/conflict, indicating that uniform random/conflict outperforms bandit/conflict if only the +final accuracy is compared. This is because, in bandit/conflict, most agents decide to update cell +A as they proceed with learning; therefore, other cells are not updated, thus resulting in residual +action-value function errors for those cells. Figure 8 shows an example of the number of agents that +choose each state-action pair (s, a) in the final time step when the number of agents is a hundred for +14 + +the bandit/conflict case. Approximately 90% of the agents update cell A and most of the remaining +agents update cell B, which has the second highest reward. Therefore, ensuring the conflict-free +property is crucial to avoiding most agents getting “stuck” when using a bandit-derived algorithm. +Figure 8: Number of agents that choose each state-action pair (s, a) at the final time step for the +bandit-based algorithm when conflicts are allowed. Approximately 90% of the agents update cell A, +which generated the highest reward, while the other cells remain largely unexplored. +4 +Discussion +This study proposed a photonic reinforcement learning scheme, which required both a novel algo- +rithm and photonic system, and demonstrated its performance. We employed the grid world problem, +a frequently used model in reinforcement learning, with the aim to learn the optimal action-value +functions for all state-action pairs (s, a) with high precision, involving multiple agents. The details +presented in the study are summarized as follows. +First, we proposed DBQL, a learning algorithm in which each agent selected one of all the state- +action pairs (s, a) in the environment in each time step t and updated Q(s, a) based on the same +formula as in the original Q-learning. The decision-making problem of selecting a state-action pair +(s, a) was similar to the bandit problem. This is because if we define the amount of Q(s, a)-update +at each time step as ∆Q, the agent must strike a balance between the demand to select state-action +pairs (s, a) with large ∆Q to accelerate the learning process (“exploitation” in the context of the +bandit problem) and the demand to investigate the values of ∆Q for other state-action pairs (s, a) +(“exploration” in the bandit problem). We compared the case in which the bandit algorithm was +used as the decision-making criteria for Alg. 2 with that in which a uniform random selection was +used, to validate the effectiveness of DBQL. The former resulted in a faster learning process as +described in Sec. 3. +Second, we proposed a multi-agent architecture where multiple agents make conflict-free decisions +in the learning, and then quantitatively evaluated the impact of the conflict avoidance on learning. +As demonstrated in Sec. +3, the learning was indeed accelerated by avoiding selection conflicts +15 + +particularly when the number of agents increased. Moreover, cooperative decision-making is essential +when multiple agents follow DBQL to avoid getting stuck in the later stage of learning. Without +coordinating the selection, most agents would be highly likely to select the cell with largest reward +in the latter parts of the learning. Although the particular configuration of the system is not yet +established, a photonic system with cascaded spatial light modulators and beam splitters is expected +to enable cooperative decision-making by three or more agents for avoiding selection conflicts. Once +this system is conceived in the future, it can be incorporated into our proposed scheme and obviate +the necessity for the agents to communicate with each other to coordinate their selections. +While Amakasu et al. provided the concept of the base idea that addressed the competitive +bandit problem, this study addresses a general reinforcement learning problem with the grid world +problem as an example. The two problems differ in that, in the bandit problem, the machines’ reward +probabilities are invariant regardless of the agent’s action; however, in the grid world problem, state +transitions, which correspond to the changes in the reward probabilities in the context of the bandit +problem, occur because of the agent’s action. Our proposed scheme applies to such challenging +problems in a dynamic environment. +Next, we discuss some of the limitations of this study and how they may be addressed in the +future. First, in DBQL, the agent’s actions are discontinuous. This can be resolved by restricting +the possible state-action pairs (s, a) that can be chosen by the agent at each time step to those in +the current cell. However, with this method, if more than four agents end up in a particular cell, at +least two will have to choose the same state-action pair (s, a) in the next time step. This requires +rule making for exception handling. Second, when the number of state-action pairs is sufficiently +larger than the number of agents, conflicts of choice occur less frequently, and the advantage of +conflict avoidance by quantum interference may be reduced. Regarding this concern, as indicated +in the birthday paradox, the probability that the choices of two agents overlap is greater than our +intuition even if the number of agents is such smaller than that of pairs. For example, suppose 100 +state-action pairs exist and 10 agents are to make uniform choices at random, the probability that +at least two agents make the same choice is over 37%. Furthermore, as mentioned in Sec. 3, conflict +avoidance is essential in DBQL because the probability of multiple agents intending to make the +same choice increases significantly as learning proceeds. +In the future, our first priority is to design a system that allows conflict-free decision-making +by three or more agents. Additionally, we would like to develop algorithms that allow agents to +take continuous actions and apply DBQL to other reinforcement learning problems that are more +complex than the grid world. +To the best of our knowledge, this study is the first to connect +the notion of photonic cooperative decision-making with Q-learning and apply it to a dynamic +environment. We believe this study makes a valuable contribution to the field of decision-making +using physical processes. +Data Availability +Data used in this study are available from the corresponding author upon request. +16 + +Conflicts of Interest +The authors declare that there are no conflicts of interest regarding the publication of this paper. +Acknowledgments +This study was supported in part by the CREST project (JPMJCR17N2) funded by the Japan +Science and Technology Agency, Grants-in-Aid for Scientific Research (JP20H00233) and Trans- +formative Research Areas (A) (JP22H05197) funded by the Japan Society for the Promotion of +Science. AR was funded by the Japan Society for the Promotion of Science as an International +Research Fellow. +References +[1] +R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction. MIT press, 2018. doi: +10.1109/TNN.1998.712192. +[2] +D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, +I. Antonoglou, V. Panneershelvam, M. Lanctot, et al., “Mastering the game of go with deep +neural networks and tree search,” Nature, vol. 529, no. 7587, pp. 484–489, 2016. doi: 10.1038/ +nature16961. +[3] +D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, +D. Kumaran, T. Graepel, et al., “A general reinforcement learning algorithm that masters +chess, shogi, and go through self-play,” Science, vol. 362, no. 6419, pp. 1140–1144, 2018. doi: +10.1126/science.aar6404. +[4] +F. Flamini, A. Hamann, S. Jerbi, L. M. Trenkwalder, H. P. Nautrup, and H. J. Briegel, +“Photonic architecture for reinforcement learning,” New Journal of Physics, vol. 22, no. 4, +p. 045 002, 2020. doi: 10.1088/1367-2630/ab783c. +[5] +G. R. Steinbrecher, J. P. Olson, D. Englund, and J. Carolan, “Quantum optical neural net- +works,” npj Quantum Information, vol. 5, no. 1, p. 60, 2019. doi: 10.1038/s41534-019- +0174-7. +[6] +V. Saggio, B. E. Asenbeck, A. Hamann, T. Str¨omberg, P. Schiansky, V. Dunjko, N. Friis, N. C. +Harris, M. Hochberg, D. Englund, et al., “Experimental quantum speed-up in reinforcement +learning agents,” Nature, vol. 591, no. 7849, pp. 229–233, 2021. doi: 10.1038/s41586-021- +03242-7. +[7] +M. Bukov, A. G. Day, D. Sels, P. Weinberg, A. Polkovnikov, and P. Mehta, “Reinforcement +learning in different phases of quantum control,” Physical Review X, vol. 8, no. 3, p. 031 086, +2018. doi: 10.1103/PhysRevX.8.031086. +[8] +J. Bueno, S. Maktoobi, L. Froehly, I. Fischer, M. Jacquot, L. Larger, and D. Brunner, “Rein- +forcement learning in a large-scale photonic recurrent neural network,” Optica, vol. 5, no. 6, +pp. 756–760, 2018. doi: 10.1364/OPTICA.5.000756. +17 + +[9] +M. Naruse, M. Berthel, A. Drezet, S. Huant, M. Aono, H. Hori, and S.-J. Kim, “Single-photon +decision maker,” Scientific Reports, vol. 5, no. 1, pp. 1–9, 2015. doi: 10.1038/srep13253. +[10] +N. Chauvet, D. Jegouso, B. Boulanger, H. Saigo, K. Okamura, H. Hori, A. Drezet, S. Huant, +G. Bachelier, and M. Naruse, “Entangled-photon decision maker,” Scientific Reports, vol. 9, +no. 1, p. 4832, 2019. doi: 10.1038/s41598-019-48647-7. +[11] +N. Chauvet, G. Bachelier, S. Huant, H. Saigo, H. Hori, and M. Naruse, “Entangled n-photon +states for fair and optimal social decision making,” Scientific Reports, vol. 10, no. 1, p. 20 420, +2020. doi: 10.1038/s41598-020-77340-3. +[12] +T. Amakasu, N. Chauvet, G. Bachelier, S. Huant, R. Horisaki, and M. Naruse, “Conflict- +free collective stochastic decision making by orbital angular momentum of photons through +quantum interference,” Scientific Reports, vol. 11, no. 1, p. 21 117, 2021. doi: 10.1038/s41598- +021-00493-2. +[13] +N. D. Daw, J. P. O’doherty, P. Dayan, B. Seymour, and R. J. Dolan, “Cortical substrates +for exploratory decisions in humans,” Nature, vol. 441, no. 7095, pp. 876–879, 2006. doi: +10.1038/nature04766. +[14] +S. Maghsudi and E. Hossain, “Multi-armed bandits with application to 5g small cells,” IEEE +Wireless Communications, vol. 23, no. 3, pp. 64–73, 2016. doi: 10.1109/MWC.2016.7498076. +[15] +J. G. March, “Exploration and exploitation in organizational learning,” Organization Science, +vol. 2, no. 1, pp. 71–87, 1991. doi: 10.1287/orsc.2.1.71. +[16] +L. Lai, H. El Gamal, H. Jiang, and H. V. Poor, “Cognitive medium access: Exploration, ex- +ploitation, and competition,” IEEE Transactions on Mobile Computing, vol. 10, no. 2, pp. 239– +253, 2010. doi: 10.1109/TMC.2010.65. +[17] +S.-J. Kim, M. Naruse, and M. Aono, “Harnessing the computational power of fluids for op- +timization of collective decision making,” Philosophies, vol. 1, no. 3, pp. 245–260, 2016. doi: +10.3390/philosophies1030245. +[18] +L. Besson and E. Kaufmann, “Multi-player bandits revisited,” in Algorithmic Learning The- +ory, PMLR, 2018, pp. 56–92. [Online]. Available: https://proceedings.mlr.press/v83/ +besson18a.html. +[19] +H. Shinkawa, N. Chauvet, G. Bachelier, A. R¨ohm, R. Horisaki, and M. Naruse, “Optimal +preference satisfaction for conflict-free joint decisions,” arXiv preprint arXiv:2205.00799, 2022. +doi: 10.48550/arXiv.2205.00799. +[20] +H. Shinkawa, N. Chauvet, A. R¨oohm, T. Mihana, R. Horisaki, G. Bachelier, and M. Naruse, +“Conflict-free joint sampling for preference satisfaction through quantum interference,” Physi- +cal Review Applied, vol. 18, no. 6, p. 064 018, 2022. doi: 10.1103/PhysRevApplied.18.064018. +[21] +R. S. Sutton, “Integrated architectures for learning, planning, and reacting based on ap- +proximating dynamic programming,” in Machine Learning Proceedings 1990, Elsevier, 1990, +pp. 216–224. doi: 10.1016/B978-1-55860-141-3.50030-4. +[22] +C. J. C. H. Watkins, “Learning from delayed rewards,” 1989. +18 + +[23] +C. J. Watkins and P. Dayan, “Q-learning,” Machine learning, vol. 8, no. 3, pp. 279–292, 1992. +doi: 10.1007/BF00992698. +[24] +B. Jang, M. Kim, G. Harerimana, and J. W. Kim, “Q-learning algorithms: A comprehensive +classification and applications,” IEEE Access, vol. 7, pp. 133 653–133 667, 2019. doi: 10.1109/ +ACCESS.2019.2941229. +[25] +R. Bellman, “Dynamic programming,” Science, vol. 153, no. 3731, pp. 34–37, 1966. doi: 10. +1126/science.153.3731.34. +[26] +——, “The theory of dynamic programming,” Bulletin of the American Mathematical Society, +vol. 60, no. 6, pp. 503–515, 1954. doi: 10.1090/S0002-9904-1954-09848-8. +[27] +A. G. Barto, R. S. Sutton, and C. W. Anderson, “Neuronlike adaptive elements that can solve +difficult learning control problems,” IEEE Transactions on Systems, Man, and Cybernetics, +no. 5, pp. 834–846, 1983. doi: 10.1109/TSMC.1983.6313077. +[28] +A. L. Samuel, “Some studies in machine learning using the game of checkers,” IBM Journal of +Research and Development, vol. 44, no. 1.2, pp. 206–226, 2000. doi: 10.1147/rd.441.0206. +[29] +M. ˙Zukowski, A. Zeilinger, and M. A. Horne, “Realizable higher-dimensional two-particle en- +tanglements via multiport beam splitters,” Physical Review A, vol. 55, no. 4, p. 2564, 1997. +doi: 10.1103/PhysRevA.55.2564. +[30] +R. A. Campos, “Three-photon hong-ou-mandel interference at a multiport mixer,” Physical +Review A, vol. 62, no. 1, p. 013 809, 2000. doi: 10.1103/PhysRevA.62.013809. +[31] +M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. De Guise, R. Heilmann, S. Nolte, A. +Szameit, and P. Walther, “Generalized multiphoton quantum interference,” Physical Review +X, vol. 5, no. 4, p. 041 015, 2015. doi: 10.1103/PhysRevX.5.041015. +19 + diff --git a/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/2301.01158v1.pdf.txt b/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/2301.01158v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..afd9e4b02bbbec7282b9296849f7f58059d66f12 --- /dev/null +++ b/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/2301.01158v1.pdf.txt @@ -0,0 +1,790 @@ +arXiv:2301.01158v1 [math.NT] 3 Jan 2023 +Values of E-functions are not Liouville numbers +S. Fischler and T. Rivoal +January 3, 2023 +Abstract +Shidlovskii has given a linear independence measure of values of E-functions with +rational Taylor coefficients at a rational point not a singularity of the underlying +differential system satisfied by these E-functions. His measure holds as well for E- +functions with coefficients in an imaginary quadratic field, but not for other number +fields. Recently, Beukers has proved a remarkable qualitative linear independence +theorem for the values at an algebraic point of E-functions with arbitrary algebraic +Taylor coefficients. But no quantitative analogue of Shidlovskii’s measure has been +given in Beukers’ setting. The goal of this paper it to obtain such a measure, in an +even more general setting where the point can be a singularity. This enables us to +solve a long standing problem: the value of an E-function at an algebraic point is +never a Liouville number, a result which had been obtained before only under addi- +tional assumptions. We deduce various explicit irrationality measures, in particular +for values of the exponential and Bessel’s J0 function at non-zero algebraic points. +We also prove that the values at rational points of E-functions with rational Taylor +coefficients are linearly independent over Q if and only if they are linearly indepen- +dent over Q. Our methods rest upon improvements of results recently obtained by +Andr´e and Beukers by means of the theory of E-operators. +1 +Introduction +Siegel [14] defined in 1929 the class of E-functions in order to generalize the Diophantine +properties of the exponential function (namely the Lindemann-Weierstrass Theorem) to +other special functions such as Bessel’s function J0(z) := �∞ +n=0(−1)n(z/2)2n/n!2 or hyper- +geometric series pFp with rational parameters. A power series �∞ +n=0 +an +n! zn ∈ Q[[z]] is said +to be an E-function when it is solution of a linear differential equation over Q(z) (i.e., +holonomic), and |σ(an)| (for any σ ∈ Gal(Q/Q)) and the least common denominator of +a0, a1, . . . , an all grow at most exponentially in n. Note that Siegel’s original definition of +E-functions is more general: see the end of this introduction. Throughout this paper we +fix an embedding of Q in C. +A lot of important qualitative results are known on the arithmetic nature of the values +taken by E-functions at algebraic points, amongst which we cite the celebrated Siegel- +Shidlovskii Theorem. It has been improved in [11]. However, these results are not always +1 + +strong enough, in particular they do not imply the linear independence of values of E- +functions solutions of a differential system of order 1 and evaluated at a non-singular +point. Quantitative versions of certain of these results exist, but only under very strict +assumptions on algebraic independence or rationality of the coefficients of the E-functions. +The main result in this direction, in the setting of linear independence, is the following +one. It is due to Shidlovskii [13, p. 358, Theorem 1, Eq. (32)]. +Theorem A (Shidlovskii). Let f = t(f1, . . . , fN) ∈ Q[[z]]N be a vector of E-functions +solution of a differential system f ′ = Af for some A ∈ MN(Q(z)). Assume that f1, . . . , fN +are linearly independent over Q(z) and that z0 ∈ Q∗ is not a pole of an entry of A. Then +for any ε > 0, there exists c = c(ε, z0, f1, . . . , fN) > 0 such that for all λ1, . . . , λN ∈ Z not +all zero, we have +���� +N +� +j=1 +λjfj(z0) +���� > cH−N+1−ε +where H := max +1≤j≤N |λj|. +This theorem holds verbatim with Q replaced by an imaginary quadratic number field +and Z replaced by its ring of integers. However no such result is known for other number +fields K. The point is that all known quantitative results are based on the Siegel-Shidlovskii +method only, which provides linear independence of the full set of the values of E-functions +in Theorem A only when K is either Q or imaginary quadratic. Even the qualitative part +of Theorem A (namely, �N +j=1 λjfj(z0) ̸= 0) has been proved only recently by Beukers [4, +Corollary 1.4] for arbitrary number fields, using Andr´e’s theory of E-operators [1]. +Theorem B (Beukers). Let f = t(f1, . . . , fN) ∈ Q[[z]]N be a vector of E-functions solution +of a differential system f ′ = Af for some A ∈ MN(Q(z)). Assume that f1, . . . , fN are line- +arly independent over Q(z) and that z0 ∈ Q +∗ is not a pole of an entry of A. Then the +numbers f1(z0), . . . , fN(z0) are linearly independent over Q. +The purpose of this paper is to prove new Diophantine results using this approach of +Andr´e and Beukers. Our first main result is the following theorem, where we generalize +Theorem A to any E-functions, by removing the rationality assumption on the coefficients +and also the non-singularity assumption on z0. We recall that for a non-zero algebraic +number α, its house α is the maximum of the moduli of α and of all its Galois conjugates +over Q. We also denote by OK the ring of integers of a number field K. +Theorem 1. Let K be a number field of degree d over Q, z0 ∈ K, and f = t(f1, . . . , fN) be +a vector of E-functions with coefficients in K such that f ′ = Af for some A ∈ MN(K(z)). +Then for any ε > 0, there exists c = c(ε, K, z0, f1, . . . , fN) > 0 with the following property. +For any λ1, . . . , λN ∈ OK, if Λ := λ1f1(z0) + . . . + λNfN(z0) is non-zero, then +|Λ| > cH−dd+1Nd+1−ε +where H := max +1≤j≤N λj . +2 + +Remarks. – Given arbitrary E-functions f1, . . . , fN and any z0 ∈ Q, this theorem applies +because there exists a number field K containing z0 and all coefficients of f1, . . . , fN, and +the family (f1, . . . , fN) can always be enlarged to satisfy a first-order differential system. +This shows, without any assumption on f1, . . . , fN and z0, the existence of κ, c > 0 such +that |Λ| > cH−κ provided Λ ̸= 0. +– Note that in Theorem 1, and even in the previous remark, we do not assume that +f1(z0), . . . , fN(z0) are linearly independent over K, and we do not wonder whether z0 is +a singularity or not of a differential system involving the fj. +Therefore with K = Q, +Theorem 1 is a generalization of Theorem A: we obtain the same lower bound under +milder assumptions (using Beukers’ results in [4] and improvements of the latter). +– We shall prove that if λ1, . . . , λN ∈ Z, the lower bound in Theorem 1 can be refined +to cH−dNd+1−ε. +– To our knowledge, Theorem 1 provides the first quantitative version of Beukers’ The- +orem B, when it is further assumed in Theorem 1 that f1, . . . , fN are linearly independent +over K(z) and that z0 ∈ K∗ is not a pole of A, ensuring that Λ ̸= 0. +An important consequence of Theorem 1 is the following result, which completely settles +the problem of deciding whether (real) values of E-functions can be Liouville numbers or +not. +We recall that a Liouville number is a real number ξ such that there exist two +sequences of rational integers pn, qn such that 0 < |qnξ −pn| < 1/qn +n for all sufficiently large +integers n (a fortiori pnqn ̸= 0). +Corollary 1. Let f be an E-function, and z0 be an algebraic number. Then f(z0) is not +a Liouville number. +The proof of Corollary 1 runs as follows: in Theorem 1, let z0 ∈ Q, take f1 := 1, f2 := f +and consider a vector t(f1, f2 . . . , fN) of E-functions with coefficients in a number field K +such that f ′ = Af for some A ∈ MN(K(z)), where K is large enough to contain z0. If +f(z0) ∈ Q, then f(z0) is not a Liouville number. If f(z0) /∈ Q, then λ1 + λ2f(z0) ̸= 0 for +all λ1, λ2 ∈ Z not both zero, so that Theorem 1 yields |λ1 + λ2f(z0)| > c max(|λ1|, |λ2|)−κ +for some c, κ > 0. This rules out the possibility that f(z0) is a Liouville number. +Of course Corollary 1 is interesting only when f(z0) ∈ R. If we do not assume this, +note however that the real and imaginary parts of f(z0) are values of E-functions (see the +remark in §2.1 below), so that none of them is a Liouville number. +Let us also mention another interesting corollary, which is a consequence of the third +remark that follows Theorem 1 with N = 2 and N = 3 respectively (recall that J0(z) is +solution of zy′′(z) + y′(z) + zy(z) = 0). +Corollary 2. For any algebraic number α ∈ Q +∗ of degree d over Q and any ε > 0, there +exists c = c(α, ε) such that, for all (p, q) ∈ Z × N with q ̸= 0, +���eα − p +q +��� > +c +qd2d+ε, +respectively +���J0(α) − p +q +��� > +c +qd3d+ε. +A general transcendence measure for E-functions due to Lang and Galochkin [6, p. 238, +Theorem 5.29 and remarks] (applied with m = 1 and m = 2 respectively) gives 4d2 + 1 +3 + +instead of d2d, and 16d3 + 1 instead of d3d; see also [13, p. 403]. This is of course much +better than Corollary 2 for large d but our bounds turn out to be smaller for d ∈ {1, 2, 3, 4} +and d ∈ {1, 2, 3, 4, 5} respectively. +A less classical example is the following: for any α ∈ Q +∗ and any integers p, q ≥ +1, the value at z = α of the function Ap,q(z) := �∞ +n=0(�n +k=0 +�n +k +�p�n+k +n +�q)zn/n! is not +a Liouville number; Ap,q(α) is proved to be a transcendental number in [5, §4.6] when +(p, q) ∈ {1, 2, 3, 4}2, the situation in general being unknown. More specifically, it is also +proved in [5, §4.6, Table 1] that when (p, q) = (2, 2), the minimal inhomogeneous differential +equation over Q(z) satisfied by A2,2 is of order 4 and 0 is its only singularity. Hence, the +following holds by the third remark that follows Theorem 1 with N = 5: for all α ∈ Q +∗ of +degree d over Q and all ε > 0, there exists c = c(ε, α) > 0 such that for all λj ∈ Z not all +0, we have +���λ4 + +3 +� +j=0 +λjA(j) +2,2(α) +��� > c max +0≤j≤4 |λj|−d5d+1−ε. +In particular, for any α ∈ Q +∗ of degree d and ε > 0, there exists c = c(ε, α) > 0 such that +for any (p, q) ∈ Z × N with q ̸= 0, +���A2,2(α) − p +q +��� > +c +qd5d+ε. +(1.1) +It seems reasonable to make the following conjecture, which probably belongs to folk- +lore. This conjecture is Roth’s Theorem if f(z0) is algebraic. +Conjecture 1. Let f be an E-function and z0 ∈ Q. For any ε > 0, there exists c > 0 such +that for any (p, q) ∈ Z × N with q ̸= 0, either qf(z0) − p = 0 or +���f(z0) − p +q +��� > +c +q2+ε. +Zudilin has proved this conjecture in [15] in a stronger form but under additional +assumptions (which even imply f(z0) /∈ Q), namely: f is an E-function with rational +coefficients, z0 ∈ Q∗ is not a singularity of a differential system satisfied by 1, f, f2, . . . , fN +over Q(z) (for some N ≥ 2) and f, f2, . . . , fN are algebraically independent over Q(z). +It would be interesting to know if A2,2, A′ +2,2, A′′ +2,2, A′′′ +2,2 are algebraically independent over +Q(z), in which case the exponent 5 could be improved to 2 in (1.1) when α ∈ Q∗. +The second goal of this paper is to understand the structure of the ring E of all values +at algebraic points of E-functions; this algebraic point can always be assumed to be 1 +because if f(z) is an E-function, so is f(αz) for any α ∈ Q. Elements of E are related to +exponential periods (see [9, §4.3]). +For any subfield K of Q, we shall also consider the subring EK of E which consists of +the evaluations f(1) where f is an E-function with coefficients in K (the number 1 could +be replaced by any non-zero element of K without changing EK). Note that E is the union +of all EK, where K is a number field, since for any E-function f the holonomy property +implies the existence of a number field that contains all coefficients of f. +4 + +We have defined and studied [7] analogous rings GK with G-functions instead of E- +functions; it turns out that GK is nearly independent from K (precisely, GK = GQ if +K ⊂ R, and GK = GQ(i) otherwise). The situation is completely different for E-functions. +A first hint in this direction was given in [8, Theorem 4] (stated as Lemma 1 in §4 below). +The way EK depends on K is completely described by the following result. +Theorem 2. Elements of EQ are linearly independent over Q if, and only if, they are +linearly independent over Q. In other words, the Q-algebras EQ and Q are linearly disjoint, +and the natural map EQ ⊗Q Q → E (sending ξ ⊗ z to ξz) is a Q-algebra isomorphism. +We refer to [3, Chapter V, §2, No. 5] for the definition and properties of linearly disjoint +algebras, which imply the following. +Corollary 3. Let K be a number field, and (ω1, . . . , ωd) be a basis of the Q-vector space +K. Then +EK = ω1EQ ⊕ . . . ⊕ ωdEQ. +In other words, for any ξ ∈ EK there exists a unique d-tuple (ξ1, . . . , ξd) ∈ Ed +Q such that +ξ = ω1ξ1 + . . . + ωdξd. +Theorems 1 and 2 may seem disjoint at first sight but their proofs share many com- +mon aspects, in particular both use Proposition 1 stated in §2.1. Another consequence of +Proposition 1 is the existence of an action of Gal(Q/Q) on E, of which the fixed points +are exactly the elements of EQ. For instance, with the notation of Corollary 3, we have +σ(ξ) = σ(ω1)ξ1 + . . . + σ(ωd)ξd for any σ ∈ Gal(Q/Q). As a first application of this Galois +action, we explain in §6 why our proof of Theorem 1 is similar to Liouville’s proof that +Liouville numbers are transcendental. This makes sense because Theorem 1 is a gener- +alization of this result, since Q ⊂ E. We hope this action can have other Diophantine +applications. +The original definition of E-functions, given by Siegel [14], is slightly less restrictive: +instead of geometric bounds, he allowed growths bounded by n!ε (for any given ε > 0, +provided n is large enough with respect to ε). +Shidlovskii’s Theorem A holds for E- +functions in Siegel’s sense, and Beukers’ Theorem B was later proved by Andr´e [2] in +this general setting, by a different method. All other results in Beukers’ paper [4] have +been adapted by Lepetit [10]. Therefore all the results of the present paper also hold for +E-functions in Siegel’s sense. +The structure of this paper is as follows. In §2 we prove the main tool of this paper, +namely Proposition 1, and use it to obtain a version of Beukers’ desingularization process +over a number field. This enables us to prove Theorem 1 in §3, and also to obtain in §4 +a decomposition of an E-function over a number field, involving an E-function that takes +only transcendental values at non-zero algebraic points. At last we apply the previous +results in §5 to study the structure of EK and prove Theorem 2. We conclude in §6 with +the above-mentioned Galois action on values of E-functions. +5 + +2 +Main tools +2.1 +Conjugates of E-functions +Let f(z) = �∞ +n=0 anzn be an E-function with coefficients an ∈ Q. For any σ ∈ Gal(Q/Q) +we let f σ(z) = �∞ +n=0 σ(an)zn. Then f σ is also an E-function, and if g is an E-function +then for any σ, τ we have (f + g)σ = f σ + gσ, (fg)σ = f σgσ and (f σ)τ = f τ◦σ. Moreover if +f has coefficients in a number field K, then f σ has coefficients in the number field σ(K). +Remark. Denoting by σ the complex conjugation, for any E-function f we can consider +1 +2(f + f σ) and +1 +2i(f − f σ). These E-functions have real coefficients, which are respectively +the real and imaginary parts of those of f. In particular, the real and imaginary parts of +any element of E belong to E. +The following result is central in the present paper; we refer to [5, Proposition 3.5] for +a similar result. Throughout this paper we agree that minimal polynomials of algebraic +elements have leading coefficient 1. +Proposition 1. Let f be an E-function with coefficients in a number field K, and z0 ∈ Q +∗. +Then the following assertions are equivalent: +(i) f vanishes at z0. +(ii) There exists σ ∈ Gal(Q/Q) such that f σ vanishes at σ(z0). +(iii) For any σ ∈ Gal(Q/Q), f σ vanishes at σ(z0). +(iv) There exists an E-function g with coefficients in K such that +f(z) = D(z)g(z) where D is the minimal polynomial of z0 over K. +In particular, if z0 is rational and f vanishes at z0, then all conjugates f σ of f also +vanish at z0. Also, if an E-function f with rational coefficients vanishes at some z0 ∈ Q +∗, +then it vanishes at all Galois conjugates of z0. +We remark that with z0 = 1, the implication (i) ⇒ (iii) is used already in the proof of +[4, Proposition 4.1], which is the main result Proposition 1 is based on. +Proof of Proposition 1. (iv) ⇒ (iii) Let σ ∈ Gal(Q/Q). Then f σ(z) = Dσ(z)gσ(z), and +Dσ(σ(z0)) = σ(D(z0)) = 0. Therefore f σ(σ(z0)) = 0. +(iii) ⇒ (ii) is trivial. +(ii) ⇒ (i) Enlarging K if necessary, we may assume the extension K/Q to be Galois and +to contain z0. Then f σ has coefficients in K, and σ(z0) ∈ K∗. Using [4, Proposition 4.1] +there exists an E-function g such that f σ(z) = (z − σ(z0))g(z). Then g has coefficients in +K; applying σ−1 yields f(z) = (z − z0)gσ−1(z) so that f(z0) = 0. +6 + +(i) ⇒ (iv) Using [4, Proposition 4.1] there exists an E-function h such that f(z) = +(z − z0)h(z). +Let σ ∈ Gal(Q/K), that is: σ is a field automorphism of Q such that +σ(x) = x for any x ∈ K. Then f(z) = f σ(z) = (z −σ(z0))hσ(z) so that f vanishes at σ(z0). +Let z1 := z0, z2, . . . , zℓ denote the (pairwise distinct) Galois conjugates of z0 over K, i.e. +the elements of the form σ(z0) with σ ∈ Gal(Q/K); we have proved that f vanishes at z1, +. . . , zℓ. Applying [4, Proposition 4.1] yields, by induction on j ∈ {0, . . . , ℓ}, the existence +of an E-function gj such that f(z) = gj(z) �j +i=1(z − zi). Since D(z) = �ℓ +i=1(z − zi), we +have f(z) = D(z)gℓ(z). Now D(z) ∈ K[z] \ {0} so that all coefficients of gℓ belong to K. +This concludes the proof of Proposition 1. +2.2 +Beukers’ desingularization process +In the proof of Theorem 1 we shall use the following version of Beukers’ desingularization +theorem (namely [4, Theorem 1.5]). The new point is that e1, . . . , eN and the coefficients +of B and M have coefficients in the number field K (whereas in [4, Theorem 1.5] these +coefficients are simply algebraic numbers). +Proposition 2. Let K be a number field, and f1, . . . , fN be E-functions with coefficients +in K, linearly independent over C(z). Assume that the vector f = t(f1, . . . , fN) satisfies a +first-order differential system f ′ = Af with A ∈ Mn(K(z)). +Then there exist E-functions e1, . . . , eN with coefficients in K, linearly independent +over C(z), a matrix B ∈ Mn(K[z, 1/z]) and a matrix M ∈ Mn(K[z]), such that with +e = t(e1, . . . , eN): +e′ = Be +and +f = Me. +The proof follows [4, p. 378], using also the additional details given in [5]. Actually +Proposition 2 is already proved implicitly (for K = Q) by the implementation described in +[5]. +In what follows we simply mention the parts of the proof where a special attention has +to be paid. Let α be a singularity of the differential system Y ′ = AY , and Q ∈ K[X] +denote the minimal polynomial of α over K. Let k ≥ 1 be the maximal order of α as a +pole of a coefficient of A, and (i0, j0) be such that Ai0,j0 has a pole of order exactly k at +α. Then QkAf = Qkf ′ vanishes at α; the i0-th coordinate of this vector provides a linear +relation +N +� +j=1 +(QkAi0,j)(α)fj(α) = 0. +Note that for any j, the rational function Q(z)kAi0,j(z) ∈ K(z) is holomorphic at α, and +for j = j0 it does not vanish at that point. Multiplying by a suitable element of K[z] which +does not vanish at α, we obtain P1, . . . , PN ∈ K[z] such that +Pj0(α) ̸= 0 +and +N +� +j=1 +Pj(α)fj(α) = 0. +7 + +Upon dividing by their gcd, we may assume the polynomials P1, . . . , PN to be coprime. +If N = 1 we let P1,1 = 1; otherwise there exist polynomials Pi,j ∈ K[z], for 2 ≤ i ≤ N +and 1 ≤ j ≤ N, such that letting P1,j = Pj, the matrix S = (Pi,j)1≤i,j≤N ∈ MN(K[z]) +has determinant 1. Then Sf is a vector of E-functions, with coefficients in K, of which +the first coordinate �N +j=1 Pj(z)fj(z) vanishes at α. Using Proposition 1, we deduce that +�N +j=1 Pj(z)fj(z) vanishes at σ(α) for any σ ∈ Gal(Q/K). Denoting by F a fundamental +system of solutions of which f is the first column, and considering the differential Galois +group as in [4], yields a matrix F1 with coefficients holomorphic at α, such that SF = DF1 +where D is the diagonal matrix with diagonal coefficients Q(z), 1, . . . , 1. This concludes +the proof as in [4]. +3 +Proof of Theorem 1 +The proof of Theorem 1 falls into 3 steps. +Step 1. Let us prove Theorem 1 in the special case where K = Q (i.e., d = 1). In other +words, we assume that z0 ∈ Q, λ1, . . . , λN ∈ Z, and f1, . . . , fN have coefficients in Q. +If z0 = 0 then f1(z0), . . . , fN(z0) are algebraic numbers, so the conclusion follows from +Schmidt’s subspace theorem (see for instance [6, Chapter 1, §8.2, Theorem 1.37]). +Therefore we assume z0 ̸= 0, and even z0 = 1 by considering the E-functions fj(z0z) +instead of fj(z). +Recall that we do not assume that f1(1), . . . , fN(1) are linearly independent over Q. +Denoting by N′ the maximal number of linearly independent numbers among them, we may +assume (up to a permutation of the indices) that f1(1), . . . , fN′(1) are linearly independent +over Q, and fN′+1(1), . . . , fN(1) belong to the Q-vector space they span. +There exist +rational numbers ̺i,j such that fj(1) = �N′ +i=1 ̺i,jfi(1) for any 1 ≤ j ≤ N, so that +Λ = +N +� +j=1 +λjfj(1) = +N′ +� +i=1 +µifi(1) +with +µi := +N +� +j=1 +λj̺i,j ∈ Q. +(3.1) +Observe that the E-functions f1, . . . , fN′ are linearly independent over C(z). Indeed, +otherwise they would be linearly dependent over Q(z) (since they have coefficients in Q), +and a relation �N′ +j=1 Sj(z)fj(z) = 0 would exist with S1, . . . , SN′ ∈ Q(z) not all zero. Upon +multiplying by (z − 1)k for a suitable k ∈ Z, we may assume that none of the Sj has a +pole at 1, and that at least of them does not vanish at 1. This provides a non-trivial linear +relation �N′ +j=1 Sj(1)fj(1) = 0, which contradicts the definition of N′. +Therefore f1, . . . , fN′ are linearly independent over C(z). Denote by N′′ the dimension +of the vector space generated over C(z) by f1, . . . , fN; we have N′ ≤ N′′ ≤ N. Notice +that it could happen that N′′ > N′, for instance if fN′+1(1) = 0. Up to a permutation of +the indices, we may assume that f1, . . . , fN′′ are linearly independent over C(z), and that +fN′′+1, . . . , fN belong to the vector space they span over C(z). +8 + +Since f1, . . . , fN′′ are linearly independent over C(z), and satisfy a linear differential +system of order 1 by definition of N′′, Proposition 2 (applied with K = Q) provides +E-functions e1, . . . , eN′′ with rational coefficients and matrices B and M = (Pi,j) with +Pi,j ∈ Q[z] such that fi(z) = �N′′ +j=1 Pi,j(z)ej(z). Since N′ ≤ N′′, Eq. (3.1) yields +Λ = +N′′ +� +j=1 +νjej(1) +with +νj := +N′ +� +i=1 +µiPi,j(1) ∈ Q. +(3.2) +Now Shidlovskii’s lower bound stated as Theorem A in the introduction applies to the +E-functions e1, . . . , eN′′ with rational coefficients, which are linearly independent over +C(z) and solution of a linear differential system of order 1 of which 1 is not a singularity. +Denoting by δ a common denominator of the rational numbers Pi,j(1) and ̺i,j (appearing +in Eqns. (3.1) and (3.2)), we obtain that δ2Λ is a Z-linear combination of e1(1), . . . , eN′′(1) +with coefficients bounded (in absolue value) by cH, where c > 0 and δ depend only on f1, +. . . , fN. For any ε > 0, Theorem A yields |δ2Λ| > c0H−N′′+1−ε ≥ c0H−N+1−ε for some +c0 > 0 which depends only on f1, . . . , fN. This concludes the proof of Theorem 1 in the +case where K = Q. +Step 2. Let us prove Theorem 1 for any number field K, in the case where λj ∈ Z for any +j. As announced in the introduction (after the statement of Theorem 1), we shall refine +the lower bound H−dd+1Nd+1−ε to H−dNd+1−ε in this case. +For simplicity of the exposition, we assume K to be a Galois extension of Q; see the +end of Step 2 for the general case. We denote by G the Galois group of K/Q, and consider +the complex number +̟ = +� +σ∈G +� +N +� +j=1 +λjf σ +j (1) +� +. +(3.3) +To begin with, let us prove that ̟ ̸= 0. +Indeed, consider the E-function g(z) = +�N +j=1 λjfj(z); it has coefficients in K (because f1, . . . , fN do), and g(1) = Λ ̸= 0. For any +σ ∈ G, Proposition 1 yields gσ(1) ̸= 0. Now gσ(1) = �N +j=1 λjf σ +j (1) since λj ∈ Z, so that +̟ = � +σ∈G gσ(1) ̸= 0. +Now we are going to expand the product in the definition (3.3) of ̟. Denote by N +the set of all tuples n = (n1, . . . , nN) of non-negative integers such that n1 + . . . + nN = d. +For any n ∈ N , we denote by I(n) the set of all families i = (iσ)σ∈G consisting in integers +iσ ∈ {1, . . . , N} such that for any j ∈ {1, . . . , N} we have: +Card{σ ∈ G, iσ = j} = nj. +Then Eq. (3.3) yields +̟ = +� +n∈N +λn1 +1 · . . . · λnN +N ϕn(1) +upon letting +ϕn(z) := +� +i∈I(n) +� +σ∈G +f σ +iσ(z). +(3.4) +9 + +Let us prove that ϕn(z), which is an E-function with coefficients in K, actually has +coefficients in Q for any n ∈ N . For any τ ∈ G we have: +ϕτ +n = +� +i∈I(n) +� +σ∈G +� +f σ +iσ +�τ += +� +i∈I(n) +� +σ∈G +f τ◦σ +iσ += +� +i∈I(n) +� +σ′∈G +f σ′ +iτ−1◦σ′ +by letting σ′ = τ ◦ σ += +� +i′∈I(n) +� +σ′∈G +f σ′ +i′ +σ′ +where the last equality comes from letting i′ +σ = iτ −1◦σ for any σ ∈ G; indeed this defines a +bijective map I(n) → I(n). Therefore ϕτ +n = ϕn for any τ ∈ G, and the E-function ϕn(z) +has coefficients in Q. +We denote by E the vector space spanned over Q(z) by the functions � +σ∈G f σ +iσ for all +families i = (iσ)σ∈G consisting in integers iσ ∈ {1, . . . , N}. There are Nd such families, so +dim E ≤ Nd. Moreover we have g′ ∈ E for any g ∈ E. +Let δ denote the dimension of the vector space spanned over Q(z) by the functions ϕn +for n ∈ N . We can choose δ functions h1, . . . , hδ among the ϕn which are linearly indepen- +dent, and span the same Q(z)-vector space. Choosing among the successive derivatives of +h1, . . . , hδ it is possible to find an integer δ′ ≥ δ and functions hi, for δ + 1 ≤ i ≤ δ′, such +that h1, . . . , hδ′ are linearly independent over Q(z) and satisfy a linear differential system +of order 1. Since they have rational coefficients, they are also linearly independent over +Q(z); now they all belong to E, so we have δ′ ≤ dim E ≤ Nd. +Proposition 2 with K = Q yields a vector of E-functions e1, . . . , eδ′ with rational co- +efficients, solution of a first-order differential system with no finite non-zero singularity, +such that each hi is a linear combination of e1, . . . , eδ′ with coefficients in Q[z]. There exist +Rn,i, Sn,i ∈ Q(z) for n ∈ N and 1 ≤ i ≤ δ′ such that, for any n, +ϕn(z) = +δ′ +� +i=1 +Rn,i(z)hi(z) = +δ′ +� +i=1 +Sn,i(z)ei(z). +If no Sn,i has a pole at z = 1, we can take z = 1 in this equation. To deal with the +general case, we expand the right hand side as a polynomial in 1/(z −1), up to an additive +term which is holomorphic and vanishes at z = 1. Since ϕn(z) is holomorphic at 1, all +polar contributions cancel out and the value at z = 1 is given by the constant term of the +above-mentioned polynomial. This provides an expression of the form +ϕn(1) = +δ′ +� +i=1 +J +� +j=0 +an,i,je(j) +i (1) +10 + +with an,i,j ∈ Q. +Since t(e1, . . . , eδ′) is solution of a first-order differential system with +coefficients in Q[z, 1/z], hence with no finite non-zero singularity, we obtain finally +ϕn(1) = +δ′ +� +i=1 +bn,iei(1) +(3.5) +with bn,i ∈ Q (where simply bn,i := Sn,i(1) in the “no pole at z = 1” case considered above). +Using Eq. (3.5) into Eq. (3.4) yields +̟ = +δ′ +� +i=1 +µiei(1) +with +µi = +� +n∈N +λn1 +1 · . . . · λnN +N bn,i ∈ Q. +This enables us to apply the special case of Theorem 1 where K = Q, proved in +Step 1, with N replaced with δ′ ≤ Nd. Indeed we denote by α ∈ Z a common positive +denominator of the rational numbers bn,i; then we have αµ1, . . . , αµδ′ ∈ Z. Since ̟ ̸= 0 +we obtain |α̟| > cH′−Nd+1−ε where +H′ = max +1≤i≤δ |αµi| ≤ β max +n∈N λn1 +1 · . . . · λnN +N +≤ βHd +where β > 0 depends only on f1, . . . , fN and K. Now Eq. (3.3) yields |̟| ≤ c′Hd−1|Λ| by +bounding trivially the factors corresponding to all σ ̸= Id; here c′ is a positive constant that +depends only on f1, . . . , fN and K. Combining these estimates yields |Λ| > c′′H−dNd+1−dε +for some constant c′′; this concludes Step 2 in the case where K is a Galois extension of Q. +If K/Q (of degree d) is not assumed to be Galois, we consider a finite Galois extension +L of Q such that K ⊂ L. We let G0 = Gal(L/Q) and H = Gal(L/K). In the definition +of ̟, namely Eq. (3.3), the product is now taken over the d cosets σ ∈ G0/H; indeed f σ +j +is the same for all σ in a given coset, because fj has coefficients in K. The same remark +holds with Eq. (3.4). Then we have ϕτ +n = ϕn for any τ ∈ G0, so that ϕn has coefficients in +Q. The other parts of the proof of Step 2 remain unchanged. +Step 3. Let us conclude the proof of Theorem 1. +Let (ω1, . . . , ωd) be a Z-basis of OK. +Each λj may be written as �d +t=1 λj,tωt with +λj,t ∈ Z such that |λj,t| ≤ c1 λj ≤ c1H, where c1 depends only on K (see [13, Chapter 3, +Lemma 12]). Then Λ = �N +j=1 +�d +t=1 λj,tωtfj(1), and the conclusion of Theorem 1 follows +from the special case proved in Step 2, applied to the dN E-functions ωtfj(z) with λj,t ∈ Z. +4 +Decomposition of E-functions over a number field +In the same spirit as Proposition 2, it is possible to prove the following result. The weaker +version with K replaced by Q was first proved in the unpublished note [12], and the special +case K = Q in [5]. +11 + +Proposition 3. Let f be an E-function with coefficients in a number field K. Then there +exist polynomials P, Q ∈ K[z], and an E-function g with coefficients in K, such that +f(z) = P(z) + Q(z)g(z) and g(z0) is transcendental for any z0 ∈ Q +∗. +In this setting, the non-zero algebraic numbers z at which a transcendental f takes an +algebraic value are exactly the roots of Q. Moreover, replacing P with its remainder in its +Euclidean division by Q, we may assume deg P < deg Q provided Q ̸= 0 (i.e., when f is +not a polynomial); see [5, Proposition 3.3]. +Proposition 3 is a generalization of the following result, which will be used in the proof. +It is stated as [8, Theorem 4] and its proof is due to the referee of [7]: +Lemma 1. Let f be an E-function with coefficients in a number field K, and α ∈ Q be +such that f(α) is algebraic. Then f(α) ∈ K(α). +For the convenience of the reader, let us deduce this lemma from Proposition 1. Let +β = f(α), and L be a finite Galois extension of K(α) such that β ∈ L. Since f(z) − β +vanishes at α, Proposition 1 shows that for any σ ∈ Gal(L/K(α)) the E-function f(z) − +σ(β) = f σ(z) − σ(β) vanishes at σ(α) = α, so that σ(β) = f(α) = β. This concludes the +proof of Lemma 1. +Proof of Proposition 3. To prove Proposition 3, we first remark that the result is obvious +if f is algebraic, hence a polynomial: we simply take P = f and Q = 0. Let us now assume +that f is transcendental. We argue by induction on the number of non-zero algebraic points +α such that f(α) ∈ Q; this number is finite because f is transcendental and of Beukers’ +theorem: any such α is a singularity of the first-order differential system satisfied by 1, f, +f ′, . . . , f (µ−1) (where µ ≥ 1 is the minimal order of an inhomogeneous linear differential +equation with coefficients in K(z) satisfied by f). +If this number is 0, one may choose P = 0 and Q = 1. Now if f(α) ∈ Q, Lemma 1 proves +that f(α) belongs to K(α): there exists P0 ∈ K[X] such that f(α) = P0(α). Therefore the +E-function f −P0, with coefficients in K, vanishes at α. Proposition 1 yields an E-function +g0 with coefficients in K such that f = P0 + Dg0 where D is the minimal polynomial of α +over K. If g0(α) ∈ Q, the same procedure can be carried out with g0, leading to P1 ∈ K[z] +and an E-function g1 with coefficients in K such that g0 = P1 + Dg1. After finitely many +steps, this procedure terminates and provides gℓ such that gℓ(α) ̸∈ Q (see the proof of [5, +Theorem 3.4]). This concludes the proof of Proposition 3. +5 +Structure of EK +Let K be a subfield of Q. As in the introduction, we denote by EK the ring of all values +f(1) where f is an E-function with coefficients in K; in particular EQ = E. +The following result contains Theorem 2 stated in the introduction (as a special case +K = Q). +12 + +Theorem 3. Elements of EK are linearly independent over Q if, and only if, they are +linearly independent over K. In other words, the K-algebras EK and Q are linearly disjoint, +and the natural map EK ⊗K Q → E is a K-algebra isomorphism. +This result implies EK ∩ Q = K, which is an equivalent form of Lemma 1 stated in §4. +Proof of Theorem 3. Let f1, . . . , fN be E-functions with coefficients in K. If f1(1), . . . , +fN(1) are linearly independent over Q, then obviously they are linearly independent over +K. +Conversely, let us assume that they are linearly independent over K. +Let λ1, . . . , +λN be algebraic numbers, not all zero, such that λ1f1(1) + . . . + λNfN(1) = 0. Up to a +permutation of the indices we may assume that λ1 ̸= 0; then dividing by λ1 we assume +that λ1 = 1. Let us consider a finite Galois extension L of K that contains λ2, . . . , λN. +Then g(z) = �N +i=1 λifi(z) is an E-function with coefficients in L, and it vanishes at z = 1. +For any σ ∈ Gal(L/K), Proposition 1 yields gσ(1) = 0, that is �N +i=1 σ(λi)fi(1) = 0 since +all fi have coefficients in K. Summing these relations, as σ varies, yields +N +� +i=1 +TrL/K(λi)fi(1) = 0 +with TrL/K(λi) = � +σ∈Gal(L/K) σ(λi) ∈ K and TrL/K(λ1) = TrL/K(1) = [L : K] ̸= 0. This +is a non-trivial linear relation, with coefficients in K, between f1(1), . . . , fN(1). +This +contradiction concludes the proof that elements of EK are linearly independent over Q if, +and only if, they are linearly independent over K. +What remains of Theorem 3 follows directly from this property (see [3, Chapter V, §2, +No. 5]). +6 +A Galois action on values of E-functions +In this section we define and study an action of the absolute Galois group of Q, namely +Gal(Q/Q), on the set E of values of E-functions. +The definition is as follows: given σ ∈ Gal(Q/Q) and ξ ∈ E, there exists an E-function +f such that ξ = f(1). Then we let σ(ξ) = f σ(1). The crucial point is to prove that σ(ξ) +depends only on σ and ξ, not on the choice of f. Indeed if g is another E-function such +that ξ = g(1), then f −g vanishes at the point 1. Proposition 1 shows that f σ−gσ vanishes +at 1 too, so that gσ(1) = f σ(1): this concludes the proof. +Proposition 4. Let K be a number field, and ξ ∈ E. Then ξ belongs to EK if, and only +if, σ(ξ) = ξ for any σ ∈ Gal(Q/K). +In other words, the fixed points of E under Gal(Q/K) are exactly the elements of EK. +13 + +Proof of Proposition 4. If ξ ∈ EK and σ ∈ Gal(Q/K) then σ(ξ) = ξ by definition, since +f σ = f for any E-function f with coefficients in K. +Now let ξ ∈ E be such that σ(ξ) = ξ for any σ ∈ Gal(Q/K). +Let f be an E- +function such that f(1) = ξ, and L denote a finite Galois extension of K that contains all +coefficients of f. Let σ ∈ Gal(L/K); then σ is the restriction to L of an element, again +denoted by σ, of Gal(Q/K). We have σ(ξ) = ξ by assumption, and also σ(ξ) = f σ(1) +by definition. Summing the identity ξ = f σ(1) over all σ yields ξ = g(1) where g(z) = +1 +[L:K] +� +σ∈Gal(L/K) f σ(z) is an E-function with coefficients in K. +Therefore ξ ∈ EK; this +concludes the proof of Proposition 4. +This Galois action sheds a new light on the proof of Theorem 1 (see §3): it is very +similar to Liouville’s proof that irrational algebraic numbers are not too well approximated +by rationals (i.e., are not Liouville numbers). +Indeed let us recall briefly Liouville’s proof, stated in terms of Galois action. Let ξ +be an algebraic number of degree d ≥ 2, and assume (for simplicity) that the extension +Q(ξ)/Q is Galois. To bound from below |qξ − p| for (p, q) ∈ Z2 \ {(0, 0)}, consider +̟ := +� +σ∈Gal(Q(ξ)/Q) +� +qσ(ξ) − p +� +. +Then ̟ ̸= 0 (since σ(ξ) is irrational for any σ), and ̟ ∈ Q (since it is the norm of qξ − p +with respect to the extension Q(ξ)/Q). Letting δ ∈ Z denote a positive integer such that +δξ is an algebraic integer, we have δd̟ ∈ Z\ {0} since OQ(ξ) ∩Q = Z. Therefore |δd̟| ≥ 1 +so that +δ−d ≤ |̟| ≤ |qξ − p|( ξ + 1)d−1Hd−1 +by bounding |qσ(ξ) − p| trivially for σ ̸= Id, where H = max(|p|, |q|). Dividing by q yields +|ξ − p/q| ≥ cH−d where c > 0 depends only on ξ. +Step 2 of the proof of Theorem 1 is very similar, except that elements of K ⊂ Q are +replaced with values of E-functions in EK; the lower bound used by Liouville (namely, +δd̟ ∈ Z \ {0} implies |δd̟| ≥ 1) is replaced accordingly by Shidlovskii’s lower bound +recalled in Theorem A. +References +[1] Y. Andr´e, S´eries Gevrey de type arithm´etique I. Th´eor`emes de puret´e et de dualit´e, +Annals of Math. 151.2 (2000), 705–740. +[2] Y. Andr´e, Solution algebras of differential equations and quasi-homogeneous varieties: +a new differential Galois correspondence, Ann. Sci. ´Ec. Norm. Sup´er. 47.2 (2014), +449—467. +[3] N. Bourbaki, Elements of Mathematics, Algebra II, Chapters 4–7, Springer, 2003. +14 + +[4] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of Math. 163 +(2006), 369–379. +[5] A. Bostan, T. Rivoal, B. Salvy, Minimization of differential equations and algebraic +values of E-functions, preprint (2022), 37 pages. +[6] N. I. Feldman, Yu. V. Nesterenko, Transcendental Numbers, in Encyclopaedia of Math- +ematical Sciences, Vol. 44: Number Theory IV (Springer, 1998). +[7] S. Fischler, T. Rivoal, On the values of G-functions, Commentarii Math. Helv. 89.2 +(2014), 313–341. +[8] S. Fischler, T. Rivoal, Arithmetic theory of E-operators, Journal de l’´Ecole polytech- +nique – Math´ematiques 3 (2016), 31–65. +[9] S. Fischler, T. Rivoal, Microsolutions of differential operators and values of arithmetic +Gevrey series, American J. of Math. 140.2 (2018), 317–348. +[10] G. Lepetit, Le th´eor`eme d’Andr´e-Chudnovsky-Katz au sens large, North-West. Eur. +J. Math. 7 (2021), 83–149. +[11] Yu. V. Nesterenko, A. B. Shidlovskii, On the linear independence of values of E- +functions, Sb. Math. 187 (1996), 1197–1211; translation from the russian Math. Sb. +187 (1996), 93–108. +[12] T. Rivoal, Valeurs alg´ebriques de E-fonctions aux points alg´ebriques, unpublished note +(2016), 4 pages, available at https://hal.archives-ouvertes.fr/hal-03676576 +[13] A. B. Shidlovskii, Transcendental numbers, de Gruyter Studies in Math., no. 12, de +Gruyter, Berlin, 1989. +[14] C. Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, vol. 1 S. Ab- +handlungen Akad., Berlin, 1929. +[15] W. Zudilin, On rational approximations of values of a certain class of entire functions, +Sb. Math. 186.4, 555–590 (1995); translation from the russian Mat. Sb. 186.4, 89–124 +(1995). +S. Fischler, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, 91405 +Orsay, France. +T. Rivoal, Institut Fourier, CNRS et Universit´e Grenoble Alpes, CS 40700, 38058 Grenoble +cedex 9, France. +Keywords: E-functions, Andr´e-Beukers Theorems, Linear independence measures, Irra- +tionality measures, Liouville numbers, Shidlovskii’s Theorem. +MSC 2020: 11J82 (Primary), 11J91 (Secondary) +15 + diff --git a/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/load_file.txt b/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1bb077a8ae9d452a7820b1b817e6ae58de5d99c8 --- /dev/null +++ b/mtAzT4oBgHgl3EQfN_t_/content/tmp_files/load_file.txt @@ -0,0 +1,732 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf,len=731 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='01158v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='NT] 3 Jan 2023 Values of E-functions are not Liouville numbers S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Fischler and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal January 3, 2023 Abstract Shidlovskii has given a linear independence measure of values of E-functions with rational Taylor coefficients at a rational point not a singularity of the underlying differential system satisfied by these E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' His measure holds as well for E- functions with coefficients in an imaginary quadratic field, but not for other number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Recently, Beukers has proved a remarkable qualitative linear independence theorem for the values at an algebraic point of E-functions with arbitrary algebraic Taylor coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' But no quantitative analogue of Shidlovskii’s measure has been given in Beukers’ setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The goal of this paper it to obtain such a measure, in an even more general setting where the point can be a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This enables us to solve a long standing problem: the value of an E-function at an algebraic point is never a Liouville number, a result which had been obtained before only under addi- tional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We deduce various explicit irrationality measures, in particular for values of the exponential and Bessel’s J0 function at non-zero algebraic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We also prove that the values at rational points of E-functions with rational Taylor coefficients are linearly independent over Q if and only if they are linearly indepen- dent over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Our methods rest upon improvements of results recently obtained by Andr´e and Beukers by means of the theory of E-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 1 Introduction Siegel [14] defined in 1929 the class of E-functions in order to generalize the Diophantine properties of the exponential function (namely the Lindemann-Weierstrass Theorem) to other special functions such as Bessel’s function J0(z) := �∞ n=0(−1)n(z/2)2n/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 or hyper- geometric series pFp with rational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' A power series �∞ n=0 an n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' zn ∈ Q[[z]] is said to be an E-function when it is solution of a linear differential equation over Q(z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', holonomic), and |σ(an)| (for any σ ∈ Gal(Q/Q)) and the least common denominator of a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , an all grow at most exponentially in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Note that Siegel’s original definition of E-functions is more general: see the end of this introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Throughout this paper we fix an embedding of Q in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' A lot of important qualitative results are known on the arithmetic nature of the values taken by E-functions at algebraic points, amongst which we cite the celebrated Siegel- Shidlovskii Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' It has been improved in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' However, these results are not always 1 strong enough, in particular they do not imply the linear independence of values of E- functions solutions of a differential system of order 1 and evaluated at a non-singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Quantitative versions of certain of these results exist, but only under very strict assumptions on algebraic independence or rationality of the coefficients of the E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The main result in this direction, in the setting of linear independence, is the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' It is due to Shidlovskii [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 358, Theorem 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (32)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Theorem A (Shidlovskii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f = t(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) ∈ Q[[z]]N be a vector of E-functions solution of a differential system f ′ = Af for some A ∈ MN(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Assume that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN are linearly independent over Q(z) and that z0 ∈ Q∗ is not a pole of an entry of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then for any ε > 0, there exists c = c(ε, z0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) > 0 such that for all λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN ∈ Z not all zero, we have ���� N � j=1 λjfj(z0) ���� > cH−N+1−ε where H := max 1≤j≤N |λj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This theorem holds verbatim with Q replaced by an imaginary quadratic number field and Z replaced by its ring of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' However no such result is known for other number fields K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The point is that all known quantitative results are based on the Siegel-Shidlovskii method only, which provides linear independence of the full set of the values of E-functions in Theorem A only when K is either Q or imaginary quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Even the qualitative part of Theorem A (namely, �N j=1 λjfj(z0) ̸= 0) has been proved only recently by Beukers [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4] for arbitrary number fields, using Andr´e’s theory of E-operators [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Theorem B (Beukers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f = t(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) ∈ Q[[z]]N be a vector of E-functions solution of a differential system f ′ = Af for some A ∈ MN(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Assume that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN are line- arly independent over Q(z) and that z0 ∈ Q ∗ is not a pole of an entry of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then the numbers f1(z0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(z0) are linearly independent over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The purpose of this paper is to prove new Diophantine results using this approach of Andr´e and Beukers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Our first main result is the following theorem, where we generalize Theorem A to any E-functions, by removing the rationality assumption on the coefficients and also the non-singularity assumption on z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We recall that for a non-zero algebraic number α, its house α is the maximum of the moduli of α and of all its Galois conjugates over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We also denote by OK the ring of integers of a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let K be a number field of degree d over Q, z0 ∈ K, and f = t(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) be a vector of E-functions with coefficients in K such that f ′ = Af for some A ∈ MN(K(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then for any ε > 0, there exists c = c(ε, K, z0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN ∈ OK, if Λ := λ1f1(z0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' + λNfN(z0) is non-zero, then |Λ| > cH−dd+1Nd+1−ε where H := max 1≤j≤N λj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 2 Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' – Given arbitrary E-functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN and any z0 ∈ Q, this theorem applies because there exists a number field K containing z0 and all coefficients of f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN, and the family (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) can always be enlarged to satisfy a first-order differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This shows, without any assumption on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN and z0, the existence of κ, c > 0 such that |Λ| > cH−κ provided Λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' – Note that in Theorem 1, and even in the previous remark, we do not assume that f1(z0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(z0) are linearly independent over K, and we do not wonder whether z0 is a singularity or not of a differential system involving the fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore with K = Q, Theorem 1 is a generalization of Theorem A: we obtain the same lower bound under milder assumptions (using Beukers’ results in [4] and improvements of the latter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' – We shall prove that if λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN ∈ Z, the lower bound in Theorem 1 can be refined to cH−dNd+1−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' – To our knowledge, Theorem 1 provides the first quantitative version of Beukers’ The- orem B, when it is further assumed in Theorem 1 that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN are linearly independent over K(z) and that z0 ∈ K∗ is not a pole of A, ensuring that Λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' An important consequence of Theorem 1 is the following result, which completely settles the problem of deciding whether (real) values of E-functions can be Liouville numbers or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We recall that a Liouville number is a real number ξ such that there exist two sequences of rational integers pn, qn such that 0 < |qnξ −pn| < 1/qn n for all sufficiently large integers n (a fortiori pnqn ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E-function, and z0 be an algebraic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f(z0) is not a Liouville number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The proof of Corollary 1 runs as follows: in Theorem 1, let z0 ∈ Q, take f1 := 1, f2 := f and consider a vector t(f1, f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) of E-functions with coefficients in a number field K such that f ′ = Af for some A ∈ MN(K(z)), where K is large enough to contain z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If f(z0) ∈ Q, then f(z0) is not a Liouville number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If f(z0) /∈ Q, then λ1 + λ2f(z0) ̸= 0 for all λ1, λ2 ∈ Z not both zero, so that Theorem 1 yields |λ1 + λ2f(z0)| > c max(|λ1|, |λ2|)−κ for some c, κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This rules out the possibility that f(z0) is a Liouville number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Of course Corollary 1 is interesting only when f(z0) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If we do not assume this, note however that the real and imaginary parts of f(z0) are values of E-functions (see the remark in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1 below), so that none of them is a Liouville number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us also mention another interesting corollary, which is a consequence of the third remark that follows Theorem 1 with N = 2 and N = 3 respectively (recall that J0(z) is solution of zy′′(z) + y′(z) + zy(z) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any algebraic number α ∈ Q ∗ of degree d over Q and any ε > 0, there exists c = c(α, ε) such that, for all (p, q) ∈ Z × N with q ̸= 0, ���eα − p q ��� > c qd2d+ε, respectively ���J0(α) − p q ��� > c qd3d+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' A general transcendence measure for E-functions due to Lang and Galochkin [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 238, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='29 and remarks] (applied with m = 1 and m = 2 respectively) gives 4d2 + 1 3 instead of d2d, and 16d3 + 1 instead of d3d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' see also [13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 403].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This is of course much better than Corollary 2 for large d but our bounds turn out to be smaller for d ∈ {1, 2, 3, 4} and d ∈ {1, 2, 3, 4, 5} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' A less classical example is the following: for any α ∈ Q ∗ and any integers p, q ≥ 1, the value at z = α of the function Ap,q(z) := �∞ n=0(�n k=0 �n k �p�n+k n �q)zn/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' is not a Liouville number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Ap,q(α) is proved to be a transcendental number in [5, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='6] when (p, q) ∈ {1, 2, 3, 4}2, the situation in general being unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' More specifically, it is also proved in [5, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='6, Table 1] that when (p, q) = (2, 2), the minimal inhomogeneous differential equation over Q(z) satisfied by A2,2 is of order 4 and 0 is its only singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Hence, the following holds by the third remark that follows Theorem 1 with N = 5: for all α ∈ Q ∗ of degree d over Q and all ε > 0, there exists c = c(ε, α) > 0 such that for all λj ∈ Z not all 0, we have ���λ4 + 3 � j=0 λjA(j) 2,2(α) ��� > c max 0≤j≤4 |λj|−d5d+1−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In particular, for any α ∈ Q ∗ of degree d and ε > 0, there exists c = c(ε, α) > 0 such that for any (p, q) ∈ Z × N with q ̸= 0, ���A2,2(α) − p q ��� > c qd5d+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1) It seems reasonable to make the following conjecture, which probably belongs to folk- lore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This conjecture is Roth’s Theorem if f(z0) is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E-function and z0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any ε > 0, there exists c > 0 such that for any (p, q) ∈ Z × N with q ̸= 0, either qf(z0) − p = 0 or ���f(z0) − p q ��� > c q2+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Zudilin has proved this conjecture in [15] in a stronger form but under additional assumptions (which even imply f(z0) /∈ Q), namely: f is an E-function with rational coefficients, z0 ∈ Q∗ is not a singularity of a differential system satisfied by 1, f, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN over Q(z) (for some N ≥ 2) and f, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN are algebraically independent over Q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' It would be interesting to know if A2,2, A′ 2,2, A′′ 2,2, A′′′ 2,2 are algebraically independent over Q(z), in which case the exponent 5 could be improved to 2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1) when α ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The second goal of this paper is to understand the structure of the ring E of all values at algebraic points of E-functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' this algebraic point can always be assumed to be 1 because if f(z) is an E-function, so is f(αz) for any α ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Elements of E are related to exponential periods (see [9, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any subfield K of Q, we shall also consider the subring EK of E which consists of the evaluations f(1) where f is an E-function with coefficients in K (the number 1 could be replaced by any non-zero element of K without changing EK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Note that E is the union of all EK, where K is a number field, since for any E-function f the holonomy property implies the existence of a number field that contains all coefficients of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 4 We have defined and studied [7] analogous rings GK with G-functions instead of E- functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' it turns out that GK is nearly independent from K (precisely, GK = GQ if K ⊂ R, and GK = GQ(i) otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The situation is completely different for E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' A first hint in this direction was given in [8, Theorem 4] (stated as Lemma 1 in §4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The way EK depends on K is completely described by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Elements of EQ are linearly independent over Q if, and only if, they are linearly independent over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In other words, the Q-algebras EQ and Q are linearly disjoint, and the natural map EQ ⊗Q Q → E (sending ξ ⊗ z to ξz) is a Q-algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We refer to [3, Chapter V, §2, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 5] for the definition and properties of linearly disjoint algebras, which imply the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let K be a number field, and (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , ωd) be a basis of the Q-vector space K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then EK = ω1EQ ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' ⊕ ωdEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In other words, for any ξ ∈ EK there exists a unique d-tuple (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , ξd) ∈ Ed Q such that ξ = ω1ξ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' + ωdξd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Theorems 1 and 2 may seem disjoint at first sight but their proofs share many com- mon aspects, in particular both use Proposition 1 stated in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Another consequence of Proposition 1 is the existence of an action of Gal(Q/Q) on E, of which the fixed points are exactly the elements of EQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For instance, with the notation of Corollary 3, we have σ(ξ) = σ(ω1)ξ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' + σ(ωd)ξd for any σ ∈ Gal(Q/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' As a first application of this Galois action, we explain in §6 why our proof of Theorem 1 is similar to Liouville’s proof that Liouville numbers are transcendental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This makes sense because Theorem 1 is a gener- alization of this result, since Q ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We hope this action can have other Diophantine applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The original definition of E-functions, given by Siegel [14], is slightly less restrictive: instead of geometric bounds, he allowed growths bounded by n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='ε (for any given ε > 0, provided n is large enough with respect to ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Shidlovskii’s Theorem A holds for E- functions in Siegel’s sense, and Beukers’ Theorem B was later proved by Andr´e [2] in this general setting, by a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' All other results in Beukers’ paper [4] have been adapted by Lepetit [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore all the results of the present paper also hold for E-functions in Siegel’s sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In §2 we prove the main tool of this paper, namely Proposition 1, and use it to obtain a version of Beukers’ desingularization process over a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This enables us to prove Theorem 1 in §3, and also to obtain in §4 a decomposition of an E-function over a number field, involving an E-function that takes only transcendental values at non-zero algebraic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' At last we apply the previous results in §5 to study the structure of EK and prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We conclude in §6 with the above-mentioned Galois action on values of E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 5 2 Main tools 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1 Conjugates of E-functions Let f(z) = �∞ n=0 anzn be an E-function with coefficients an ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any σ ∈ Gal(Q/Q) we let f σ(z) = �∞ n=0 σ(an)zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f σ is also an E-function, and if g is an E-function then for any σ, τ we have (f + g)σ = f σ + gσ, (fg)σ = f σgσ and (f σ)τ = f τ◦σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Moreover if f has coefficients in a number field K, then f σ has coefficients in the number field σ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denoting by σ the complex conjugation, for any E-function f we can consider 1 2(f + f σ) and 1 2i(f − f σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' These E-functions have real coefficients, which are respectively the real and imaginary parts of those of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In particular, the real and imaginary parts of any element of E belong to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The following result is central in the present paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' we refer to [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='5] for a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Throughout this paper we agree that minimal polynomials of algebraic elements have leading coefficient 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E-function with coefficients in a number field K, and z0 ∈ Q ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then the following assertions are equivalent: (i) f vanishes at z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (ii) There exists σ ∈ Gal(Q/Q) such that f σ vanishes at σ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (iii) For any σ ∈ Gal(Q/Q), f σ vanishes at σ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (iv) There exists an E-function g with coefficients in K such that f(z) = D(z)g(z) where D is the minimal polynomial of z0 over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In particular, if z0 is rational and f vanishes at z0, then all conjugates f σ of f also vanish at z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Also, if an E-function f with rational coefficients vanishes at some z0 ∈ Q ∗, then it vanishes at all Galois conjugates of z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We remark that with z0 = 1, the implication (i) ⇒ (iii) is used already in the proof of [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1], which is the main result Proposition 1 is based on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (iv) ⇒ (iii) Let σ ∈ Gal(Q/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f σ(z) = Dσ(z)gσ(z), and Dσ(σ(z0)) = σ(D(z0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore f σ(σ(z0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (iii) ⇒ (ii) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (ii) ⇒ (i) Enlarging K if necessary, we may assume the extension K/Q to be Galois and to contain z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f σ has coefficients in K, and σ(z0) ∈ K∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Using [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1] there exists an E-function g such that f σ(z) = (z − σ(z0))g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then g has coefficients in K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' applying σ−1 yields f(z) = (z − z0)gσ−1(z) so that f(z0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 6 (i) ⇒ (iv) Using [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1] there exists an E-function h such that f(z) = (z − z0)h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let σ ∈ Gal(Q/K), that is: σ is a field automorphism of Q such that σ(x) = x for any x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f(z) = f σ(z) = (z −σ(z0))hσ(z) so that f vanishes at σ(z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let z1 := z0, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , zℓ denote the (pairwise distinct) Galois conjugates of z0 over K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' the elements of the form σ(z0) with σ ∈ Gal(Q/K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' we have proved that f vanishes at z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Applying [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1] yields, by induction on j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , ℓ}, the existence of an E-function gj such that f(z) = gj(z) �j i=1(z − zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since D(z) = �ℓ i=1(z − zi), we have f(z) = D(z)gℓ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now D(z) ∈ K[z] \\ {0} so that all coefficients of gℓ belong to K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This concludes the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 Beukers’ desingularization process In the proof of Theorem 1 we shall use the following version of Beukers’ desingularization theorem (namely [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The new point is that e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN and the coefficients of B and M have coefficients in the number field K (whereas in [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='5] these coefficients are simply algebraic numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let K be a number field, and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN be E-functions with coefficients in K, linearly independent over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Assume that the vector f = t(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN) satisfies a first-order differential system f ′ = Af with A ∈ Mn(K(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then there exist E-functions e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN with coefficients in K, linearly independent over C(z), a matrix B ∈ Mn(K[z, 1/z]) and a matrix M ∈ Mn(K[z]), such that with e = t(e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN): e′ = Be and f = Me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The proof follows [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 378], using also the additional details given in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Actually Proposition 2 is already proved implicitly (for K = Q) by the implementation described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In what follows we simply mention the parts of the proof where a special attention has to be paid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let α be a singularity of the differential system Y ′ = AY , and Q ∈ K[X] denote the minimal polynomial of α over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let k ≥ 1 be the maximal order of α as a pole of a coefficient of A, and (i0, j0) be such that Ai0,j0 has a pole of order exactly k at α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then QkAf = Qkf ′ vanishes at α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' the i0-th coordinate of this vector provides a linear relation N � j=1 (QkAi0,j)(α)fj(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Note that for any j, the rational function Q(z)kAi0,j(z) ∈ K(z) is holomorphic at α, and for j = j0 it does not vanish at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Multiplying by a suitable element of K[z] which does not vanish at α, we obtain P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , PN ∈ K[z] such that Pj0(α) ̸= 0 and N � j=1 Pj(α)fj(α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 7 Upon dividing by their gcd, we may assume the polynomials P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , PN to be coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If N = 1 we let P1,1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' otherwise there exist polynomials Pi,j ∈ K[z], for 2 ≤ i ≤ N and 1 ≤ j ≤ N, such that letting P1,j = Pj, the matrix S = (Pi,j)1≤i,j≤N ∈ MN(K[z]) has determinant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then Sf is a vector of E-functions, with coefficients in K, of which the first coordinate �N j=1 Pj(z)fj(z) vanishes at α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Using Proposition 1, we deduce that �N j=1 Pj(z)fj(z) vanishes at σ(α) for any σ ∈ Gal(Q/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denoting by F a fundamental system of solutions of which f is the first column, and considering the differential Galois group as in [4], yields a matrix F1 with coefficients holomorphic at α, such that SF = DF1 where D is the diagonal matrix with diagonal coefficients Q(z), 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This concludes the proof as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 3 Proof of Theorem 1 The proof of Theorem 1 falls into 3 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us prove Theorem 1 in the special case where K = Q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', d = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In other words, we assume that z0 ∈ Q, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN ∈ Z, and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN have coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If z0 = 0 then f1(z0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(z0) are algebraic numbers, so the conclusion follows from Schmidt’s subspace theorem (see for instance [6, Chapter 1, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore we assume z0 ̸= 0, and even z0 = 1 by considering the E-functions fj(z0z) instead of fj(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Recall that we do not assume that f1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(1) are linearly independent over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denoting by N′ the maximal number of linearly independent numbers among them, we may assume (up to a permutation of the indices) that f1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN′(1) are linearly independent over Q, and fN′+1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(1) belong to the Q-vector space they span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' There exist rational numbers ̺i,j such that fj(1) = �N′ i=1 ̺i,jfi(1) for any 1 ≤ j ≤ N, so that Λ = N � j=1 λjfj(1) = N′ � i=1 µifi(1) with µi := N � j=1 λj̺i,j ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1) Observe that the E-functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN′ are linearly independent over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Indeed, otherwise they would be linearly dependent over Q(z) (since they have coefficients in Q), and a relation �N′ j=1 Sj(z)fj(z) = 0 would exist with S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , SN′ ∈ Q(z) not all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Upon multiplying by (z − 1)k for a suitable k ∈ Z, we may assume that none of the Sj has a pole at 1, and that at least of them does not vanish at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This provides a non-trivial linear relation �N′ j=1 Sj(1)fj(1) = 0, which contradicts the definition of N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN′ are linearly independent over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denote by N′′ the dimension of the vector space generated over C(z) by f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' we have N′ ≤ N′′ ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Notice that it could happen that N′′ > N′, for instance if fN′+1(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Up to a permutation of the indices, we may assume that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN′′ are linearly independent over C(z), and that fN′′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN belong to the vector space they span over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 8 Since f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN′′ are linearly independent over C(z), and satisfy a linear differential system of order 1 by definition of N′′, Proposition 2 (applied with K = Q) provides E-functions e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN′′ with rational coefficients and matrices B and M = (Pi,j) with Pi,j ∈ Q[z] such that fi(z) = �N′′ j=1 Pi,j(z)ej(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since N′ ≤ N′′, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1) yields Λ = N′′ � j=1 νjej(1) with νj := N′ � i=1 µiPi,j(1) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2) Now Shidlovskii’s lower bound stated as Theorem A in the introduction applies to the E-functions e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN′′ with rational coefficients, which are linearly independent over C(z) and solution of a linear differential system of order 1 of which 1 is not a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denoting by δ a common denominator of the rational numbers Pi,j(1) and ̺i,j (appearing in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2)), we obtain that δ2Λ is a Z-linear combination of e1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eN′′(1) with coefficients bounded (in absolue value) by cH, where c > 0 and δ depend only on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any ε > 0, Theorem A yields |δ2Λ| > c0H−N′′+1−ε ≥ c0H−N+1−ε for some c0 > 0 which depends only on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This concludes the proof of Theorem 1 in the case where K = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us prove Theorem 1 for any number field K, in the case where λj ∈ Z for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' As announced in the introduction (after the statement of Theorem 1), we shall refine the lower bound H−dd+1Nd+1−ε to H−dNd+1−ε in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For simplicity of the exposition, we assume K to be a Galois extension of Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' see the end of Step 2 for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We denote by G the Galois group of K/Q, and consider the complex number ̟ = � σ∈G � N � j=1 λjf σ j (1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3) To begin with, let us prove that ̟ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Indeed, consider the E-function g(z) = �N j=1 λjfj(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' it has coefficients in K (because f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN do), and g(1) = Λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any σ ∈ G, Proposition 1 yields gσ(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now gσ(1) = �N j=1 λjf σ j (1) since λj ∈ Z, so that ̟ = � σ∈G gσ(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now we are going to expand the product in the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3) of ̟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Denote by N the set of all tuples n = (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , nN) of non-negative integers such that n1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' + nN = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any n ∈ N , we denote by I(n) the set of all families i = (iσ)σ∈G consisting in integers iσ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , N} such that for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , N} we have: Card{σ ∈ G, iσ = j} = nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3) yields ̟ = � n∈N λn1 1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' · λnN N ϕn(1) upon letting ϕn(z) := � i∈I(n) � σ∈G f σ iσ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4) 9 Let us prove that ϕn(z), which is an E-function with coefficients in K, actually has coefficients in Q for any n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any τ ∈ G we have: ϕτ n = � i∈I(n) � σ∈G � f σ iσ �τ = � i∈I(n) � σ∈G f τ◦σ iσ = � i∈I(n) � σ′∈G f σ′ iτ−1◦σ′ by letting σ′ = τ ◦ σ = � i′∈I(n) � σ′∈G f σ′ i′ σ′ where the last equality comes from letting i′ σ = iτ −1◦σ for any σ ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' indeed this defines a bijective map I(n) → I(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore ϕτ n = ϕn for any τ ∈ G, and the E-function ϕn(z) has coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We denote by E the vector space spanned over Q(z) by the functions � σ∈G f σ iσ for all families i = (iσ)σ∈G consisting in integers iσ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' There are Nd such families, so dim E ≤ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Moreover we have g′ ∈ E for any g ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let δ denote the dimension of the vector space spanned over Q(z) by the functions ϕn for n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We can choose δ functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , hδ among the ϕn which are linearly indepen- dent, and span the same Q(z)-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Choosing among the successive derivatives of h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , hδ it is possible to find an integer δ′ ≥ δ and functions hi, for δ + 1 ≤ i ≤ δ′, such that h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , hδ′ are linearly independent over Q(z) and satisfy a linear differential system of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since they have rational coefficients, they are also linearly independent over Q(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' now they all belong to E, so we have δ′ ≤ dim E ≤ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 2 with K = Q yields a vector of E-functions e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eδ′ with rational co- efficients, solution of a first-order differential system with no finite non-zero singularity, such that each hi is a linear combination of e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eδ′ with coefficients in Q[z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' There exist Rn,i, Sn,i ∈ Q(z) for n ∈ N and 1 ≤ i ≤ δ′ such that, for any n, ϕn(z) = δ′ � i=1 Rn,i(z)hi(z) = δ′ � i=1 Sn,i(z)ei(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If no Sn,i has a pole at z = 1, we can take z = 1 in this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' To deal with the general case, we expand the right hand side as a polynomial in 1/(z −1), up to an additive term which is holomorphic and vanishes at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since ϕn(z) is holomorphic at 1, all polar contributions cancel out and the value at z = 1 is given by the constant term of the above-mentioned polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This provides an expression of the form ϕn(1) = δ′ � i=1 J � j=0 an,i,je(j) i (1) 10 with an,i,j ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since t(e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , eδ′) is solution of a first-order differential system with coefficients in Q[z, 1/z], hence with no finite non-zero singularity, we obtain finally ϕn(1) = δ′ � i=1 bn,iei(1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='5) with bn,i ∈ Q (where simply bn,i := Sn,i(1) in the “no pole at z = 1” case considered above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4) yields ̟ = δ′ � i=1 µiei(1) with µi = � n∈N λn1 1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' · λnN N bn,i ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This enables us to apply the special case of Theorem 1 where K = Q, proved in Step 1, with N replaced with δ′ ≤ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Indeed we denote by α ∈ Z a common positive denominator of the rational numbers bn,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' then we have αµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , αµδ′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since ̟ ̸= 0 we obtain |α̟| > cH′−Nd+1−ε where H′ = max 1≤i≤δ |αµi| ≤ β max n∈N λn1 1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' · λnN N ≤ βHd where β > 0 depends only on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3) yields |̟| ≤ c′Hd−1|Λ| by bounding trivially the factors corresponding to all σ ̸= Id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' here c′ is a positive constant that depends only on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Combining these estimates yields |Λ| > c′′H−dNd+1−dε for some constant c′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' this concludes Step 2 in the case where K is a Galois extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If K/Q (of degree d) is not assumed to be Galois, we consider a finite Galois extension L of Q such that K ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We let G0 = Gal(L/Q) and H = Gal(L/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In the definition of ̟, namely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3), the product is now taken over the d cosets σ ∈ G0/H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' indeed f σ j is the same for all σ in a given coset, because fj has coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The same remark holds with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then we have ϕτ n = ϕn for any τ ∈ G0, so that ϕn has coefficients in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The other parts of the proof of Step 2 remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , ωd) be a Z-basis of OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Each λj may be written as �d t=1 λj,tωt with λj,t ∈ Z such that |λj,t| ≤ c1 λj ≤ c1H, where c1 depends only on K (see [13, Chapter 3, Lemma 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then Λ = �N j=1 �d t=1 λj,tωtfj(1), and the conclusion of Theorem 1 follows from the special case proved in Step 2, applied to the dN E-functions ωtfj(z) with λj,t ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 4 Decomposition of E-functions over a number field In the same spirit as Proposition 2, it is possible to prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The weaker version with K replaced by Q was first proved in the unpublished note [12], and the special case K = Q in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 11 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E-function with coefficients in a number field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then there exist polynomials P, Q ∈ K[z], and an E-function g with coefficients in K, such that f(z) = P(z) + Q(z)g(z) and g(z0) is transcendental for any z0 ∈ Q ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In this setting, the non-zero algebraic numbers z at which a transcendental f takes an algebraic value are exactly the roots of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Moreover, replacing P with its remainder in its Euclidean division by Q, we may assume deg P < deg Q provided Q ̸= 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', when f is not a polynomial);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' see [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 3 is a generalization of the following result, which will be used in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' It is stated as [8, Theorem 4] and its proof is due to the referee of [7]: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E-function with coefficients in a number field K, and α ∈ Q be such that f(α) is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then f(α) ∈ K(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For the convenience of the reader, let us deduce this lemma from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let β = f(α), and L be a finite Galois extension of K(α) such that β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Since f(z) − β vanishes at α, Proposition 1 shows that for any σ ∈ Gal(L/K(α)) the E-function f(z) − σ(β) = f σ(z) − σ(β) vanishes at σ(α) = α, so that σ(β) = f(α) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This concludes the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' To prove Proposition 3, we first remark that the result is obvious if f is algebraic, hence a polynomial: we simply take P = f and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us now assume that f is transcendental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We argue by induction on the number of non-zero algebraic points α such that f(α) ∈ Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' this number is finite because f is transcendental and of Beukers’ theorem: any such α is a singularity of the first-order differential system satisfied by 1, f, f ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , f (µ−1) (where µ ≥ 1 is the minimal order of an inhomogeneous linear differential equation with coefficients in K(z) satisfied by f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If this number is 0, one may choose P = 0 and Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now if f(α) ∈ Q, Lemma 1 proves that f(α) belongs to K(α): there exists P0 ∈ K[X] such that f(α) = P0(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore the E-function f −P0, with coefficients in K, vanishes at α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 1 yields an E-function g0 with coefficients in K such that f = P0 + Dg0 where D is the minimal polynomial of α over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If g0(α) ∈ Q, the same procedure can be carried out with g0, leading to P1 ∈ K[z] and an E-function g1 with coefficients in K such that g0 = P1 + Dg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' After finitely many steps, this procedure terminates and provides gℓ such that gℓ(α) ̸∈ Q (see the proof of [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This concludes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 5 Structure of EK Let K be a subfield of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' As in the introduction, we denote by EK the ring of all values f(1) where f is an E-function with coefficients in K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' in particular EQ = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The following result contains Theorem 2 stated in the introduction (as a special case K = Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 12 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Elements of EK are linearly independent over Q if, and only if, they are linearly independent over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In other words, the K-algebras EK and Q are linearly disjoint, and the natural map EK ⊗K Q → E is a K-algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This result implies EK ∩ Q = K, which is an equivalent form of Lemma 1 stated in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN be E-functions with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If f1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(1) are linearly independent over Q, then obviously they are linearly independent over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Conversely, let us assume that they are linearly independent over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN be algebraic numbers, not all zero, such that λ1f1(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' + λNfN(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Up to a permutation of the indices we may assume that λ1 ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' then dividing by λ1 we assume that λ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let us consider a finite Galois extension L of K that contains λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then g(z) = �N i=1 λifi(z) is an E-function with coefficients in L, and it vanishes at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' For any σ ∈ Gal(L/K), Proposition 1 yields gσ(1) = 0, that is �N i=1 σ(λi)fi(1) = 0 since all fi have coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Summing these relations, as σ varies, yields N � i=1 TrL/K(λi)fi(1) = 0 with TrL/K(λi) = � σ∈Gal(L/K) σ(λi) ∈ K and TrL/K(λ1) = TrL/K(1) = [L : K] ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This is a non-trivial linear relation, with coefficients in K, between f1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' , fN(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This contradiction concludes the proof that elements of EK are linearly independent over Q if, and only if, they are linearly independent over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' What remains of Theorem 3 follows directly from this property (see [3, Chapter V, §2, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 6 A Galois action on values of E-functions In this section we define and study an action of the absolute Galois group of Q, namely Gal(Q/Q), on the set E of values of E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The definition is as follows: given σ ∈ Gal(Q/Q) and ξ ∈ E, there exists an E-function f such that ξ = f(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then we let σ(ξ) = f σ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' The crucial point is to prove that σ(ξ) depends only on σ and ξ, not on the choice of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Indeed if g is another E-function such that ξ = g(1), then f −g vanishes at the point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 1 shows that f σ−gσ vanishes at 1 too, so that gσ(1) = f σ(1): this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let K be a number field, and ξ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then ξ belongs to EK if, and only if, σ(ξ) = ξ for any σ ∈ Gal(Q/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' In other words, the fixed points of E under Gal(Q/K) are exactly the elements of EK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 13 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' If ξ ∈ EK and σ ∈ Gal(Q/K) then σ(ξ) = ξ by definition, since f σ = f for any E-function f with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Now let ξ ∈ E be such that σ(ξ) = ξ for any σ ∈ Gal(Q/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let f be an E- function such that f(1) = ξ, and L denote a finite Galois extension of K that contains all coefficients of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let σ ∈ Gal(L/K);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' then σ is the restriction to L of an element, again denoted by σ, of Gal(Q/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' We have σ(ξ) = ξ by assumption, and also σ(ξ) = f σ(1) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Summing the identity ξ = f σ(1) over all σ yields ξ = g(1) where g(z) = 1 [L:K] � σ∈Gal(L/K) f σ(z) is an E-function with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore ξ ∈ EK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' this concludes the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' This Galois action sheds a new light on the proof of Theorem 1 (see §3): it is very similar to Liouville’s proof that irrational algebraic numbers are not too well approximated by rationals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', are not Liouville numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Indeed let us recall briefly Liouville’s proof, stated in terms of Galois action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Let ξ be an algebraic number of degree d ≥ 2, and assume (for simplicity) that the extension Q(ξ)/Q is Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' To bound from below |qξ − p| for (p, q) ∈ Z2 \\ {(0, 0)}, consider ̟ := � σ∈Gal(Q(ξ)/Q) � qσ(ξ) − p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Then ̟ ̸= 0 (since σ(ξ) is irrational for any σ), and ̟ ∈ Q (since it is the norm of qξ − p with respect to the extension Q(ξ)/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Letting δ ∈ Z denote a positive integer such that δξ is an algebraic integer, we have δd̟ ∈ Z\\ {0} since OQ(ξ) ∩Q = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Therefore |δd̟| ≥ 1 so that δ−d ≤ |̟| ≤ |qξ − p|( ξ + 1)d−1Hd−1 by bounding |qσ(ξ) − p| trivially for σ ̸= Id, where H = max(|p|, |q|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Dividing by q yields |ξ − p/q| ≥ cH−d where c > 0 depends only on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Step 2 of the proof of Theorem 1 is very similar, except that elements of K ⊂ Q are replaced with values of E-functions in EK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' the lower bound used by Liouville (namely, δd̟ ∈ Z \\ {0} implies |δd̟| ≥ 1) is replaced accordingly by Shidlovskii’s lower bound recalled in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Andr´e, S´eries Gevrey de type arithm´etique I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Th´eor`emes de puret´e et de dualit´e, Annals of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 (2000), 705–740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Andr´e, Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 (2014), 449—467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Bourbaki, Elements of Mathematics, Algebra II, Chapters 4–7, Springer, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 14 [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 163 (2006), 369–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Bostan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Salvy, Minimization of differential equations and algebraic values of E-functions, preprint (2022), 37 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Feldman, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Nesterenko, Transcendental Numbers, in Encyclopaedia of Math- ematical Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 44: Number Theory IV (Springer, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Fischler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, On the values of G-functions, Commentarii Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 (2014), 313–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Fischler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, Arithmetic theory of E-operators, Journal de l’´Ecole polytech- nique – Math´ematiques 3 (2016), 31–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Fischler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, Microsolutions of differential operators and values of arithmetic Gevrey series, American J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='2 (2018), 317–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Lepetit, Le th´eor`eme d’Andr´e-Chudnovsky-Katz au sens large, North-West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 7 (2021), 83–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [11] Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Nesterenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Shidlovskii, On the linear independence of values of E- functions, Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 187 (1996), 1197–1211;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' translation from the russian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 187 (1996), 93–108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, Valeurs alg´ebriques de E-fonctions aux points alg´ebriques, unpublished note (2016), 4 pages, available at https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='fr/hal-03676576 [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Shidlovskii, Transcendental numbers, de Gruyter Studies in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 12, de Gruyter, Berlin, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Ab- handlungen Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=', Berlin, 1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Zudilin, On rational approximations of values of a certain class of entire functions, Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4, 555–590 (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' translation from the russian Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content='4, 89–124 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Fischler, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, 91405 Orsay, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Rivoal, Institut Fourier, CNRS et Universit´e Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' Keywords: E-functions, Andr´e-Beukers Theorems, Linear independence measures, Irra- tionality measures, Liouville numbers, Shidlovskii’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} +page_content=' MSC 2020: 11J82 (Primary), 11J91 (Secondary) 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtAzT4oBgHgl3EQfN_t_/content/2301.01158v1.pdf'} diff --git a/n9FST4oBgHgl3EQfMDgV/content/tmp_files/2301.13742v1.pdf.txt b/n9FST4oBgHgl3EQfMDgV/content/tmp_files/2301.13742v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..07ccb71d163677375ad9e8ebd66cfadf1dfa47f8 --- /dev/null +++ b/n9FST4oBgHgl3EQfMDgV/content/tmp_files/2301.13742v1.pdf.txt @@ -0,0 +1,1356 @@ +Ultrafast Umklapp-assisted electron-phonon cooling in magic-angle twisted bilayer +graphene +Jake Dudley Mehew,1 Rafael Luque Merino,2, 3, 4 Hiroaki Ishizuka,5 Alexander +Block,1 Jaime Díez Mérida,2, 3, 4 Andrés Díez Carlón,2, 3, 4 Kenji Watanabe,6 Takashi +Taniguchi,7 Leonid S. Levitov,8 Dmitri K. Efetov,3, 4 and Klaas-Jan Tielrooij1, 9, ∗ +1Catalan Institute of Nanoscience and Nanotechnology (ICN2), +BIST and CSIC, Campus UAB, 08193 Bellaterra (Barcelona), Spain +2ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology (BIST), Castelldefels 08860, Spain +3Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4, München 80799, Germany +4Munich Center for Quantum Science and Technology (MCQST), München, Germany +5Department of Physics, Tokyo Institute of Technology, Tokyo, Japan +6Research Center for Functional Materials, National Institute for Material Sciences, Tsukuba, Japan +7International Center for Materials Nanoarchitectonics, +National Institute for Material Sciences, Tsukuba, Japan +8Department of Physics, Massachusetts Institute of Technology, Cambridge, 02139 MA, USA +9Department of Applied Physics, TU Eindhoven, +Den Dolech 2, Eindhoven, 5612 AZ, The Netherlands. +Carrier relaxation measurements in moiré materials offer a unique probe of the microscopic in- +teractions, in particular the ones that are not easily measured by transport. Umklapp scattering +between phonons is a ubiquitous momentum-nonconserving process that governs the thermal con- +ductivity of semiconductors and insulators. +In contrast, Umklapp scattering between electrons +and phonons has not been demonstrated experimentally. Here, we study the cooling of hot elec- +trons in moiré graphene using time- and frequency-resolved photovoltage measurements as a direct +probe of its complex energy pathways including electron-phonon coupling. We report on a dramatic +speedup in hot carrier cooling of twisted bilayer graphene near the magic angle: the cooling time +is a few picoseconds from room temperature down to 5 K, whereas in pristine graphene coupling +to acoustic phonons takes nanoseconds. Our analysis indicates that this ultrafast cooling is a com- +bined effect of the formation of a superlattice with low-energy moiré phonons, spatially compressed +electronic Wannier orbitals, and a reduced superlattice Brillouin zone, enabling Umklapp scattering +that overcomes electron-phonon momentum mismatch. These results demonstrate a way to engineer +electron-phonon coupling in twistronic systems, an approach that could contribute to the fundamen- +tal understanding of their transport properties and enable applications in thermal management and +ultrafast photodetection. +INTRODUCTION +Moiré superlattices provide a novel material platform in which twist angle controls the effective lattice constant. +As the twist angle decreases, the larger moiré unit cell corresponds to a smaller electron momentum. This tunes the +relative strength of the kinetic energy of electrons and the interaction energy between them. In magic-angle twisted +bilayer graphene (MATBG), these interactions result in a rich phase diagram that includes superconductors, [1–4] +correlated insulators [5, 6] and orbital magnets. [2, 7] In transition metal dichalcogenides (TMDs), correlated insulating +[8, 9] and ferromagnetic states [10] are observed over a broad range of angles, with moiré excitons [11, 12] providing +a test-bed for exploring Hubbard model physics. [9] In addition, the moiré potential modifies the phonon spectra +for small twist angles. [13] This results in phonon renormalization in MoS2 homobilayers [14] and the emergence of +phonon minibands in twisted bilayer graphene. [15] Theoretical studies predict that the moiré potential strongly affects +electron-phonon coupling, [16–18] which has important implications for electrical transport, excited-state relaxation +dynamics, and beyond. +Excited-state relaxation measurements are particularly well-suited probes to quantitatively assess electron-phonon +coupling. The relaxation dynamics in graphene after excitation involve thermalization of high-energy carriers through +carrier-carrier scattering within tens of femtoseconds, [19] creating a hot carrier distribution that subsequently cools +via phonons. Inelastic electron-phonon scattering allows electrons to gain (lose) energy by the absorption (emission) +of a phonon. In graphene, cooling typically occurs via the emission of optical and acoustic graphene phonons, and +near-field coupling to substrate phonons. [20–27] Importantly, in all cases, cooling becomes increasingly slow for lower +lattice temperatures, with predicted electron-phonon cooling times in the nanosecond regime for the case of pristine +arXiv:2301.13742v1 [cond-mat.mes-hall] 31 Jan 2023 + +2 +graphene at cryogenic temperatures. [20] +Experimental studies of the relaxation dynamics of twisted bilayer graphene have so far been limited to large +twist angles (θ > 5◦). In these systems, a dark exciton state emerges between van Hove singularities, leading to +slower dynamics. [28, 29] At such relatively large angles, the moiré potential has limited influence on electron-phonon +coupling. [15, 18] Recent Raman spectroscopy measurements suggest an enhanced electron-phonon coupling strength +for small twist angles around the magic angle (θ ≈ 1.1◦). [30] However, direct experimental measurements of moiré- +enhanced electron-phonon coupling and its implications for cooling dynamics, are lacking, nor is there any clear +understanding of the origin of the enhanced coupling. +In this paper, we report the observation of ultrafast cooling in magic-angle twisted bilayer graphene (MATBG) +through Umklapp-assisted electron-phonon scattering. We directly probe the electron-phonon interaction by mea- +suring carrier cooling dynamics using two well-established optoelectronic techniques – time-resolved photovoltage +microscopy [19, 31, 32] and continuous-wave photomixing [33, 34]. We make a direct comparison between a non- +twisted Bernal bilayer graphene sample (BLG, θ = 0◦, see Fig. 1a) and a near-magic-angle twisted bilayer graphene +sample (MATBG, θ = 1.24◦, see Fig. 1b). At low temperature, the cooling dynamics are much faster in MATBG than +in non-twisted bilayer graphene, see Fig. 1c. This unexpected result highlights the crucial role of the moiré pattern +and suggests the emergence of an enhanced electron-phonon interaction in small twist angle systems. We explain the +observed relaxation dynamics using a theoretical model based on Umklapp-assisted electron-phonon scattering, which +can occur in both the dispersive and flat bands of MATBG, see Fig. 1d. The Umklapp processes are enabled by the +presence of compressed electronic Wannier orbitals (see Fig.1e), and the superlattice with reduced Brillouin zone (see +Fig. 1f). +RELAXATION DYNAMICS +We study relaxation dynamics in hBN-encapsulated MATBG and BLG Hall bar devices as shown in Fig. 1a-b +(see Methods for details on the device fabrication and characterization). These devices enable both electrical and +optoelectronic measurements, as they are equipped with a split gate that we employ to create a photoactive pn- +junction region. The resistance map as a function of the gate voltage applied to each of the two sides of the split +gate, shown in Extended Data Figure 1, displays clear peaks at the usual Dirac points with vanishing carrier density. +The MATBG device exhibits additional peaks at integer fillings of the superlattice unit cell. By illuminating the +pn-junction with light, a photovoltage is generated via the photothermoelectric effect. This effect has a characteristic +six-fold symmetry in dual-gate photovoltage maps, as shown in Extended Data Fig. 2 for both devices. This indicates +that the measured photovoltage is a direct probe of the electron temperature. [35] +We study hot electron relaxation using ultrafast time-resolved photovoltage microscopy (TrPV) as implemented in +Refs. [19, 24] and continuous-wave heterodyne photomixing (CW-PM) as implemented in Ref. [34] In the former, +ultrashort laser pulses are incident upon the pn-junction whereas for the latter two continuous wave lasers are used. In +both cases we probe the generated photovoltage. These two techniques allow us to obtain directly the carrier cooling +dynamics – in the time domain by varying the time delay between two ultrashort laser pulses, and in the frequency +domain by varying the spectral detuning of two spectrally narrow laser beams. Both techniques independently show +that charge carriers cool much faster in MATBG than in BLG at low temperature, see Fig. 2a (and Extended Data +Figs. 3-6). In BLG the cooling time increases from 3 ps to 25 ps as the temperature decreases from 300 K to 5 K, +which is expected as it takes longer for hot carriers to couple to phonons at lower temperature due to the reduced +phonon occupation. [20, 21] Surprisingly, for MATBG the cooling time remains short, around 3 ps, across a broad +temperature range (5-300 K). This suggests the involvement of low-energy phonons that still have occupation at such +low temperature, which are likely phonons related to the superlattice. Indeed, the moiré potential breaks the original +linear phonon dispersion into minibands with enhanced density of states. [15] The energy of the lowest band is below +1 meV corresponding to temperatures below 10 K. +In order to understand the origin of the observed cooling dynamics, we first consider the case of relaxation through +energy transfer to phonons in non-twisted BLG. Coupling to optical phonons is highly inefficient at low temperature +due to the large optical phonon energy, which is > 160 meV, corresponding to T > 2000 K. [26] Coupling between +electrons and acoustic phonons is normally also inefficient due to the reduced phase space available for scattering, +and would give cooling times well above a nanosecond below 25 K. [20] The presence of defects can help overcome +the electron-phonon momentum mismatch through disorder-assisted cooling, which speeds up this acoustic phonon +cooling process. [21–23] However, even with this mechanism, we expect cooling times between 10−10 s and 10−8 s for +the lowest temperatures, depending on the electron mean free path (see Methods). We therefore consider diffusive +cooling, where electronic heat diffuses out of the initially excited hot spot, thus leading to a lower average electron + +3 +FIG. 1. Excited carrier relaxation in MATBG. a-b, Illustration of the hBN-encapsulated BLG device with 0◦ twist +angle (a) and the hBN-encapsulated MATBG device with twist angle 1.24◦ (b), each equipped with split gates. By applying +voltages of opposite sign (±V ) to the split gates, we create a pn-junction (the interface between yellow and orange regions). +Illuminating the junction generates a photovoltage via the photothermoelectric effect, which is proportional to the electron +temperature (Te). We obtain the temperature dynamics either by using two ultrashort laser pulses separated in time by a +variable temporal delay, [19, 31, 32] or by using two spectrally narrow laser beams with variable frequency detuning. [33, 34] c, +Photovoltage as a function of time delay for a lattice temperature of 25 K. The decay, which represents the cooling dynamics, is +much faster in MATBG (blue pluses) than BLG (red circles). d, Schematic of the MATBG band structure. Umklapp scattering +processes (solid arrow) allow for efficient electron (black circle) relaxation via coupling to moiré phonons (wiggly lines). These +Umklapp processes can occur in both the flat and the dispersive bands. The dashed arrows represent the equivalent final state +in the first Brillouin zone. e, Schematic of the compressed Wannier orbitals of radius ξ. Electrons are localised to AA sites in +the reconstructed superlattice. f, Umklapp scattering processes (blue arrows) couple electrons in the first Brillouin zone (white +hexagon) to large-momentum phonons in higher-order Brillouin zones (blue hexagons). +temperature. [25] In this diffusive cooling mechanism, the cooling time will thus depend on laser spot size. Indeed, +for non-twisted BLG, we observe an increase in cooling time for larger spot sizes, which is largest for the lowest +temperatures (25 K, 50 K), see Fig. 2b-c. At 100 K, the cooling length is shorter and therefore diffusive cooling +has a smaller contribution. We thus understand the cooling dynamics for non-twisted BLG from a combination of +disorder-assisted and diffusive cooling. Indeed, our calculations of the cooling time based on these two mechanisms +are close to the experimentally observed ones (see Methods for details on the calculations). Importantly, for MATBG +we observe no dependence of the cooling time on spot size (see Fig. 2b-c), which suggests that diffusive cooling does +not play a role for this system. +We next study the effect of changing the laser power and therefore initial electron temperature, see Fig. 3a. This +corresponds to increasing the population of the dispersive band (Fig. 3b). The peak power density is roughly five +orders of magnitude larger for our pulsed laser experiment (TrPV) than our continuous wave experiment (CW-PM). +For the non-twisted BLG device, we observe somewhat slower cooling at higher initial electron temperature, which has +also been observed for high-quality monolayer graphene samples and was ascribed to a bottleneck involving optical +and acoustic phonons. [26] Interestingly, the role of electron temperature is minor for the relaxation dynamics of +MATBG, suggesting that there are no electron-phonon or phonon-phonon bottlenecks. This result indicates that + +a +c +人e +(a.u.) +Photovoltage (a.u.) +BLG +0° +hBN +1.24° ++ +-75 +-100 +-50 +-25 +0 +25 +50 +75 +100 +Delay time (ps) +f +d +e +Umklapp-assisted cooling +Localised +MATBG +electrons +Moire phonons +AA +2 +Moiréphonons +AB +dns +Normal +1.24°4 +FIG. 2. Relaxation mechanisms in MATBG and BLG. a, Cooling time as a function of lattice temperature. In MATBG +(1.24◦, blue pluses), the cooling time is constant between 5 K and 300 K (3 ps, blue line). For BLG (0◦, red circles), it is +greater at lower temperatures. b, Laser spot size dependence of the cooling time. The strong dependence in BLG at 25 K and +50 K is a signature of diffusive cooling. This effect is weaker at 100 K, where disorder-assisted cooling becomes significant. The +effect is absent in MATBG for these spot sizes. The filled (open) shapes are measured using the TrPV (CW-PM) technique. +Error bars represent the statistical spread across different gate voltages. The thick blue line in a and b represents the cooling +time obtained from the low temperature model of Umklapp-assisted cooling (see Main text). c Schematics of diffusive cooling +for BLG (upper) and its absence for MATBG (lower). +direct optical phonon emission does not play a role in MATBG. The much faster cooling for MATBG compared to +BLG at low temperatures thus suggests that a completely different mechanism is responsible for cooling, outcompeting +all currently known cooling mechanisms in non-twisted graphene. +We therefore explore the effect of the superlattice on electron cooling by examining the cooling time in MATBG +as a function of filling factor (ν) which represents the electronic occupation of the superlattice unit cell. For most +filling factors (|ν| < 4), we observe a nearly constant cooling time of 3 ps across a wide temperature range (5-300 +K). However, at ν = ±4 the cooling time increases dramatically. Low-temperature transport measurements on the +same device reveal an increase in resistance at the same voltages, see Fig. 3d, which confirms the full filling of the +superlattice unit cell. In Extended Data Figure 7, we show that the cooling time increases strongly upon increasing +laser power at full filling for a second MATBG device (θ = 1.08◦). The strong dependence of cooling upon the flat +band filling - with the cooling rates high at partial filling and lower at full filling - indicates that the moiré pattern +and its low-energy phonons are crucial for explaining the ultrafast cooling dynamics observed in MATBG. +ORIGIN OF ENHANCED COOLING +To gain insight into the different mechanisms that govern electron-lattice cooling pathways, we consider in detail +the microscopic electron-phonon scattering processes in the MATBG system. To this end, we consider a four-band +model consisting of two nearly-flat and two dispersive bands (Fig. 1d). There are two main types of electron-phonon +scattering in this model, interband and intraband. The intraband processes for the intra-dispersive band and intra- +flat-band transitions are different and must be evaluated separately. At temperatures higher than the bandgap, which +corresponds to the highest temperatures in our measurement, the electrons are thermally excited to the dispersive +bands allowing both dispersive and flat bands to contribute to cooling. To the contrary, when the electron temperature +is low, all carriers reside in the flat band. +Therefore, we consider two regimes: i) the high temperature regime +(T ∼ 150 − 300 K), where the dispersive bands contribute to the cooling process, and ii) the low temperature regime +(T ∼ 10 K), wherein cooling is dominated by intra-flat-band processes. In both cases, we consider both the Umklapp +and normal scattering contributions, finding that at the temperatures of interest (T > 10 K) Umklapp scattering +consistently wins over normal scattering. +For the first regime (high temperatures) we consider a four-band model consisting of two flat bands of bandwidth + +a +b +c +30 +40 +0. +0° +35 +1 +25 +30 +: +(sd) +25K +25 +time ( +time +50K +Diffusive cooling +15 +20 +Cooling +100K +Disorder-assisted cooling +15 +TOI +0 +1.24° +10 +2K +5I +O+ +100K +5 +1.24° +0 +Umklapp-assisted cooling +-diffusive cooling +57 +$ +Non +± +0 +00 +1.0 +1.5 +2.0 +2.5 +100 +200 +300 +3.0 +Lattice temperature (K) +Spot size (μm)5 +FIG. 3. Origin of enhanced cooling in MATBG. a, Dependence of cooling time on peak power density for BLG (red +circles) and MATBG (blue pluses). The filled (open) shapes are measured using the TrPV (CW-PM) technique. The error +bars signify the one sigma confidence interval from the fitting algorithm. b, e Schematics of cooling power in MATBG for part +filling (b) and full filling (e) of the flat bands. For part filling, the interband transition is not rate-limiting as evidenced by the +absence of a power dependence in a. At full filling, cooling times are longer due to the interband bottleneck effect illustrated +in panel (e). c-d, Gate dependence of cooling time, c, and four terminal resistance acquired at T = 35 mK (Rxx), d. Orange +shaded region highlights full filling of the moiré unit cell, where Rxx and cooling time increase. The thick blue line in a and c +represents the cooling time obtained from the low temperature model of Umklapp-assisted cooling (see Main text) +. +W and two dispersive bands with the eigenstate energies ε > ∆ and ε < −∆ (∆ > W), see Fig. 4b. The dispersive +bands are separated from the flat bands by a gap ∆ − W (see Methods for details). A direct analysis based on +Boltzmann theory yields cooling rates dominated by the intra-band processes in the dispersive bands, whereas the +interband processes have a minor contribution. Accounting for the Umklapp processes, we estimate the cooling rates +as τ −1 = +6ρ1 +πTel +� +m(∥g1,1 +m ∥2 + ∥g−1,−1 +m +∥2)ω2 +m, where ρ1 is the density of states of the dispersive particle and hole bands +labeled by n = ±1, Tel is the electron temperature, gn,n +m +is the electron-phonon coupling constant in the nth band and +ωm is the phonon energy in the mth phonon band. Direct calculation gives cooling rates that are independent of the +lattice temperature Tph, in agreement with the observed dynamics, see Fig. 2a. +For the regime of low temperatures, we describe the system using a model of a flat band with electron and hole +subbands (see Methods for a detailed description of the model). For a quantitative comparison with the experimental +results shown in Fig. 4a, we calculate the cooling power J accounting for the Umklapp processes assuming the Wannier +function radius ξ = a/6 where a is the lattice parameter for the moiré structure [36], see Fig. 1c. The cooling rate +τ −1 is estimated from the calculated cooling power and specific heat using τ −1 = J/C(Tel − Tph); here we calculate +the specific heat C using the fluctuation formula, Eq. 2 in the Methods section. In that temperature values are not +constrained by the flat-band width and can be as large as the bandgap. The filling dependence of the cooling rate is +shown in Fig. 4a. The calculated Umklapp-assisted cooling times as a function of the filling factor are seen to be in + +a +30 +0° + 25K +25 +50K +Cooling time (ps) +20 +100K +15 +1O +10 +25K +Part filling, v<±4 +50K +1.24° +5 +F +100K +0 +0 +1 +2 +3 +4 +5 +Peak power density (GW cm-2) +C +e +30 +25 +Cooling time (ps) +20 +5K +15 +300K +10 +５ +0 +102 +Full filling, V=±4 +(kΩ2) +101 +XX +100 +R +10-1 +-4-3-2-101 +3 +Filling factor, v6 +FIG. 4. Quantitative comparison with Umklapp-assisted cooling. a, Comparison between calculated (solid line) and +experimental (symbols) cooling times for MATBG at 5 K and 10 K (upper and lower panels). The grey shaded region allows +for uncertainty in the value of the deformation potential (D = 16 ± 4 eV). b, Schematic of the model used for the calculations +with two dispersive and two flat bands separated by an energy gap (∆ − W). γ1 and γ0 represent intra-dispersive-band and +intra-flat-band scattering processes, respectively. The low temperature calculations shown in (a) consider only γ0. +agreement with the experimental results. For the calculated cooling times, we used a deformation potential of 16 eV. +This is close to the values reported for single-layer graphene (10-30 eV). [22, 37–39] We therefore conclude that the +Umklapp-assisted carrier cooling model reproduces the main experimental findings. +We note that, here, we did not take account of the disorder-assisted cooling processes. [21] In pristine graphene, the +bottleneck due to limited phase space due to the small Fermi surface is relieved by disorder scattering. The situation in +MATBG differs from that in pristine graphene in two ways. First, as the superlattice provides additional momentum +recoil, MATBG does not require defects and/or disorder for electron-lattice cooling. Second, the formation of highly +localized Wannier orbitals at AA sites in the moiré pattern modulates the electron-phonon interaction. These effects +produce strong coupling of the electrons to moiré phonons even in the absence of disorder. [36] +OUTLOOK +Importantly, the cooling measurement is predominantly sensitive to the electron-phonon interactions, and is less +sensitive to the electron-electron interactions. This presents a unique window of opportunity for probing underlying +physics, and an advantage compared to other measurements types that do not easily separate these two interactions. +The finding that electron-phonon Umklapp scattering dominates ultrafast electron-phonon cooling is likely to have +important implications for MATBG physics. Electron-phonon scattering plays an important role in charge transport, +limiting the carrier mobility at high temperatures. This interaction also mediates the pairing interaction in Bardeen- +Cooper-Schrieffer superconductors. Understanding the electron-phonon coupling could give important insights into +the origin of superconductivity in MATBG. [16, 40] For metals, electron-electron Umklapp scattering gives rise to finite +electrical resistance at low temperatures. In graphene/hBN superlattices and MATBG, this effect dominates transport +at temperatures up to 10 K or higher, leading to excess resistivity and degradation of charge carrier mobility. [41–43] +In MATBG, electron-phonon Umklapp scattering could explain some of the open questions from electrical transport +measurements, such as the strange metal phase or the role of phonons in superconductivity. [16, 40] Finally, the +ultrafast Umklapp-assisted electron-phonon cooling, enhanced density of states, and rich phase diagram are appealing +for single-photon detection in the highly sought after mid-IR wavelength range. [44, 45] +METHODS +Device fabrication The MATBG devices were fabricated using a cut and stack technique. All flakes were first +exfoliated on a Si/SiO2 (285 nm) substrate and later picked up using a polycarbonate (PC)/polydimethylsiloxane +(PDMS) stamp. All the layers were picked up at a temperature of ∼ 100◦C. We used an AFM tip to cut the graphene +in order to avoid strain during the pick-up process. The PC/PDMS stamp picks up first the top graphite layer, the + +a +b +T=5 K +Cooling time (ps) +20 +Yi +E(k) +10 +D=16±4 eV +W +T=10 K +Cooling time (ps) +15 +10 +5 +0 +1 +2 +m +4 +Filling factor, Iv7 +top hBN and the first graphene layer. Before picking up the second graphene layer, we rotate the stage by an angle +of 1.1 − 1.2◦. Finally, the stamp picks up the bottom hBN and bottom graphite gates. We drop the finalized stack +on a Si/SiO2 substrate by melting the PC at 180◦C, see Supplementary Figure S1a. The resulting stack is etched +into a Hall bar using a CHF3/O2 plasma and a 1D contact is formed by evaporating Cr (5 nm)/Au (50 nm), see +Supplementary Figure S1b. We etch a narrow channel of ∼ 150 nm in the top gate using an O2 plasma. Before +etching the top gate, the device was characterized at T = 35 mK to identify the pair of contacts closest to the magic +angle (θ ∼ 1.1◦). The junction was made in between this pair of contacts. +Twist angle extraction The twist angle θ is extracted from the superlattice carrier density of the full band ns +by applying the relation ns = 8θ2/ +√ +3a2, where a = 0.246 nm is the graphene lattice constant. First, we calibrate +the gate induced carrier density using the Hall effect data at ±1 T. In the carrier density region close to charge +neutrality, the Hall carrier density nH = −B/eRxy should closely follow the gate induced carrier density nH = n, +see Supplementary Figure S2. By plotting nH vs Vg and fitting this slope around charge neutrality we can obtain +the capacitance of the device and therefore extract the real carrier density n. Then we extract the carrier density +corresponding to a fully filled superlattice unit cell, in this case we find it to be ns = (3.58 ± 0.10) × 1012 cm−2. +Finally using the above relation we extract a twist angle θ = 1.24◦ ± 0.02◦. In Supplementary Note 1, we verify that +there is minimal twist angle disorder in the junction region. +Transport Measurements Low-temperature transport measurements were carried out in a dilution refrigerator +(Bluefors SD250) with a base temperature of 20 mK. Standard low-frequency lock-in techniques (Stanford Research +SR860 amplifiers) were used to measure Rxx with an excitation current of 10 nA at a frequency of 13.11 Hz. +Optoelectronic measurements In time-resolved photovoltage (TrPV) experiments, we vary the delay time (dt) +between the arrival of two ultrafast pulses. [31] [32] [19] Due to the non-linear relationship between carrier temperature +and optical heating, we observe a dip in the photovoltage when the two pulses arrive at the same time (dt = 0), see +Fig. 1c and Extended Data Figs. 3 and 4. At longer delay times, the signal recovers to its maximal value. We obtain +the cooling time by describing the observed dynamics with an exponential function. For heterodyne photomixing +(CW-PM) experiments, the wavelength detuning between the two continuous wave lasers creates an optical beating. +[33, 34] The photovoltage oscillates at the beating frequency. Due to the competition between beat frequency (Ω) and +the characteristic cooling time (τe), we observe a peak for Ω = 0 whereas the oscillations are damped when Ω−1 ≪ τe, +see Extended Data Figs. 5 and 6. The frequency response takes the form of a Lorentzian function of width Γ, from +which we extract the cooling time as: Γ = 1/πτe. [33] +Estimating cooling times in untwisted graphene The hot electron cooling time for energy transfer to acoustic +phonons in monolayer graphene is given by τAP ≈ 848/(D2T 2 +L) [µs], [20] where D is the deformation potential +in eV. This expression is valid in the neutral limit (TF < Te) and close to equilibrium (Te ≳ TL). +Te/L/F is +the electron/lattice/Fermi temperature. [20] Taking D = 20 eV, we calculate a cooling time of τAP = 3.4 ns for +TL = 25 K. +In disorder-assisted or supercollision cooling, [21–23] the dependence on lattice temperature is given by: +τSC = +α +3ATL +, with +α = 2πEF k2 +B +3ℏ2v2 +F +and A = 9.62g2ν2(EF )k3 +B +ℏkF ℓ +. +Here, g is the electron-phonon coupling, ν(EF ) is the density of states at the Fermi level per valley/spin flavour, kF is +the Fermi wavevector and ℓ is the mean free path. In high-quality samples and at cryogenic temperatures, the device +size typically limits the latter. For low doping levels (1012 cm−2), 0.1 < ℓ < 2 µm and T = 25 K, τSC = 0.5 − 11 ns. +Cooling due to lateral diffusion The lateral diffusion of photoexcited carriers reduces the hot electron temper- +ature when the cooling length is greater than the laser spot size. This effect is particularly relevant in high-mobility +samples, as the Wiedemann-Franz law relates electrical to thermal conductivity. [25] At low lattice temperatures +efficient heat conduction manifests in our experiments as a shorter cooling time. By considering the spatial evolution +of a Gaussian heat spot induced by the laser pulse, [26] we describe the temperature dynamics by: +Te(t) = 2πApuApr +σ2 +puσ2 +pr +σ2pu + σ2pr + 2Dt, +where A and σ are the peak intensity of the pump (pu) and probe (pr). Clearly, this effect is greater for smaller spot +sizes and larger electronic heat diffusivities (D). Using a diffusivity of D = 750 cm2s−1, and pump-probe spot sizes +of σ ≈ 0.9 µm we find a cooling time of τdiff ≈ 18 ps. For σ ≈ 1.4 µm, τdiff ≈ 45 ps, see Fig. 2b. + +8 +Cooling rate at low temperatures The cooling rate in Fig. 3f is estimated by +J(Tel,Tph) +C(Tel)(Tel−Tph), where J is the +cooling power, C(Tel) is the electron specific heat, and Tel (Tph) is the electron (phonon) temperature. To evaluate J +and C, we consider an effective two-band model similar to pristine graphene used in Ref. [36]. Following the previous +study, we use the electron-phonon interaction for the Wannier orbital radius ξ = a/6 where a is the lattice parameter. +In the Boltzmann theory, the cooling power J by electron-phonon scattering reads [36] +J = +� +n,n′ +Jn,n′, +Jn,n′ = 2π +V 2 +� +m,⃗k,⃗k′ +∥gnn′ +⃗k−⃗k′,m∥2ω2 +⃗k−⃗k′,mN⃗k−⃗k′,m +× +� +f⃗k′n′[1 − f⃗kn]e +βphω⃗k−⃗k′,m − f⃗kn[1 − f⃗k′n′] +� +× δ(εn′ − εn − ω⃗k−⃗k′,m), +(1) +where Jn,n′ is the contribution from the scattering between nth and n′th bands, V is the volume of the system, +gnn′ +⃗k−⃗k′,m is the coupling constant, ε⃗kn is the one-particle eigenenergy of the eigenstate in nth band with momentum +⃗k, ω⃗qm is the phonon eigenenergy in the mth band with momentum ⃗q, and βel = 1/kBTel (βph = 1/kBTph) is the +inverse temperature of electrons (phonons) with kB being the Boltzmann constant, respectively; f⃗kn = +1 +eβe(ε⃗kn−µ)+1 +and N⃗qm = +1 +eβphω⃗km−1 are respectively the Fermi and Bose distribution functions. The estimation of specific heat uses +the fluctuation formula +C(T) = kB +� +⟨ε2 +n⃗k⟩ − ⟨εn⃗k⟩2 +⟨1⟩ +� +, +(2) +⟨On⃗k⟩ = +� +n +� +dkd +(2π)d +β2On⃗k +4 cosh2 � β(εn⃗k−µ) +2 +�. +(3) +Note that the common formula for Fermi-degenerate electron systems does not apply here as the temperature +exceeds the Fermi energy at T ≳ 100 K. This model gives a good approximation when the temperature is much lower +than the energy gap separating the flat band from high-energy dispersive bands. +Cooling rate at high temperatures At high temperatures, we cannot neglect the high-energy bands because the +electron temperature exceeds the band gap. In such a case, the Umklapp scattering involving high-energy phonons +contributes to electron cooling due to a large number of high-energy phonons. Hence, we also expect that Umklapp +scattering plays a key role in the high temperature regime. +To study the electron-lattice cooling involving the interband processes, we assume the electrons only couple to +phonons with energies below a cutoff Λph. This assumption is justifiable in a system where the electron-phonon +coupling between the electrons and the acoustic phonons reduces exponentially as the momentum increases. In a +system with compact Wannier orbitals, Λph becomes a few times higher than the energy of folded acoustic bands. +Hence, a large Λph, considerably larger than the phonon bandwidth of the folded acoustic phonons, represents the +enhanced coupling by compact Wannier orbitals. Below, we label the folded acoustic bands by an integer m and +define the high-temperature limit as Tel > Tph ≫ Λph. +At high temperatures, the cooling power in Eq. (1) reads +Jnn′ = π +V +� +m +∥gnn′ +m ∥ω2 +mρnρ′ +n [Tel − Tph] × +� +tanh(β(bm +nn′ − µ) +2 +) − tanh(β(am +nn′ − µ) +2 +) +� +, +where ρn is the density of states (DOS) for the nth band (we assume a constant DOS with the bandwidth Wn), and +am +nn′ = max(ε− +n − ε− +n′ − ωm) [bm +nn′ = min(ε+ +n − ε+ +n′ − ωm)] with ε± +n being the energy of the top and bottom edge of the +electron band. Here, we approximated the phonon energy as ωn⃗k ∼ ωn considering the small Brillouin zone, and the +coupling constant gnn′ +m (⃗k) ∼ gnn′ +m +which is valid in the small orbital radius limit. +We apply the above formula to a four-band model consisting of two flat and two dispersive bands. The two flat +bands are at energies 0 ≤ ε ≤ W and −W ≤ ε ≤ 0 with DOS ρ0, and the two dispersive bands are W < ∆ ≤ ε ≤ Λ + +9 +and −Λ ≤ ε ≤ −∆ < −W with DOS ρ1 (Fig. 4b). To the leading order in Tel, the cooling power reads +J = π +� +m +(∥g1,1 +m ∥2 + ∥g−1,−1 +m +∥2)ω2 +m[Tel − Tph]ρ2 +1. +Hence, the cooling rate becomes τ −1 = +6ρ1 +πV Tel +� +m(∥g1,1 +m ∥2 + ∥g−1,−1 +m +∥2)ω2 +m, independent of phonon temperature, Tph. +SUPPLEMENTARY INFORMATION +This article has an accompanying supplementary file. +ACKNOWLEDGEMENTS +We would like to thank Nick Feldman for his contribution to preliminary experiments. ICN2 was supported by the +Severo Ochoa program from Spanish MINECO Grant No. SEV-2017-0706. R.L.M. acknowledges that this project has +received funding from the “Secretaria d’Universitats I Recerca de la Generalitat de Catalunya, as well as the European +Social Fund (L’FSE inverteix en el teu futur)—FEDER. H.I. acknowledges support from JSPS KAKENHI (Grant +Numbers JP19K14649). J.D.M. acknowledges support from the INphINIT ‘la Caixa’ Foundation (ID 100010434) +fellowship programme (LCF/BQ/DI19/11730021). K.W. and T.T. acknowledge support from the JSPS KAKENHI +(Grant Numbers 19H05790 and 20H00354 and 21H05233). K.J.T. acknowledges funding from the European Union’s +Horizon 2020 research and innovation program under Grant Agreement No. 804349 (ERC StG CUHL), RYC fellowship +No. RYC-2017-22330 and IAE project PID2019-111673GB-I00. +∗ Correspondence to: klaas.tielrooij@icn2.cat +[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43 (2018). +[2] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al., Nature +574, 653 (2019). +[3] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Science +363, 1059 (2019). +[4] P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K. +Efetov, Nature 583, 375 (2020). +[5] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, +E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature 556, 80 (2018). +[6] D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, +Nature 582, 198 (2020). +[7] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, +Science 365, 605 (2019). +[8] E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta, et al., +Nature 579, 359 (2020). +[9] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan, et al., Nature +579, 353 (2020). +[10] X. Wang, C. Xiao, H. Park, J. Zhu, C. Wang, T. Taniguchi, K. Watanabe, J. Yan, D. Xiao, D. R. Gamelin, et al., Nature +604, 468 (2022). +[11] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray, D. G. Mandrus, J. Yan, W. Yao, and X. Xu, Nature 567, 66 +(2019). +[12] E. M. Alexeev, D. A. Ruiz-Tijerina, M. Danovich, M. J. Hamer, D. J. Terry, P. K. Nayak, S. Ahn, S. Pak, J. Lee, J. I. +Sohn, et al., Nature 567, 81 (2019). +[13] M.-L. Lin, Q.-H. Tan, J.-B. Wu, X.-S. Chen, J.-H. Wang, Y.-H. Pan, X. Zhang, X. Cong, J. Zhang, W. Ji, P.-A. Hu, K.-H. +Liu, and P.-H. Tan, ACS Nano 12, 8770 (2018), pMID: 30086224, https://doi.org/10.1021/acsnano.8b05006. +[14] J. Quan, L. Linhart, M.-L. Lin, D. Lee, J. Zhu, C.-Y. Wang, W.-T. Hsu, J. Choi, J. Embley, C. Young, et al., Nature +materials 20, 1100 (2021). +[15] M. Koshino and Y.-W. Son, Phys. Rev. B 100, 075416 (2019). +[16] F. Wu, A. H. MacDonald, and I. Martin, Phys. Rev. Lett. 121, 257001 (2018). +[17] Y. W. Choi and H. J. Choi, PHYSICAL REVIEW B 98, 10.1103/PhysRevB.98.241412 (2018). +[18] M. Koshino and N. N. T. Nam, PHYSICAL REVIEW B 101, 10.1103/PhysRevB.101.195425 (2020). + +10 +[19] K.-J. Tielrooij, L. Piatkowski, M. Massicotte, A. Woessner, Q. Ma, Y. Lee, K. S. Myhro, C. N. Lau, P. Jarillo-Herrero, +N. F. van Hulst, et al., Nature nanotechnology 10, 437 (2015). +[20] R. Bistritzer and A. H. MacDonald, Phys. Rev. Lett. 102, 206410 (2009). +[21] J. C. W. Song, M. Y. Reizer, and L. S. Levitov, PHYSICAL REVIEW LETTERS 109, 10.1103/PhysRevLett.109.106602 +(2012). +[22] M. W. Graham, S.-F. Shi, D. C. Ralph, J. Park, and P. L. McEuen, NATURE PHYSICS 9, 103 (2013). +[23] J. F. Kong, L. Levitov, D. Halbertal, and E. Zeldov, Phys. Rev. B 97, 245416 (2018). +[24] K.-J. Tielrooij, N. C. H. Hesp, A. Principi, M. B. Lundeberg, E. A. A. Pogna, L. Banszerus, Z. Mics, M. Massicotte, +P. Schmidt, D. Davydovskaya, D. G. Purdie, I. Goykhman, G. Soavi, A. Lombardo, K. Watanabe, T. Taniguchi, M. Bonn, +D. Turchinovich, C. Stampfer, A. C. Ferrari, G. Cerullo, M. Polini, and F. H. L. Koppens, NATURE NANOTECHNOLOGY +13, 41+ (2018). +[25] M. Massicotte, G. Soavi, A. Principi, and K.-J. Tielrooij, Nanoscale 13, 8376 (2021). +[26] E. A. A. Pogna, X. Jia, A. Principi, A. Block, L. Banszerus, J. Zhang, X. Liu, T. Sohier, S. Forti, K. Soundarapandian, +B. Terres, J. D. Mehew, C. Trovatello, C. Coletti, F. H. L. Koppens, M. Bonn, H. Wang, I, N. van Hulst, M. J. Verstraete, +H. Peng, Z. Liu, C. Stampfer, G. Cerullo, and K.-J. Tielrooij, ACS NANO 15, 11285 (2021). +[27] L. Kim, S. Kim, P. K. Jha, V. W. Brar, and H. A. Atwater, NATURE MATERIALS 20, 805+ (2021). +[28] H. Patel, R. W. Havener, L. Brown, Y. Liang, L. Yang, J. Park, and M. W. Graham, NANO LETTERS 15, 5932 (2015). +[29] H. Patel, L. Huang, C.-J. Kim, J. Park, and M. W. Graham, NATURE COMMUNICATIONS 10, 10.1038/s41467-019- +09097-x (2019). +[30] A. C. Gadelha, D. A. A. Ohlberg, C. Rabelo, E. G. S. Neto, T. L. Vasconcelos, J. L. Campos, J. S. Lemos, V. Ornelas, +D. Miranda, R. Nadas, F. C. Santana, K. Watanabe, T. Taniguchi, B. van Troeye, M. Lamparski, V. Meunier, V.-H. +Nguyen, D. Paszko, J.-C. Charlier, L. C. Campos, L. G. Cançado, G. Medeiros-Ribeiro, and A. Jorio, Nature 590, 405 +(2021). +[31] A. Urich, K. Unterrainer, and T. Mueller, NANO LETTERS 11, 2804 (2011). +[32] D. Sun, G. Aivazian, A. M. Jones, J. S. Ross, W. Yao, D. Cobden, and X. Xu, NATURE NANOTECHNOLOGY 7, 114 +(2012). +[33] M. M. Jadidi, R. J. Suess, C. Tan, X. Cai, K. Watanabe, T. Taniguchi, A. B. Sushkov, M. Mittendorff, J. Hone, H. D. +Drew, M. S. Fuhrer, and T. E. Murphy, PHYSICAL REVIEW LETTERS 117, 10.1103/PhysRevLett.117.257401 (2016). +[34] M. A. Aamir, J. N. Moore, X. Lu, P. Seifert, D. Englund, K. C. Fong, and D. K. Efetov, Nano Letters 21, 5330 (2021), +pMID: 34101476, https://doi.org/10.1021/acs.nanolett.1c01553. +[35] N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and +P. Jarillo-Herrero, Science 334, 648 (2011), https://www.science.org/doi/pdf/10.1126/science.1211384. +[36] H. +Ishizuka, +A. +Fahimniya, +F. +Guinea, +and +L. +Levitov, +Nano +Letters +21, +7465 +(2021), +pMID: +34515488, +https://doi.org/10.1021/acs.nanolett.1c00565. +[37] D. K. Efetov and P. Kim, PHYSICAL REVIEW LETTERS 105, 10.1103/PhysRevLett.105.256805 (2010). +[38] J.-H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, NATURE NANOTECHNOLOGY 3, 206 (2008). +[39] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, +and J. Hone, NATURE NANOTECHNOLOGY 5, 722 (2010). +[40] T. J. Peltonen, R. Ojajärvi, and T. T. Heikkilä, Phys. Rev. B 98, 220504 (2018). +[41] J. R. Wallbank, R. K. Kumar, M. Holwill, Z. Wang, G. H. Auton, J. Birkbeck, A. Mishchenko, L. A. Ponomarenko, +K. Watanabe, T. Taniguchi, K. S. Novoselov, I. L. Aleiner, A. K. Geim, and V. I. Fal’ko, NATURE PHYSICS 15, 32+ +(2019). +[42] A. Jaoui, I. Das, G. Di Battista, J. Diez-Merida, X. Lu, K. Watanabe, T. Taniguchi, H. Ishizuka, L. Levitov, and D. K. +Efetov, NATURE PHYSICS 18, 633+ (2022). +[43] H. Ishizuka and L. Levitov, New J. Phys. 24, 052001 (2022), pMID: 34515488, https://doi.org/10.1088/1367-2630/ac688c. +[44] G. Di Battista, P. Seifert, K. Watanabe, T. Taniguchi, K. C. Fong, A. Principi, and D. K. Efetov, Nano Letters 22, 6465 +(2022), pMID: 35917225, https://doi.org/10.1021/acs.nanolett.1c04512. +[45] B. Deng, C. Ma, Q. Wang, S. Yuan, K. Watanabe, T. Taniguchi, F. Zhang, and F. Xia, NATURE PHOTONICS 14, 549+ +(2020). + +11 +EXTENDED DATA FIGURES +Extended Data Fig. 1. +Dual gate map of the four-probe resistance of MATBG (θ = 1.24◦) at T=3.6 K. The maxima in +resistance correspond to the charge neutrality points (CNPs) and integer filling factors (ν = ±2, ±3, ±4). + +V=-2 +V=+2 +V=+4 +Resistance (kQ) +10 +V=+4 .. +4 +V=+3... +8 +2 +6 +gate B (V) +0 +OCNP +4 +-2 - +2 +-4 +0 +-2 CNP...0. +2 +4 +4 +gate A (V)12 +Extended Data Fig. 2. Dual gate photovoltage maps for; a MATBG (θ = 1.24◦, T = 10 K) and b BLG (θ = 0◦, T = 100 K). + +r +Phobovoltage (μM) +OT +1.5 +75 +1.0 + 50 +0.5. +25 +M +0.0 +-25 +-0.5 +-50 +-1.0 - +-75 +-1.5 +DT- +-1.5 -1.0 -0.50.00.5 +1.0 +1.5 +gate A (M) +0 +Phobovoltage (uV) +2.0 +15 +1.5 - +10 +1.0 - +5 +0.5 +(I I +0 +0"0 +s- +-0.5 +ot-. +o't- +-15 +-1.5 +oz- +o'z- +-1.5 -1.0 -0.5 0.00.5 +1.0 +1.5 +gate A [M]13 +Extended Data Fig. 3. TrPV dips for the MATBG (θ = 1.24◦) device as a function of DU vector (indicated by arrow) and +temperature (see plot title). Each time trace has been offset for clarity. + +1.24, T=5 K +1.24, T=10 K +1.24, T=15 K +1.24, T=20 K +oF +-4.5 V +10 +-10 +10 +-10 +() +(Ar) +-20 +-20 + +-20 +-20 +abeal +voltage +hotovol +QE- +DE- +30 +-30 +40 +40 +-40 +-50 +-50 +50 +*+4.5 V +-30 +-20 +-10 +10 +20 +30 +-30 +-20 +-10 +10 +20 +30 +-20 +-10 +10 +20 +30 +-30 +-20 +-10 +10 +20 +30 +t (ps) +(sd) +t (ps) +(sd) * +1.24, T=25 K +1.24, T=50 K +1.24, T=100 K +1.24, T=150 K +0 +-2 +(An) +(Ar) +(A) +-6 +Photovoltage +6 +-6 +++ +-30 +-8 +8 +-8 +-10 +-10 +-10 +40+ +12 +-12μ +-12H +-50 +14 +-14F +-14 +++++ +-16E +-16 E +Om- +20 +o1- +0 +10 +20 +30 +DE- +-20 +-10 +10 +20 +30 +20 +-10 +10 +20 +30 +-30 +-20 +-10 +10 +20 +30 +t (ps) +t (ps) +t (ps) +t (ps) +1.24, T=200 K +1.24, T=250 K +1.24, T=300 K +0] +-2 +-2 +2 +(Ar) +4 +(Ar) +(A) : +voltage +abe +6 +-8 +8 +-8 +lotov +-10 +-10 +-10 +. +-12μ +-12H +-12 +-14 +-14 +-16 +16 +16 E +-10 +10 +20 +30 +20 +ot- +10 +Oml +OE- +20 +Om- +20 +-10 +10 +20 +30 +t (ps) +t (ps) +(sd) 14 +Extended Data Fig. 4. TrPV dips for the BLG (θ = 0◦) device as a function of DU vector (indicated by arrow) and temperature +(see plot title). Each time trace has been offset for clarity. The slower cooling at low temperatures produces a broader dip. + +0°. T=5 K +0, T=15 K +0, T=25 K +-2.0V +0 +-10 +-10 +(A) +-20 +(Ar) +-20 +-10 +M +Photovoltage I +Photovoltage +-30 +Photovoltage +-30 +-15 +40 +40 +-20 +-50 +-50 +25 +-60 +-60 +OE- ++2.0V +70 +-70 +-35 +-150 +-100 +-50 +50 +100 +150 +-150 +-100 +-50 +50 +100 +150 +-150 +-100 +-50 +50 +100 +150 +t (ps) +t (ps) +t (ps) +09, T=50 K +0, T=100 K +02. T=150 K +5 +(A) +-10 +(Ar) +(A) +Photovoltage +Photovoltage + Photovoltage I +-15 +-10 +-10 +20 +25 +-15 +-15 +30 +20 +20 +35 +-150 +-100 +-50 +0 +50 +100 +150 +150 +-100 +-50 +50 +100 +150 +-150 +-100 +-50 +0 +50 +100 +150 +t (ps) +t (ps) +t (ps) +0, T=200 K +0, T=250 K +0, T=300 K +-2 +-5 + Photovoltage (μV) +(μV) +(uV) +Photovoltage + Photovoltage +-10 +-10 +15 +-15 +-8 +-20 +-20 +10 +-150 +o0T- +s- +0 +50 +100 +150 +-150 +-100 +-50 +0 +50 +100 +150 +-150 +-100 +-50 +0 +50 +100 +150 +t (ps) +t (ps) +(sd) 15 +Extended Data Fig. 5. CW-PM peaks for the MATBG (θ = 1.24◦) device as a function of DU vector (indicated by arrow) and +temperature (see plot title). Each frequency sweep has been offset for clarity. + +1.24, T=25K +1.24, T=50K +1.24, T=100K +2.5 +1 +-1.65 V +1F +0.0 +Q +2.5 +Photovoltage (μV) +-1 +(ar) +-1 +Photovoltage ( +-2 +5.0 +-2 +-7.5 +E- +-4 +10.0 +m4 +-5 +-12.5 ++1.65 V +-5 +-6 +15.0 +-7E +-300 +-200 +-100 +0 +100 +200 +300 +-200 +-100 +0 +100 +200 +00E +-300 +-200 +-100 +0 +100 +200 +300 +freq. offset (GHz) +freq. offset (GHz) +freq. offset (GHz) +1.24, T=150K +1.24*, T=200K +1.24, T=300K +1 +0.2 +0 +0.00 +0.0 +Photovoltage (μV) +-1 +(Ar) +(μV) +z'- +Photovoltage ( +-0.05 + Photovoltage +-0.4 +-3 +-0.6 +-0.10 +-4 +0.8 +0.15 +-5 +1.0 +-300 +-200 +-100 +0 +100 +200 +QOE +00E- +-200 +-100 +0 +100 +200 +00E +00m- +-200 +001- +0 +100 +200 +DOE +freq. offset (GHz) +freq. offset (GHz) +freq. offset (GHz)16 +Extended Data Fig. 6. CW-PM peaks for the BLG (θ = 0◦) device as a function of DU vector (indicated by arrow) and +temperature (see plot title). Each frequency sweep has been offset for clarity. The slower cooling at low temperatures produces +a narrower peak. + +0, T=25K +0, T=50K +0, T=75K +0, T=100K +5 +-1.65 V +Photovoltage (μM) +Photovoltage (μV) +5 +-5 +-10 +OT- +15 +15 +-15 +-20 +-20 +20 +-25 +-25 +-25 ++1.65\ +-30 +-30 +200 -150 +150 +200 +-200- +100 +200 +100 +150 +200 +50 +100 +150 +200 +freq. ffset (GHz) +freq. ffset (GHz) +freq. offset (GHz) +freq. offset (GHz) +09, T=150K +09, T=200K +09, T=300K +0.2 +0.0 +0'0 +(Ar) +0.5 +Phatovoltage +Photovoltage +0.4 +0.6 +-1.0 +0.8 +1.5 +-1.0# +1.2 +200 150 -100 +-50 +50 +100 +150 +200 +-200 -150 -100 50 +50 +100 +150 +200 +200 -150 -100-50 +50 +100 +150 +200 +0 +freq- offset (GHz) +freq. offset (GHz) +freq- offset (GHz)17 +Extended Data Fig. 7. Power dependence of cooling time for a second MATBG device (θ = 1.08◦). The electron relaxation +bottleneck at full filling (ν = ±4) leads to slower cooling time for higher laser powers. The orange line is a guide to eye. + +MATBG, 0=1.08 +70 +60 +V=±4 +Cooling time (ps) +50 +40 +30 +20 +V=±2 +10 +0 +0 +1 +2 +3 +4 +5 +Peak power density (GW cm-2)18 +SUPPLEMENTARY INFORMATION +Supplementary Note 1 +In Supp. Fig. 3, we investigate the influence of twist angle disorder on the electrical transport at T = 35 mK. +At the junction contact, the twist angle is θ = 1.24◦ and we observe sharp resistance peaks at ν = ±2 arising from +correlated insulating states. The contacts at the top of the junction display a shoulder around ν = −2 that indicates +a mixing of two angles (θ = 1.24−1.28◦). For the contacts at the bottom of the junction the angle is θ = 1.24◦. From +this we conclude that there is minimal twist angle disorder in the proximity of the pn-junction. +Supplementary Fig. 1. Optical images of the device before and after nanofabrication. a, Heterostructure stack dropped on a +Si/SiO2 substrate. b, Finalised device after etching Hall bar and metallisation. Both scale bars are 5µm. + +a +b19 +Supplementary Fig. 2. Low field Hall effect at 1.8 K. Hall carrier density nH vs n. In the region close to charge neutrality +nH = n, which allows us to calibrate the relationship between Vg and n to extract the twist angle. +Supplementary Fig. 3. Longitudinal resistance (Rxx) vs. filling factor (ν) for contacts around the junction. + +4 +2 +nH +0 +H +n +-4 +-2 +0 +2 +4 +n (1012 cm-2)102 +Top of junction +Junction Contact +Bottom of junction +101 +(kΩ2) +Rxx +100 +-4 +-3 +-2 +-1 +0 +2 +3 +1 +4 +Filling factor v \ No newline at end of file diff --git a/n9FST4oBgHgl3EQfMDgV/content/tmp_files/load_file.txt b/n9FST4oBgHgl3EQfMDgV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cd0ded49045f382a7c7f1be22082b3e64a03f6a --- /dev/null +++ b/n9FST4oBgHgl3EQfMDgV/content/tmp_files/load_file.txt @@ -0,0 +1,1159 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf,len=1158 +page_content='Ultrafast Umklapp-assisted electron-phonon cooling in magic-angle twisted bilayer graphene Jake Dudley Mehew,1 Rafael Luque Merino,2, 3, 4 Hiroaki Ishizuka,5 Alexander Block,1 Jaime Díez Mérida,2, 3, 4 Andrés Díez Carlón,2, 3, 4 Kenji Watanabe,6 Takashi Taniguchi,7 Leonid S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov,8 Dmitri K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4 and Klaas-Jan Tielrooij1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ∗ 1Catalan Institute of Nanoscience and Nanotechnology (ICN2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' BIST and CSIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Campus UAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 08193 Bellaterra (Barcelona),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Spain 2ICFO - Institut de Ciencies Fotoniques,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The Barcelona Institute of Science and Technology (BIST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Castelldefels 08860,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Spain 3Fakultät für Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ludwig-Maximilians-Universität,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Schellingstrasse 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' München 80799,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Germany 4Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Germany 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tokyo Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Japan 6Research Center for Functional Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' National Institute for Material Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Japan 7International Center for Materials Nanoarchitectonics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' National Institute for Material Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Japan 8Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 02139 MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' USA 9Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' TU Eindhoven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Den Dolech 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Eindhoven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 5612 AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Carrier relaxation measurements in moiré materials offer a unique probe of the microscopic in- teractions, in particular the ones that are not easily measured by transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Umklapp scattering between phonons is a ubiquitous momentum-nonconserving process that governs the thermal con- ductivity of semiconductors and insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In contrast, Umklapp scattering between electrons and phonons has not been demonstrated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Here, we study the cooling of hot elec- trons in moiré graphene using time- and frequency-resolved photovoltage measurements as a direct probe of its complex energy pathways including electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We report on a dramatic speedup in hot carrier cooling of twisted bilayer graphene near the magic angle: the cooling time is a few picoseconds from room temperature down to 5 K, whereas in pristine graphene coupling to acoustic phonons takes nanoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Our analysis indicates that this ultrafast cooling is a com- bined effect of the formation of a superlattice with low-energy moiré phonons, spatially compressed electronic Wannier orbitals, and a reduced superlattice Brillouin zone, enabling Umklapp scattering that overcomes electron-phonon momentum mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' These results demonstrate a way to engineer electron-phonon coupling in twistronic systems, an approach that could contribute to the fundamen- tal understanding of their transport properties and enable applications in thermal management and ultrafast photodetection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' INTRODUCTION Moiré superlattices provide a novel material platform in which twist angle controls the effective lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' As the twist angle decreases, the larger moiré unit cell corresponds to a smaller electron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This tunes the relative strength of the kinetic energy of electrons and the interaction energy between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In magic-angle twisted bilayer graphene (MATBG), these interactions result in a rich phase diagram that includes superconductors, [1–4] correlated insulators [5, 6] and orbital magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [2, 7] In transition metal dichalcogenides (TMDs), correlated insulating [8, 9] and ferromagnetic states [10] are observed over a broad range of angles, with moiré excitons [11, 12] providing a test-bed for exploring Hubbard model physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [9] In addition, the moiré potential modifies the phonon spectra for small twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [13] This results in phonon renormalization in MoS2 homobilayers [14] and the emergence of phonon minibands in twisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [15] Theoretical studies predict that the moiré potential strongly affects electron-phonon coupling, [16–18] which has important implications for electrical transport, excited-state relaxation dynamics, and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Excited-state relaxation measurements are particularly well-suited probes to quantitatively assess electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The relaxation dynamics in graphene after excitation involve thermalization of high-energy carriers through carrier-carrier scattering within tens of femtoseconds, [19] creating a hot carrier distribution that subsequently cools via phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Inelastic electron-phonon scattering allows electrons to gain (lose) energy by the absorption (emission) of a phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In graphene, cooling typically occurs via the emission of optical and acoustic graphene phonons, and near-field coupling to substrate phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20–27] Importantly, in all cases, cooling becomes increasingly slow for lower lattice temperatures, with predicted electron-phonon cooling times in the nanosecond regime for the case of pristine arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='13742v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='mes-hall] 31 Jan 2023 2 graphene at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20] Experimental studies of the relaxation dynamics of twisted bilayer graphene have so far been limited to large twist angles (θ > 5◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In these systems, a dark exciton state emerges between van Hove singularities, leading to slower dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [28, 29] At such relatively large angles, the moiré potential has limited influence on electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [15, 18] Recent Raman spectroscopy measurements suggest an enhanced electron-phonon coupling strength for small twist angles around the magic angle (θ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [30] However, direct experimental measurements of moiré- enhanced electron-phonon coupling and its implications for cooling dynamics, are lacking, nor is there any clear understanding of the origin of the enhanced coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In this paper, we report the observation of ultrafast cooling in magic-angle twisted bilayer graphene (MATBG) through Umklapp-assisted electron-phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We directly probe the electron-phonon interaction by mea- suring carrier cooling dynamics using two well-established optoelectronic techniques – time-resolved photovoltage microscopy [19, 31, 32] and continuous-wave photomixing [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We make a direct comparison between a non- twisted Bernal bilayer graphene sample (BLG, θ = 0◦, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1a) and a near-magic-angle twisted bilayer graphene sample (MATBG, θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At low temperature, the cooling dynamics are much faster in MATBG than in non-twisted bilayer graphene, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This unexpected result highlights the crucial role of the moiré pattern and suggests the emergence of an enhanced electron-phonon interaction in small twist angle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We explain the observed relaxation dynamics using a theoretical model based on Umklapp-assisted electron-phonon scattering, which can occur in both the dispersive and flat bands of MATBG, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The Umklapp processes are enabled by the presence of compressed electronic Wannier orbitals (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1e), and the superlattice with reduced Brillouin zone (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' RELAXATION DYNAMICS We study relaxation dynamics in hBN-encapsulated MATBG and BLG Hall bar devices as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1a-b (see Methods for details on the device fabrication and characterization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' These devices enable both electrical and optoelectronic measurements, as they are equipped with a split gate that we employ to create a photoactive pn- junction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The resistance map as a function of the gate voltage applied to each of the two sides of the split gate, shown in Extended Data Figure 1, displays clear peaks at the usual Dirac points with vanishing carrier density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The MATBG device exhibits additional peaks at integer fillings of the superlattice unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' By illuminating the pn-junction with light, a photovoltage is generated via the photothermoelectric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This effect has a characteristic six-fold symmetry in dual-gate photovoltage maps, as shown in Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2 for both devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This indicates that the measured photovoltage is a direct probe of the electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [35] We study hot electron relaxation using ultrafast time-resolved photovoltage microscopy (TrPV) as implemented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [19, 24] and continuous-wave heterodyne photomixing (CW-PM) as implemented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [34] In the former, ultrashort laser pulses are incident upon the pn-junction whereas for the latter two continuous wave lasers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In both cases we probe the generated photovoltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' These two techniques allow us to obtain directly the carrier cooling dynamics – in the time domain by varying the time delay between two ultrashort laser pulses, and in the frequency domain by varying the spectral detuning of two spectrally narrow laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Both techniques independently show that charge carriers cool much faster in MATBG than in BLG at low temperature, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2a (and Extended Data Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In BLG the cooling time increases from 3 ps to 25 ps as the temperature decreases from 300 K to 5 K, which is expected as it takes longer for hot carriers to couple to phonons at lower temperature due to the reduced phonon occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20, 21] Surprisingly, for MATBG the cooling time remains short, around 3 ps, across a broad temperature range (5-300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This suggests the involvement of low-energy phonons that still have occupation at such low temperature, which are likely phonons related to the superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Indeed, the moiré potential breaks the original linear phonon dispersion into minibands with enhanced density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [15] The energy of the lowest band is below 1 meV corresponding to temperatures below 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In order to understand the origin of the observed cooling dynamics, we first consider the case of relaxation through energy transfer to phonons in non-twisted BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Coupling to optical phonons is highly inefficient at low temperature due to the large optical phonon energy, which is > 160 meV, corresponding to T > 2000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [26] Coupling between electrons and acoustic phonons is normally also inefficient due to the reduced phase space available for scattering, and would give cooling times well above a nanosecond below 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20] The presence of defects can help overcome the electron-phonon momentum mismatch through disorder-assisted cooling, which speeds up this acoustic phonon cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [21–23] However, even with this mechanism, we expect cooling times between 10−10 s and 10−8 s for the lowest temperatures, depending on the electron mean free path (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We therefore consider diffusive cooling, where electronic heat diffuses out of the initially excited hot spot, thus leading to a lower average electron 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Excited carrier relaxation in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a-b, Illustration of the hBN-encapsulated BLG device with 0◦ twist angle (a) and the hBN-encapsulated MATBG device with twist angle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦ (b), each equipped with split gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' By applying voltages of opposite sign (±V ) to the split gates, we create a pn-junction (the interface between yellow and orange regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Illuminating the junction generates a photovoltage via the photothermoelectric effect, which is proportional to the electron temperature (Te).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We obtain the temperature dynamics either by using two ultrashort laser pulses separated in time by a variable temporal delay, [19, 31, 32] or by using two spectrally narrow laser beams with variable frequency detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [33, 34] c, Photovoltage as a function of time delay for a lattice temperature of 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The decay, which represents the cooling dynamics, is much faster in MATBG (blue pluses) than BLG (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' d, Schematic of the MATBG band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Umklapp scattering processes (solid arrow) allow for efficient electron (black circle) relaxation via coupling to moiré phonons (wiggly lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' These Umklapp processes can occur in both the flat and the dispersive bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The dashed arrows represent the equivalent final state in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' e, Schematic of the compressed Wannier orbitals of radius ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Electrons are localised to AA sites in the reconstructed superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' f, Umklapp scattering processes (blue arrows) couple electrons in the first Brillouin zone (white hexagon) to large-momentum phonons in higher-order Brillouin zones (blue hexagons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [25] In this diffusive cooling mechanism, the cooling time will thus depend on laser spot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Indeed, for non-twisted BLG, we observe an increase in cooling time for larger spot sizes, which is largest for the lowest temperatures (25 K, 50 K), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2b-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At 100 K, the cooling length is shorter and therefore diffusive cooling has a smaller contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We thus understand the cooling dynamics for non-twisted BLG from a combination of disorder-assisted and diffusive cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Indeed, our calculations of the cooling time based on these two mechanisms are close to the experimentally observed ones (see Methods for details on the calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Importantly, for MATBG we observe no dependence of the cooling time on spot size (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2b-c), which suggests that diffusive cooling does not play a role for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We next study the effect of changing the laser power and therefore initial electron temperature, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This corresponds to increasing the population of the dispersive band (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The peak power density is roughly five orders of magnitude larger for our pulsed laser experiment (TrPV) than our continuous wave experiment (CW-PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For the non-twisted BLG device, we observe somewhat slower cooling at higher initial electron temperature, which has also been observed for high-quality monolayer graphene samples and was ascribed to a bottleneck involving optical and acoustic phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [26] Interestingly, the role of electron temperature is minor for the relaxation dynamics of MATBG, suggesting that there are no electron-phonon or phonon-phonon bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This result indicates that a c 人e (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=') Photovoltage (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=') BLG 0° hBN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24° + 75 100 50 25 0 25 50 75 100 Delay time (ps) f d e Umklapp-assisted cooling Localised MATBG electrons Moire phonons AA 2 Moiréphonons AB dns Normal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24°4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Relaxation mechanisms in MATBG and BLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a, Cooling time as a function of lattice temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In MATBG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦, blue pluses), the cooling time is constant between 5 K and 300 K (3 ps, blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For BLG (0◦, red circles), it is greater at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' b, Laser spot size dependence of the cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The strong dependence in BLG at 25 K and 50 K is a signature of diffusive cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This effect is weaker at 100 K, where disorder-assisted cooling becomes significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The effect is absent in MATBG for these spot sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The filled (open) shapes are measured using the TrPV (CW-PM) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Error bars represent the statistical spread across different gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The thick blue line in a and b represents the cooling time obtained from the low temperature model of Umklapp-assisted cooling (see Main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' c Schematics of diffusive cooling for BLG (upper) and its absence for MATBG (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' direct optical phonon emission does not play a role in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The much faster cooling for MATBG compared to BLG at low temperatures thus suggests that a completely different mechanism is responsible for cooling, outcompeting all currently known cooling mechanisms in non-twisted graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We therefore explore the effect of the superlattice on electron cooling by examining the cooling time in MATBG as a function of filling factor (ν) which represents the electronic occupation of the superlattice unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For most filling factors (|ν| < 4), we observe a nearly constant cooling time of 3 ps across a wide temperature range (5-300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' However, at ν = ±4 the cooling time increases dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Low-temperature transport measurements on the same device reveal an increase in resistance at the same voltages, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3d, which confirms the full filling of the superlattice unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In Extended Data Figure 7, we show that the cooling time increases strongly upon increasing laser power at full filling for a second MATBG device (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='08◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The strong dependence of cooling upon the flat band filling - with the cooling rates high at partial filling and lower at full filling - indicates that the moiré pattern and its low-energy phonons are crucial for explaining the ultrafast cooling dynamics observed in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ORIGIN OF ENHANCED COOLING To gain insight into the different mechanisms that govern electron-lattice cooling pathways, we consider in detail the microscopic electron-phonon scattering processes in the MATBG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' To this end, we consider a four-band model consisting of two nearly-flat and two dispersive bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' There are two main types of electron-phonon scattering in this model, interband and intraband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The intraband processes for the intra-dispersive band and intra- flat-band transitions are different and must be evaluated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At temperatures higher than the bandgap, which corresponds to the highest temperatures in our measurement, the electrons are thermally excited to the dispersive bands allowing both dispersive and flat bands to contribute to cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' To the contrary, when the electron temperature is low, all carriers reside in the flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Therefore, we consider two regimes: i) the high temperature regime (T ∼ 150 − 300 K), where the dispersive bands contribute to the cooling process, and ii) the low temperature regime (T ∼ 10 K), wherein cooling is dominated by intra-flat-band processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In both cases, we consider both the Umklapp and normal scattering contributions, finding that at the temperatures of interest (T > 10 K) Umklapp scattering consistently wins over normal scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For the first regime (high temperatures) we consider a four-band model consisting of two flat bands of bandwidth a b c 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 0° 35 1 25 30 : (sd) 25K 25 time ( time 50K Diffusive cooling 15 20 Cooling 100K Disorder-assisted cooling 15 TOI 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24° 10 2K 5I O+ 100K 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24° 0 Umklapp-assisted cooling diffusive cooling 57 $ Non ± 0 00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 100 200 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 Lattice temperature (K) Spot size (μm)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Origin of enhanced cooling in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a, Dependence of cooling time on peak power density for BLG (red circles) and MATBG (blue pluses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The filled (open) shapes are measured using the TrPV (CW-PM) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The error bars signify the one sigma confidence interval from the fitting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' b, e Schematics of cooling power in MATBG for part filling (b) and full filling (e) of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For part filling, the interband transition is not rate-limiting as evidenced by the absence of a power dependence in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At full filling, cooling times are longer due to the interband bottleneck effect illustrated in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' c-d, Gate dependence of cooling time, c, and four terminal resistance acquired at T = 35 mK (Rxx), d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Orange shaded region highlights full filling of the moiré unit cell, where Rxx and cooling time increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The thick blue line in a and c represents the cooling time obtained from the low temperature model of Umklapp-assisted cooling (see Main text) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W and two dispersive bands with the eigenstate energies ε > ∆ and ε < −∆ (∆ > W), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The dispersive bands are separated from the flat bands by a gap ∆ − W (see Methods for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A direct analysis based on Boltzmann theory yields cooling rates dominated by the intra-band processes in the dispersive bands, whereas the interband processes have a minor contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Accounting for the Umklapp processes, we estimate the cooling rates as τ −1 = 6ρ1 πTel � m(∥g1,1 m ∥2 + ∥g−1,−1 m ∥2)ω2 m, where ρ1 is the density of states of the dispersive particle and hole bands labeled by n = ±1, Tel is the electron temperature, gn,n m is the electron-phonon coupling constant in the nth band and ωm is the phonon energy in the mth phonon band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Direct calculation gives cooling rates that are independent of the lattice temperature Tph, in agreement with the observed dynamics, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For the regime of low temperatures, we describe the system using a model of a flat band with electron and hole subbands (see Methods for a detailed description of the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For a quantitative comparison with the experimental results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4a, we calculate the cooling power J accounting for the Umklapp processes assuming the Wannier function radius ξ = a/6 where a is the lattice parameter for the moiré structure [36], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The cooling rate τ −1 is estimated from the calculated cooling power and specific heat using τ −1 = J/C(Tel − Tph);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' here we calculate the specific heat C using the fluctuation formula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2 in the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In that temperature values are not constrained by the flat-band width and can be as large as the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The filling dependence of the cooling rate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The calculated Umklapp-assisted cooling times as a function of the filling factor are seen to be in a 30 0° 25K 25 50K Cooling time (ps) 20 100K 15 1O 10 25K Part filling, v<±4 50K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24° 5 F 100K 0 0 1 2 3 4 5 Peak power density (GW cm-2) C e 30 25 Cooling time (ps) 20 5K 15 300K 10 ５ 0 102 Full filling, V=±4 (kΩ2) 101 XX 100 R 10-1 4-3-2-101 3 Filling factor, v6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Quantitative comparison with Umklapp-assisted cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a, Comparison between calculated (solid line) and experimental (symbols) cooling times for MATBG at 5 K and 10 K (upper and lower panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The grey shaded region allows for uncertainty in the value of the deformation potential (D = 16 ± 4 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' b, Schematic of the model used for the calculations with two dispersive and two flat bands separated by an energy gap (∆ − W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' γ1 and γ0 represent intra-dispersive-band and intra-flat-band scattering processes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The low temperature calculations shown in (a) consider only γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' agreement with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For the calculated cooling times, we used a deformation potential of 16 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This is close to the values reported for single-layer graphene (10-30 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [22, 37–39] We therefore conclude that the Umklapp-assisted carrier cooling model reproduces the main experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We note that, here, we did not take account of the disorder-assisted cooling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [21] In pristine graphene, the bottleneck due to limited phase space due to the small Fermi surface is relieved by disorder scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The situation in MATBG differs from that in pristine graphene in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' First, as the superlattice provides additional momentum recoil, MATBG does not require defects and/or disorder for electron-lattice cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Second, the formation of highly localized Wannier orbitals at AA sites in the moiré pattern modulates the electron-phonon interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' These effects produce strong coupling of the electrons to moiré phonons even in the absence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [36] OUTLOOK Importantly, the cooling measurement is predominantly sensitive to the electron-phonon interactions, and is less sensitive to the electron-electron interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This presents a unique window of opportunity for probing underlying physics, and an advantage compared to other measurements types that do not easily separate these two interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The finding that electron-phonon Umklapp scattering dominates ultrafast electron-phonon cooling is likely to have important implications for MATBG physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Electron-phonon scattering plays an important role in charge transport, limiting the carrier mobility at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This interaction also mediates the pairing interaction in Bardeen- Cooper-Schrieffer superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Understanding the electron-phonon coupling could give important insights into the origin of superconductivity in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [16, 40] For metals, electron-electron Umklapp scattering gives rise to finite electrical resistance at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In graphene/hBN superlattices and MATBG, this effect dominates transport at temperatures up to 10 K or higher, leading to excess resistivity and degradation of charge carrier mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [41–43] In MATBG, electron-phonon Umklapp scattering could explain some of the open questions from electrical transport measurements, such as the strange metal phase or the role of phonons in superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [16, 40] Finally, the ultrafast Umklapp-assisted electron-phonon cooling, enhanced density of states, and rich phase diagram are appealing for single-photon detection in the highly sought after mid-IR wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [44, 45] METHODS Device fabrication The MATBG devices were fabricated using a cut and stack technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' All flakes were first exfoliated on a Si/SiO2 (285 nm) substrate and later picked up using a polycarbonate (PC)/polydimethylsiloxane (PDMS) stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' All the layers were picked up at a temperature of ∼ 100◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We used an AFM tip to cut the graphene in order to avoid strain during the pick-up process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The PC/PDMS stamp picks up first the top graphite layer, the a b T=5 K Cooling time (ps) 20 Yi E(k) 10 D=16±4 eV W T=10 K Cooling time (ps) 15 10 5 0 1 2 m 4 Filling factor, Iv7 top hBN and the first graphene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Before picking up the second graphene layer, we rotate the stage by an angle of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Finally, the stamp picks up the bottom hBN and bottom graphite gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We drop the finalized stack on a Si/SiO2 substrate by melting the PC at 180◦C, see Supplementary Figure S1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The resulting stack is etched into a Hall bar using a CHF3/O2 plasma and a 1D contact is formed by evaporating Cr (5 nm)/Au (50 nm), see Supplementary Figure S1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We etch a narrow channel of ∼ 150 nm in the top gate using an O2 plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Before etching the top gate, the device was characterized at T = 35 mK to identify the pair of contacts closest to the magic angle (θ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The junction was made in between this pair of contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Twist angle extraction The twist angle θ is extracted from the superlattice carrier density of the full band ns by applying the relation ns = 8θ2/ √ 3a2, where a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='246 nm is the graphene lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' First, we calibrate the gate induced carrier density using the Hall effect data at ±1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In the carrier density region close to charge neutrality, the Hall carrier density nH = −B/eRxy should closely follow the gate induced carrier density nH = n, see Supplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' By plotting nH vs Vg and fitting this slope around charge neutrality we can obtain the capacitance of the device and therefore extract the real carrier density n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Then we extract the carrier density corresponding to a fully filled superlattice unit cell, in this case we find it to be ns = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='10) × 1012 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Finally using the above relation we extract a twist angle θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='02◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In Supplementary Note 1, we verify that there is minimal twist angle disorder in the junction region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Transport Measurements Low-temperature transport measurements were carried out in a dilution refrigerator (Bluefors SD250) with a base temperature of 20 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Standard low-frequency lock-in techniques (Stanford Research SR860 amplifiers) were used to measure Rxx with an excitation current of 10 nA at a frequency of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='11 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Optoelectronic measurements In time-resolved photovoltage (TrPV) experiments, we vary the delay time (dt) between the arrival of two ultrafast pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [31] [32] [19] Due to the non-linear relationship between carrier temperature and optical heating, we observe a dip in the photovoltage when the two pulses arrive at the same time (dt = 0), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1c and Extended Data Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At longer delay times, the signal recovers to its maximal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We obtain the cooling time by describing the observed dynamics with an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For heterodyne photomixing (CW-PM) experiments, the wavelength detuning between the two continuous wave lasers creates an optical beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [33, 34] The photovoltage oscillates at the beating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Due to the competition between beat frequency (Ω) and the characteristic cooling time (τe), we observe a peak for Ω = 0 whereas the oscillations are damped when Ω−1 ≪ τe, see Extended Data Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The frequency response takes the form of a Lorentzian function of width Γ, from which we extract the cooling time as: Γ = 1/πτe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [33] Estimating cooling times in untwisted graphene The hot electron cooling time for energy transfer to acoustic phonons in monolayer graphene is given by τAP ≈ 848/(D2T 2 L) [µs], [20] where D is the deformation potential in eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This expression is valid in the neutral limit (TF < Te) and close to equilibrium (Te ≳ TL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Te/L/F is the electron/lattice/Fermi temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20] Taking D = 20 eV, we calculate a cooling time of τAP = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='4 ns for TL = 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In disorder-assisted or supercollision cooling, [21–23] the dependence on lattice temperature is given by: τSC = α 3ATL , with α = 2πEF k2 B 3ℏ2v2 F and A = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='62g2ν2(EF )k3 B ℏkF ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Here, g is the electron-phonon coupling, ν(EF ) is the density of states at the Fermi level per valley/spin flavour, kF is the Fermi wavevector and ℓ is the mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In high-quality samples and at cryogenic temperatures, the device size typically limits the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For low doping levels (1012 cm−2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1 < ℓ < 2 µm and T = 25 K, τSC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 − 11 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cooling due to lateral diffusion The lateral diffusion of photoexcited carriers reduces the hot electron temper- ature when the cooling length is greater than the laser spot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This effect is particularly relevant in high-mobility samples, as the Wiedemann-Franz law relates electrical to thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [25] At low lattice temperatures efficient heat conduction manifests in our experiments as a shorter cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' By considering the spatial evolution of a Gaussian heat spot induced by the laser pulse, [26] we describe the temperature dynamics by: Te(t) = 2πApuApr σ2 puσ2 pr σ2pu + σ2pr + 2Dt, where A and σ are the peak intensity of the pump (pu) and probe (pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Clearly, this effect is greater for smaller spot sizes and larger electronic heat diffusivities (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Using a diffusivity of D = 750 cm2s−1, and pump-probe spot sizes of σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='9 µm we find a cooling time of τdiff ≈ 18 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For σ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='4 µm, τdiff ≈ 45 ps, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 8 Cooling rate at low temperatures The cooling rate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3f is estimated by J(Tel,Tph) C(Tel)(Tel−Tph), where J is the cooling power, C(Tel) is the electron specific heat, and Tel (Tph) is the electron (phonon) temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' To evaluate J and C, we consider an effective two-band model similar to pristine graphene used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Following the previous study, we use the electron-phonon interaction for the Wannier orbital radius ξ = a/6 where a is the lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In the Boltzmann theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' the cooling power J by electron-phonon scattering reads [36] J = � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='n′ Jn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='n′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='n′ = 2π V 2 � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='⃗k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='⃗k′ ∥gnn′ ⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='m∥2ω2 ⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='mN⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='m × � f⃗k′n′[1 − f⃗kn]e βphω⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='m − f⃗kn[1 − f⃗k′n′] � × δ(εn′ − εn − ω⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' (1) where Jn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='n′ is the contribution from the scattering between nth and n′th bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' V is the volume of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' gnn′ ⃗k−⃗k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='m is the coupling constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ε⃗kn is the one-particle eigenenergy of the eigenstate in nth band with momentum ⃗k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ω⃗qm is the phonon eigenenergy in the mth band with momentum ⃗q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' and βel = 1/kBTel (βph = 1/kBTph) is the inverse temperature of electrons (phonons) with kB being the Boltzmann constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' f⃗kn = 1 eβe(ε⃗kn−µ)+1 and N⃗qm = 1 eβphω⃗km−1 are respectively the Fermi and Bose distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The estimation of specific heat uses the fluctuation formula C(T) = kB � ⟨ε2 n⃗k⟩ − ⟨εn⃗k⟩2 ⟨1⟩ � , (2) ⟨On⃗k⟩ = � n � dkd (2π)d β2On⃗k 4 cosh2 � β(εn⃗k−µ) 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' (3) Note that the common formula for Fermi-degenerate electron systems does not apply here as the temperature exceeds the Fermi energy at T ≳ 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This model gives a good approximation when the temperature is much lower than the energy gap separating the flat band from high-energy dispersive bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cooling rate at high temperatures At high temperatures, we cannot neglect the high-energy bands because the electron temperature exceeds the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In such a case, the Umklapp scattering involving high-energy phonons contributes to electron cooling due to a large number of high-energy phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hence, we also expect that Umklapp scattering plays a key role in the high temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' To study the electron-lattice cooling involving the interband processes, we assume the electrons only couple to phonons with energies below a cutoff Λph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' This assumption is justifiable in a system where the electron-phonon coupling between the electrons and the acoustic phonons reduces exponentially as the momentum increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In a system with compact Wannier orbitals, Λph becomes a few times higher than the energy of folded acoustic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hence, a large Λph, considerably larger than the phonon bandwidth of the folded acoustic phonons, represents the enhanced coupling by compact Wannier orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Below, we label the folded acoustic bands by an integer m and define the high-temperature limit as Tel > Tph ≫ Λph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At high temperatures, the cooling power in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' (1) reads Jnn′ = π V � m ∥gnn′ m ∥ω2 mρnρ′ n [Tel − Tph] × � tanh(β(bm nn′ − µ) 2 ) − tanh(β(am nn′ − µ) 2 ) � , where ρn is the density of states (DOS) for the nth band (we assume a constant DOS with the bandwidth Wn), and am nn′ = max(ε− n − ε− n′ − ωm) [bm nn′ = min(ε+ n − ε+ n′ − ωm)] with ε± n being the energy of the top and bottom edge of the electron band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Here, we approximated the phonon energy as ωn⃗k ∼ ωn considering the small Brillouin zone, and the coupling constant gnn′ m (⃗k) ∼ gnn′ m which is valid in the small orbital radius limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' We apply the above formula to a four-band model consisting of two flat and two dispersive bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The two flat bands are at energies 0 ≤ ε ≤ W and −W ≤ ε ≤ 0 with DOS ρ0, and the two dispersive bands are W < ∆ ≤ ε ≤ Λ 9 and −Λ ≤ ε ≤ −∆ < −W with DOS ρ1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' To the leading order in Tel, the cooling power reads J = π � m (∥g1,1 m ∥2 + ∥g−1,−1 m ∥2)ω2 m[Tel − Tph]ρ2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hence, the cooling rate becomes τ −1 = 6ρ1 πV Tel � m(∥g1,1 m ∥2 + ∥g−1,−1 m ∥2)ω2 m, independent of phonon temperature, Tph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' SUPPLEMENTARY INFORMATION This article has an accompanying supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank Nick Feldman for his contribution to preliminary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ICN2 was supported by the Severo Ochoa program from Spanish MINECO Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' SEV-2017-0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' acknowledges that this project has received funding from the “Secretaria d’Universitats I Recerca de la Generalitat de Catalunya, as well as the European Social Fund (L’FSE inverteix en el teu futur)—FEDER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' acknowledges support from JSPS KAKENHI (Grant Numbers JP19K14649).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' acknowledges support from the INphINIT ‘la Caixa’ Foundation (ID 100010434) fellowship programme (LCF/BQ/DI19/11730021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' acknowledge support from the JSPS KAKENHI (Grant Numbers 19H05790 and 20H00354 and 21H05233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' acknowledges funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 804349 (ERC StG CUHL), RYC fellowship No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' RYC-2017-22330 and IAE project PID2019-111673GB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' ∗ Correspondence to: klaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='tielrooij@icn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='cat [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cao, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fatemi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kaxiras, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jarillo-Herrero, Nature 556, 43 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Stepanov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Aamir, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Das, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Urgell, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature 574, 653 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yankowitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Polshyn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Graf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Young, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Dean, Science 363, 1059 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Stepanov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Das, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fahimniya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Koppens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lischner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov, Nature 583, 375 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cao, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fatemi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Demir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tomarken, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sanchez-Yamagishi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kaxiras, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ashoori, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jarillo-Herrero, Nature 556, 80 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nuckolls, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Oh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jeon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Bernevig, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yazdani, Nature 582, 198 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sharpe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fox, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Barnard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Finney, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kastner, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Goldhaber-Gordon, Science 365, 605 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Regan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Bakti Utama, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yumigeta, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature 579, 359 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Barmak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' MacDonald, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Shan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature 579, 353 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xiao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Gamelin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature 604, 468 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Seyler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rivera, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wilson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mandrus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yao, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xu, Nature 567, 66 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Alexeev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ruiz-Tijerina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Danovich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hamer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Terry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nayak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Pak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sohn, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature 567, 81 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Pan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ji, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Liu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tan, ACS Nano 12, 8770 (2018), pMID: 30086224, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='8b05006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Quan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Linhart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hsu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Embley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Young, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature materials 20, 1100 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Koshino and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Son, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' B 100, 075416 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [16] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' MacDonald, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Martin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 121, 257001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Choi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Choi, PHYSICAL REVIEW B 98, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='241412 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Koshino and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nam, PHYSICAL REVIEW B 101, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='195425 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 10 [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tielrooij, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Piatkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Massicotte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Woessner, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Myhro, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jarillo-Herrero, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' van Hulst, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=', Nature nanotechnology 10, 437 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Bistritzer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' MacDonald, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 102, 206410 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Song, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Reizer, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, PHYSICAL REVIEW LETTERS 109, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='106602 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Graham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Shi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ralph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Park, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' McEuen, NATURE PHYSICS 9, 103 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Halbertal, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zeldov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' B 97, 245416 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tielrooij, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hesp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Principi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lundeberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Pogna, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Banszerus, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mics, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Massicotte, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Davydovskaya, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Purdie, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Goykhman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Soavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lombardo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Bonn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Turchinovich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Stampfer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ferrari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cerullo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Polini, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Koppens, NATURE NANOTECHNOLOGY 13, 41+ (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Massicotte, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Soavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Principi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tielrooij, Nanoscale 13, 8376 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Pogna, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Principi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Block, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Banszerus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sohier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Forti, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Soundarapandian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Terres, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mehew, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Trovatello, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Coletti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Koppens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Bonn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, I, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' van Hulst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Verstraete, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Peng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Stampfer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cerullo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tielrooij, ACS NANO 15, 11285 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jha, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Brar, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Atwater, NATURE MATERIALS 20, 805+ (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Patel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Havener, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Brown, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Park, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Graham, NANO LETTERS 15, 5932 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Patel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Park, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Graham, NATURE COMMUNICATIONS 10, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1038/s41467-019- 09097-x (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Gadelha, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ohlberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rabelo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Neto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Vasconcelos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Campos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lemos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ornelas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Miranda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nadas, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Santana, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' van Troeye, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lamparski, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Meunier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Paszko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Charlier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Campos, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cançado, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Medeiros-Ribeiro, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jorio, Nature 590, 405 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Urich, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Unterrainer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mueller, NANO LETTERS 11, 2804 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Aivazian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ross, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cobden, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xu, NATURE NANOTECHNOLOGY 7, 114 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jadidi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Suess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Tan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Cai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sushkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mittendorff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hone, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Drew, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fuhrer, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Murphy, PHYSICAL REVIEW LETTERS 117, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='257401 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Aamir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Moore, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Seifert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Englund, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fong, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov, Nano Letters 21, 5330 (2021), pMID: 34101476, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1c01553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [35] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Gabor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Song, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ma, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Nair, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taychatanapat, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jarillo-Herrero, Science 334, 648 (2011), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1211384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ishizuka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fahimniya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Guinea, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, Nano Letters 21, 7465 (2021), pMID: 34515488, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1c00565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kim, PHYSICAL REVIEW LETTERS 105, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='256805 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ishigami, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fuhrer, NATURE NANOTECHNOLOGY 3, 206 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Dean, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Young, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Meric, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Sorgenfrei, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Shepard, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hone, NATURE NANOTECHNOLOGY 5, 722 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Peltonen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ojajärvi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Heikkilä, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' B 98, 220504 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wallbank, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Kumar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Holwill, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Auton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Birkbeck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Mishchenko, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ponomarenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Novoselov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Aleiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Geim, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fal’ko, NATURE PHYSICS 15, 32+ (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Jaoui, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Das, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Di Battista, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Diez-Merida, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Lu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ishizuka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov, NATURE PHYSICS 18, 633+ (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [43] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ishizuka and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Levitov, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 24, 052001 (2022), pMID: 34515488, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1088/1367-2630/ac688c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Di Battista, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Seifert, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Principi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Efetov, Nano Letters 22, 6465 (2022), pMID: 35917225, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='nanolett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='1c04512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Deng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Yuan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Taniguchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Zhang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Xia, NATURE PHOTONICS 14, 549+ (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 11 EXTENDED DATA FIGURES Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Dual gate map of the four-probe resistance of MATBG (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦) at T=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The maxima in resistance correspond to the charge neutrality points (CNPs) and integer filling factors (ν = ±2, ±3, ±4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' V=-2 V=+2 V=+4 Resistance (kQ) 10 V=+4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='. 4 V=+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 8 2 6 gate B (V) 0 OCNP 4 2 - 2 4 0 2 CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2 4 4 gate A (V)12 Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Dual gate photovoltage maps for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a MATBG (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦, T = 10 K) and b BLG (θ = 0◦, T = 100 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' r Phobovoltage (μM) OT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 25 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 - 75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 DT- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 gate A (M) 0 Phobovoltage (uV) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 - 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 - 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 (I I 0 0"0 s- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 ot-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=" o't- 15 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content="5 oz- o'z- 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 gate A [M]13 Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' TrPV dips for the MATBG (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦) device as a function of DU vector (indicated by arrow) and temperature (see plot title).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Each time trace has been offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=5 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=10 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=15 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=20 K oF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 V 10 10 10 10 () (Ar) 20 20 + 20 20 abeal voltage hotovol QE- DE- 30 30 40 40 40 50 50 50 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='5 V 30 20 10 10 20 30 30 20 10 10 20 30 20 10 10 20 30 30 20 10 10 20 30 t (ps) (sd) t (ps) (sd) * 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=25 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=50 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=100 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=150 K 0 2 (An) (Ar) (A) 6 Photovoltage 6 6 ++ 30 8 8 8 10 10 10 40+ 12 12μ 12H 50 14 14F 14 ++++ 16E 16 E Om- 20 o1- 0 10 20 30 DE- 20 10 10 20 30 20 10 10 20 30 30 20 10 10 20 30 t (ps) t (ps) t (ps) t (ps) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=200 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=250 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24, T=300 K 0] 2 2 2 (Ar) 4 (Ar) (A) : voltage abe 6 8 8 8 lotov 10 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 12μ 12H 12 14 14 16 16 16 E 10 10 20 30 20 ot- 10 Oml OE- 20 Om- 20 10 10 20 30 t (ps) t (ps) (sd) 14 Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' TrPV dips for the BLG (θ = 0◦) device as a function of DU vector (indicated by arrow) and temperature (see plot title).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Each time trace has been offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The slower cooling at low temperatures produces a broader dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T=5 K 0, T=15 K 0, T=25 K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0V 0 10 10 (A) 20 (Ar) 20 10 M Photovoltage I Photovoltage 30 Photovoltage 30 15 40 40 20 50 50 25 60 60 OE- +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='0V 70 70 35 150 100 50 50 100 150 150 100 50 50 100 150 150 100 50 50 100 150 t (ps) t (ps) t (ps) 09, T=50 K 0, T=100 K 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T=150 K 5 (A) 10 (Ar) (A) Photovoltage Photovoltage Photovoltage I 15 10 10 20 25 15 15 30 20 20 35 150 100 50 0 50 100 150 150 100 50 50 100 150 150 100 50 0 50 100 150 t (ps) t (ps) t (ps) 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T=200 K 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T=250 K 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' T=300 K 2 5 Photovoltage (μV) (μV) (uV) Photovoltage Photovoltage 10 10 15 15 8 20 20 10 150 o0T- s- 0 50 100 150 150 100 50 0 50 100 150 150 100 50 0 50 100 150 t (ps) t (ps) (sd) 15 Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' CW-PM peaks for the MATBG (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦) device as a function of DU vector (indicated by arrow) and temperature (see plot title).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Each frequency sweep has been offset for clarity.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content="0 Photovoltage (μV) 1 (Ar) (μV) z'- Photovoltage ( 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='05 Photovoltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='4 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' offset (GHz)16 Extended Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' CW-PM peaks for the BLG (θ = 0◦) device as a function of DU vector (indicated by arrow) and temperature (see plot title).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Each frequency sweep has been offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The slower cooling at low temperatures produces a narrower peak.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Power dependence of cooling time for a second MATBG device (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='08◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The electron relaxation bottleneck at full filling (ν = ±4) leads to slower cooling time for higher laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The orange line is a guide to eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' MATBG, 0=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='08 70 60 V=±4 Cooling time (ps) 50 40 30 20 V=±2 10 0 0 1 2 3 4 5 Peak power density (GW cm-2)18 SUPPLEMENTARY INFORMATION Supplementary Note 1 In Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3, we investigate the influence of twist angle disorder on the electrical transport at T = 35 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' At the junction contact, the twist angle is θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦ and we observe sharp resistance peaks at ν = ±2 arising from correlated insulating states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' The contacts at the top of the junction display a shoulder around ν = −2 that indicates a mixing of two angles (θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='28◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' For the contacts at the bottom of the junction the angle is θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='24◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' From this we conclude that there is minimal twist angle disorder in the proximity of the pn-junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Optical images of the device before and after nanofabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a, Heterostructure stack dropped on a Si/SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' b, Finalised device after etching Hall bar and metallisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Both scale bars are 5µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' a b19 Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Low field Hall effect at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Hall carrier density nH vs n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' In the region close to charge neutrality nH = n, which allows us to calibrate the relationship between Vg and n to extract the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' Longitudinal resistance (Rxx) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' filling factor (ν) for contacts around the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} +page_content=' 4 2 nH 0 H n 4 2 0 2 4 n (1012 cm-2)102 Top of junction Junction Contact Bottom of junction 101 (kΩ2) Rxx 100 4 3 2 1 0 2 3 1 4 Filling factor v' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf'} diff --git a/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/2301.02553v1.pdf.txt b/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/2301.02553v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4738f892a4e73b7209abd72000f16dce3c0c4887 --- /dev/null +++ b/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/2301.02553v1.pdf.txt @@ -0,0 +1,6908 @@ +Pushing the Limits of the Periodic Table – A Review on Atomic +Relativistic Electronic Structure Theory and Calculations for +the Superheavy Elements∗ +O. R. Smitsa,1, P. Indelicatob,2, W. Nazarewiczc,3, +M. Piibeleht1, P. Schwerdtfegerd,1 +1 Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey +University Auckland, 0632 Auckland, New Zealand +2Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Research University, Collège de +France, Case 74; 4, place Jussieu, F-75005 Paris, France +3Facility for Rare Isotope Beams and Department of Physics and Astronomy, Michigan State +University, East Lansing, Michigan 48824, USA +In memoriam to two of the pioneers in this field, Jean-Paul Desclaux (Grenoble) and Sigurd +Hofmann (Darmstadt) +Received: date / Accepted: date +Contents +1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2 +The Dirac Equation in Strong Coulomb Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.1 +The QED Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +The Many-Electron Dirac-Coulomb-Breit Hamiltonian . . . . . . . . . . . . . . . . . . . . +5 +2.3 +The one-particle Dirac equation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3.1 +Point nucleus and self-adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3.2 +Nuclear Recoil and Uehling terms . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3.3 +Finite nuclear charge distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.3.4 +1s energy level reaching the negative energy continuum . . . . . . . . . . . . . . . . +13 +2.4 +Electron states in the super-critical region . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +2.4.1 +Energy-projected Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +2.4.2 +Hartree-Fock-Bogoliubov equation analogy . . . . . . . . . . . . . . . . . . . . . . +16 +2.4.3 +Perturbative approach +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.4.4 +Analytical continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.4.5 +Gamow states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +2.5 +Positron production in the super-critical regime . . . . . . . . . . . . . . . . . . . . . . . . +23 +2.6 +Experimental perspective: heavy-ion collisions +. . . . . . . . . . . . . . . . . . . . . . . . +25 +3 +Multi-Configuration Dirac-Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4 +Quantum Electrodynamic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.1 +S-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +4.2 +Two-times Green’s function method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +4.3 +Covariant evolution-operator procedure +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +4.4 +Calculation of QED corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +4.4.1 +One-electron radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +4.4.2 +Two-electron radiative and non-radiative corrections +. . . . . . . . . . . . . . . . . +40 +4.4.3 +Effective QED Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +5 +Electron Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +ae-mail: smits.odile.rosette@gmail.com +be-mail: paul.indelicato@lkb.upmc.fr +ce-mail: witek@frib.msu.edu +de-mail: peter.schwerdtfeger@gmail.com +arXiv:2301.02553v1 [physics.atom-ph] 6 Jan 2023 + +2 +6 +Atomic Structure Calculations of the Superheavy Elements . . . . . . . . . . . . . . . . . . . . . +46 +6.1 +Dominant Electron Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +6.2 +Ionization potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +48 +6.3 +QED effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +6.4 +Atomic static dipole polarizabilities +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +6.5 +Electron Localization Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +6.6 +Examples of relativistic effects on the chemistry of SHE +. . . . . . . . . . . . . . . . . . . +53 +7 +General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +7.1 +Placing new elements on the periodic table . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +7.2 +Dominant ground state configurations predicted by electronic structure calculations . . . . . +56 +7.3 +Periodic Table - How far can we go? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +8 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +58 +9 +Appendix A: The Self-Adjointness of the Dirac-Coulomb Hamiltonian . . . . . . . . . . . . . . . +58 +10 Appendix B: The Rigged Hilbert Space Formalism +. . . . . . . . . . . . . . . . . . . . . . . . . +60 +Abstract We review the progress in atomic structure theory with a focus on superheavy ele- +ments and the aim to predict their ground state configuration and element’s placement in the +periodic table. To understand the electronic structure and correlations in the regime of large +atomic numbers, it is important to correctly solve the Dirac equation in strong Coulomb +fields, and also to take into account quantum electrodynamic effects. We specifically fo- +cus on the fundamental difficulties encountered when dealing with the many-particle Dirac +equation. We further discuss the possibility for future many-electron atomic structure calcu- +lations going beyond the critical nuclear charge Zcrit ≈ 170, where levels such as the 1s shell +dive into the negative energy continuum (Enκ < −mec2). The nature of the resulting Gamow +states within a rigged Hilbert space formalism is highlighted. +1 Introduction +The periodic table (PT) of the elements, introduced by Dmitri Mendeleev and Lothar Meyer, +is based on the Pauli and Aufbau (building-up) principle [1]. Arguably, the PT is the most +important and useful tool concerning the electronic structure of atoms and molecules [2–5]. +Chemical and physical similarities between the elements within a group or period obtained +from their measurable properties is often hailed as a building block of the PT, but these +patterns also follow from the underlying electronic shell structure of the atoms. Despite +many controversies concerning the PT, for example, the starting and ending points of the +f-block elements, the placement of the lightest elements hydrogen and helium, observed +anomalies in chemical behavior or even the shape and visual representation [4, 6–8], it is +still going strong after 150 years. Furthermore, with the nuclear synthesis of the 7p block +elements up to oganesson with nuclear charge Z = 118 [9, 10], the full 7th period of the PT is +now complete. Hence, what remains to be solved is how the PT can successfully be extended +both theoretically and experimentally into the superheavy element region beyond Z = 118 +[11–15]. A progress in this direction has been made by placing the unknown elements up to +nuclear charge Z = 172 into the Periodic Table [16, 17], see for example Fig. 1. +The existence and properties of new superheavy elements beyond oganesson depends on +both nuclear and electronic structure properties [18]. There are, however, a number of open +questions and major challenges to both electronic and nuclear structure theory concerning +the accurate prediction of physical and chemical properties of the superheavy elements.* +*Here we define the starting point of the superheavy element region at the transactinides, Z ≥ 103 + +3 +For example, to correctly place an element into the PT and predict its basic properties, one +should gain knowledge of its atomic shell structure, such as ground and excited electronic +states and underlying dominant configurations [11, 12]. In the case of dense spectra, which +are prominent in open-shell systems as well as in the superheavy element region where high +principal quantum number and angular momentum states are occupied, detailed knowledge +of low-lying excited electronic states are required within a window of a few eV. This is +often a very challenging task as both relativistic and electron correlation effects play a major +role requiring sophisticated multi-reference methods at the relativistic Dirac-Coulomb-Breit +level of theory. Currently, the heaviest element for which it is possible to compare theory +and experiment is lawrencium (Z = 103) [19, 20]. +Moreover, the Dirac-Coulomb Hamiltonian has its limits in strong Coulomb fields as +beyond the critical nuclear charge of Zcrit ≈ 170 for finite-size nuclei, the 1s electron level +dives into the negative energy continuum below E = −mec2 [21–32]. At the single-particle +level of theory, the correct description and interpretation of the resulting resonances can +be given in terms of Gamow states [33–37], but how such diving states can correctly and +accurately be described within a multi-electron framework, and how the PT can be extended +beyond the critical nuclear charge, are open questions. +At high nuclear charge, the PT is ultimately limited by the nuclear stability, not by +its electronic shell structure [18, 38]. For nuclear structure theory and corresponding pre- +dictions of nuclear stability of isotopes see for example Refs. [18, 38, 39] and references +Fig. 1 Pyykkö’s periodic table extended to Z = 172 (with permission from PCCP [17]). + +Group +Period +1 +2345678 +９101112 +13 +14 +15 +16 +17 +18 +Orbital +1 +2 +1 +H +He +1s +3 +b +5 +6 +7 +8 +9 +10 +2 +2s2p +Li +Be +B +C +N +0 +F +Ne +11 +12 +13 +14 +15 +16 +17 +18 +3 +Na +Mg +Al +Si +P +s +CI +Ar +3s3p +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +4 +Co +4s3d4p +K +Ca +Sc +Ti +V +Cr +Mn +Fe +Ni +Cu +Zn +Ga +Ge +As +Se +Br +Kr +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +5 +5s4d5p +Rb +Sr +Y +Zr +Nb +Mo +Tc +Ru +Rh +Pd +Ag +Cd +In +Sn +Sb +Te +1 +Xe +55 +56 +57-71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +6 +Hg +6s5d6p +Cs +Ba +Hf +Ta +W +Re +Os +Ir +Pt +Au +TI +Pb +Bi +Po +At +Rn +87 +88 +89-103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +7 +Sg +Mc +7s6d7p +Fr +Ra +Rf +Db +Bh +Hs +Mt +Ds +Rg +Cn +Nh +FI +LV +Ts +Og +121- +8 +119 +120 +156 +157 +158 +159 +160 +161 +162 +163 +164 +139 +140 +169 +170 +171 +172 +8s7d8p +9 +165 +166 +167 +168 +9s9p +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +6 +Pm +Dy +Ho +4f +La +Ce +Pr +Nd +Sm +Eu +Gd +Tb +Er +Tm +Yb +Lu +89 +06 +91 +92 +93 +94 +95 +96 +97 +98 +66 +100 +101 +102 +103 +7 +Ac +Cm +5f +Th +Pa +U +Np +Pu +Am +Bk +Cf +Es +Fm +Md +No +Lr +8 +141 +142 +143 +144 +145 +146 +147 +148 +149 +150 +151 +152 +153 +154 +155 +6f +8 +121 +122 +123 +124 +125 +126 +127 +128 +129 +130 +131 +132 +133 +134 +135 +136 +137 +138 +5g4 +therein. Here we focus solely on the discussion of relativistic electronic structure theory in +the superheavy element region [14, 40–42]. +The outline of this Review is as follows. We first discuss the Dirac equation and its pe- +culiarities compared to the non-relativistic Schrödinger equation, specifically for electrons +in strong Coulomb fields. We discuss the critical nuclear charge in detail to clarify the region +of validity of the Dirac-Coulomb Hamiltonian and discuss how states embedded in the neg- +ative energy continuum should be interpreted. The process of spontaneous pair creation in +a supercritical field is analyzed including most recent references. The importance of quan- +tum electrodynamics (QED) effects and how these can be treated in strong Coulomb fields +is outlined. The major problem of correctly describing electron correlation for the accurate +prediction of electronic spectra in the superheavy element region is addressed. We review +the current status of electronic structure calculations for the transactinides and discuss the +placement of the elements beyond oganesson into the PT based on quantum theoretical pre- +dictions. The literature on this topic is vast [28, 43–45], including a rigorous mathematical +treatment of the Dirac equation and its generalizations [29, 46–50]. +2 The Dirac Equation in Strong Coulomb Fields +2.1 The QED Lagrangian +Electronic structure theory is based on the QED sector of the Standard Model of particle +physics. Within the Standard Model, electrons are spin-1/2 Dirac fermions, and their dy- +namics is described by the QED Lagrangian density +LQED = +i¯hc ¯ψ(x)γµ∂µψ(x)−mec2 ¯ψ(x)ψ(x) +−1 +4FµνFµν −e ¯ψ(x)γµAµ(x)ψ(x), +(1) +where ψ(x) is the field operator and γµ are the Dirac matrices. The first two terms in (1) +are the kinetic and mass terms describing the free electrons with mass me, whereas the third +term describes the photon field Aµ = (φ,AAA), corresponding to the electromagnetic scalar +and vector potentials (with Fµν = ∂µAν −∂νAµ). The last term corresponds to the interaction +between electrons and photons, with the elementary charge e acting as the coupling constant. +The interaction picture represented by Eq. (1) has been extensively used in quantum field +theory and it has been demonstrated to work to astonishingly high accuracy. +It would be highly desirable to treat the QED Lagrangian for a many-electron system in +an external Coulomb field to avoid divergencies that appear in perturbative treatments [51]. +Such a direct treatment could in principle be performed through lattice gauge theory which +is mathematically well defined [52, 53]. However, the long-range nature of the Coulomb +potential, related to the zero rest-mass of the photon, currently prevents any accurate com- +putational treatment using lattice gauge theory in finite boxes [54]. Treating the required +large boxes is currently computationally too demanding. However, progress in this field has +recently been made on the nuclear length scale. For instance, a combined lattice QCD+QED +approach has been used to successfully calculate hadron and meson mass differences, such +as the proton-to-neutron mass splitting, and its dependence on both the strong and electro- +magnetic coupling constants [55, 56]. + +5 +2.2 The Many-Electron Dirac-Coulomb-Breit Hamiltonian +Atomic physics calculations are performed in the Hamiltonian formalism derived from the +Langrangian (1) by a Legendre transformation [57–60]. The resulting first-quantized N- +particle Hamiltonian can be written in atomic units (i.e., ¯h = 1,e = 1,me = 1) as [45, 61, 62]: +HD = +N +∑ +k=1 +hk + +N +∑ +k 0, (b) free particle (eZ=0), and (c) the charge +conjugated case of an antiparticle of charge +e and mass m in a Coulomb potential with eZ < 0. +seen, however, as more of a technical problem than a fundamental one† discussed in more +detail in Sec. 3. +2.3 The one-particle Dirac equation +In order to solve the many-electron problem, one must first understand the single-particle +case. Thus, in the following, we consider the stationary Dirac equation for a single particle. +2.3.1 Point nucleus and self-adjointness +In strong Coulomb fields, a difficulty arises for the Dirac equation modelled with a point +nuclear charge (PNC). To illustrate this, it suffices to consider the radial form of the one- +particle Dirac-Coulomb equation: +� mec2 +V(r)−Enκ +c +� +− d +dr + κ +r +� +c +� d +dr + κ +r +� +−mec2 +V(r)−Enκ +�� Pnκ(r) +Qnκ(r) +� += 0, +(4) +with the corresponding four-component orbital spinor +ψnκµ(r) = 1 +r +� Pnκ(r)χκµ(θ,φ) +iQnκ(r)χ−κµ(θ,φ) +� +, +(5) +†We distinguish between problems of fundamental nature as those where knowledge to solve a particular +problem is not yet available (such as problems involving physics beyond the standard model, the foundation +of quantum field theory and Haag’s theorem [86], etc.) and those where knowledge is in principle available +but the solution of the problem can be very hard to obtain (such as electron correlation and QED to all orders) +or can be solved based on existing theory (such as resonant states embedded in the scattering continuum). + +- E=+mc2 +- E=-mc2 +eZ>0 +O=a +0>za +(a) +(b) +(c)7 +-1 + 0 + 1 + 0 + 20 + 40 + 60 + 80 + 100 + 120 + 140 + 160 + 180 +Energy 1s (mc2) +Atomic number Z +PNC +PNC + recoil +PNC + recoil + VP + +FNC +FNC + VP + SE +FNC Ar + +NR PNC +Fig. 3 Nuclear charge dependence of the 1s energy levels for hydrogen-like atoms at various levels of theory +using the Dirac equation. If not otherwise stated the results are from Ref. [31]. The models considered are : +PNC - point nuclear charge; FNC - finite nuclear charge distribution; recoil - nuclear recoil effects according +to Eq. (8) [94]; recoil+VP - includes the Uehling vacuum polarization term [94]; VP+SE - includes major +QED corrections from vacuum polarization and self-energy; NR - nonrelativistic results. Results are also +shown for the Ar-like system with FNC. +where κ = ±(j+ 1 +2) for j = ℓ∓ 1 +2. The bound state eigenvalues for the point nuclear charge, +V(r) = −Z/r are [87–89] +Enκ(Z) = mec2 +� +� +�1+ +(Zα)2 +� +n−|κ|+ +� +κ2 −(Zα)2 +�2 +� +� +� +−1/2 +, +(6) +where α is the fine-structure constant (α−1 = 137.035999206(11) [90]). The solution (6) +is known as the Sommerfeld fine-structure formula [91]. A historical overview is given in +Weinberg’s book on the quantum theory of fields [92]. +It is apparent that a problem occurs when Z > Zcp = |κ|/α, as Enκ(Z) becomes imag- +inary [93]. The range of such large Z-values is usually referred to as the critical nuclear +charge region. At the onset of the imaginary solutions, Eq. (6) simplifies to +En,κ(Zcp) = mec2(n−|κ|) +� +n2 −2|κ|n+2κ2�−1/2 ≥ 0, +(7) +and one obtains E1,−1 = 0 for 1s, E2,−1 = E2,1 = mec2/ +√ +2 for 2s and 2p1/2 at Zcp = 1/α ≃ +137.036, and E2,−2 = 0 for 2p3/2 at Zcp = 2/α ≃ 274.072. The difference between the +behaviour of the nonrelativistic and relativistic 1s energies with increasing nuclear charge is +shown in Fig. 3. +The presence of the critical charge distinguishes the Dirac equation from the standard +Schrödinger equation with a Coulomb potential of a point nuclear charge, where all values +Z ≥ 1 are allowed, although one would run into similar problems with the Schrödinger +equation for potentials of the form V(r) = −Z/rn with n ≥ 2 [95]. + +8 +To treat atoms with nuclear charges beyond a certain critical charge, Z > Znsa, where the +Dirac operator becomes non-self-adjoint (nsa), one has to carefully choose an appropriate +self-adjoint extension to the basic Dirac-Coulomb operator together with the correct operator +domain [29, 30, 46, 96–100]. For example, this can be done by adding additional operators +such as the nuclear recoil and Uehling terms, discussed in section 2.3.2, or by removing +the problematic singularity in the Coulomb term at zero by working with a realistic finite- +size nuclear charge distribution to regularize the Coulomb interaction. The mathematical +problem arises due to the singularity of the Coulomb operator −Z/r at the origin. As a result, +the Dirac operator is not (essentially) self-adjoint anymore in the critical nuclear charge +region. In fact, HD becomes non-self-adjoint [98] for a j-state at Z ≥ Znsa = +� +j( j +1)/α. +For the 1s level this corresponds to Z ≥ +√ +3/(2α) ≃ 118.677 [96, 97, 101], which lies just +above the nuclear charge of oganesson (Z = 118). This was pointed out as early as in 1928 +by Gordon [89]. For a more rigorous mathematical analysis on the self-adjointness of the +point-charge Dirac-Coulomb operator we refer the reader to Sec. 9 and the literature cited +therein. +On a historical note, the onset of imaginary solutions for the Dirac equation with the +bare Coulomb operator led Feynman to the conclusion that elements above Z = 137 should +not exist. Hence, the element with nuclear charge 137 is sometimes (jokingly) called Feyn- +manium [102]. +2.3.2 Nuclear Recoil and Uehling terms +For a point-like nucleus, the nuclear recoil operator can be approximated by [94] +HNRB = − 1 +2M ∆ +i(Zα) +2Mr +� +ααα ·∇∇∇+ 1 +r2 (ααα ·rrr)(rrr ·∇∇∇) +� +, +(8) +where M is the mass of the nucleus (for a more concise QED treatment see [103]). This +recoil operator can be added to the one-particle Dirac-Coulomb operator. For a more detailed +discussion of nuclear recoil effects see Refs. [104, 105]. +In Ref. [94], both the recoil correction and the Uehling potential VU for a point nu- +cleus were included in the Dirac equation to see how that would change the Zcp. A value +of Zcp(1s) = 144 is then obtained. These additional operators do not necessarily secure the +self-adjointness of HD +HNRB in the critical charge region, hence, a careful analysis of the +1s eigenfunction at the origin is still required. More details on vacuum polarization (VP) +and on the Uehling potential can be found in Sec. 4. +Nevertheless, numerical calculations show that the critical charge for the 1s state before +reaching the onset of the negative energy continuum is Zc(1s) ≈ 144.75 due to the nuclear +recoil, and Zcp(1s) ≈ 143.95 due to the nuclear recoil-plus-Uehling term as shown in Fig. 3 +[94]. Moreover, the diving of the 2p1/2, 2s and 3s levels comes at nuclear charges of Zcp ≈ +146, 165, and 193 respectively. The lifting of the 2s/2p1/2 level degeneracy due to the +nuclear recoil becomes thus quite sizable at high-Z values. For Zα < 1, the results including +nuclear recoil and Uehling terms are close to the point nuclear charge (PNC) case, as is the +steep descent of the energy levels towards the critical nuclear charge. On the other hand, +Fig. 4 demonstrates that around Z = 120 the finite nuclear size correction becomes more +important than that originating from the nuclear recoil and Uehling terms. This is addressed +in the following section. + +9 +10-6 +10-4 +10-2 +100 +102 +104 +106 + 0 + 20 40 60 80 100 120 140 +Contributions to F(Zα) +Nuclear charge Z +Total Energy +FNS +Self-Energy +Recoil +Uehling VP +Error Nuc. Size +W and K VP +H.O. order VP +Loop-after-Loop VP +SE-SE +SE-VP +S(VP)E +Total Lamb Shift +Light-by-Light scattering +Fig. 4 Different contributions to the 1s energy level for hydrogen-like atoms, evaluated using the MCDFGME +code [106]. Higher-order VP includes the Wichmann and Kroll (WK) correction (order α(Zα)3) as well as +approximation to the α(Zα)5 and α(Zα)7 potential contributions. Two-loop self-energy corrections SE-SE, +SE-VP and S(VP)E are from Refs. [107–113]. Loop-after-loop VP is approximated by solving the Dirac +equation including the Uelhing potential. Finite nuclear size correction and uncertainties on nuclear size are +from Ref. [114]. See also [115, 116] and references therein. +2.3.3 Finite nuclear charge distributions +By considering a finite nuclear charge distribution, ρN(rrr), the problematic singularity at zero +is removed. As a result HD becomes self-adjoint for Z > Zcp with real eigenvalues and real +radial functions for the discrete spectrum, and thus represents the most natural self-adjoint +extension to the PNC Dirac Hamiltonian. This was already realized by Schiff, Snyder and +Weinberg as early as in 1939 [93]: In all these cases where the energy cannot be brought +to diagonal form, one must take into account either existing deviations from the assumed +potential, such as the breakdown of the Coulomb law at small distances, or the reaction of +the pair field itself on the external field. +The potential for an electron interacting with a nuclear charge distribution is given by +V(rrr) = − +� +dRRR ρN(RRR) +|rrr −RRR|. +(9) +The nuclear charge densities should in principle be obtained using nuclear density func- +tional theory (DFT) based on realistic energy density functionals, see Sec. 2.4.2. To obtain +the nuclear charge density from computed proton and neutron density distributions, several +corrections have to be considered [117–119]. The nucleon structure is taken into account by +folding with the intrinsic form factor of the free nucleons expressed in terms of the Sachs +form factors [120]. The spurious center-of-mass motion can be corrected by an unfolding +with the width of the centre-of-mass vibrations. Finally, one should include the contribution +from the spin-orbit currents [121]. Note that, for the deformed nuclei, the spin-orbit con- +tributions change gradually as the single-particle spin-orbit strength becomes highly frag- +mented by deformation and nucleonic pairing (nucleonic superconductivity) [119]. Precise +nuclear charge densities are essential for interpreting atomic experiments searching for new +physics [122, 123] or for studying effects related to fundamental symmetry violations [124]. + +10 +10 +5 +0 +5 +10 +r (fm) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +ρ (fm−3) +ρn +ρp +48Ca +208Pb +302Og +472164 +Fig. 5 +Radial proton (left) and neutron (right) densities of doubly-magic nuclei 48Ca, 208Pb, 302Og, and +472164 obtained in nuclear DFT with three different energy density functionals. The shaded areas indicate the +spread of DFT predictions. (Modified from [125].) +Realistic nuclear modeling of charge densities is particularly important for the super- +heavy nuclei, the existence of which depends on the interplay between the short-ranged +attractive nuclear force and long-ranged electrostatic repulsion, which rapidly grows with +Z. Since the Coulomb repulsion minimizes the total binding energy of the nucleus by in- +creasing the average distance between protons, the total energy is significantly lowered by +pushing protons toward the nuclear surface. This mismatch between interaction ranges in +superheavy nuclei results in Coulomb frustration effects [18, 38], which are expected to +produce exotic topologies of nucleonic densities, such as voids (bubbles) or tori. Figure 5 +shows the proton and neutron density distributions of several nuclei predicted by nuclear +DFT [125]. The superheavy nuclei such as 302Og, and 472164 show a clear central depres- +sion in the proton density distributions resulting in a semi-bubble structure. The properties +of Coulomb-frustrated superheavy nuclei, including their characteristic density distributions +and shell structure, have been investigated in numerous studies, see Refs. [125–127] and +references cited therein. +In the absence of predictions based on realistic nuclear models, schematic approxima- +tions for ρN are often applied. These are sufficient for most applications in heavy element +research. There is a range of nuclear charge models in use and, for several of these models, +analytical expressions for the integral (9) in terms of standard functions can be found in +Ref. [128]. Most implementations in numerical atomic structure programs apply the (spher- +ical) Fermi two-parameter model [129, 130] +ρN(R) = +ρ0 +1+e(R−R0)/a , +(10) +where R0 is the half-density radius, a is the diffuseness parameter, and ρ0 is a normalization +constant such that +� ρN(RRR)dRRR = Z. For many nuclei, this model reasonably agrees with +nuclear DFT calculations. It is to be noted, however, that a simple model like (10) is bound +to fail for superheavy nuclei that exhibit appreciable Coulomb frustration effects, see Fig. 5. +Nevertheless, for the valence shell this nuclear charge model should perform reasonably +well even for the superheavy elements. For example, the Fermi charge distribution have been +used for electronic structure calculations of Ref. [14] in the superheavy element region up +to Z = 173. + +11 +For the homogeneous nuclear charge distribution analytical expressions for the radial +Dirac components of the wave function exist. In that category of nuclear models, the simplest +one is the uniformly charged spherical shell or top slice (TS) model, with the nuclear charge +being smeared out over a spherical nuclear surface at radius R0 [128, 131, 132] +ρ(r) = +Z +4πr2 δ(r −R0). +(11) +It results in a potential of the form V(r) = −Z/R0 for 0 ≤ r ≤ R0 together with the usual +Coulomb term V(r) = −Z/r at r > R0. This approximation cuts off the problematic singular- +ity of the Coulomb potential at nuclear radius R0 and therefore secures the self-adjointness +in the region |E| < mec2 of the discrete spectrum [21]. To express the radial Dirac compo- +nents analytically, one divides the solution of the Dirac equation into the two regions [0,R0] +and [R0,∞) with an additional boundary condition at R0 to match the two wave functions +(see also Sec. 2.4.4) [28]. +The potential V(r) for the TS model is, however, discontinuous in its first derivative and +is therefore often extended to the homogeneously charged sphere (HCS) model of the form +[133] +ρ(r) = ρ0Θ(1−r/R0), +(12) +where ρ0 = 3Z/4πR3 +0 and Θ(x) is the Heaviside step function [128, 132]. The resulting HCS +potential is of the form V(r) = V0 +V2r2, where V0 = −3Z/2R0 and V2 = V0/2R2 +0. This po- +tential is discontinuous in its second derivative.‡ It is clear that by choosing V0 = −Z/R0 and +V2 = 0 the TS model is recovered. This results in a special case of a Fuchs-type differential +equation for which analytical solutions to the Dirac equation can be formulated similarly +to the procedure used for the TS model [135]. The HCS model was recently used to study +isotope shifts using a modified nuclear parameter δ⟨r2⟩ → δ⟨r2γ⟩, such that the electronic +structure factor ˜Fi becomes isotope independent [136–138]. Note that this approximate ex- +pression for the energy shift is valid only when αZR0 ≪ 1 [139], where R0 is expressed in +atomic units. For nuclear charge Z > 137 this expression is manifestly wrong as γ becomes +imaginary. +The HCS model can be further generalized by using a Taylor expansion for the nuclear +density around the origin [128] +ρ(x) = Θ(1−x) +n +∑ +i=0 +aixi , +(13) +with x = r/R0 resulting in a power series for V(r). Breit introduced the simple potential +V(r) = V0 +V2rn, where V0 = −(n+1)Z/nR0 and V2 = Z/nR(n+1) +0 +[133]. For these nuclear +models one can derive the radial Dirac wave function from a polynomial expansion. [128, +135]. In the region r < R0 for κ > 0 the radial wave function is expressed as [140] +Pnκ(r) = Nnκrκ +� +r − +�(Enκ −V0)(Enκ +2c2 −V0) +2c2(3+2κ) ++ +V2(1+2κ) +(Enκ +2c2 −V0)(3+2κ) +� +r3 +··· +� +Qnκ(r) = Nnκrκ +� +c(1+2κ) +(Enκ +2c2 −V0) − Enκ −V0 +2c +r2 +··· +� +, +(14) +‡Because of the discontinuity in the potential at R0, one has to set one of the grid points in numerical +program packages at the nuclear boundary to avoid numerical instabilities [134]. + +12 + 1.3 + 1.305 + 1.31 + 1.315 + 1.32 + 233 + 234 + 235 + 236 + 237 + 238 +Uranium Isotopes +ΔE (a.u.) +Mass number A +Deformed Fermi +Fitted Fermi +Uniform Charge +Fig. 6 The nuclear-size contributions to the ground-state energies of the Li-like uranium isotopes using a +deformed Fermi model for ρN, a fitted Fermi model, and a uniform charge distribution. (From [145].) +and for κ < 0 +Pnκ(r) = Nnκr|κ| +� +1− (Enκ −V0)(Enκ +2c2 +V0) +2c2(1+2|κ|) +r2 +··· +� +Qnκ(r) = Nnκr|κ| +� +− Enκ −V0 +c(1+2|κ|)r + +�(Enκ −V0)2(Enκ +2c2 +V0) +2c3(1+2|κ|)(3+2|κ|) ++ +V2 +c(3+2|κ|) +� +r3 +··· +� +. +(15) +Since the exponents of the r|κ|+i terms in (14) and (15) are integers, there is no problem at the +origin and the derivative norm exists. Furthermore, the wave function is locally absolutely +continuous, unlike for the PNC case. To show this more rigorously, one applies the Weyl- +Weidmann limit point - limit circle theorem [141] and shows that the Dirac operator is self- +adjoint in the range Enκ ∈ [−mec2,mec2], with the Sobolev space W1,2(R+)2 as the natural +domain of the Dirac operator. Hence, for the Dirac equation with a finite-size nuclear charge +distribution, the only critical charge is at the onset of the negative energy continuum at +E = −mec2. Full analytic expressions for the Dirac wave functions for TS and uniformly +charged nucleus have been derived for s1/2 and p1/2 orbitals and used in the evaluation of +the self-energy with finite size contribution [142, 143]. +Shifting from a point nucleus to a model that accounts for the finite nuclear charge +distribution leads to a noticeable contribution to the total electronic energy. The difference +in energy originating from the use of different nuclear charge models is far smaller [144]. +For example, Fig. 6 shows the calculated ground state energy shift in Li-like uranium due to +the finite nuclear charge distribution for the Fermi and uniform charge distributions [145]. +When introducing a finite nuclear charge into the Dirac equation, the degeneracy be- +tween the states of the same (n j) but with different κ quantum numbers is lifted. This is +most prominently seen between the 2s1/2 and 2p1/2 levels. This lifting of degeneracy al- +ready appears at the nonrelativistic level between levels of same n but different ℓ quantum +numbers, but to a much smaller extent compared to the relativistic case [128]. +Figure 7 shows the energy difference ∆E between the 2p1/2 and 2p3/2 orbitals and the +2s1/2 orbital for the hydrogen-like and the Be-like state [40]. The lifting of degeneracy by the + +13 +Fig. 7 Orbital energy difference ∆E (in a.u.) of the 2p1/2 (purple) and 2p3/2 (orange) states relative to the +2s1/2 state (in a.u.). The dashed lines are hydrogenic energy differences. The solid lines are multi-reference +energies for Be-like J = 0 states systems involving the major configurations 1s2 2s2,1s2 2p2 +1/2,1s2 2p2 +3/2. +finite size of the nuclear charge for the hydrogen-like system can be qualitatively explained +by perturbation theory. However, in the small region inside the nucleus, the perturbing po- +tential is so large that a first-order calculation for high nuclear charges is insufficient [146]. +In contrast to the hydrogen-like energy difference, in multi-electron systems the 2s shell +lies below the 2p shell for nuclear charges up to about Z = 120. This comes from the dif- +ferent effective screening of the nucleus for these two shells, which gave rise in the early +history of quantum theory to the Slater rules [147]. For nuclear charges beyond Z = 120, +the 1s2 2p2 +1/2 J = 0 configuration lies below the 1s2 2s2 configuration, as demonstrated for +the Be-like J = 0 state in Fig. 7 and in Ref. [40]. This is because, in strong Coulomb fields, +the Coulomb operator starts to dominate over the electron-electron repulsion and the atom +behaves more hydrogen-like. As a result of this effect, the 2p1/2 level dives into the nega- +tive energy continuum at a far earlier stage at Zc +� +2p1/2 +� +≈ 218 compared to the 2s level at +Zc (2s) ≈ 247 [31], see discussion in Sec. 2.4.4 for more details. +2.3.4 1s energy level reaching the negative energy continuum +Figure 3 shows the 1s energy level as a function of nuclear charge for hydrogen-like sys- +tems in the FNC variant, computed using the relativistic atomic program package GRASP +[148]. The calculations predict a critical charge of Zc(1s) = 170.161 (170.017 including +QED effects) before diving into the negative energy continuum [31]. +Using different models of nuclear charge distribution, the predictions for the critical +nuclear charge can vary widely between Zc = 164 to 174 for the 1s level [149, 150], but +more realistically between 168 to 172 using the uniform nuclear charge distribution and +neutron numbers varying between N = Z and N = 3Z. This is demonstrated in Fig. 8, which +shows the relation between the rms nuclear charge radius Rch and the proton number Z. +The critical charge as a function of Rch has been computed using the analytical expressions +of Ref. [151]. Filled squares mark the experimental charge radii [114]. The lines denote +the relation between Rch and Z for three different neutron to proton ratios [152, 153] and + +50 +25 +2p1/2 +2p3/2 +-25 +hydrogenic +Be-like +-50 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 100 110 120 130 140 +Nuclear charqe Z14 + 0 + 50 + 100 + 150 + 200 + 0 + 2 + 4 + 6 + 8 + 10 + 12 +Zc ≈ 170 +Nuclear charge Z +Zch (fm) +Zc(Rch) An. +Zc(Rch) Num. + +Rch(exp) + N/Z = 1 + N/Z = 2 + N/Z = 3 +Andrae +Fig. 8 The nuclear charge radius Rch as a function of Z, using phenomenological expressions with different +neutron/proton ratios (solid lines) and the expression by Andrae [128] (dashed line). Experimentally known +charge radii [114] are marked by orange squares. The critical charge as a function of nuclear radius obtained +with the analytical expression of Ref. [151] is shown by a dash-dotted line. The Zc (Rch) Numerical values +(black square) have been obtained using the MDFGME code [14, 154] with a Fermi nuclear charge model. +the semi-empirical relation [128]. From the intercept between the dashed and dash-dotted +lines, an estimate for the critical charge is Zc(1s) = 170.26, with a nuclear charge radius of +Rch = 7.19 fm. +In the context of the above discussion, it is interesting to notice that because of the +mass scaling √m of the Dirac equation (4) the critical charge for muonic atoms (mµ/me = +206.7682830(46) [155]) for a point nucleus is more than an order of magnitude larger +Zµ +cp(1s) ≈ 1966 compared to the electronic case. Taking into consideration the finite nu- +clear radius, the critical value shifts to Zµ +c (1s) ≈ 2200 [156]. As in the free-particle case, the +small component becomes large and takes over for E → −mec2. +2.4 Electron states in the super-critical region +In 1969, Pieper and Greiner [135] analyzed in detail the analytical solutions for FNC models +as the limit Enκ = −mec2 is approached for different (nκ) states. The coefficients in the r- +expansion in (14) and (15) do not exhibit any pathological behavior, but the radial functions +and eigenvalues become complex in the critical region Enκ < −mec2 and thus lie outside the +natural domain of the self-adjoint Dirac operator. As a result, the Dirac-Hamiltonian eigen- +states embedded in the continuum cannot readily be reached by standard atomic structure +theory. In the following, we discuss some of the approaches to deal with this problem. +2.4.1 Energy-projected Dirac equation +The relation between the absence of self-adjointness and the appearance of the negative en- +ergy continuum in the spectrum of the Dirac operator was studied by restricting the Hilbert + +15 +space to the subspace defined by the positive energy continuum states. This can be effec- +tively achieved by means of the projection technique, analogous to the Feshbach projec- +tion technique [157, 158] used in the context of open quantum systems. Effectively, in this +method, the negative-energy continuum space is removed [71]. The resulting single-particle +Dirac Hamiltonian, the so-called no-pair external field Dirac Hamiltonian, becomes: +ˆh+ = Λ +(hD +Vext)Λ + +(16) +where Λ + is the projection operator onto the free-particle positive energy subspace of the +free-particle Dirac Hamiltonian HFP +D . As long as HFP +D has no zero eigenvalues, the operator +Λ + can be written as +Λ + = 1 +2 +� +1+ HFP +D +|HFP +D | +� += ααα · ppp+βmc +� +ppp2 +m2c2 +(17) +where the quotient HFP +D /|HFP +D | is called the sign operator.§ The eigenvalues of the free- +particle Dirac Hamiltonian hD are |E| ≥ mec2 [29]. The projected Dirac Hamiltonian (16) +can be traced back to Bethe and Salpeter [43, 160], and is therefore sometimes referred to as +the Bethe-Salpeter operator [161]. As discussed in [162], the projection operator effectively +removes the pair creation and annihilation terms from the Dirac Hamiltonian, i.e., removes +the coupling to the pair creation/annihilation channel. +Intuitively one would expect that various mathematical problems with the Dirac equation +might disappear if the negative-energy continuum states are projected out. However, if the +external field Vext(r) is the simple 1/r potential corresponding to a point nucleus, ˆh+ also has +a critical charge at which it becomes non-self-adjoint, just like the standard Dirac operator. +In fact, the critical charge of ˆh+, +Zc = +� 2 +π + π +2 +� +α−1 ≈ 124.16, +(18) +is lower than α−1 ≈ 137 [71, 161].¶ The no-pair approach based on the free Dirac Hamil- +tonian has therefore been criticized in Ref. [82], where it is shown that it does not prevent +continuum dissolution and that projection operators from the bound Dirac Hamiltonian must +be used instead. The necessity to use projection operators for correlation orbitals is shown +in Ref. [76]. +Unlike the Dirac equation, the no-pair operator has a lower bound in the sub-critical +region. This result was further refined in Refs. [163, 164], which demonstrated that the +operator’s eigenvalues are strictly positive [163, 164], in contrast to the point nucleus Dirac +equation, for which the eigenvalues go to zero for increasing nuclear charge up to Zα = 1. +The projection equation with a finite nuclear potential was initially thought to remove all +the problems with the negative energy continuum. Table 2.4.1 benchmarks the no-pair ap- +proximation against Dirac-Coulomb calculations for the 1s1/2 ionization potential and tran- +sition energies of 238U91+. Such highly charged atoms are important for precision tests of +§The Hamiltonian HFP +D , while similar to, is not the same as the no-pair Hamiltonian often used in rela- +tivistic quantum chemistry to avoid the continuum dissolution. In that case, the projection operator is usually +constructed from the positive energy eigenstates of the full external-field Dirac Hamiltonian, and does not +span quite the same space as that of free-particle states. Furthermore, the corresponding projection operators +depends on the nuclear charge distribution [59, 76, 82, 159]. +¶As discussed in Sec. 9, the Dirac equation with a 1/r potential has another critical nuclear charge at +Zc = ( +√ +3/2)α−1 ≈ 118.68, when the condition ||HDφ||2 < ∞ is imposed to guarantee self-adjointness. The +projected equation exhibits a similar critical charge at Zc = (3/4)α−1 ≈ 102.78 [71, Eq. (2.9)]. + +16 +Ionization potential +1s1/2 → 2p1/2 +1s1/2 → 2p3/2 +E +∆Eexp +E +∆Eexp +E +∆Eexp +DC / PNC +132,279.93 +454.83 +98,064.45 +458.84 +102,630.10 +451.98 +DC / FNC +132,083.55 +258.45 +97,872.42 +266.81 +102,433.71 +255.59 +PDC/ FNC +140,474.30 +8,649.20 +105,686.10 +8,080.49 +110,767.33 +8,589.21 +Exp. +131,825.10±4.20 +97,605.61±16.00 +102,178.12±4.33 +Table 1 Comparison of Dirac-Coulomb calculations with experimental values for the 1s1/2 ionization poten- +tial and transition energies of 238U91+. All energies in eV. The rows correspond to the standard hydrogenic +Dirac-Coulomb (DC) equation with point nucleus (PNC), finite nucleus (FNC), and the free-particle pro- +jected Dirac-Coulomb (PDC) equation with a FNC approximation. The experimental values are taken from +Refs. [116, 165] by picking the values with the lowest uncertainty. The homogeneous uniformly charged +sphere model was used. The difference with experiment and calculation for the FNC value is due to QED +corrections which are not included here. +QED [165], and QED results agree with experiments to a few eV [116]. Unlike in the Dirac- +Coulomb variant, the results of the free-particle projected approach shown in Table 2.4.1 +compare poorly with experiment. This indicates that the projected Dirac Hamiltonian ap- +pears to be a far worse starting point than the standard Dirac equation for further QED +refinements. The reasonable choice of projection operators for the whole range of nuclear +charges Z remains a challenging problem. At this stage, keeping the physically relevant +negative-energy continuum and dealing with directly it seems to be a better solution. How- +ever, this requires to correctly describe resonance states with E ≤ −mec2 as discussed in +Secs. 2.4.2-2.4.5. +2.4.2 Hartree-Fock-Bogoliubov equation analogy +It is instructive to make an analogy between the one-particle Dirac-Coulomb Eq. (4) and one- +quasiparticle Hartree-Fock-Bogoliubov (HFB; or Bogoliubov-de Gennes) equation used in +the density functional theory (DFT) of superconductors and atomic nuclei. +The HFB equation in the coordinate representation [166, 167] can be written as: +� h−λ +∆ +−∆ ∗ −h∗ +λ +�� ui +vi +� += Ei +� ui +vi +� +, +(19) +where h is the single-particle Hamiltonian; ∆ is the pairing mean-field; λ is the chemical +potential (or Fermi level); Ei is the quasi-particle energy; and ui(rrr,σ) and vi(rrr,σ) are the +upper and lower components of quasi-particle wave functions, respectively, that depend on +the spatial coordinates rrr and spin σ. The main DFT ingredient is the energy density func- +tional (EDF) that depends on the particle and pair densities and currents. The mean-fields h +and ∆ are determined self-consistently from the one-body densities and the assumed EDF. +The quasiparticle vectors are two-component wave functions ui(rrr,σ) and vi(rrr,σ), which +acquire specific asymptotic properties [166–169] determining the asymptotic behavior of lo- +cal densities. As shown in Fig. 9, the quasiparticle energy spectrum Ei of HFB consists of +discrete bound states, resonances, and non-resonant continuum states. The bound HFB solu- +tions exist only in the energy region |E| < −λ. The quasiparticle continuum with |E| > −λ +consists of non-resonant (scattering) continuum and quasiparticle resonances. +The HFB equation (19) possesses the quasiparticle-quasihole symmetry. Namely, for +each quasiparticle state (ui,vi) and energy Ei there exists a conjugate quasihole state (v∗ +i ,u∗ +i ) +of opposite energy −Ei. That is, the spectrum is composed of pairs of states with opposite + +17 +energies, see Fig. 9. The conjugate states can be related through a discrete symmetry, such +as time reversal [170]. In the HFB vacuum, corresponding to even number of fermions, all +negative-energy eigenstates are occupied by quasiparticles. This set of quasihole states is +referred to as the Bogoliubov sea [171, 172]. It follows from the projection property of the +generalized HFB density matrix that if a positive-energy one-quasiparticle state is occupied, +its conjugated negative-energy partner is empty [173], and vice-versa. +The Bogoliubov sea is infinitely deep, in a full analogy with the sea of negative-energy +states of the Dirac equation. In practice, since infinite sums over the Bogoliubov sea cannot +be carried out when computing local HFB densities, the number of HFB-active states must +be truncated. Two different ways of achieving this goal are most often implemented, namely, +solution of the HFB equations in a finite Hartree-Fock space [174] and truncation of the +quasiparticle space. The second method corresponds to truncating directly the quasiparticle +space and using a renormalization or regularization technique to account for the truncated +states [167, 169, 175–179]. +bound quasiparticles +one-quasiparticle HFB energy +bound quasiholes +quasipaticle continuum +quasihole continuum +(chemical potential) +E < λ +E > −λ +−λ +λ +resonances +Fig. 9 One-quasiparticle HFB spectrum. The bound states exist in the energy region |E| < −λ, where λ is +the chemical potential (negative for a particle-bound system). +The proper treatment of nuclear quasi-particle HFB continuum is important for accurate +description of ground-state properties and excitations [169, 171, 177, 180, 181]. Within the +real-energy HFB framework, the HFB equations must be solved by imposing the scattering +boundary conditions on the quasiparticle vectors. If the outgoing boundary conditions are +imposed, the unbound HFB eigenstates have complex energies; within such Gamow HFB +(GHFB) approach [182] the imaginary energies are related to the particle decay width. + +18 +The quasi-particle HFB continuum can also be treated in an approximate way by means +of a discretization method. The commonly used approach is to impose the box boundary +conditions [169, 177, 183, 184], in which HFB eigenvectors (ui,vi) are spanned by a basis +of L 2-integrable orthonormal functions defined on a lattice in coordinate space and van- +ish at box boundaries. In this approach, referred to as the L 2 discretization, quasi-particle +continuum of HFB is represented by a finite number of box states. The structure of the dis- +cretized continuum depends on the size and geometry of the box [185]. In the context of +the Dirac equation, scalar confinements at the level of strong Coulomb fields need to be +explored, for example within a finite element approach [186, 187].|| +There are two kinds of quasiparticle HFB resonances. The particle resonances represent +metastable states that have large particle (upper) component, i.e., the normalization of ui +is much larger than that of vi. The deep-hole resonances are associated with excitations +of low-lying hole states of the s.p. Hamiltonian h. For those states, the lower component +vi dominates. The deep-hole resonances acquire decay width through the coupling to the +pairing channel [168, 169]. +Quasiparticle resonances can be directly calculated using coordinate-space Green’s func- +tion technique [189, 190] and GHFB [182]. For approaches based on the L 2-discretization, +approximate methods have been developed to deal with HFB resonances. Since the HFB +quasiparticle resonances are highly-localized states whose energies are weakly affected +by the box size, the stabilization method based on box solutions with different box sizes +[177, 191] can be used to obtain the resonance energies and widths. Besides the stabiliza- +tion method, a straightforward smoothing and fitting technique that utilizes the smoothed +occupation numbers obtained from the dense spectrum of box states has been successfully +used [177]. +Summarizing this section, there are many similarities between the single-particle Dirac +problem and one-quasiparticle HFB problem: +– The corresponding equations have a similar two-component form. +– In both cases, the energy spectra are symmetric with respect to zero energy. In the +Dirac case, this is related to charge conjugation. In the HFB case, this is due to the +quasiparticle-quasihole symmetry. For a recent discussion of particle–hole symmetries +of multi-fermion systems (such as band insulators or superconductors) and the charge- +conjugation symmetry of relativistic Dirac fermions, see Ref. [192]. +– In both cases, the resonances can be divided into particle resonances with the upper +component dominating over the lower component and the hole resonances, for which +the lower component dominates. At Z ≈ Zc, the diving states resemble hole resonances +of HFB. +– In both cases, one deals with spectra that are partly discrete and partly continuous. The +continuum space contains metastable states (resonances) that are embedded in the non- +resonant background. +– The Dirac and HFB spectra are bound neither from above nor from below. This leads to +a variational collapse (Dirac) and difficulties with the use of the imaginary time method +(for both Dirac and HFB), see, e.g., Ref. [193] for a remedy. +– In both cases, one has to deal with continuum-space truncations. +Those analogies can be helpful when tackling similar problems or interpreting similar phe- +nomena with the Dirac equation. See also Refs. [192, 194] for relevant examples. +||Confinement potentials need to be introduced in scalar form, i.e. added to the mass term. Adding a +confinement to the potential term causes the spectrum to become completely continuous [28, 29, 188]. + +19 +2.4.3 Perturbative approach +For narrow resonances with energies close to E = −mec2, the energy eigenstates can be +obtained perturbatively. To this end, one can employ the two-potential approach [195] to +the decay of a metastable state [196, 197]. Within this method, the potential describing +the decaying system can be decomposed into V = V0 +V ′, where V0 represents the bound- +state potential of a closed quantum system and V ′ is the closing potential. When applied +to the diving states, one can assume V0 in the form of the Coulomb potential of the finite +nuclear charge distribution with Z0 < Zc and V ′ = (Z′/Z0)V0, where Z > Zc and Z′ = Z −Z0 +[24, 28]. This decomposes the overcritical Dirac Hamiltonian into HD = HD0 +V ′. Seeking +for an expression of the discrete 1s state as a solution to the overcritical Hamiltonian, the +approximate eigenvector is chosen to be +ψE(xxx) = a(E)ψ0 +1s(xxx)+ +� −mec2 +−∞ +dE′ b(E′,E)ψ0 +E′(xxx), +(20) +where ψ0 +1s(xxx), the 1s bound state, and ψ0 +E′(xxx), a continuum state with energy E′, are the +solutions of the total Dirac equation just before diving. a(E) and b(E′,E) are coefficients +to be determined [198]. This leads to the perturbative expression for the 1s state energy +embedded in the continuum +Ecr +1s = E0 +1s +∆E1s +F1s(E), +(21) +where +∆E1s = ⟨ψ0 +1s(xxx)|V ′(xxx)|ψ0 +1s(xxx)⟩ ∝ Z′ +(22) +and +F1s(E) = − +� +dE′ |⟨ψ0 +E′(xxx)|V ′(xxx)|ψ0 +1s(xxx)⟩|2 +E −E′ +∝ −Z′2 . +(23) +The dash in the integral (23) indicates the Cauchy principal value. The function F1s(E) in +(21) introduces an energy distribution to E0 +1s +∆E1s with a width of +ΓE = 2π|VE|2 ∝ γZ′2. +(24) +The width can be interpreted in terms of the positron escape width [24, 28]. +2.4.4 Analytical continuation +One-particle resonances embedded in the negative energy continuum can be found by ex- +tending the Dirac eigenvalue problem into the complex domain. One approach is based +on solving the Dirac-equation eigenproblem with the incoming boundary condition. The +resulting discrete resonant (Gamow) states have complex energies E = E0 + iΓ /2 with a +positive imaginary part Γ , termed supercritical in the remainder of this review. This in- +terpretation differs from the usual complex-energy description of decaying Gamow states +for which E = E0 −iΓ /2. Indeed, the supercritical negative-energy electron resonances can +be interpreted in terms of resonances in scattering of positive-energy positron propagating +backwards in time according to the Feynman-Stückelberg interpretation [199, 200]. +Complex-energy solutions for the differential equation are obtained using the appropri- +ate boundary conditions, analogous to states in the discrete region. This has, for example, +been studied for the spectrum of the Dirac equation with a spherical well potential [201– +203] and for a Coulomb cut-off potential [28, 151, 194, 203, 204], for which the solutions + +20 +can be analytically expressed. Alternatively, solutions to the Dirac equation can be analyt- +ically continued into the complex plane by complex scaling or by introducing a complex +absorbing potential [205–209]. In the following, we discuss the complex-energy solutions +following the analytical continuation approach of Ref. [204], in which the nuclear potential +is assumed to be constant inside the sphere of radius R0. +At distances up to a cut-off radius R0, the solution to the radial Dirac equation is given +by the Bessel functions: +�P(r) +Q(r) +� += C +� +βr +� +∓J∓(1/2+κ)(βr) +J±(1/2−κ)(βr) +β +E+mec2+ Zα +R0 +� +, +(25) +where β = +� +(E +Zα/R0)2 −m2c4 and upper (lower) signs correspond to κ < 0 (κ > 0). +For r > R0, the solutions are given by the Dirac equation with a Coulomb potential. A +combination of exponential and confluent hypergeometric functions satisfy the boundary +conditions [210]: +� P(E,r) +Q(E,r) +� += +� � +mec2 +E +− +� +mec2 −E +� +eikrρiτ +� f1(E,r) +f2(E,r) +� +(26) +Here, τ = +� +(Zα)2 −κ2,ρ = −2ikr,−ik = +� +(mec2 −E)(mec2 +E), and the functions fi +contain Kummer’s confluent hypergeometric functions. The full analytical form can be +found in Ref. [204]. ** The poles of the S-matrix correspond to the resonant states; these are +found by matching the P/Q ratio of (25) and (26) at R0. This results in real eigenenergies +for solutions in the domain E0 ∈ [mec2,−mec2]. Solutions with E0 ≤ −mec2 are embed- +ded in the negative energy continuum, and are of the form E = E0 + i +2Γ with real energies +E0 < −mec2 and widths Γ > 0. The states in the continuum diverge as r → ∞ and are iden- +tified as Gamow wave functions, see Sec. 2.4.5 below for a detailed description. Note that +at the critical energy E = −mec2 the upper Dirac component P in Eq. (26) vanishes at large +distances. This means that close to Zc the diving resonances resemble the deep-hole HFB +states discussed in Sec. 2.4.2. +Energies of several single-particle states, obtained by the exact approach as detailed +above are shown in Fig. 10. The energies are similar to the perturbative result of Sec. 2.4.3 +at close vicinity to Zc but deviate at larger Z values as expected. +Figures 11 and 12 show the (outgoing Gamow) wave function of a 1s resonant state +embedded in the negative energy continuum for (hypothetical) nuclei with charges Z = 185 +and Z = 300, respectively. The wave function at short range is localized close to the nucleus. +At large distances from the nucleus (panels (b) and (d)) the wave function is dominated by +the term eikr and shows an exponential increasing oscillatory behaviour. +2.4.5 Gamow states +The narrow resonances embedded in the continuum are essentially Gamow resonant states. +Gamow states are generalized eigenfunctions of linear operators with complex eigenvalues, +which do not belong to the natural domain of a self-adjoint operators in the standard Hilbert +**A linear combination of the two-parameter Tricomi function and the exponential terms eikr and e−ikr, is +given in Ref. [151]. + +21 +-16 +-14 +-12 +-10 +-8 +-6 +-4 +-2 + 0 + 120 + 140 + 160 + 180 + 200 + 220 + 240 + 260 + 280 + 300 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 +Zc(1s) +Zc(2p1/2) +Zc(2s) +Ε0 (mc2) +Γ/2 (mc2) +Nuclear charge Z +1s +2s +2p1/2 +3s +Fig. 10 Single particle energy levels as a function of the nuclear charge Z. Solid lines corresponds to the real +part of the energy E0, the dashed lines to the complex contribution i +2Γ . Energies are obtained by analytical +continuation for a nuclear cut-off of Rcut = 0.031 +in units of ¯h/(mc). The critical charges are highlighted +with a vertical dash-dotted line (Zc(1s1/2) ≈ 177, Zc(2p1/2) ≈ 218, Zc(2s1/2) ≈ 247). +space formalism. The mathematical foundation lies in a rigged Hilbert space (RHS) formal- +ism [36], which is outlined in Sec. 10. In scattering theory, Gamow states describe captur- +ing or decaying states corresponding to the poles of the scattering matrix in the complex- +momentum space. +Gamow states have been extensively used in nuclear and atomic physics for describing +resonances and other quasi-stationary states [37, 182, 211–219]. They were originally in- +troduced in 1928 as resonance states by Gamow to describe α decay of nuclei [33, 34] and +by Siegert [35]†† to describe scattering cross sections. For a detailed discussion of Gamow +states in nuclear physics see Refs. [220, 221]. +Asymptotically, the resonant states un(En,r) obey the outgoing (or incoming) boundary +condition +un(En,r)−−−→ +r→∞ Ol(knr) ∼ eiknr +(27) +where kn = γn − iκn (for details see Ref.[212]). As shown in Fig. 13, the bound states with +kn = iκn (κn > 0) lie on the positive imaginary k-axis while the antibound (or virtual) states +with κn < 0 lie on the negative imaginary k-axis. The decaying resonant states with (κn,γn > +0) lie in the fourth quadrant of the complex k-plane while the capturing resonant states with +(κn > 0,γn < 0) lie in the third quadrant. The resonant-state trajectories in complex k-plane +near the continuum thresholds E = ±mec2 have been analysed in Ref. [203]. +The single-particle resonant states, augmented by complex-energy scattering continuum +states u(k,r) lying on the contour L obey the Berggren completeness relation [212]: +∑ +n +|un⟩⟨un|+ +� +L |u(k)⟩⟨u(k)|dk = 1. +(28) +Since complex-energy continuum states belong to the RHS, the metric has to be general- +ized by introducing a biorthogonal basis for the radial wave functions. In particular, contrary +††Gamow states are sometimes also called Siegert states. + +22 +-8 +-6 +-4 +-2 + 0 + 2 + 4 + 6 + 8 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +(a) +Re(P) +Im(P) +Re(Q) +Im(Q) +-0.4 +-0.2 + 0 + 0.2 + 0.4 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 +(b) +-120 +-100 +-80 +-60 +-40 +-20 + 0 + 20 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +(c) +Re(P2 + Q2) +Im(P2 + Q2) +-0.1 +-0.05 + 0 + 0.05 + 0.1 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 +(d) +distance r (-h/(mc)) +Fig. 11 Individual unnormalized P and Q components (a), (b) and the unnormalized density of the 1s wave +function (c), (d) for an atom with Z = 185 and Rcut = 0.031 in units of ¯h/(mc). +to the Hilbert space situation, no complex conjugation appears in the radial wave functions +of bra vectors [212, 221]. That is why the radial densities of 1s states shown in Figs. 11 +and 12 are defined through squared upper and lower Dirac components [182]. Moreover, +the radial integrals must be regularized as the Gamow states with Im(k) < 0 exponentially +diverge as r → ∞, see Figs. 11 and 12. This can be done by various techniques [224–226], +including the external complex scaling method [227]. The very reason for the asymptotic +growth of the Gamow state wave function at large r is the fact that such a state represents +the stationary approach to the intrinsically time-dependent problem of decay. Indeed, the +exponential temporal decrease of the wave function amplitude must be complemented by its +exponential spatial increase, and this assures that the particle number is conserved [228]. +It is important to note, that due to the charge conjugation property of the Dirac equation, +the appearance of the electron Gamow state in the negative-energy continuum results in +the presence of a positron resonant state in the positive-energy continuum [203, 204]. This +suggest an interpretation of diving electron states in terms of positron scattering resonances, +see Sec. 2.5. +As discussed in Sec. 2.4.2, resonances can also be described within the real-energy +framework of standard quantum mechanics. The commonly used approach is based on the +dense continuum discretization, elimination of the smooth non-resonant background, and +fitting the resonance peaks [177]. Another approach is the stabilization method, in which +resonances are extracted from phase shifts obtained from box solutions obtained by as- +suming different box sizes [177, 191, 229, 230]. For very narrow resonances, perturbative +methods, such as the two-potential method of Sec. 2.4.3 can also be used. +Despite some work on resonances embedded in the continuum [28, 151, 203, 204], a +direct utilization of Dirac Gamow states in atomic many-body calculations is practically + +23 +-200 +-150 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +(a) +Re(P) +Im(P) +Re(Q) +Im(Q) +-600 +-400 +-200 + 0 + 200 + 400 + 600 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 +(b) + 0 + 10000 + 20000 + 30000 + 40000 + 50000 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 +(c) +Re(P2 + Q2) +Im(P2 + Q2) +-60000 +-40000 +-20000 + 0 + 20000 + 40000 + 60000 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 +(d) +distance r (-h/(mc)) +Fig. 12 Similar as in Fig. 11 but for an atom with Z = 300. +nonexistent. The basic mathematical formulation rests on the rigged Hilbert space structure +which comes with its own challenges. To be of use in atomic structure calculations of the su- +perheavy elements, Dirac Gamow states need be studied within a multi-electron framework. +Computing Dirac Berggren ensemble defined in Eq. (28), which can be used in a numerical +atomic structure program packages, will offer many exciting avenues. +2.5 Positron production in the super-critical regime +The QED vacuum is unstable in the presence of a strong electromagnetic field above the +Schwinger field limit, ES = m2 +ec3/e¯h = 1.32×10−18 Vm−1 (or the equivalent intensity of +IS =2.3×1029 Wm−2) [231], and decays by emitting electron-positron pairs [232–234]. In +the case of a potential barrier, it results in the much discussed and debated Klein’s paradox. +As pointed out in Ref. [235], pair production cannot be described within a one-body +Dirac theory: it requires quantum field theoretical treatment within a time-dependent rigged +Fock-space formalism that dynamically couples particles (electrons) and holes (positrons) +in the Dirac continuum. A close non-relativistic analogy to this problem is a two-nucleon +nuclear decay of a Gamow resonance [236]. A concise mathematical treatment in terms of +incoming and outgoing electron/positron states is given by Rumpf, where the outgoing basis +may be connected with the ingoing one by a unitary Bogoliubov transformation [237–239]. +For further details see Ref. [240]. +As discussed in Sec. 2.4.5, the resonance states of the supercritical Dirac equation are +the Gamow states. The physical interpretation of an electron state embedded in the contin- +uum was extensively studied by the Frankfurt group [28]. According to these works, if an +empty level is embedded in the negative energy continuum, the initially neutral vacuum can +spontaneously decay into a positron and a bound electron with a supercritical energy. In such + +24 +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +-2 +-1 +0 +1 +2 +bound states (b) +antibound states (a) +decaying states (d) +capturing states (c) +Im(k) +Re(k) +(b) +(a) +(c) +(d) +L +decaying states +capturing states +Fig. 13 Location of resonant states in the complex momentum plane. The Berggren completeness relation, +Eq. (28), used in the decay context involves the bound states (b) lying on the imaginary k-axis, scattering +states on the L contour (solid thick line), and resonant decaying states (d) in the fourth quadrant of the +complex k-plane lying between the real axis and L . For problems involving capture, the capturing resonant +states (c) need to be considered and the scattering contour needs to be moved to the third quadrant. The +antibound states (a) can be included in the generalized completeness relation, see For general expansions +of the resolvent, see Refs. [214, 215]. The antibound (virtual) states (a) can be included in the generalized +completeness relation; in this case the scattering contour has to be slightly deformed [222, 223]. +a case, an empty level in the Dirac sea is interpreted as a positronic state, with the positron +escaping the supercritical field. After two positrons are emitted, the supercritical K-shell has +been successively filled with two electrons, and the Pauli principle prevents further decay +[23, 28, 135, 241–245].‡‡ The resonance’s width has been interpreted as the positron escape +width with the characteristic time τE = ¯h/Γ for the pair creation process. +This picture of pair creation was debated [151, 203, 246] on the basis of the unitarity of +the S-matrix. Indeed, the unitarity of the partial scattering matrix is equivalent to the absence +of inelastic channels, in particular, the absence of spontaneous electron-positron creation. in +which it has been proven that for a static external field the probability of pair creation is ex- +actly zero [29, p. 298]. However, the probability for pair creation does not go exactly to zero +as the time derivative of the external field approaches zero. Instead, in the adiabatic limit one +observes a sudden jump in the probability of adiabatic pair creation for critical fields which +may be defined as spontaneous pair creation [29]. That is, one requires only a weak time de- +pendence to trigger pair creation. Consequently, rather than to talk about “spontaneous pair +creation”, it has been recommended to use “adiabatic pair creation” [247, 248]. Recently, +the vacuum polarization energy decline and spontaneous positron emission in QED under +Coulomb supercriticality were explored within the Dirac-Coulomb problem with an external +static or adiabatically slowly varying spherically symmetric Coulomb potential created by +‡‡We could, in principle, excite an electron from the filled Gamow 1s1/2 state in the continuum into one +of the discrete states above −mec2. This creates another hole in the state embedded in the negative energy +continuum and the possibility for yet another pair creation. + +25 +a uniformly charged sphere [249]. It was found that in the supercritical region the vacuum +polarization energy is a decreasing function of the Coulomb charge, resulting in a decay, +with a vacuum polarization energy EVPren ∼ −Z4/R(Z), which provides the required energy +for positron emission ([249] Eq. (104)). Here R(Z) is the nuclear radius. The vacuum po- +larization and its effect on the value of supercritical Z are also studied in [250]. This debate +could, however, have been avoided by referring to Thaller’s work on scattering operators +[29]. +In principle, one could initiate pair creation using intensive laser fields above the Schwinger +limit IS (see Ref. [251] for a recent review). One proposal is by using multiple§§ focused +beams from x-ray free electron lasers [252]. Repeated cycles of particle creation and anni- +hilation can take place in tune with the laser frequency and the production of a few hundred +particle pairs per laser period can occur. As an analogous approach, Ref. [253] proposed a +model of the quantum Dirac field realized by ultra–cold fermionic atoms in an optical lat- +tice. Here, numerical simulations demonstrate the effect of spontaneous pair creation in the +optical analogue system. Yet another possibility is to use a strong laser beam coupled to an +atomic or molecular system with a strong Coulomb field as found for example in graphene +[246, 254–256]. A Schwinger-like production of hot electron-hole plasma in semi-metallic +graphene has been claimed to be observed for the first time only very recently [257]. +2.6 Experimental perspective: heavy-ion collisions +It was proposed, that pair creation should occur in the collision between two bare nuclei with +total charge number exceeding the critical value, such as the case for two U92+ ions with a +combined nuclear charge of Z = 184 [28, 244]. The collision system will have a supercrit- +ical regime time for ∼2.3×10−21 s for the U92++U92+ collision at center-of-mass energy +of Ecm =740 MeV [245] as shown in Fig. 14. The expected lifetime of the supercritical +resonance state is ∼392×10−21 s, which is two orders of magnitude shorter than the time +required for vacuum decay. The probability of pair production is therefore estimated to be +around 1% for the 1s level [209]. Early attempt to observe this effect [258] using ions with- +out a 1s1/2 hole failed. The use of cooled U92+ ions in the ESR storage ring of GSI/FAIR +[259] could allow to observe this effect for the first time. A test experiment using collisions +of a Xe54+ beam on a Xe gas jet target is underway [260]. +The spontaneous emission is not the only process that can occur during the collision.¶¶ +It is generally masked by a dynamical positron emission, which is induced by the time- +dependent potential of the colliding nuclei above the Coulomb barrier [209, 263–266]. In +this mechanism, the two colliding nuclei create a strong electromagnetic field, strong enough +to generate electron-positron pairs. The pair creation in heavy atom collisions is visualized +by the Feynman diagrams in Fig. 15 [202]. +While the spontaneous pair creation works only in the supercritical regime, the dynami- +cal pair creation takes place in both subcritical [264] and supercritical modes if the collision +energy is high enough [268, 269]. Experimental verification of spontaneous pair creation and +the distinction from the dynamical process is however challenging as the energy-differential +§§An electromagnetic plane wave that fulfills E2 = B2 and EEE · BBB = 0 cannot produce electron-positron +pairs. +¶¶One should not forget possible weak decay processes such as the electron capture that is a common +decay mode of proton rich nuclei, albeit the time frame for weak decays is much longer than for nuclear or +electronic transitions [261]. Take for example the work on relativistic quantum dynamic calculations of the +probability of K-vacancy production in the Xe-Xe54+ collision at 30 MeV [262]. + +26 +Fig. 14 +The low-lying energy levels formed by the collision of two uranium nuclei as functions of time. +The arrows a, b, and c denote different dynamical pair-creation mechanisms and the arrow d indicates the +spontaneous pair creation. The 1s state dives into the negative-energy continuum for about 1×10−21 s. Figure +taken from [209]; see also [202]. +e+ +e− +(A1Z1) +(A2Z2) +(A′ +1Z′ +1) +(A′ +2Z′ +2) +Fig. 15 Schematic Feynman diagram for the dynamical pair creation for the (inelastic) collision of two heavy +nuclei with mass numbers and nuclear charges (A1,Z1) and (A2,Z2) respectively, where the outgoing nucleus +binds an electron (Z +′ +2 + e−). Two colliding nuclei create a strong electromagnetic field, strong enough to +generate an electron positron pair [267]. Colliding nuclei are represented by normal lines while wavy lines +refer to virtual photons and the lines with arrows correspond to leptons (electrons and positron). The double +line represents a bound electron. +spectra of emitted positrons by spontaneous vacuum decay are indistinguishable from the +spectra of positrons emitted by the dynamical process. There are however a range of dif- +ferent approaches that should make vacuum decay observable. One example is by collisions +with nuclear sticking, in which nuclei are bound to each other for some period of time by nu- +clear forces allowing for few nuclear rotations. In this very short time frame, typically of the +order of 1×10−21 s to 1×10−20 s [270, 271], there is an increase in pair creation probabil- +ity that can only be explained with the spontaneous pair creation mechanism [27, 244, 266]. +Additionally, it has been shown that the pair-production probability varies as a function of +nuclear collision velocities in the supercritical and subcritical region, allowing for the detec- +tion of vacuum decay experimentally [209, 272]. The impact of the vacuum polarization on + +E +positive-energy continuum +mc2 +2s0 +0 +2p1 /20 +time +1so +-mc2 +negative-energy continuum +occupied with electrons27 +the value of Zc in the case of heavy ions collisions is considered in Ref. [250]. Moreover, it +has been argued [209] that the positron spectra for symmetric collisions of heavy ions with +83 ≤ Z ≤ 96 as a function of the collision energy should show a signature of the transition +to the supercritical regime. +3 Multi-Configuration Dirac-Hartree-Fock +With very few exceptions [273, 274], one treats the multi-electron Dirac equation within +mean-field theory, that is either at the D-HF (Dirac-Hartree-Fock) level or by using D-DFT +(Dirac density functional theory) [275, 276], with the latter method being more popular +in molecular calculations. It is fair to say that the accuracy of current density functional +approximations cannot compete with wave-function-based methods (for a recent critical +analysis on DFT see Ref. [277]), especially when QED effects need to be included. At +an early stage of atomic structure calculations, however, DFT in the form of D-HF-Slater +theory did play an important role as electron correlation is approximately included in such a +scheme [278]. Here, we focus on modern multi-reference D-HF electronic structure theory +for static correlation describing correctly the states of a given Jπ symmetry, with J being the +total angular momentum and π the parity. Dynamic electron correlation and its effects on +atomic structure is described in Sec. 5 below. +Like in the nonrelativistic HF case, to obtain the correct ground state symmetry and +low-lying electronic transitions in open-shell cases, one requires the correct description of +static correlations. In finite basis-set calculations this requires a set of Slater determinants +in a multi-reference treatment within a nonrelativistic or relativistic coupling scheme. In +relativistic atomic numerical program packages such as GRASP [60, 148, 279–281] or MD- +FGME [14, 154], this is done through linear combinations of multi-shell configurational +state functions (CSF’s) within a j j-coupling scheme [45]: +Ψi(Jπ,MJ) = ∑ +r +criΦr (γνJπ,MJ), +(29) +where the Φr wavefunctions share the same overall total angular momentum J, correspond- +ing MJ, and parity π. The quantity γν stands for all other values such as angular momentum +recoupling and seniority numbers [45]. Each CSF Φr is a linear combination of Slater de- +terminants +Φr (γνJπ,MJ) = ∑ +i +di +������� +φ i +1(r1) ... φ i +N(r1) +... +... +... +φ i +1(rN) ... φ i +N(rN) +������� +, +(30) +where φ are the Dirac four-component orbital spinors defined in Eq. (5), and the coefficients +di are determined such that the CSF is an eigenstate to both J2 and Jz. The eigenvalues and +eigenvectors (configuration mixing coefficients cri) are then obtained by diagonalizing the +Hamiltonian matrix Hij = ⟨Ψi(Jπ,MJ)|HD +��Ψj(Jπ,MJ) +� +. +Multi-reference methods (including complete active space SCF) used in the quantum +chemistry community have been reviewed extensively [282–284]. A comprehensive account +on MCSCF theory in relativistic atomic structure calculations (usually termed MCDHF) has +been provided in a textbook [45] and several publications [285, 286]. The construction of +these multi-reference functions can be a formidable task if many high angular momentum +open-shell j-states are involved [45]. The multi-reference treatment, therefore, provides a +challenge for superheavy element calculations where the electronic spectrum becomes very + +28 +dense and, as a result, the multi-reference space becomes huge.*** In addition, SCF con- +vergence problems can arise for nearly-degenerate states. Nonetheless, for few-electron sys- +tems, high-accuracy in excitation energies can be achieved if both QED and dynamic corre- +lation effects are included, see Secs. 4.4 and 5, respectively. High-accuracy atomic structure +calculations are also required, for example, in the search of physics beyond the standard +model (BSM) [116, 122, 287–296]. +Numerical program packages, such as MCHF for the nonrelativistic [297] case or GRASP +and MDFGME for the relativistic case, apply the finite difference method (FDM) [298]. Al- +ternatively, the finite element method (FEM) employing, for example, B-splines (piecewise +polynomials) [186, 299–301] can be used, as implemented for example in the program AM- +BiT [302]. The use of B-splines has certain advantages in relativistic atomic structure calcu- +lations [300]. As the radial wave functions are restricted to an interval [0,Rc], the atoms are +spherically confined within a radius Rc set large enough (usually around 40 au) to achieve +accurate numerical results. This discretizes the positive and negative real-energy continuum. +It thus allows for an easy implementation of projection operators [76]. This method could +therefore be well suited to approximately describe diving occupied levels with E < −mec2 at +charges Z > Zc.††† The virtual space created can be used for a successive electron correlation +procedure, such as configuration interaction or coupled cluster or MCDF. In all these numer- +ical procedures one usually chooses exponentially spaced grid points (called knots in FEM) +with r = r0et,t > 0, to describe the radial wave function φ(r) accurately in the near nuclear +region. We note that the correct description of the wave function in the inner core region is +mandatory for the accurate treatment of relativistic effects [303, 304]. B-splines have also +been used to create basis sets to perform many-body perturbation theory [300, 305, 305, 306] +or to do MCDF calculations, as they can be used to implement projection operators with the +nucleus and electrons average potential, and obtain correlation orbitals [76]. More recently +an improved method, the dual kinetic balance [186], has been proposed to obtain basis sets +free of spurious states. +The systems of coupled integro-differential equations obtained in multi-configuration +methods are intrinsically very non-linear. In particular exchange potentials for correlation +orbitals are inversely proportional to the square of the configuration weight, and can then +be huge. Initial configuration state functions for an SCF calculation are usually obtained +from either the Thomas-Fermi model or from single-particle Dirac-Coulomb solutions us- +ing screened nuclear charges [45]. However, severe convergence problems can be experi- +enced when, for example, diffuse orbitals are involved such as for high angular momentum +functions or negatively charged atoms, or when doing correlation calculations with highly- +excited configurations. In such cases, choosing the right initial guess becomes important. +Convergence issues within the MCDF procedure have been discussed in Refs. [40, 45, 307, +308]. In some cases the problem occurs due to the relativistic nature of the atom or ion being +studied. When going to very high-Z the angular coupling goes from LSJ coupling to almost +pure JJ coupling. In that case the weight of some of the configurations contributing to a +given LSJ level becomes very small and severe convergence problems are observed [40]. +When the four components of the spinor in relativistic methods are each allowed to +vary independently, the matrix representations of the Dirac operator will fail to give the +right formal nonrelativistic limit, resulting in an energy below the true numerical value, +***This is similar to the strong correlation problem in solid state physics to describe, for example, metallic +systems. +†††The accuracy of such a discretization procedure has been shown to be poor, when it comes to the de- +scription of narrow resonance states [177]. In such a case, the preferred method to deal with these resonances +is the Gamow-state framework. + +29 +known as variational collapse or finite basis set disease [79]. It arises whenever one wants to +expand wave functions in a given basis replacing operators by their matrix representation. +To prevent such an unwanted effect, certain boundary conditions such as the kinetic balance +(which is automatically considered in numerical calculations) have to be imposed which +ensures the correct relation between the large and small component [45, 79, 85]. In finite +basis set treatments of the D-HF equations, using for example Slater or Gaussian type basis +sets, small errors may nevertheless occur due to variational problems (prolapse). Since the +kinetic balance condition implicitly projects onto the positive energy states, it is possible +that, due to the incompleteness of the basis set, the total energy lies below the one obtained +from numerical DHF calculations [159]. This can be avoided by freezing the inner core +functions such that core orbitals are sufficiently well described, or by restricting the size of +the s and p basis sets, or by making use of specifically derived prolapse-free Gaussian basis +sets [309–312]. +Upon inclusion of the Breit operator in Eq. (2), coupling of the positive and negative +continuum states occurs due to electron-electron interaction, leading to the non-existence +of a discrete spectrum. This is known as continuum dissolution or the Brown–Ravenhall +disease [77], and can be avoided by removing all Slater determinants containing negative- +energy orbitals using a projection operator, effectively eliminating electron-positron pair +contributions [59]. The projection operator is usually constructed from the positive energy +eigenstates of the full external field Dirac Hamiltonian, leading to the no-pair Hamiltonian of +Sec. 2.4.1. (For a recent discussion on this topic see [313].) For the case where photon-matter +field interactions are removed, a single Slater determinant (D-HF solution) automatically +includes the HF projection operators on positive energy states [314], i.e., the low-frequency +Breit interaction has been shown to cause no variational failure when included in the iterative +solutions of the D-HF equations [315, 316]. Thus, the Breit interaction has been successfully +applied perturbatively [74] as well as in variational treatments [45, 76, 316–321], where +the solutions of the Dirac-Breit-HF equations serve as a starting point for further electron +correlation and QED treatments. +An other issue with Dirac-Fock codes is the fact that for levels originating from the same +LS level, they may give wrong values. It was shown in [322] that the 2p1/2 − 2p3/2 fine +structure energy in B-like ions and the 2p5 J = 3/2−2p5 J = 1/2 one in F-like ions did not +provide the right value for light elements. The non-relativistic limit obtained by setting the +speed of light to a high value was not zero as it should have been. At the time the proposed +solution was to remove the energy splitting obtained for c → ∞ from the relativistic value. +More recently it was shown that this effect could be handled by doing large scale correlation +calculations to obtain those level energies, including all single excitations, even the ones +obeying the Brillouin theorem [323]. The same issue was also identified in the evaluation of +forbidden transitions probabilities [324]. +4 Quantum Electrodynamic Effects +Besides the corrections stemming from relativistic electron correlation described in Sec. 3, +corrections issued from bound-state quantum electrodynamics must be added to get accurate +predictions. The need for such corrections was demonstrated by two famous experimental +discoveries. The first discovery, made by Lamb and Retherford, was the non-degeneracy +between the 2p1/2 and 2s1/2 states, in contradiction to the Dirac equation, which gives de- +generate levels [325]. The second discovery, made by Kusch and Foley, was that the electron +Landé g-factor is not exactly equal to 2 in Na and Ga [326], later understood to be due to the + +30 +anomalous magnetic moment of the electron. The experimental discoveries were followed +by the theoretical work of Bethe [327], Feynman [200, 328], Schwinger [234, 329–331] +and Tomonaga [332], which lead to the foundation of QED, the principle of which remains +unchallenged up to now [116]. +The derivation of the different QED contributions starts from the QED Lagrangian (1). +Several methods have been proposed to calculate all-order QED corrections which are nec- +essary for applications to high-Z elements. However, it is not trivial to define physical parti- +cle states in the presence of an external gauge field within the framework of gauge invariant +quantum field theory [240]. Pioneering works on all-order vacuum polarization [333–335] +have led to the modern calculations. The first accurate all-order calculation of the 1s self- +energy [336, 337] showed that Zα expansions used up to that time were non-convergent at +medium- and high-Z. A first attempt to evaluate the 1s1/2 state self-energy in superheavy +elements was done in Ref. [338]. It was followed by the calculation of the self-energy con- +tribution of the 1s1/2 level for finite nuclei up to Z = 170 [339]. This evaluation has recently +been extended to all states up to n = 5 and J = 5/2 [340]. The method described in Ref. +[336] is based on the S-Matrix formalism, which allow a full treatment of QED corrections +in one-electron systems and to calculate corrections to the electron-electron interaction in +few-electron systems beyond the no-pair approximation [341, 342], provided there is a well +isolated reference system. A review of QED corrections in low-Z one-electron systems can +be found in Ref. [343]. +The Bethe-Salpeter equation [43] is a real two-body equation that has been used to de- +rive, for example, higher-order recoil corrections in hydrogen [344] beyond what can be +obtained with the Breit equation (see, e.g., Ref. [343] and references therein). The Bethe- +Salpeter equation has, however, some fundamental problems [345, 346] and it becomes +soon intractable for many-electron systems. For the efficient treatment of many-electron +systems one requires a Hamiltonian approach (e.g., the Dirac-Coulomb-Breit Hamiltonian +as a starting point) with additional effective QED perturbation terms that describe the multi- +electron system to the required accuracy. The Bethe-Salpeter equation can in principle be +transformed into two independent equations that match the equations of Hamiltonian rela- +tivistic quantum mechanics [347]; there is also the quasipotential approach [348]). +Three methods have been developed to deal with bound state QED (BSQED) calcula- +tions, in particular in heavy-elements. The original one is based on the S-matrix formalism. +A detailed description of the S-matrix formalism and review of QED calculations based on +it can be found in Refs. [143, 349, 350]. An overview of this method is given in subsection +4.1. Approaches capable of dealing with quasi-degenerate reference states have been pro- +posed by using (i) a method based on the two-times Green function [351–353] (Subsec. 4.2) +and (ii) a covariant version of RMBPT based on the time-evolution operator, which allow to +treat more easily degenerate and quasi-degenerate states [354–357] (Subsec. 4.3). In prac- +tice, the complexity of the involved calculation is the main limitation to the use of any of +these approaches, and approximate methods had to be devised. +BSQED is usually based on the Furry bound picture [358]. The unperturbed Dirac +Hamiltonian HD contains the Coulomb field of the nucleus, such that the Coulomb potential +is included to all orders. The electron-electron interaction is treated as a perturbation given +by the potential +Vε,g = gHIe−ε|t|, +(31) +where g is a formal expansion parameter and the interaction Hamiltonian is +HI = jµAµ −δM(x), +(32) + +31 +which contains a mass renormalization term. As the electromagnetic interactions can act at +an infinite distance, the term e−ε|t| is added to turn off adiabatically the interaction at t = ±∞ +to recover the unperturbed states before and after the interaction. +The electron-positron field operators defined on an appropriate Fock-space are expanded +in terms of electron and positron annihilation and creation operators, +ψ(x) = +∑ +En>−mec2 +anφn(x)+ +∑ +Em<−mec2 +b† +mφm(x), +(33) +while the BSQED Hamiltonian is given as [240], +H0 = +∑ +En>−mec2 +Ena† +nanφn(x)− +∑ +Em<−mec2 +Emb† +mbmφm(x), +(34) +where an an electron annihilation operator for an electron in state n, with energy En > −mec2 +and b† a positron creation operator for a positron in state m with energy Em < −mec2. For +the Gamow states that dive into the negative energy continuum and have complex energies, +the formalism has to be further extended. It should be noted that the formalism in Eqs.(33) +and (34) is the proper quantum-field theory replacement for the Hamiltonian with projection +operators given in Eq. (16), which is based on the Dirac sea definition of the positrons. Yet, +for practical applications in many-electron systems, the BSQED formalism is too difficult +to use, and has not been used beyond second-order corrections. +The expressions (33) and (34) are usually formulated in terms of the positive (En > +0) and negative (En < 0) spectrum of the Dirac operator, loosely termed electronic and +positronic states [143]. Such terminology originates from a free-particle QED formalism +[359, 360], and was later adopted for Coulomb fields describing a point nuclear charge +where the lower part of the discrete spectrum terminates at En = 0 at Zα = 1. As already +pointed out, for the general case of a finite nucleus the energy can become negative and +eventually the state can dive below E = −mec2 for Z ≈ 170. Hence the terminology of +positive and negative energy states makes only sense if one shifts the spectrum up by mec2 +where the lower continuum starts then at E < 0. +4.1 S-matrix formalism +The evaluation of the energy shift in QED for an isolated q-electron state with no real pho- +tons +��Nq;0 +� += +��n1,...,nq;0 +� +, is made through the Gell-Mann and Low theorem [361, 362], +symmetrized by Sucher [363] +∆ENq = lim +ε→0 +g→1 +iεg +2 +∂ +∂g log +� +Nq;0 +��Sε,g +��Nq;0 +� +, +(35) +where the adiabatic S-matrix is given by +Sε,g = lim +t→∞Uε,g(−t,t), +(36) +and Uε,g is the adiabatic evolution operator defined as +Uε,g (t1,t2) = Te−i +� t2 +t1 dtVε,g(t) , +(37) +where T is the time ordering operator. + +32 +The next step is to expand the connected adiabatic S-matrix in power of the coupling +constant g as done in Refs. [143, 349, 360, 364, 365] +g ∂ +∂g log +� +Sε,g +� +C +���� +g=1 += +� +S(1) +ε,1 +� +C +2 +� +S(2) +ε,1 +� +C +3 +� +S(3) +ε,1 +� +C +··· +1+ +� +S(1) +ε,1 +� +C + +� +S(2) +ε,1 +� +C + +� +S(3) +ε,1 +� +C +··· += +� +S(1) +ε,1 +� +C +2 +� +S(2) +ε,1 +� +C − +� +S(1) +ε,1 +�2 +C ++ 3 +� +S(3) +ε,1 +� +C −3 +� +S(1) +ε,1 +� +C +� +S(2) +ε,1 +� +C + +� +S(1) +ε,1 +�3 +C ++ 4 +� +S(4) +ε,1 +� +C −4 +� +S(1) +ε,1 +� +C +� +S(3) +ε,1 +� +C −2 +� +S(2) +ε,1 +�2 +C ++ 4 +� +S(1) +ε,1 +�2 +C +� +S(2) +ε,1 +� +C − +� +S(1) +ε,1 +�4 +C , +(38) +where the connected S-matrix is defined by +� +Sε,g +� +C = +� +Nq;0 +��Sε,g +��Nq;0 +� +C = ∑ +j +� +S(j) +ε,1 +� +C , +(39) +with +� +S(j) +ε,1 +� +C = +� +Nq;0 +��S(j) +ε,1 +��Nq;0 +� +C . +(40) +Connected diagrams are diagrams with external legs, which are bound-state wave func- +tions like the ones in Figs. 16, 18 and 19. The disconnected diagrams, which have only +closed loops, only contribute to the energy of the vacuum. Examples of disconnected dia- +grams for one- and two-electron systems are shown in Fig. 17. Each order in Eq. (38) has +poles at ε j, which cancel out only if all terms of a given order are calculated simultaneously. +For example, for j = 2 both terms of 2 +� +S(2) +ε,1 +� +C − +� +S(1) +ε,1 +�2 +C must be calculated together to +cancel the 1/ε2 pole. +The S-matrix formalism is not limited to the evaluation of QED energy shifts. It can also +be used for the evaluation of radiative corrections to one- [366, 367] and two-photon [368] +emission probability for example, and line shapes [369]. +From the definition of the S-matrix (36) and the evolution operator (37) one obtains for +the matrix element of order j: +S(j) +ε,g = (−ig) j +j! +� +d4x j ... +� +d4x1e−ε|tj| ...e−ε|t1|T [HI (x j)...HI (x1)]. +(41) +These matrix elements can be expressed in terms of the electron propagator and photon prop- +agator. The electron propagator is connected to the Dirac bound electron Green’s function +by +SF (x,y) = ⟨0|T [ψ (x) ¯ψ (y)]|0⟩ += +� +∑En>0 φn (x) ¯φn (y) tx > ty +−∑En<0 φn (x) ¯φn (y) tx < ty += −i +2π +� +CF +dzG(xxx2,xxx1,z(1+iδ))γ0e−iz(t2−t1). +(42) + +33 +(a) +DF +Ψnℓ j +¯Ψnℓ j +SF +eγµ +eγν +(b) +SF +DF +Ψnℓj +¯Ψnℓj +eγν +eγµ +Fig. 16 Bound state QED corrections of lowest order with the usual labelling of Feynman diagrams. (a) +vacuum polarisation; (b) one-electron self-energy. Elementary charge e is included for clarity. DF and SF are +the Dyson (photon) and Feynman (electron/positron) propagators respectively. The double line represents a +propagator in the field of the nucleus. Ψnℓ j represents a bound electron wave function. +The electron Green’s function in (42) is the solution of [143, 349] +(−iααα ·∇∇∇2 +V (|xxx2|)+βm−z)G(xxx2,xxx1,z(1+iδ)) = δ (xxx2 −xxx1) . +(43) +The energies of the bound states are given by the poles of the Green’s function along the +real axis. +The contraction of the two photon field operators in (41) gives +⟨0|Aµ (x2)Aν (x1)|0⟩ = gµνDF (x2 −x1) +(44) +where +DF (x2 −x1) = − +i +(2π)4 +� +d4qe−iq·(x2−x1) +q2 +iδ += +1 +(2πi) +� +∞ +−∞ dq0H (xxx2 −xxx1,q0)e−iq0(t2−t1) +(45) +In Eq. (45), H (xxx2 −xxx1,q0) is the photon Green’s function, given by +H (xxx2 −xxx1,q0) = −e−bx21 +4πx21 +x21 = |xxx2 −xxx1|, +b = −i +� +q2 +0 +iδ +� 1 +2 , ℜ(b) > 0. +(46) +As noted by Dyson, the expansion in power of α of Eq. (35) has a radius of convergence +equal to zero [52]. The series in α is thus only an asymptotic series that diverges for n ≥ 1/α. +Thanks to the small value of α, this is not an issue unlike for the strong interactions. +The first-order contribution in Eq. (38), the mass renormalization term, can be written +as: +∆E(1) +n += lim +ε→0 +1 +2iε +� +S(1) +ε,1 +� +c = −δm +� +dxxxφ † +n (xxx)γ0φn (xxx) . +(47) + +34 +A +B +Fig. 17 Example of disconnected diagrams of order α, which only contribute to the vacuum energy. (a): one +electron case, (b): two-electron case. +(a) +DF +Ψnℓj +Ψn′ℓ′ j′ +¯Ψnℓj +¯Ψn′ℓ′ j′ +eγµ +eγν +Fig. 18 Similar as in Fig. 16 but for the electron-electron interaction Feynman diagram. +The three possible second-order connected diagrams are shown in Figs. 16 (first order, +one-electron QED corrections) and 18 (electron-electron interaction). They originate from +the second-order term in Eq. (38) which can be explicitly written (in natural units) as +� +S(2a) +ε,1 +� +c = +1 +(4πi) +� +∞ +−∞ dq0 +� +d4x2 +� +d4x1e−ε(|t2|+|t1|)e−iq0(t2−t1) e−bx21 +4πx21 +� +∑ +m2n2m1n1 +ei(En2−Em2)t2ei(En1−Em1)t1 +×φ † +n2 (xxx2)γ0γµφm2 (xxx2)φ † +n1 (xxx1)γ0γνφm1 (xxx1) +× +� +Nq;0 +�� : a† +n2am2a† +n1am1 : +��Nq;0 +� +−2Tr +� +γµ −i +2π +� +∞ +−∞ dzG(xxx2,xxx2,z(1+iδ))γ0 +� +×∑ +nm +ei(En−Em)t1φ † +n (xxx1)γ0γνφm (xxx1) +� +Nq;0 +��a† +nam +��Nq;0 +� ++−i +π +� +∞ +−∞ dz∑ +nm +ei(Ent2−Emt1−iz(t2−t1)) +×φ † +n (xxx2)γ0γµG(xxx2,xxx1,z(1+iδ))γ0γνφm (xxx1) +× +� +Nq;0 +��a† +nam +��Nq;0 +�� +. +(48) +Those diagrams are of the order of α/π since they have two vertices. + +35 +(a) +(b) +DF +DF +Ψnℓj +Ψn′ℓ′ j′ +¯Ψnℓj +¯Ψn′ℓ′ j′ +eγµ +eγµ +eγν +eγν +Fig. 19 Similar as in Fig. 16 but for the second-order electron-electron interaction Feynman diagrams: (a) +ladder diagram; (b) crossed diagram. +4.2 Two-times Green’s function method +This method is based on the generalisation of the Green’s function (42) to a system of N +electrons [351, 353]. The 2N−times Green’s function is defined as +G(x′ +1 ···x′ +N;x1 ···xN) = +� +0|T +� +ψ(x′ +1)···ψ(x′ +N) ¯ψ(x1)··· ¯ψ(xN) +� +|0 +� +. +(49) +It can be expressed as +G(x′ +1,...x′ +N;x1,...xN) +(50) += ⟨0|T [ψin(x′ +1)···ψin(x′ +N)ψin(xN)···ψin(x1)]exp{−i +� d4z HI(z)}|0⟩ +⟨0|T [exp{−i +� d4z HI(z)}]|0⟩ += +� ∞ +∑ +m=0 +(−i)m +m! +� +d4y1 ···d4ym ⟨0| +� +Tψin(x′ +1)···ψin(x′ +N)ψin(xN)···ψin(x1) +× HI(y1)···HI(ym)]|0⟩ +�� ∞ +∑ +l=0 +(−i)l +l! +� +d4z1 ···d4zl ⟨0|T [HI(z1)···HI(zl)]|0⟩ +�−1 +. +Two-times Green’s function method starts by keeping only two times in Eq. (50), setting +t1 ≡ t2 ··· ≡ tN ≡ t and t′ +1 ≡ t′ +2 ··· ≡ t′ +N ≡ t′. This operation does not lead to any loss of +information. For an isolated level a of an N-electron atom, with an unperturbed energy E(0) +a , +the energy shift is given by +∆Ea = +1 +2πi +� +Γ dE +� +E −E(0) +a +� +∆Gaa(E) +1+ +1 +2πi +� +Γ dE∆Gaa(E) +, +(51) +where Gaa(E) is the mean value for state a of the Fourier transform of the two-times Green’s +function (50). A perturbation expansion of Gaa(E) in powers of the fine structure constant +α generates results similar to those shown in section 4.1. +In the case of two quasi-degenerate levels, one can use the 4-times Green’s function in +a similar manner. +G(x′ +1x′ +2;x1x2) = ⟨0|T [ψin(x′ +1)ψin(x′ +2)ψin(x2)ψin(x1)]exp{−i +� d4z HI(z)}|0⟩ +⟨0|T [exp{−i +� d4z HI(z)}]|0⟩ +. +(52) +In this case, the perturbation expansion is realized on the subspace containing the two +quasi-degenerate levels. This procedure can formally be extended to any number of quasi- +degenerate states. + +36 +4.3 Covariant evolution-operator procedure +The covariant evolution-operator method has been developed in Refs. [357, 370–373]. The +method also starts from the evolution operator, and applies Relativistic Many-Body Pertur- +bation Theory methods (RMBPT). It is closely related to the two-times Green’s function +method discussed in the previous subsection. In contrast to the S-matrix formalism, which +cannot handle quasi-degenerate states due to the energy-conservation condition caused by +the integration over all times, the covariant evolution operator procedure has been success- +fully applied to quasi-degenerate states. The method is based on the fact that at t = 0, the +Green’s operator is equivalent to the RMBPT wave operator, which is obtained as solution +of a generalized Bloch equation. Additionally, in the standard evolution operator, time runs +only in the forward direction and is therefore not relativistically covariant. By allowing the +time to evolve forwards as well as backward, the relativistic covariance is restored. This +method allows to evaluate non-QED many body effects to high-order and to take into ac- +count first and second order QED diagrams as well, at least in simple systems. +4.4 Calculation of QED corrections +There are several kinds of QED corrections that need to be evaluated for computing transi- +tion energies in superheavy elements. In the first category, there are one-electron corrections +like the self-energy and the vacuum polarization shown in Fig. 16. These corrections con- +cern all atoms. The Feynman diagram for the electron-electron interaction is shown in Fig. +18. It contains the Breit interaction and all-order retardation corrections. This diagram can +be iterated to provide higher-order corrections to the electron-electron interaction as shown +in Fig. 19. The latter diagrams and similar ones with more photons provide QED corrections +to the correlation energy. The full ladder diagram in Fig. 19(a) contains both the correlation +contribution present in many-body theories like RMBPT, MCDF or RCI, and pure QED cor- +rections, which involve positrons. The crossed diagrams in Fig. 19(b) provides pure QED +corrections not included in many-body calculations. A last category of Feynman diagrams +contains electron-electron interaction corrections to one-electron correction, like self-energy +screening presented in Fig. 20. These two last categories concern atoms with at least two +electrons. +4.4.1 One-electron radiative corrections +The energy shift due to one-electron radiative corrections of order i, corresponding to an +ensemble of diagrams with 2i vertices, is formally of order (α/π)imec2, but after renormal- +ization it can be written as: +∆E(i) +(n,κ) = +�α +π +�i (Zα)4 +n3 +F(i) +n,κ (Zα)mec2, +(53) +where F(i) +n,κ(Zα) is a slowly varying function of Z for a level of quantum numbers (n,κ). +For low Z, one can write an expansion of F(i) +n,κ(Zα) as an expansion in powers of Zα +and log +� +(Zα)−2� +. The lower-order coefficients of this expansion can be found in Refs. +[374, 375]. As shown in [336], this expansion is not convergent at medium to high-Z. For +superheavy elements, we will thus only consider results evaluated to all orders in Zα. + +37 +(a) +SF +DF +SF +DF +Ψnℓj +Ψn′l′ j′ +¯Ψnℓj +¯Ψn′l′ j′ +eγν +eγµ +eγµ +eγµ +(b) +SF +DF +SF +DF +Ψnℓj +Ψn′l′ j′ +¯Ψnℓj +¯Ψn′l′ j′ +eγµ +eγν +eγµ +eγµ +Fig. 20 Self-energy screening diagrams of order (α/π)2.The other notations are defined in the legend of Fig. +16. +We now discuss the evaluation of the self-energy diagram 16(b). The 1s1/2 self-energy +has been evaluated to all-orders for 5 ≤ Z ≤ 120 for point nucleus [68, 337, 338, 376, 377]. +The finite nuclear-size correction is important for high-Z and small values of the principal +quantum number and for s and p states. It is negligible for larger values of |κ|. The self- +energy for the 1s1/2 state has been evaluated in [142, 143, 338, 378]. In Ref. [338], it was +evaluated up to Z = 160 and in Ref. [339] up to Z = 170. An extension of the evaluation of +F(1) +1s (Zα) for i = 1 for point nuclei up to Z = 137 has been performed recently [379] and for +uniformly charged nuclei up to Z = 135 [380] with radii in the range 1.5 fm to 7.3 fm. The +work from Ref. [68] has been extended recently to Z = 170 [340]. In both works, the self- +energy for a given Z value is calculated for a specific nuclear size, using the Fermi model. +A comparison between the different values of F(1) +1s (Zα) for i = 1 and finite nuclear size is +shown in Fig. 21. All calculations are in good agreement with each other, that is within the +differences of the nuclear model and size applied. +In the case of excited states, the self-energy has been evaluated for ns, np1/2, np3/2 and +nd3/2 states for 5 ≤ Z ≤ 110 and 2 ≤ n ≤ 5 in Refs. [336, 376, 377, 381, 382]. The point- +nucleus self-energy of ns, np and nd states up to n = 5 can also be found in Ref. [68] for +10 ≤ Z ≤ 120. Reference [383] contains the values of F(1) +n,κ (Zα) for nd3/2 to ng9/2 up to +n = 5 for a point nucleus. The finite size of correction for 2s states and 2p1/2 states can be +found in [142, 143, 378] for 26 ≤ Z ≤ 100, for 10 ≤ Z ≤ 120 in [68] and for 100 ≤ Z ≤ 170 +in [340]. The comparison between different theoretical values with and without finite size +correction for 2s, 2p1/2 and 2p3/2 states is shown in Fig. 21. The divergence of the point- +nucleus values when Z → α−1 ≈ 137 for states with |κ| = 1 is visible for both 2s and 2p1/2 +states. It is in fact even more pronounced than for the 1s state. +For larger values of n and Z > 120 there are no published results for F(1) +n,κ (Zα). A large +scale effort has been recently undertaken to provide values of F(1) +n,κ (Zα) to cover the range of +interest for superheavy elements. The values of F(1) +n,κ (Zα) with all possible κ for all 1 ≤ n ≤ +10 and Z up to 137 have been evaluated for point nuclei [379]. The point nucleus values for +all possible |κ| > 1 for n = 5 are plotted in Fig. 22, together with values from Refs. [68, 340] +for p and d states. The functions F(1) +n,κ (Zα,R) including finite nuclear size correction for +3 ≤ n ≤ 6 and |κ| = 1 (s and p1/2 states) have been evaluated for Z up to 135, with values +of R, the mean spherical radius in the 1.5 fm to 7.3 fm [380]. + +38 +80 +100 +120 +140 +160 +1.5 +2.0 +2.5 +3.0 +3.5 +1s, finite nuclear size +[a] +[b] +[c] +[d] +[e] +[a] +[a1] +[d] +[e] +80 +100 +120 +140 +160 +0 +20 +40 +60 +Atomic number Z +F(Zα) +point nucleus +finite nuclear size +2s1/2 +2p1/2 +2s1/2 +2p1/2 +2p3/2 +2s1/2 +2p1/2 +2p3/2 +2s1/2 +2p1/2 +2p3/2 +Fig. 21 Values of the F(1)(Zα) function in the high-Z and supercritical region. Top: comparison between +finite size values for 1s1/2. Bottom: comparison between finite-size and point nucleus values for the n = 2 +shell. References: [a]=[380], [a1]=[379], [b]=[338], [c]=[339], [d]=[68], [e]=[340]. +The vacuum polarization correction of order one in (α/π), presented in Fig. 16(a) +can be evaluated with good enough accuracy by using an expansion in power of Zα. The +potential of order α(Zα), with only one interaction with the nucleus is called the the Uehling +potential VU. The next term in the expansion, of order α(Zα)3 is called the Wichmann-Kroll +potential VWK. All orders calculations of the vacuum polarization have been performed in +[384, 385]. The Uehling potential [386] is evaluated as +VU(r) = −2α +3π +Z +r +� +[1,∞) exp +� +−2α−1ξr +�� +1+ 1 +2ξ 2 +� � +ξ 2 −1 +ξ 2 +dξ +(54) +if one treats the nucleus as a point charge. An analytical formula for the Uehling potential in +terms of modified Bessel functions has been provided in Ref. [387]. The expression in (54) +can be extended to a finite nuclear charge distribution [388, 389]. The Uehling potential can +be added to the Dirac equation potential, providing an easy way to include the loop-after- +loop vacuum polarization correction to all orders [81]. + +39 +80 +100 +120 +140 +160 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Atomic number Z +F(Zα) +point nucleus [a1] +5p3/2 +5d3/2 +5d5/2 +5f5/2 +5f7/2 +5g7/2 +5g9/2 +5p3/2 +5d3/2 +5d5/2 +5p3/2 +5d3/2 +5d5/2 +[d] +[e] +finite nuclear size +Fig. 22 Similar as in Fig. 21 but for F(1) +5p3/2(Zα), F(1) +5d3/2(Zα), F(1) +5d5/2(Zα), F(1) +5 f5/2(Zα), F(1) +5 f7/2(Zα), F(1) +5g7/2(Zα), +and F(1) +5g9/2(Zα) functions in the high-Z and supercritical region. +The Wichmann-Kroll contribution VWK to the vacuum polarization is of order α(Zα)3; +it can be written approximately for r → 0 as [333], +VWK(r) ≈α(Zα)3 +π +�� +−3 +2ζ(3)+ π2 +6 − 7 +9 +� 1 +r +2πζ(3) +−π3 +4 + +� +−6ζ(3)+ π4 +16 − π2 +6 +� +r +O(r2) +� +(55) +where ζ(n) is the Riemann zeta function. For more details see for example Refs. [385, 390]. +An efficient numerical method to evaluate VWK without low-r expansion is given in [391]. +Additional terms on this expansion of order α(Zα)5 and α(Zα)7, corresponding to 5 and +7 interactions with the nucleus in the vacuum-polarization loop, are approximately known. +They have been used in muonic atoms for many years [333, 392–394]. Numerical methods +to evaluate them can be found in Ref. [391]. +One should also add next-order terms with i = 2 in (53). These terms are of the order of +(α/π) ≈ 2×10−3 compared to the two-vertex terms (see Fig. 4) but the leading coefficients +in their Zα expansion can be large. These terms represent, e.g., two-loop self-energy, two- +loop vacuum-polarization corrections, and mixed self-energy vacuum-polarization terms. +The corresponding Feynman diagrams are shown for example in Refs. [110, 321, 349]. Some +of these terms can be easily calculated, such as the Källen-Sabry contribution to the vacuum +polarization [395] for which a potential is known [388]. The two-loop self-energy terms have +been evaluated for one- [107, 108, 111–113, 396, 397] and three-electron atoms [398] for +30 ≤ Z ≤ 100, but only for n = 1 and n = 2 states. Mixed self-energy vacuum polarization +diagrams have been evaluated in [110, 399, 400]. Whilst these diagrams are important for +inner shell electron energies, they are not expected to contribute significantly to the outer- +shell energies of superheavy elements compared to correlation effects. The evaluation of the +diagrams of order (α/π)2, which are easier to calculate, like the Källen-Sabry term or the +loop-after-loop Uehling contribution, can provide the needed order of magnitude to assess +the importance of the uncalculated ones on specific cases. + +40 +Dirac Energy − mc2 +Self Energy +Uehling VP +W&K VP +total order α QED +Total energy +10- 5 +0.001 +0.100 +10 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +1 −Z +Energy (mc2) +α +Fig. 23 Dirac energy, self-energy, and vacuum polarization near Zc = 1/α for a point nucleus. The Uehling +and Wichmann and Kroll vacuum polarization contributions have been evaluated with the MDFGME code +and the self-energy is from [379] and evaluated to all order in Zα. +In Fig. 23 we show the evolution of the QED contributions of order α and of the Dirac +energy for the 1s state as a function of 1/α − Z with non-integer values of Z, to show +what happens near the critical Zc = 1/α in the case of a point nucleus. It shows that the +self-energy becomes nearly independent of Z and that the vacuum polarization becomes the +dominant contribution almost one order of magnitude larger than the self-energy. The total +energy to that order becomes close to −mec2. Although the Wichmann-Kroll contribution is +very small, it remains to be checked how the all-order vacuum polarization and the sum of +higher-order contributions would behave. The same comparison for finite size nucleus does +not show the same effect: the self-energy and vacuum polarization remain of the same size +and their values are strongly reduced. +4.4.2 Two-electron radiative and non-radiative corrections +Concerning the calculation of atomic spectra of heavy and superheavy elements, the bot- +tleneck in terms of accuracy in such many-electron systems still lies in the treatment of +electron correlation (see discussion in Sec. 5). There are, however, mixed terms between ra- +diative corrections and the electron-electron interaction that need to be considered. The main +one is known as self-energy screening (see Fig. 20). It has been evaluated by direct calcu- +lation of the Feynman diagrams only for n = 1 and n = 2 states [401, 402]. These terms +containing self-energy loops cannot be put into the form of an exact potential, and thus can- +not be easily generalized to arbitrary atoms. It would therefore be useful to formulate an +approximate QED potential that could describe the screened Lamb-shift and other atomic +properties sufficiently accurately and could be successfully used in molecular calculations. +One could then introduce a (model) perturbation Hamiltonian to represent the radiative part +of QED corrections of the form +∆ ˜HQED = VU +VWK +hSE +hh.o.t., +(56) + +41 +where VU is the Uehling potential, VWK is the Wichmann and Kroll potential, hSE the self en- +ergy model potential and hh.o.t. represents two-loops contributions. The aim of this operator +is to include approximate QED corrections to the electron-electron operators, with negligi- +ble errors compared to the electron correlation treatment. In addition, the matrix elements +of this QED Hamiltonian can be added to the CI matrix or to the Hamiltonian matrix and +differential equation in the MCDF procedure. +The non-radiative part in Eq. (2) is dominated by the electron-electron interaction, which +is obtained by evaluating the Feynman diagram of Fig. 18. It is given in atomic units and in +the Coulomb gauge by +V (rij,ωij) = 1 +rij +− αααi ·ααα j +rij +− αααi ·ααα j +rij +(cos(αωijrij)−1) ++ (αααi ·∇∇∇i)(ααα j ·∇∇∇j) cos(αωijrij)−1 +(αωij)2rij +, +(57) +where ωij = Ei −Ej is the energy of the photon exchanged between the two electrons. The +∇∇∇ operator acts only on rij and not on the following wave function. The Breit operator +[403–405] in Eq. (3) corresponds to the expansion of Eq. (57) in powers of α = 1/c up +to the second order. It is then independent of ωij. The frequency-dependent part is called +higher-order retardation. This finite frequency contribution becomes important at high nu- +clear charges [321]. The frequency dependent Breit interaction has been explored in detail in +many works [66, 76, 314, 406–408]. The main difficulty lies in the definition of ωij in CI or +MCDF calculations, where the energy of an individual orbitals is not physical and can also +reach very negative values, much lower than −mec2 [66, 76, 408]. The gauge dependence +of the resulting energy shift has been discussed in detail in Refs. [74, 409]. +The second-order diagrams of Fig. 19 have been evaluated in Refs. [341, 342, 410, 411] +for the ground state and n = 2 excited states of two-electron atoms. They contains specific +QED corrections beyond what can be obtained by many-body treatment of the interaction +in Eq. (57) with the necessary projection operators. These corrections contains the positron +part of the ladder diagram 19 (a) and the contribution from the cross-ladder diagram 19 (b). +4.4.3 Effective QED Hamiltonians +To bring the self-energy term into a useful effective Hamiltonian form, hSE, is the most chal- +lenging part as this operator is inherently non-local. Nevertheless, many attempts were made +to estimate the self-energy shift in atomic spectra by approximations. Earlier ones were +summarized in [412]. Approximations based on effective Z values to account for electron +screening were introduced in the early versions of GRASP [413]. The Welton approxima- +tion [414] was introduced in the MDFGME code in 1987 [65] for s-states and generalized +to ℓ ≥ 0 in [66]. Effective operators directly based on BSQED have been introduced more +recently [68, 340]. +In Ref. [412], a local self-energy potential was introduced in a simple Gaussian form +hSE(r) = +� +b0 +b1Z +b2Z2� +exp +� +−(β0 +β1Z +β2Z2)r2� +, +(58) +which serves as a rough estimate. Here bi and βi are adjustable parameters and the Gaussian +is located close to the nucleus. +A far more accurate expression for an effective self-energy Hamiltonian has been pro- +posed in Ref. [67] to be +hSE(r) = Φmag(r)+Φel(r)+Φlow(r), +(59) + +42 +with the magnetic form factor +Φmag(r) = +α +4πmiγ ·∇ +� +φ(r) +�� ∞ +1 dt +e2trm +t2√ +t2 −1 +−1 +�� +, +(60) +where φ(r) is the electric potential of the nucleus. The last two terms are contributions from +the electric form factor decomposed into a high- and a low-frequency part, +Φel(r) =A(Z)α +π φ(r) +� ∞ +1 dt e−2trm +√ +t2 −1 +�� +1− 1 +2t2 +� +� +log(t2 −1)+4log +� 1 +Zα + 1 +2 +�� +− 3 +2 + 1 +t2 +� +(61) +The (long-range) low-frequency contribution is given by +Φlow(r) = −B(Z)Z4α5me−Zr/aB, +(62) +where B(Z) = 0.074+0.35Zα is a coefficient adjusted to reproduce the radiative shifts for +the high Coulomb p-levels [67], and aB is the Bohr radius. This expression of the self- +energy operator was implemented into the program GRASP [280] by simply replacing the +Coulomb potential −Z/r by its extension to the finite nucleus case [321]. Later, this has +been correctly folded into the self-energy potential leading to more complicated expressions, +which slightly improves the self-energy shifts [415]. To improve the SE corrections for the s- +levels, and especially for the 1s1/2 level, in multi-electron systems the prefactor A(Z) in (61) +was chosen to be dependent on the principal quantum number n [321]. More recently, both +coefficients A(Z) and B(Z) were refitted and made dependent on the angular momentum ℓ +[415]. +To go beyond models with adjustable parameters, one can in principle go back to first +principle QED and use a spectral decomposition of the self-energy operator +hnl +SE = ∑ +i, j +|ψi⟩DSE +i, j +� +ψj +�� +(63) +where {ψi} represents a complete set of hydrogenic wave functions (including both con- +tinua), and the matrix elements DSE +i, j need to come from accurate self-energy calculations. +This has been explored [416] with limited success because of the basis set restrictions im- +posed and the underlying slow convergence of this sum, which is well known from direct +exact QED evaluations [143, 336, 337]. Furthermore, the off-diagonal elements DSE +i,j with +i ̸= j are crucial and cannot be neglected. To this end, an additional exponential type semi- +local operator has been added to reduce the matrix elements Dij in size for the subsequent +spectral decomposition [68, 417], +hSE = hsl +SE +hnl +SE, +(64) +with +hsl +SE(r) = ∑ +κ +Vκ(r)Pκ. +(65) +The semi-local operator +Vκ(r) = Aκe−r/λc +(66) +differentiates between the different κ states through the projection operator Pκ (for the def- +inition see Ref. [417]). Here λc is the Compton wavelength. For details see Refs.[68, 417]. + +43 + 0 + 1 + 2 + 3 + 4 + 5 +Ni +Pd +Pt +Ds +3F4 +3F3 +3F2 +1D2 +3P2 +3P1 +3P0 +1G4 +3D3 +3D2 +3D1 +1D2 +1S0 +Energy [eV] +[(n-1)d8ns2] +[(n-1)d9ns1] +[3d10] +Fig. 24 +Energy levels for the dominant configurations of the Group 10 elements Ni, Pd, Pt, and Ds. The +values for Ni, Pd, and Pt are from the NIST database [420]. The Ds levels are from Ref. [42]. Different +colors are used to distinguish between the three different configurations: green [(n − 1)d8 ns2], blue [(n − +1)d9 ns] and black [3d10]. For Pd, there are intruder states (not shown here) arising from the [(n−1)d9 np] +configuration (for Pt from the [(n − 1)d9 np] and [(n − 1)d8 ns np] configurations), which mix with several +of the low energy states shown. Thus, some configuration assignments (especially for the 3P0 level) are +approximate at best. For Ds, a dense spectrum arising from the [6d7 7s2 7p] configuration intrudes into the +normal spectrum and only few predicted lines of even parity are shown [42]. Adapted from Ref. [1]. +A recent extension to superheavy elements up to nuclear charge Z = 170 has been carried +out in Ref. [340]. This scheme gives very accurate results for the self-energy. One wonders +if a semilocal ansatz in the same form of a pseudopotential applied commonly in electronic +structure theory could be efficiently used as well [418, 419] for all-electron QED treatments +in molecules for example. It would certainly be an improvement to the original local ansatz +[412] and possibly of sufficient accuracy in molecular calculations. +5 Electron Correlation +The accurate computational treatment of both static and dynamic electron correlation in +atomic open-shell multi-electron systems is a daunting task. Even for the lightest elements +such as nickel, a correct description of the many low-lying states arising from the 3d8 4s2, +3d9 4s and 3d10 configurations is currently not available. For example, using Gaussian type +basis sets (GTOs), Ref. [421] applied large-scale complete active space second-order pertur- +bation theory (CASPT2) calculations including relativistic corrections for nickel correlat- +ing 18 electrons within 14 orbitals. These calculations resulted in the following excitation +energies with respect to the 3D(d9s) ground state ( j-averaged experimental values set in +parentheses): 3F −0.08 eV (0.03 eV), 1D 0.32 eV (0.33 eV), 1S 1.77 eV (1.74 eV). Figure +24 shows the energy levels for the Group 10 elements. From this it is clear that the correct +prediction of the ground state symmetry is difficult for atoms with dense spectra. This prob- +lem will become worse when degenerate high angular momentum states are involved such +as in the lanthanides and actinides and for the superheavy elements. + +44 +One of the main workhorses in relativistic atomic structure theory is the configuration +interaction (CI) method with a predetermined set of CSFs where the radial shapes of the +one-electron orbital spinors remain unchanged. In a typical CI procedure, the active virtual +and core space are systematically increased and higher angular momentum functions added +to test convergence against the final value. The resulting CI wave functions are then used +for calculating QED effects, albeit QED matrix elements can be directly added to the CI +matrix resulting in correlated QED calculations (this still needs to be explored for the SHEs). +These CI techniques are invaluable for obtaining accurate properties to, for example, test +the standard model [422]. However, as the size of a CI calculation scales exponentially +with the excitation level (the number of determinants is Ndet ∼ nmNm +v /(m!)2 with n being +the number of electrons, Nv the number of virtual orbitals and m the excitation level), the +CI method is often combined with many-body perturbation theory for electron correlation +(CI+MBPT) to allow for an efficient treatment of important core excitations [423]. Again, +because of the large computer time involved one rarely goes beyond second-order MBPT, +although calculations for atoms with one valence electron (Cs and Tl for example) have been +performed up to third order [424, 425]. A mix of MBPT and CC methods has also been used +for evaluating electron affinities for Ca and Sr [426]. +A very popular electron correlation method within the quantum chemistry community +is coupled cluster (CC) theory originally proposed by Coester and Kuemmel [427] for nu- +clear interactions, and subsequently brought into electronic structure theory [428]. There +are several excellent papers, books and reviews on CC applications [429–436]. In CC the- +ory, the ground state wave function is related to the DHF ground state configuration by an +exponential operator containing the cluster operator T, +Ψ0 = eTΨ DHF +0 +, +(67) +with T = T1 + T2 + ... and Tn are the n-particle excitation operators. For example, if one +restricts to double excitations only (T = T2, CCSD), the cluster operator is +T2 = ∑ +i 120 region, which will help designing future atom-at-a-time chemistry +experiments [552]. + +57 + 148 152 156 160 164 168 172 176 180 184 188 +Neutron number N + + 108 + 112 + 116 + 120 + 124 +Proton number Z +(b) log10Tα/s +6 +4 +2 +0 +-2 +-4 +-6 +-8 +-10 +-4 +-6 +-8 + 108 + 112 + 116 + 120 + 124 +(a) log10TSF/s +8 +6 +42 +0 +-2 +-4 +-6 +-8 +-10 +-2 +-4 +-6 +-8 +-10 +Fig. 32 +Summary of theoretical predictions [559] for decay modes of superheavy nuclei obtained with +nuclear DFT: (a) SF half-lives; (b) α-decay half-lives. +7.3 Periodic Table - How far can we go? +Oganesson was the heaviest element and nihonium the last element to be added officially +into the PT [553]. Thus the 7th period of elements is now complete. The question arises if +one can go much further in the atomic number. From an electronic point of view, there is no +limitation to Z. While the correct description of multi-electron systems with Z > Zc ≈ 170 is +still a difficult problem, and the inclusion of Gamow states for multi-electron systems needs +to be addressed, the real limitation to the PT comes from the nuclear stability [18, 554]. +In the transactinide region, Z > 103, in early days known as the sea of instability [555], +the half-life of the elements varies between hours +�266 +103Lr +� +and seconds +�294 +118Og +� +or below. +Although there is a small predicted region of increased stability between Z=114-126 and +N = 184 with predicted lifetimes of hours or even days [18, 556–558], it is currently not +clear how far the PT can be extended from the nuclear point of view. Indeed, the IUPAC +defines an element to exist if its lifetime is longer than Tel≈1×10−14 s, which is the time it +takes for electron cloud to form around the nucleus. This means that for atomic nuclei living +shorter than Tel it makes no sense to talk about atoms and chemistry [18, 554]. + +58 +The lifetimes of most known superheavy nuclei are governed by the competition be- +tween α-decay and spontaneous fission (SF). The corresponding lifetimes predicted by a +particular DFT model [559] are shown in Fig. 32. For a survey of various predictions of α- +decay and SF lifetimes, see Refs. [261] and [560], respectively. The shortest SF half-lives, +reaching down to 10−10 s, are predicted for nuclei from a narrow corridor formed by 280Hs, +284Fl, and 284Og. This corridor of fission instability separates the regions of superheavy nu- +clei synthesized in hot- and cold-fusion reactions. Moving towards more neutron-rich nuclei +beyond N = 184, dramatic decrease of SF lifetimes, below 10−15 s is expected, see Fig. 32(b) +and Ref. [561]. +It is to be noted that predictions of nuclear models in the region of superheavy nuclei +are very sensitive to both input (forces, functionals) and theoretical framework used. Con- +sequently, theoretical lifetime estimates, especially for SF, often differ by many orders of +magnitude [560]. The heaviest nuclei synthesized so far are all proton-rich; hence, they can +in principle decay by means of electron capture or β +/EC process. So far, no such decay +modes have been observed in the upper superheavy region, and this indicates that they can- +not compete with the dominant α -decay and SF modes. Indeed, according to theory β +/EC +lifetimes shorter than 1 s are expected in nuclei that lie rather far from the current superheavy +region [261]. +Experiments to synthesize new superheavy nuclei and elements beyond Og are under- +way [562, 563]. If discovered, these systems will be crucial for testing many-body nuclear +structure theories [18]. To explore their chemistry will be very challenging, however. +8 Conclusions +Atomic structure theory developed enormously over the past decades to the extend where +QED can be tested to high precision for few-electron systems. However, for many-electron +systems the accurate description of both QED and electron correlation effects remains a +major challenge [480]. But even here progress has been made for elements with large +atomic numbers [14]. While the negative energy continuum is required for QED, is creates a +formidable conceptual and computational challenge, especially when bound states approach +the negative energy continuum threshold. The correct description of diving (Gamow) states +within a multi-electron formalism including QED effects still needs to be explored. It is clear +that effects associated with the negative energy continuum distinguishes the Dirac from the +Schrödinger equation in both mathematical and in physical terms [29]. Once these problems +are solved, there is no limitation to the treatment of atoms beyond the critical nuclear charge. +The PT, seen as the foundation for chemistry, is therefore not limited to a certain nuclear +charge region, but limited by nuclear stability [18]. The future looks bright for superheavy +element synthesis and associated chemistry experiments, which require the support of accu- +rate electronic and nuclear structure theory. +9 Appendix A: The Self-Adjointness of the Dirac-Coulomb Hamiltonian +In the two appendices we address some of the more mathematical features of the Dirac +equation, the self-adjointness problem and (in the next section) the rigged Hilbert space +formalism for Gamow states. Both aspects often lead to some misunderstandings in the +community and are therefore discussed briefly here. + +59 +Table 5 +Norm and expectation value existences for different ranges of nuclear charges Z and parameter +±γ(Z) = ± +� +1−(Zα)2 appearing in the radial 1s function of hydrogenic atoms with potential V(r) = −Z/r. +The symbol S stands for the Sobolev norm (S = 1 for Dirac and 2 for Schrödinger) which requires the gradient +norm to exist for the Dirac-Coulomb operator (first and second derivatives for the Schrödinger case). +System +range +∥φ∥2 +⟨φ|H |φ⟩ +∥φ∥(S) +2 +∥Hφ∥2 +Schrödinger +Z > 0 +yes +yes +yes +yes +Dirac +γ(Z) +0 < Zα < +√ +3/2 +yes +yes +yes +yes +Dirac +γ(Z) +√ +3/2 ≤ Zα < 1 +yes +yes +no +no +Dirac −γ(Z) +0 < Zα < +√ +3/2 +no +no +no +no +Dirac −γ(Z) +√ +3/2 ≤ Zα < 1 +yes +no +no +no +To explain the non-self-adjointness at critical charge in correct mathematical terms, we +require the L2-norm of the derivative || d +drφ||2 to exist (or the gradient norm for the three- +dimensional case) for the eigensolutions as the Dirac equation is a first-order differential +equation (see discussion of Sobolev spaces further below). The radial solutions for a point +charge nucleus φ(r) have the general form φ(r) = anκ(2Zr)γe−Zr f P,Q +nκ,Z(r) with the exponent +γ = ± +� +κ2 −(Zα)2 and f P,Q +nκ,Z(r) containing expressions of Pnκ(r) and Qnκ(r) in terms of +confluent hypergeometric functions [89]. Unlike for the Schrödinger equation, γ is a non- +integer and derivatives lead to negative r-exponents. As a result, both || d +drφ||2 and ||HDφ||2 +become infinite if γ ≤ 1 +2 leading to Zc1α = +√ +3/2 as discussed in Sec. 2.4.1. In contrast, for +the nonrelativistic radial Schrödinger equation, all derivative norms exist, i.e., || dn +drn φNR||2 < +∞, as only integers appear in the r-exponent. This leads often to misunderstandings as the L2 +Hilbert space only requires the norm ||φ||2 to exist, but as soon as we introduce an unbound +differential operator such as HD we have to deal with the domain of such an operator and the +existence of certain expectation values and derivative norms. For a more detailed analysis +using the Weyl’s limit point - limit circle theorem, generalized to the Dirac equation by +Weidmann [141], the reader is referred to a recent paper by Gallone [99]. The various norm +existences for different nuclear charge regions are summarized in Table 5. +To rephrase the self-adjointness condition in different terms, a self-adjoint extension +of the radial Dirac operator should have the following domain [29, 141]: dom(HD) = {φ ∈ +L2(R+)2| each component of φ is locally absolutely continuous; HDφ ∈ L2(R+)2; deficiency +indices d{φ(r = 0)} = (0,0)} (see also Hogreve [98]), where L2(F)n ≡ L2(F) ⊗ Cn over a +field F (F ≡ R+ and n = 2 for the radial Dirac equation) [29]. This allows us to select the +Sobolev space W1,2(R+)2 [29, 46] as the natural domain for the (unbound) Dirac operator (or +similarly for a four-component wave function in the three-dimensional case W1,2(R3)2) lying +dense in L2(R+)2, such that dom(HD) ⊆ W1,2(R3)2 ⊆ L2(R3)2. The domain problem of the +Dirac-Coulomb operator has been very recently discussed and critically analyzed by Estaban +[564]. In the subcritical nuclear charge region ( +√ +3/2 ≤ Zα ≤ 1), φ is an eigenfunction to +a non-self-adjoint Dirac-Coulomb Hamiltonian with real eigenvalues and norm ||φ||2 < ∞, +but does not belong to dom(HD) (HD self-adjoint)! More generally, one looks for the largest +subdomain of the Hilbert space that remains invariant under the action of certain powers of +required operators (observables) including the Hamiltonian of the system, which is known +as the maximal invariant subspace of the algebra generated by these operators [565]. +A note of caution should be added here. If we eliminate the small component and fo- +cus on the resulting second-order differential equation, the underlying Sobolev space is now +W2,2(R3)2, which makes the conditions more stringent for the norm existence. In any case, + +60 +we seek for an appropriate self-adjoint extension of HD [48, 101, 566] as physics does not re- +strict atoms to a maximum critical charge (except for nuclear instability which is an entirely +different matter [39]). The necessary boundary condition for securing the self-adjointness of +the Dirac operator at the origin has been discussed in detail by Kuleshov [151] and Gitman +[48, 567]. The physical realization of this boundary condition is the introduction of a finite +nuclear charge density [21]. +Last we mention that mathematically, there are many self-adjoint extensions which can +be realized for the Dirac operator. For example, Kato showed that a potential energy ma- +trix of the form Vik = a/2r + b with a < 1 and b > 0 makes the Dirac operator essentially +self-adjoint on certain domains [568]. For a more recent discussion on possible self-adjoint +extensions we refer to [566]. +10 Appendix B: The Rigged Hilbert Space Formalism +If we consider the spectrum of the Dirac-Coulomb operator, σ(HD), we need to include +the discrete (d) and both the positive (+) and negative (−) continuum states (c) (cf. Figure +2), i.e., σ(HD) = {φ d} ∪ {φ c ++} ∪ {φ c +−}. The continuum states are important in scattering +(resonance) theory and are essential for the quantum mechanical completeness relation. It +is well known, however, that unlike the discrete states, the continuum states lead to domain +problems for unbound operators, i.e., they are not normalizable and therefore do not belong +to the quantum mechanically relevant L2 space. As a result, von Neumann’s original Hilbert +space formalism requires an extension to include such (generalized) functions. Continuum +states are properly defined within an extended or rigged Hilbert space (G ) formalism [569– +575] originally proposed for the quantum mechanical framework by Roberts, Antoine and +Bohm [576–580].‡‡‡ In fact, rigged Hilbert spaces are the structures required for both the +discrete orthonormal and continuous bases to coexist [581]. Their existence for self-adjoint +operators on separable Hilbert spaces is guaranteed by the Gelfand-Maurin nuclear spectral +theorem. +In strict mathematical terms, the rigged Hilbert space (RHS) G is defined as a triple +of topological vector spaces G = (Φ,H ,Φ×), called the Gelfand triple [582], generated +by an infinite dimensional separable Hilbert space H such that the denseness relation is +Φ ⊆ H ⊆ Φ×. Here, Φ is a (complete) nuclear Fréchet space, also called test-function +space (not necessarily a Hilbert space, which enables to use the nuclear spectral theorem of +Gelfand and Maurin [569, 582]), and Φ× is the topological dual (or topological conjugate) +of Φ, that is the complete space of continuous anti-linear functionals on Φ (also called +distribution or Schwartz space). The RHS structure includes an inductive limit of a sequence +of topological spaces Φ(n) in which the topologies get rapidly coarser with increasing n +[575, 583], e.g., we might think of a series of Sobolev spaces Wk+1,2 ⊆ Wk,2, with W1,2 ⊆ L2. +It is clear from this example that these subspaces have different norms (or semi-norms for +the more general nuclear spaces [575]). +The RHS formalism provides a correct mathematical foundation to Dirac’s original bra +and ket notation [575, 584], used extensively in quantum theory. Needless to say that the +Dirac delta “function” is a distribution required for the properties of continuum states be- +longing to Φ× rather than Φ. To cite Bohm [570]: The difference between [the rigged] +Hilbert space formulation and the usual [von Neumann] Hilbert space formulation appears +‡‡‡The term rigged Hilbert space is misleading as G is not a Hilbert space per se, but is generated from a +Hilbert space H . + +61 +to be minor from a physicists point of view, but is essential from a mathematical point of view +and leads far to tremendous mathematical simplification; in fact it justifies the mathemati- +cally undefined operations that the physicists have been accustomed to in their calculations. +The RHS formalism can easily be generalized to a rigged Fock space formalism required in +quantum field theory [585–587].§§§ +To be more specific, the Gelfand triple for the spectrum of the Dirac operator is chosen +as Φ ⊆dom(HD) ⊆ L2 ⊆ Φ×. Vectors in Φ will be complex linear, and the vectors in Φ× +are complex antilinear continuous functionals compatible with the scalar product in the un- +derlying Hilbert space, e.g., F : ψ ∈ Φ → C. For example, we define the action F ∈ Φ× on +ψ ∈ Φ as an extension to the Hilbert space product F(ψ) = ⟨ψ|F⟩ = ⟨F|ψ⟩∗. This action is +linear to the right and antilinear to the left.¶¶¶ Continuity is defined such that if ψn → ψ for +n → ∞, ψn ∈ Φ,ψ ∈ L2 then F(ψn) → F(ψ) in C. +States with given quantum numbers n,κ, when diving into the negative energy contin- +uum, have complex eigenenergies [151, 204] and therefore move out of the natural domain +of the self-adjoint operator HD. We need to give such states belonging to a subset of distri- +butions in the space Φ× a physical interpretation, however, we need first to interpret such +generalized eigenfunctions (or eigenfunctionals) from a mathematical point of view. +Let A : Φ → Φ a (closed) linear operator and A× : Φ× → Φ× its natural extension of +the usual adjoint operator A† such that F(Aψ) = A×F(ψ) = ⟨Aψ|F⟩ = ⟨ψ|A×F⟩ for all +ψ ∈ Φ,F ∈ Φ×. According to the Riesz representation theorem, for every F ∈ Φ× and lin- +ear operator A there exist a unique complex function φ with complex eigenvalue λ, Aφ = λφ +such that for all ψ ∈ Φ we have F(Aψ) = ⟨Aψ|F⟩ = ⟨ψ|A×φ⟩ = λ ∗⟨ψ|φ⟩ [570]. We call φ +the generalized eigenfunction of A with respect to F. These extended eigenstates can be used +in the normal way using Dirac’s notation keeping in mind that these may not be normaliz- +able and, in general, have complex eigenvalues. Of course, the scalar product ⟨ψ|φ⟩ always +needs to be finite which may require a specific (semi)norm definition or regularization of +integrals. +Acknowledgements We acknowledge financial support by the Program Hubert Curien Dumont d’Urville +New Zealand - France Science & Technology Support Program number 43245QC, and the Marsden Fund +of the Royal Society of New Zealand. We thank Profs. Trond Saue (Toulouse), Ephraim Eliav (Tel Aviv), +W. H. Eugen Schwarz (Siegen), James Avery (Copenhagen) and Vladimir Shabaev (St. Petersburg) for many +stimulating and critical discussions. This material is also based upon work supported by the U.S. Department +of Energy, Office of Science, Office of Nuclear Physics under award number DE-SC0013365. P.S. thanks +ENS-PSL Research University for providing a one month invited professor position during the course of this +work. P.I. is a member of the Allianz Program of the Helmholtz Association, contract no EMMI HA-216 +“Extremes of Density and Temperature: Cosmic Matter in the Laboratory”. +References +1. P. Schwerdtfeger, O.R. Smits, P. Pyykkö, Nature Rev. Chem. 4(7), 359 (2020). DOI 10. +1038/s41570-020-0195-y. URL https://doi.org/10.1038/s41570-020-0195-y +§§§As quantum operators (in first quantization) act on Hilbert spaces, quantum field operators act on Fock +spaces. +¶¶¶There is always another rigged Hilbert space Φ ⊆ H ⊆ Φ∗, where Φ∗ is the dual space of Φ containing +the continuous, linear functionals over Φ [576, 588]. Dirac’s bras and kets belong to Φ∗ and Φ×, respectively, +and both spaces are isometrically isomorph. + +62 +2. E.R. Scerri, in Philosophy of Chemistry, Handbook of the Philosophy of +Science, +vol. +6, +ed. +by +A.I. +Woody, +R.F. +Hendry, +P. +Needham +(North- +Holland, Amsterdam, 2012), pp. 329 – 338. +DOI https://doi.org/10.1016/ +B978-0-444-51675-6.50024-4. +URL http://www.sciencedirect.com/science/ +article/pii/B9780444516756500244 +3. P. Pyykkö, Chemical Reviews 112(1), 371 (2012). DOI 10.1021/cr200042e. URL +https://doi.org/10.1021/cr200042e +4. C. Cao, R.E. Vernon, W.H.E. Schwarz, J. Li, Frontiers in Chemistry 8, 813 (2021). +DOI 10.3389/fchem.2020.00813. URL https://www.frontiersin.org/article/ +10.3389/fchem.2020.00813 +5. W.H.E. Schwarz, U. Müller, F. Kraus, Zeitschrift für anorganische und allgemeine +Chemie 648(13), e202200008 (2022). DOI https://doi.org/10.1002/zaac.202200008. +URL https://onlinelibrary.wiley.com/doi/abs/10.1002/zaac.202200008 +6. G. Restrepo, Chemistry - A European Journal 25(68), 15430 (2019). DOI 10.1002/ +chem.201902802. +URL https://onlinelibrary.wiley.com/doi/abs/10.1002/ +chem.201902802 +7. S. Shaik, E. Cremades, S. Alvarez, Angew. Chem. Int. Ed. 58(38), 13194 (2019). DOI +10.1002/anie.201904584. URL https://onlinelibrary.wiley.com/doi/abs/10. +1002/anie.201904584 +8. G. Steinhauser, Nachr. Chemie 67(9), 8 (2019). +DOI 10.1002/nadc.20194086536. +URL https://onlinelibrary.wiley.com/doi/abs/10.1002/nadc.20194086536 +9. Y. Oganessian, J. Phys. G: Nucl. Part. Phys. 34(4), R165 (2007). +DOI 10.1088/ +0954-3899/34/4/r01. +URL https://doi.org/10.1088%2F0954-3899%2F34%2F4% +2Fr01 +10. Y.T. Oganessian, Radiochim. Acta 99(7-8), 429 (2011). URL https://doi.org/10. +1524/ract.2011.1860 +11. B. Fricke, W. Greiner, J.T. Waber, Theoret. Chim. Acta 21, 235 (1971). DOI 10.1007/ +BF01172015. URL 10.1007/BF01172015 +12. B. Fricke, J. McMinn, Naturwiss. 63(4), 162 (1976) +13. E. Scerri, Scientific American 308(6), 68 (2013) +14. P. Indelicato, J. Biero´n, P. Jönsson, Theor. Chem. Acc. 129(3-5), 495 (2011). URL +https://doi.org/10.1007/s00214-010-0887-3 +15. P. Pyykkö, Pure and Applied Chemistry 91(12), 1959 (2019). +DOI doi:10.1515/ +pac-2019-0801. URL https://doi.org/10.1515/pac-2019-0801 +16. V.I. Nefedov, M.B. Trzhaskovskaya, V.G. Yarzhemskii, Doklady Physical Chemistry +408(2), 149 (2006). DOI 10.1134/S0012501606060029. URL https://doi.org/10. +1134/S0012501606060029 +17. P. Pyykkö, Physical Chemistry Chemical Physics 13(1), 161 (2011). URL https: +//doi.org/10.1039/C0CP01575J +18. S.A. Giuliani, Z. Matheson, W. Nazarewicz, E. Olsen, P.G. Reinhard, J. Sad- +hukhan, B. Schuetrumpf, N. Schunck, P. Schwerdtfeger, Rev. Mod. Phys. 91, 011001 +(2019). DOI 10.1103/RevModPhys.91.011001. URL https://link.aps.org/doi/ +10.1103/RevModPhys.91.011001 +19. T.K. Sato, M. Asai, A. Borschevsky, T. Stora, N. Sato, Y. Kaneya, K. Tsukada, C.E. +Düllmann, K. Eberhardt, E. Eliav, S. Ichikawa, U. Kaldor, J.V. Kratz, S. Miyashita, +Y. Nagame, K. Ooe, A. Osa, D. Renisch, J. Runke, M. Schädel, P. Thörle-Pospiech, +A. Toyoshima, N. Trautmann, Nature 520(7546), 209 (2015). +DOI 10.1038/ +nature14342. URL https://doi.org/10.1038/nature14342 + +63 +20. T.K. Sato, M. Asai, A. Borschevsky, R. Beerwerth, Y. Kaneya, H. Makii, A. Mit- +sukai, Y. Nagame, A. Osa, A. Toyoshima, K. Tsukada, M. Sakama, S. Takeda, K. Ooe, +D. Sato, Y. Shigekawa, S.i. Ichikawa, C.E. Düllmann, J. Grund, D. Renisch, J.V. Kratz, +M. Schädel, E. Eliav, U. Kaldor, S. Fritzsche, T. Stora, J. Am. Chem. Soc. 140(44), +14609 (2018). DOI 10.1021/jacs.8b09068. URL https://doi.org/10.1021/jacs. +8b09068. PMID: 30358998 +21. I.Y. Pomeranchuk, Y.A. Smorodinsky, J. Fiz. USSR 9, 97 (1945) +22. P.G. Reinhard, W. Greiner, H. Arenhövel, Nucl. Phys. A 166, 173 (1971). DOI https: +//doi.org/10.1016/0375-9474(71)90421-0. URL http://www.sciencedirect.com/ +science/article/pii/0375947471904210 +23. Y.B. +Zeldovich, +V.S. +Popov, +Soviet +Physics +Uspekhi +14(6), +673 +(1972). +DOI 10.1070/pu1972v014n06abeh004735. +URL https://doi.org/10.1070/ +pu1972v014n06abeh004735 +24. B. Müller, H. Peitz, J. Rafelski, W. Greiner, Phys. Rev. Lett. 28, 1235 (1972). +DOI 10.1103/PhysRevLett.28.1235. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.28.1235 +25. V.S. Popov, V.D. Mur, Sov. J. Nucl. Phys. 18 (1974) +26. J. Reinhardt, W. Greiner, Rep. Prog. Phys. 40(3), 219 (1977). +DOI 10.1088/ +0034-4885/40/3/001. +URL https://doi.org/10.1088%2F0034-4885%2F40%2F3% +2F001 +27. J. Reinhardt, B. Müller, W. Greiner, Phys. Rev. A 24, 103 (1981). +DOI 10.1103/ +PhysRevA.24.103. URL https://link.aps.org/doi/10.1103/PhysRevA.24.103 +28. W. Greiner, B. Müller, J. Rafelski, Quantum Electrodynamics of Strong Fields +(Springer, Berlin, 1985) +29. B. Thaller, The Dirac equation (Springer, Berlin, 1992). URL https://doi.org/10. +1007/978-3-662-02753-0 +30. D.M. Gitman, A.D. Levin, I.V. Tyutin, B.L. Voronov, Phys. Scr. 87(3), 038104 +(2013). DOI 10.1088/0031-8949/87/03/038104. URL https://doi.org/10.1088/ +0031-8949/87/03/038104 +31. P. Schwerdtfeger, L.F. Pašteka, A. Punnett, P.O. Bowman, Nucl. Phys. A 944, 551 +(2015). DOI 10.1016/j.nuclphysa.2015.02.005. URL 10.1016/j.nuclphysa.2015. +02.005 +32. V. Shabaev, A. Bondarev, D. Glazov, Y. Kozhedub, I. Maltsev, A. Malyshev, R. Popov, +D. Tumakov, I. Tupitsyn, arXiv preprint arXiv:1910.01373 (2019) +33. G. Gamow, Z. Phys. 51(3), 204 (1928). DOI 10.1007/BF01343196. URL https: +//link.springer.com/article/10.1007/BF01343196 +34. G. Gamow, Z. Phys. 52(7), 510 (1929). DOI 10.1007/BF01339451. URL https: +//doi.org/10.1007/BF01339451 +35. A.J.F. Siegert, Phys. Rev. 56, 750 (1939). DOI 10.1103/PhysRev.56.750. URL https: +//link.aps.org/doi/10.1103/PhysRev.56.750 +36. A. Bohm, M. Gadella, J. Dollard, Dirac Kets, Gamow Vectors and Gel’fand Triplets: +The Rigged Hilbert Space Formulation of Quantum Mechanics. Lectures in Mathemat- +ical Physics at the University of Texas at Austin (Springer, berlin, 1989) +37. O. Civitarese, M. Gadella, Physics Reports 396(2), 41 (2004). DOI https://doi.org/ +10.1016/j.physrep.2004.03.001. URL https://www.sciencedirect.com/science/ +article/pii/S0370157304001085 +38. W. Nazarewicz, Nature Phys. 14(6), 537 (2018). DOI 10.1038/s41567-018-0163-3. +URL https://doi.org/10.1038/s41567-018-0163-3 + +64 +39. W. Nazarewicz, Journal of Physics G: Nuclear and Particle Physics 43(4), 044002 +(2016). DOI 10.1088/0954-3899/43/4/044002. URL https://doi.org/10.1088% +2F0954-3899%2F43%2F4%2F044002 +40. P. Indelicato, J. Santos, S. Boucard, J.P. Desclaux, Eur. Phys. J. D 45(1), 155 (2007). +URL http://link.springer.com/article/10.1140%2Fepjd%2Fe2007-00229-y +41. B.G.C. Lackenby, V.A. Dzuba, V.V. Flambaum, Phys. Rev. A 98, 042512 (2018). +DOI 10.1103/PhysRevA.98.042512. URL https://link.aps.org/doi/10.1103/ +PhysRevA.98.042512 +42. B.G.C. Lackenby, V.A. Dzuba, V.V. Flambaum, Phys. Rev. A 101, 012514 (2020). +DOI 10.1103/PhysRevA.101.012514. URL https://link.aps.org/doi/10.1103/ +PhysRevA.101.012514 +43. E.E. Salpeter, H.A. Bethe, Phys. Rev. 84, 1232 (1951). DOI 10.1103/PhysRev.84. +1232. URL https://link.aps.org/doi/10.1103/PhysRev.84.1232 +44. W. Greiner, et al., Relativistic quantum mechanics, vol. 2 (Springer, 2000) +45. I.P. Grant, Relativistic quantum theory of atoms and molecules: theory and computa- +tion, vol. 40 (Springer Science & Business Media, 2007) +46. R.D. Richtmyer, C. Burdorf, Principles of advanced mathematical physics, vol. 1 +(Springer, 1978) +47. V.G. Bagrov, D. Gitman, Exact solutions of relativistic wave equations, vol. 39 +(Springer Science & Business Media, 1990) +48. D.M. Gitman, I.V. Tyutin, B.L. Voronov, Self-adjoint extensions in quantum mechan- +ics: general theory and applications to Schrödinger and Dirac equations with singular +potentials, vol. 62 (Springer Science & Business Media, 2012) +49. I. Sargsjan, et al., Sturm—Liouville and Dirac Operators, vol. 59 (Springer Science & +Business Media, 2012) +50. V.G. Bagrov, D. Gitman, The Dirac equation and its solutions, vol. 4 (Walter de +Gruyter GmbH & Co KG, 2014) +51. G. Magnifico, T. Felser, P. Silvi, S. Montangero, Nature Communications 12(1), +3600 (2021). DOI 10.1038/s41467-021-23646-3. URL https://doi.org/10.1038/ +s41467-021-23646-3 +52. F.J. Dyson, Phys. Rev. 85(4), 631 (1952). +URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.85.631 +53. T. Heinzl, A. Ilderton, B. King, Phys. Rev. Lett. 127, 061601 (2021). +DOI +10.1103/PhysRevLett.127.061601. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.127.061601 +54. J.B. Kogut, E. Dagotto, Phys. Rev. Lett. 59, 617 (1987). DOI 10.1103/PhysRevLett. +59.617. URL https://link.aps.org/doi/10.1103/PhysRevLett.59.617 +55. S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg, L. Lellouch, +T. Lippert, A. Portelli, K.K. Szabo, B.C. Toth, Science 347(6229), 1452 (2015). +DOI 10.1126/science.1257050. URL https://science.sciencemag.org/content/ +347/6229/1452 +56. D.K. Sinclair, J.B. Kogut. Lattice QED in external electromagnetic fields (2021). DOI +10.48550/ARXIV.2111.01990. URL https://arxiv.org/abs/2111.01990 +57. B. Fricke, J. Desclaux, J. Waber, Phys. Rev. Lett. 28(12), 714 (1972). URL https: +//doi.org/10.1103/PhysRevLett.28.714 +58. J.P. Desclaux, At. Data, Nucl. Data Tab. 12(4), 311 (1973). URL https://doi.org/ +10.1016/0092-640X(73)90020-X +59. J. Sucher, Phys. Rev. A 22, 348 (1980). DOI 10.1103/PhysRevA.22.348. URL https: +//link.aps.org/doi/10.1103/PhysRevA.22.348 + +65 +60. C.F. Fischer, M. Godefroid, T. Brage, P. Jönsson, G. Gaigalas, Journal of Physics +B: Atomic, Molecular and Optical Physics 49(18), 182004 (2016). +DOI 10.1088/ +0953-4075/49/18/182004. +URL https://doi.org/10.1088/0953-4075/49/18/ +182004 +61. I.P. Grant, in Relativistic Effects in Atoms, Molecules, and Solids, ed. by G.L. Malli +(Plenum Press, New York, 1983), pp. 73–88 +62. W.R. Johnson, Atomic structure theory (Springer, 2007) +63. K.T. Cheng, M.H. Chen, W.R. Johnson, J. Sapirstein, Canadian Journal of Physics +86(1), 33 (2008). DOI 10.1139/p07-106. URL http://www.nrcresearchpress. +com/doi/10.1139/p07-106 +64. P. Indelicato, J. Phys. B: At. Mol. Opt. Phys. 19, 1719 (1986). URL https://doi. +org/10.1088/0022-3700/19/12/012 +65. P. Indelicato, O. Gorceix, J.P. Desclaux, J. Phys. B: At. Mol. Opt. Phys. 20(4), 651 +(1987). URL http://dx.doi.org/10.1088/0022-3700/20/4/007 +66. P. Indelicato, J.P. Desclaux, Phys. Rev. A 42(9), 5139 (1990). +URL http:// +journals.aps.org/pra/abstract/10.1103/PhysRevA.42.5139 +67. V.V. Flambaum, J.S.M. Ginges, Phys. Rev. A 72, 052115 (2005). +DOI 10.1103/ +PhysRevA.72.052115. URL https://link.aps.org/doi/10.1103/PhysRevA.72. +052115 +68. V.M. Shabaev, I.I. Tupitsyn, V.A. Yerokhin, Phys. Rev. A 88, 012513 (2013). +DOI 10.1103/PhysRevA.88.012513. URL https://link.aps.org/doi/10.1103/ +PhysRevA.88.012513 +69. P. Pyykkö, L.B. Zhao, Journal of Physics B: Atomic, Molecular and Op- +tical +Physics +36(8), +1469 +(2003). +DOI +10.1088/0953-4075/36/8/302. +URL +http://stacks.iop.org/0953-4075/36/i=8/a=302?key=crossref. +9362092d92024faa2ceaa5016292b8ed +70. H.M. Pilkuhn, Relativistic Quantum Mechanics: Theory of Atomic Bound States +(Kluwer Academic Publishers Group., 2008) +71. G. Hardekopf, J. Sucher, Phys. Rev. A 31, 2020 (1985). DOI 10.1103/PhysRevA.31. +2020. URL https://link.aps.org/doi/10.1103/PhysRevA.31.2020 +72. W. Greiner, B. Müller, et al., Gauge theory of weak interactions, vol. 5 (Springer, +1996) +73. J. Mourad, H. Sazdjian, Journal of Physics G: Nuclear and Particle Physics 21(3), +267 (1995). DOI 10.1088/0954-3899/21/3/004. URL https://doi.org/10.1088/ +0954-3899/21/3/004 +74. O. Gorceix, P. Indelicato, Phys. Rev. A 37, 1087 (1988). DOI 10.1103/PhysRevA.37. +1087. URL https://link.aps.org/doi/10.1103/PhysRevA.37.1087 +75. L.F. Pašteka, E. Eliav, A. Borschevsky, U. Kaldor, P. Schwerdtfeger, Phys. Rev. Lett. +118, 023002 (2017). DOI 10.1103/PhysRevLett.118.023002. URL https://link. +aps.org/doi/10.1103/PhysRevLett.118.023002 +76. P. Indelicato, Phys. Rev. A 51(2), 1132 (1995). URL https://link.aps.org/doi/ +10.1103/PhysRevA.51.1132 +77. G.E. Brown, D.G. Ravenhall, R.E. Peierls, Proceedings of the Royal Society of Lon- +don. Series A. Mathematical and Physical Sciences 208(1095), 552 (1951). DOI +10.1098/rspa.1951.0181. URL https://royalsocietypublishing.org/doi/abs/ +10.1098/rspa.1951.0181 +78. W.H.E. Schwarz, H. Wallmeier, Molecular Physics 46(5), 1045 (1982). DOI 10.1080/ +00268978200101771. URL https://doi.org/10.1080/00268978200101771 + +66 +79. W. Kutzelnigg, International Journal of Quantum Chemistry 25(1), 107 (1984). DOI +https://doi.org/10.1002/qua.560250112. URL https://onlinelibrary.wiley.com/ +doi/abs/10.1002/qua.560250112 +80. G.E. Brown, Phys. Scr. 36(1), 71 (1987). DOI 10.1088/0031-8949/36/1/011. URL +https://doi.org/10.1088/0031-8949/36/1/011 +81. P. Indelicato, Phys. Rev. A 87(2), 022501 (2013). URL http://link.aps.org/doi/ +10.1103/PhysRevA.87.022501 +82. J. Heully, I. Lindgren, E. lindroth, A. Mårtensson-Pendrill, Phys. Rev. A 33(6), 4426 +(1986). URL https://doi.org/10.1103/PhysRevA.33.4426 +83. I. Grant, J. Phys. B: At. Mol. Opt. Phys. 20, L735 (1987). URL https://doi.org/ +10.1088/0022-3700/20/22/002 +84. E. Lindroth, J.L. Heully, I. Lindgren, A.M. Martensson-Pendrill, Journal of Physics B: +Atomic and Molecular Physics 20(8), 1679 (1987). URL http://stacks.iop.org/ +0022-3700/20/i=8/a=007 +85. J. Dolbeault, M.J. Esteban, E. Séré, M. Vanbreugel, Phys. Rev. Lett. 85, 4020 (2000). +DOI 10.1103/PhysRevLett.85.4020. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.85.4020 +86. R. Haag, Kgl. Danske Videnskab. Selakab, Mat.-Fys. Medd. 29 (1955) +87. C.G. Darwin, Proceedings of the Royal Society of London. Series A, Containing +Papers of a Mathematical and Physical Character 118(780), 654 (1928). +DOI +10.1098/rspa.1928.0076. URL https://royalsocietypublishing.org/doi/abs/ +10.1098/rspa.1928.0076 +88. C.G. Darwin, Proceedings of the Royal Society of London. Series A, Containing +Papers of a Mathematical and Physical Character 120(786), 621 (1928). +DOI +10.1098/rspa.1928.0169. URL https://royalsocietypublishing.org/doi/abs/ +10.1098/rspa.1928.0169 +89. W. Gordon, Zeitschrift für Physik 48(1), 11 (1928). DOI 10.1007/BF01351570. URL +https://doi.org/10.1007/BF01351570 +90. L. Morel, Z. Yao, P. Cladé, S. Guellati-Khélifa, Nature 588(7836), 61 (2020). DOI 10. +1038/s41586-020-2964-7. URL https://doi.org/10.1038/s41586-020-2964-7 +91. A. Sommerfeld, Annalen der Physik 356(17), 1 (1916) +92. S. Weinberg, The quantum theory of fields, vol. 2 (Cambridge university press, 1995) +93. L.I. Schiff, H. Snyder, J. Weinberg, Phys. Rev. 57, 315 (1940). DOI 10.1103/PhysRev. +57.315. URL https://link.aps.org/doi/10.1103/PhysRev.57.315 +94. I.A. Aleksandrov, G. Plunien, V.M. Shabaev, The European Physical Journal D 70(1), +18 (2016). DOI 10.1140/epjd/e2015-60644-y. URL https://doi.org/10.1140/ +epjd/e2015-60644-y +95. S. Alliluev, Sov. Phys. JETP 34, 8 (1972) +96. U.W. Schmincke, Mathematische Zeitschrift 126(1), 71 (1972). +DOI 10.1007/ +BF01580357. URL https://doi.org/10.1007/BF01580357 +97. U.W. +Schmincke, +Mathematische +Zeitschrift +129(4), +335 +(1972). +URL +https://doi.org/10.1007/BF01181622https://link.springer.com/article/ +10.1007/BF01181622 +98. H. Hogreve, Journal of Physics A: Mathematical and Theoretical 46(2), 025301 +(2012). DOI 10.1088/1751-8113/46/2/025301. URL https://doi.org/10.1088/ +1751-8113/46/2/025301 +99. M. Gallone, in Advances in Quantum Mechanics: Contemporary Trends and Open +Problems, ed. by A. Michelangeli, G. Dell’Antonio (Springer International Publishing, +Cham, 2017), pp. 169–185. DOI 10.1007/978-3-319-58904-6_10. URL https:// + +67 +doi.org/10.1007/978-3-319-58904-6_10 +100. K.M. Case, Phys. Rev. 80, 797 (1950). DOI 10.1103/PhysRev.80.797. URL https: +//link.aps.org/doi/10.1103/PhysRev.80.797 +101. M.J. Esteban, M. Loss, Journal of Mathematical Physics 48(11), 112107 (2007). DOI +10.1063/1.2811950. URL https://doi.org/10.1063/1.2811950 +102. S.S. Schweber, QED and the men who made it: Dyson, Feynman, Schwinger, and +Tomonaga (Princeton University Press, 2020) +103. G.S. Adkins, S. Morrison, J. Sapirstein, Phys. Rev. A 76, 042508 (2007). DOI 10.1103/ +PhysRevA.76.042508. URL https://link.aps.org/doi/10.1103/PhysRevA.76. +042508 +104. G. Breit, G.E. Brown, Phys. Rev. 74, 1278 (1948). DOI 10.1103/PhysRev.74.1278. +URL https://link.aps.org/doi/10.1103/PhysRev.74.1278 +105. V.M. Shabaev, in The Hydrogen Atom (Springer, 2001), pp. 714–726 +106. P. +Indelicato, +J. +Desclaux. +Mcdfgme: +A +multiconfig- +uration +dirac-fock +and +general +matrix +elements +program +(2015). +http://www.lkb.upmc.fr/metrologysimplesystems/ +mdfgme-a-general-purpose-multiconfiguration-dirac-foc-program +107. V.A. Yerokhin, P. Indelicato, V.M. Shabaev, Phys. Rev. A 71(4), 040101R (2005). URL +http://link.aps.org/abstract/PRA/v71/e040101 +108. V. Yerokhin, P. Indelicato, V. Shabaev, Journal of Experimental and Theoretical +Physics 101(2), 280 (2005). URL https://doi.org/10.1134/1.2047793 +109. V.A. Yerokhin, P. Indelicato, V.M. Shabaev, Can. J. Phys. 85(5), 521 (2007). URL +http://dx.doi.org/10.1140/epjd/e2006-00064-8 +110. V.A. Yerokhin, P. Indelicato, V.M. Shabaev, Phys. Rev. A 77, 062510 (2008). +DOI 10.1103/PhysRevA.77.062510. URL https://link.aps.org/doi/10.1103/ +PhysRevA.77.062510 +111. V.A. Yerokhin, Phys. Rev. A 80(4), 040501 (R) (2009). URL http://link.aps.org/ +abstract/PRA/v80/e040501 +112. V.A. Yerokhin, Eur. Phys. J. D 58(1), 57 (2010). URL http://dx.doi.org/10.1140/ +epjd/e2010-00089-4 +113. V.A. Yerokhin, Phys. Rev. A 97(5), 052509 (2018). URL https://link.aps.org/ +doi/10.1103/PhysRevA.97.052509 +114. I. Angeli, K. Marinova, At. Data Nucl. Data Tables 99(1), 69 (2013). DOI 10.1016/j. +adt.2011.12.006. URL https://www.sciencedirect.com/science/article/pii/ +S0092640X12000265 +115. V.A. Yerokhin, V.M. Shabaev, Journal of Physical and Chemical Reference Data 44(3), +033103 (2015). URL http://scitation.aip.org/content/aip/journal/jpcrd/ +44/3/10.1063/1.4927487 +116. P. Indelicato, J. Phys. B: At. Mol. Opt. Phys. 52, 232001 (2019). URL https://doi. +org/10.1088/1361-6455/ab42c9 +117. J.L. Friar, J.W. Negele, Adv. Nucl. Phys. 8, 219 (1975). +DOI 10.1007/ +978-1-4757-4398-2\_3. URL https://doi.org/10.1007/978-1-4757-4398-2_3 +118. J. Friedrich, P.G. Reinhard, Phys. Rev. C 33, 335 (1986). DOI 10.1103/PhysRevC.33. +335. URL https://link.aps.org/doi/10.1103/PhysRevC.33.335 +119. P.G. Reinhard, W. Nazarewicz, Phys. Rev. C 103, 054310 (2021). +DOI 10.1103/ +PhysRevC.103.054310. +URL https://link.aps.org/doi/10.1103/PhysRevC. +103.054310 +120. R.G. Sachs, Phys. Rev. 126(6), 2256 (1962). URL http://link.aps.org/abstract/ +PR/v126/p2256 + +68 +121. W. Bertozzi, J. Friar, J. Heisenberg, J. Negele, Phys. Lett. B 41(4), 408 (1972). DOI +10.1016/0370-2693(72)90662-4. URL http://www.sciencedirect.com/science/ +article/pii/0370269372906624 +122. M. Safronova, D. Budker, D. DeMille, D.F.J. Kimball, A. Derevianko, C.W. Clark, +Rev. Mod. Phys. 90(2), 025008 (2018). URL https://link.aps.org/doi/10.1103/ +RevModPhys.90.025008 +123. J. Hur, D.P.L. Aude Craik, I. Counts, E. Knyazev, L. Caldwell, C. Leung, +S. Pandey, J.C. Berengut, A. Geddes, W. Nazarewicz, P.G. Reinhard, A. Kawasaki, +H. Jeon, W. Jhe, V. Vuleti´c, Phys. Rev. Lett. 128, 163201 (2022). +DOI +10.1103/PhysRevLett.128.163201. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.128.163201 +124. P.G. Reinhard, X. Roca-Maza, W. Nazarewicz, Phys. Rev. Lett. 127, 232501 (2021). +DOI 10.1103/PhysRevLett.127.232501. +URL https://link.aps.org/doi/10. +1103/PhysRevLett.127.232501 +125. B. Schuetrumpf, W. Nazarewicz, P.G. Reinhard, Phys. Rev. C 96, 024306 (2017). +DOI 10.1103/PhysRevC.96.024306. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.96.024306 +126. A.V. Afanasjev, S. Frauendorf, Phys. Rev. C 71, 024308 (2005). +DOI 10.1103/ +PhysRevC.71.024308. URL https://link.aps.org/doi/10.1103/PhysRevC.71. +024308 +127. S.E. Agbemava, A.V. Afanasjev, Phys. Rev. C 103, 034323 (2021). DOI 10.1103/ +PhysRevC.103.034323. +URL https://link.aps.org/doi/10.1103/PhysRevC. +103.034323 +128. D. Andrae, Phys. Rep. 336(6), 413 (2000). +DOI 10.1016/S0370-1573(00) +00007-7. +URL +http://www.sciencedirect.com/science/article/pii/ +S0370157300000077 +129. R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956). DOI 10.1103/RevModPhys.28.214. +URL https://link.aps.org/doi/10.1103/RevModPhys.28.214 +130. G.R. Burleson, R. Hofstadter, Phys. Rev. 112, 1282 (1958). DOI 10.1103/PhysRev. +112.1282. URL https://link.aps.org/doi/10.1103/PhysRev.112.1282 +131. L. Wilets, D.L. Hill, K.W. Ford, Phys. Rev. 91, 1488 (1953). DOI 10.1103/PhysRev. +91.1488. URL https://link.aps.org/doi/10.1103/PhysRev.91.1488 +132. D.L. Hill, K.W. Ford, Phys. Rev. 94, 1617 (1954). DOI 10.1103/PhysRev.94.1617. +URL https://link.aps.org/doi/10.1103/PhysRev.94.1617 +133. G. Breit, Rev. Mod. Phys. 30, 507 (1958). DOI 10.1103/RevModPhys.30.507. URL +https://link.aps.org/doi/10.1103/RevModPhys.30.507 +134. L. Visscher, K. Dyall, At. Data Nucl. Data Tabl. 67(2), 207 (1997). DOI https://doi. +org/10.1006/adnd.1997.0751. +URL https://www.sciencedirect.com/science/ +article/pii/S0092640X97907518 +135. W. Pieper, W. Greiner, Z. Phys. A 218, 327 (1969). DOI 10.1007/BF01670014. URL +https://link.springer.com/article/10.1007/BF01670014 +136. V.V. Flambaum, A.J. Geddes, A.V. Viatkina, Phys. Rev. A 97, 032510 (2018). +DOI 10.1103/PhysRevA.97.032510. URL https://link.aps.org/doi/10.1103/ +PhysRevA.97.032510 +137. B.G.C. Lackenby, V.A. Dzuba, V.V. Flambaum, Phys. Rev. A 99, 042509 (2019). +DOI 10.1103/PhysRevA.99.042509. URL https://link.aps.org/doi/10.1103/ +PhysRevA.99.042509 +138. L.c.v.F. Pašteka, Y. Hao, A. Borschevsky, V.V. Flambaum, P. Schwerdtfeger, Phys. +Rev. Lett. 122, 160801 (2019). DOI 10.1103/PhysRevLett.122.160801. URL https: + +69 +//link.aps.org/doi/10.1103/PhysRevLett.122.160801 +139. V. Shabaev, J. Phys. B 26(6), 1103 (1993). URL https://iopscience.iop.org/ +article/10.1088/0953-4075/26/6/011 +140. A. Mårtensson-Pendrill, M. Gustavsson, in Handbook of Molecular Physics and +Quantum Chemistry, vol. 1, ed. by S. Wilson (Wiley, Chichester, 2003), chap. 30, +pp. 477–484 +141. J. Weidmann, Mathematische Zeitschrift 180(2), 423 (1982). +DOI 10.1007/ +BF01214182. URL https://doi.org/10.1007/BF01214182 +142. P. Mohr, G. Soff, Phys. Rev. Lett. 70(2), 158 (1993). URL https://doi.org/10. +1103/PhysRevLett.70.158 +143. P. Mohr, G. Plunien, G. Soff, Physics Reports 293(5 & 6), 227 (1998). URL https: +//doi.org/10.1016/S0370-1573(97)00046-X +144. W. +Johnson, +G. +Soff, +Atomic +Data +and +Nuclear +Data +Tables +33(3), +405 +(1985). DOI https://doi.org/10.1016/0092-640X(85)90010-5. URL https://www. +sciencedirect.com/science/article/pii/0092640X85900105 +145. A. Ynnerman, J. James, I. Lindgren, H. Persson, S. Salomonson, Phys. Rev. A 50, +4671 (1994). DOI 10.1103/PhysRevA.50.4671. URL https://link.aps.org/doi/ +10.1103/PhysRevA.50.4671 +146. A.L. Schawlow, C.H. Townes, Phys. Rev. 100, 1273 (1955). DOI 10.1103/PhysRev. +100.1273. URL https://link.aps.org/doi/10.1103/PhysRev.100.1273 +147. J.C. Slater, Phys. Rev. 36, 57 (1930). DOI 10.1103/PhysRev.36.57. URL https: +//link.aps.org/doi/10.1103/PhysRev.36.57 +148. K. Dyall, I. Grant, C. Johnson, F. Parpia, E. Plummer, Comput. Phys. Commun. 55(3), +425 (1989). DOI http://dx.doi.org/10.1016/0010-4655(89)90136-7. URL http:// +www.sciencedirect.com/science/article/pii/0010465589901367 +149. S. Graf, B. Müller, E. Stein, J. Reinhardt, G. Soff, W. Greiner, in Vacuum Structure +in Intense Fields, ed. by H.M. Fried, B. Müller (Springer US, Boston, MA, 1991), pp. +119–151 +150. W. +Greiner, +in +Advances +in +Quantum +Chemistry, +vol. +30, +ed. +by +P.O. +Löwdin (Academic Press, 1998), pp. 195–208. +DOI https://doi.org/10.1016/ +S0065-3276(08)60508-0. +URL +https://www.sciencedirect.com/science/ +article/pii/S0065327608605080 +151. V.M. Kuleshov, V.D. Mur, N.B. Narozhny, A.M. Fedotov, Y.E. Lozovik, V.S. Popov, +Physics-Uspekhi 58(8), 785 (2015). DOI 10.3367/ufne.0185.201508d.0845. URL +https://doi.org/10.3367/ufne.0185.201508d.0845 +152. H.A. Bethe, Phys. Rev. 57, 1125 (1940). DOI 10.1103/PhysRev.57.1125. URL https: +//link.aps.org/doi/10.1103/PhysRev.57.1125 +153. R.D. Present, Phys. Rev. 60, 28 (1941). DOI 10.1103/PhysRev.60.28. URL https: +//link.aps.org/doi/10.1103/PhysRev.60.28 +154. J. Desclaux, Computer Physics Communications 9(1), 31 (1975) +155. P.J. Mohr, D.B. Newell, B.N. Taylor, Journal of Physical and Chemical Reference Data +45(4), 043102 (2016) +156. G. Soff, B. Müller, J. Rafelski, Zeitschrift für Naturforschung A 29(9), 1267 (1974). +URL https://doi.org/10.1515/zna-1974-0905 +157. H. Feshbach, Ann. Phys. (N.Y.) 5(4), 357 (1958). +DOI 10.1016/0003-4916(58) +90007-1. +URL +https://www.sciencedirect.com/science/article/pii/ +0003491658900071 +158. H. Feshbach, Ann. Phys. (N.Y.) 19(2), 287 (1962). +DOI 10.1016/0003-4916(62) +90221-X. +URL +https://www.sciencedirect.com/science/article/pii/ + +70 +000349166290221X +159. K.G. Dyall, K. Fægri Jr, Introduction to relativistic quantum chemistry (Oxford Uni- +versity Press, 2007) +160. H.A. Bethe, E.E. Salpeter, Quantum mechanics of one-and two-electron atoms +(Springer Science & Business Media, Berlin, 2012) +161. W.D. Evans, P. Perry, H. Siedentop, Commun. Math. Phys. 178(3), 733 (1996). DOI +10.1007/BF02108822. URL https://doi.org/10.1007/BF02108822 +162. G. Hardekopf, J. Sucher, Phys. Rev. A 30, 703 (1984). DOI 10.1103/PhysRevA.30. +703. URL https://link.aps.org/doi/10.1103/PhysRevA.30.703 +163. C. Tix, arXiv preprint hep-ph/9702259 (1997) +164. C. +Tix, +Bulletin +of +the +London +Mathematical +Society +30(3), +283 +(1998). +DOI https://doi.org/10.1112/S0024609397004256. +URL https://londmathsoc. +onlinelibrary.wiley.com/doi/abs/10.1112/S0024609397004256 +165. A. Gumberidze, T. Stöhlker, D. Bana´s, K. Beckert, P. Beller, H.F. Beyer, +F. Bosch, S. Hagmann, C. Kozhuharov, D. Liesen, F. Nolden, X. Ma, P.H. Mok- +ler, M. Steck, D. Sierpowski, S. Tashenov, Phys. Rev. Lett. 94, 223001 (2005). +DOI 10.1103/PhysRevLett.94.223001. URL https://link.aps.org/doi/10.1103/ +PhysRevLett.94.223001 +166. A. Bulgac, arXiv preprint nucl-th/9907088 (1999) +167. J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 422(1), 103 (1984). +DOI +10.1016/0375-9474(84)90433-0. URL http://www.sciencedirect.com/science/ +article/pii/0375947484904330 +168. S. Belyaev, A. Smirnov, S. Tolokonnikov, S. Fayans, Sov. J. Nucl. Phys. 45, 1263 +(1987) +169. J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.F. Berger, C.R. Chinn, J. Dechargé, +Phys. Rev. C 53, 2809 (1996). DOI 10.1103/PhysRevC.53.2809. URL https:// +link.aps.org/doi/10.1103/PhysRevC.53.2809 +170. S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001). DOI 10.1103/RevModPhys.73.463. +URL https://link.aps.org/doi/10.1103/RevModPhys.73.463 +171. J. Dobaczewski, W. Nazarewicz, in 50 Years of Nuclear BCS, ed. by R.A. Broglia, +V. Zelevinsky (World Scientific, 2013), p. 40. DOI 10.1142/8526. URL https:// +doi.org/10.1142/8526. Arxiv:1909.13041 +172. G. Bertsch, J. Dobaczewski, W. Nazarewicz, J. Pei, Phys. Rev. A 79, 043602 (2009). +DOI 10.1103/PhysRevA.79.043602. URL https://link.aps.org/doi/10.1103/ +PhysRevA.79.043602 +173. J.G. Valatin, Phys. Rev. 122, 1012 (1961). DOI 10.1103/PhysRev.122.1012. URL +https://link.aps.org/doi/10.1103/PhysRev.122.1012 +174. B. Gall, P. Bonche, J. Dobaczewski, H. Flocard, P.H. Heenen, Z. Phys. A 348(3), 183 +(1994). DOI 10.1007/BF01291916. URL https://doi.org/10.1007/BF01291916 +175. A. Bulgac, Y. Yu, Phys. Rev. Lett. 88, 042504 (2002). DOI 10.1103/PhysRevLett.88. +042504. URL https://link.aps.org/doi/10.1103/PhysRevLett.88.042504 +176. P.J. Borycki, J. Dobaczewski, W. Nazarewicz, M.V. Stoitsov, Phys. Rev. C 73, 044319 +(2006). DOI 10.1103/PhysRevC.73.044319. URL https://link.aps.org/doi/10. +1103/PhysRevC.73.044319 +177. J.C. Pei, A.T. Kruppa, W. Nazarewicz, Phys. Rev. C 84, 024311 (2011). DOI 10.1103/ +PhysRevC.84.024311. URL https://link.aps.org/doi/10.1103/PhysRevC.84. +024311 +178. L. Li, J. Meng, P. Ring, E.G. Zhao, S.G. Zhou, Phys. Rev. C 85, 024312 (2012). +DOI 10.1103/PhysRevC.85.024312. +URL https://link.aps.org/doi/10.1103/ + +71 +PhysRevC.85.024312 +179. J.C. Pei, N. Fei, Y.N. Zhang, P. Schuck, Phys. Rev. C 92, 064316 (2015). DOI 10.1103/ +PhysRevC.92.064316. URL https://link.aps.org/doi/10.1103/PhysRevC.92. +064316 +180. J. Terasaki, J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, M. Stoitsov, Phys. +Rev. C 71, 034310 (2005). DOI 10.1103/PhysRevC.71.034310. URL https://link. +aps.org/doi/10.1103/PhysRevC.71.034310 +181. K. Mizuyama, M. Matsuo, Y. Serizawa, Phys. Rev. C 79, 024313 (2009). DOI 10.1103/ +PhysRevC.79.024313. URL https://link.aps.org/doi/10.1103/PhysRevC.79. +024313 +182. N. Michel, K. Matsuyanagi, M. Stoitsov, Phys. Rev. C 78, 044319 (2008). +DOI 10.1103/PhysRevC.78.044319. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.78.044319 +183. M. Grasso, N. Sandulescu, N. Van Giai, R.J. Liotta, Phys. Rev. C 64, 064321 (2001). +DOI 10.1103/PhysRevC.64.064321. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.64.064321 +184. Y.N. Zhang, J.C. Pei, F.R. Xu, Phys. Rev. C 88, 054305 (2013). +DOI 10.1103/ +PhysRevC.88.054305. URL https://link.aps.org/doi/10.1103/PhysRevC.88. +054305 +185. M. Chen, T. Li, B. Schuetrumpf, P.G. Reinhard, W. Nazarewicz, Comput. Phys. Com- +mun. 276, 108344 (2022). +DOI 10.1016/j.cpc.2022.108344. +URL https://www. +sciencedirect.com/science/article/pii/S0010465522000625 +186. V.M. Shabaev, I.I. Tupitsyn, V.A. Yerokhin, G. Plunien, G. Soff, Phys. Rev. Lett. +93(13), 130405 (2004). URL http://link.aps.org/abstract/PRL/v93/e130405 +187. I.P. Grant, Journal of Physics B: Atomic, Molecular and Optical Physics 42(5), 055002 +(2009). DOI 10.1088/0953-4075/42/5/055002. URL https://doi.org/10.1088/ +0953-4075/42/5/055002 +188. M.S. Plesset, Phys. Rev. 41, 278 (1932). DOI 10.1103/PhysRev.41.278. URL https: +//link.aps.org/doi/10.1103/PhysRev.41.278 +189. H. Oba, M. Matsuo, Phys. Rev. C 80, 024301 (2009). DOI 10.1103/PhysRevC.80. +024301. URL https://link.aps.org/doi/10.1103/PhysRevC.80.024301 +190. Y. Zhang, M. Matsuo, J. Meng, Phys. Rev. C 83, 054301 (2011). +DOI 10.1103/ +PhysRevC.83.054301. URL https://link.aps.org/doi/10.1103/PhysRevC.83. +054301 +191. L. Zhang, S.G. Zhou, J. Meng, E.G. Zhao, Phys. Rev. C 77, 014312 (2008). +DOI 10.1103/PhysRevC.77.014312. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.77.014312 +192. M.R. Zirnbauer, Journal of Mathematical Physics 62(2), 021101 (2021). DOI 10.1063/ +5.0035358. URL https://doi.org/10.1063/5.0035358 +193. Y. Tanimura, K. Hagino, P. Ring, Phys. Rev. C 88, 017301 (2013). DOI 10.1103/ +PhysRevC.88.017301. URL https://link.aps.org/doi/10.1103/PhysRevC.88. +017301 +194. V. Popov, V. Mur, Yadernaya Fizika 18(3), 684 (1973) +195. M.L. Goldberger, K.M. Watson, Collision Theory (Wiley, New York, 1964) +196. S.A. Gurvitz, G. Kalbermann, Phys. Rev. Lett. 59, 262 (1987). +DOI 10.1103/ +PhysRevLett.59.262. +URL https://link.aps.org/doi/10.1103/PhysRevLett. +59.262 +197. S.A. Gurvitz, P.B. Semmes, W. Nazarewicz, T. Vertse, Phys. Rev. A 69, 042705 (2004). +DOI 10.1103/PhysRevA.69.042705. URL https://link.aps.org/doi/10.1103/ + +72 +PhysRevA.69.042705 +198. U. Fano, Phys. Rev. 124(6), 1866 (1961) +199. E. Stückelberg, Helvetica physica acta 15, 23 (1942) +200. R. Feynman, Phys. Rev. 76, 749 (1949). DOI 10.1103/PhysRev.76.749. URL https: +//link.aps.org/doi/10.1103/PhysRev.76.749 +201. J. Rafelski, L.P. Fulcher, A. Klein, Physics Reports 38(5), 227 (1978). URL https: +//doi.org/10.1016/0370-1573(78)90116-3 +202. N. Szpak, Journal of Physics A: Mathematical and Theoretical 41(16), 164059 +(2008). DOI 10.1088/1751-8113/41/16/164059. URL https://doi.org/10.1088/ +1751-8113/41/16/164059 +203. K.S. Krylov, V.D. Mur, A.M. Fedotov, Eur. Phys. J. C 80(3), 270 (2020). +DOI 10.1140/epjc/s10052-020-7833-x. +URL https://doi.org/10.1140/epjc/ +s10052-020-7833-x +204. S.I. Godunov, B. Machet, M.I. Vysotsky, Eur. Phys. J. C 77(11), 782 (2017). +DOI 10.1140/epjc/s10052-017-5325-4. +URL https://doi.org/10.1140/epjc/ +s10052-017-5325-4 +205. P. Šeba, Lett. Math. Phys. 16(1), 51 (1988). DOI 10.1007/BF00398170. URL https: +//doi.org/10.1007/BF00398170 +206. U. Riss, H.D. Meyer, J. Phys. B 26(23), 4503 (1993). DOI 10.1088/0953-4075/26/23/ +021. URL https://iopscience.iop.org/article/10.1088/0953-4075/26/23/ +021 +207. E. Ackad, M. Horbatsch, Phys. Rev. A 76, 022503 (2007). DOI 10.1103/PhysRevA. +76.022503. URL https://link.aps.org/doi/10.1103/PhysRevA.76.022503 +208. E. Ackad, M. Horbatsch, Phys. Rev. A 75, 022508 (2007). DOI 10.1103/PhysRevA. +75.022508. URL https://link.aps.org/doi/10.1103/PhysRevA.75.022508 +209. R.V. Popov, V.M. Shabaev, D.A. Telnov, I.I. Tupitsyn, I.A. Maltsev, Y.S. Kozhedub, +A.I. Bondarev, N.V. Kozin, X. Ma, G. Plunien, T. Stöhlker, D.A. Tumakov, V.A. Za- +ytsev, Phys. Rev. D 102, 076005 (2020). DOI 10.1103/PhysRevD.102.076005. URL +https://link.aps.org/doi/10.1103/PhysRevD.102.076005 +210. J. Slater, New York-Toronto-London (1960) +211. J. Humblet, L. Rosenfeld, Nuclear Physics 26(4), 529 (1961). DOI https://doi.org/10. +1016/0029-5582(61)90207-3. +URL https://www.sciencedirect.com/science/ +article/pii/0029558261902073 +212. T. Berggren, Nucl. Phys. A 109, 265 (1968). DOI 10.1016/0375-9474(68)90593-9. +URL https://dx.doi.org/10.1016/0375-9474(68)90593-9 +213. T. Berggren, Nucl. Phys. A 389, 261 (1982). DOI 10.1016/0375-9474(82)90519-X. +URL https://dx.doi.org/10.1016/0375-9474(82)90519-X +214. T. Berggren, P. Lind, Phys. Rev. C 47, 768 (1993). DOI 10.1103/PhysRevC.47.768. +URL https://dx.doi.org/10.1103/PhysRevC.47.768 +215. P. Lind, Phys. Rev. C 47, 1903 (1993). DOI 10.1103/PhysRevC.47.1903. URL https: +//link.aps.org/doi/10.1103/PhysRevC.47.1903 +216. C.G. Bollini, O. Civitarese, A.L. De Paoli, M.C. Rocca, Journal of Mathematical +Physics 37(9), 4235 (1996). DOI 10.1063/1.531633. URL https://doi.org/10. +1063/1.531633 +217. O.I. Tolstikhin, V.N. Ostrovsky, H. Nakamura, Phys. Rev. Lett. 79, 2026 (1997). +DOI 10.1103/PhysRevLett.79.2026. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.79.2026 +218. K. Kat¯o, T. Myo, S. Aoyama, K. Ikeda, in Resonances in Few-Body Systems (Springer, +2001), pp. 96–104 + +73 +219. J. Hinze, Electron-atom and electron-molecule collisions (Springer Science & Busi- +ness Media, 2013) +220. N. Michel, W. Nazarewicz, M. Płoszajczak, T. Vertse, J. Phys. G 36(1), 013101 +(2008). DOI 10.1088/0954-3899/36/1/013101. URL https://doi.org/10.1088/ +0954-3899/36/1/013101 +221. N. Michel, M. Płoszajczak, Gamow Shell Model (Springer, 2021). URL https:// +www.springer.com/gp/book/9783030693558 +222. N. Michel, W. Nazarewicz, M. Płoszajczak, J. Rotureau, Phys. Rev. C 74, 054305 +(2006). DOI 10.1103/PhysRevC.74.054305. URL https://link.aps.org/doi/10. +1103/PhysRevC.74.054305 +223. X. Mao, K. Fossez, W. Nazarewicz, Phys. Rev. A 98, 062515 (2018). DOI 10.1103/ +PhysRevA.98.062515. URL https://link.aps.org/doi/10.1103/PhysRevA.98. +062515 +224. Y.B. Zel’dovich, Zhur. Eksptl’. i Teoret Fiz. 39 (1960) +225. V.D. Mur, S.G. Pozdnyakov, S.V. Popruzhenko, V.S. Popov, Phys. Atom. Nucl. 66(11), +1964 (2003). DOI 10.1134/1.1625740. URL https://doi.org/10.1134/1.1625740 +226. W.J. Romo, Nucl. Phys. A 116(3), 617 (1968). +DOI 10.1016/0375-9474(68) +90395-3. +URL +https://www.sciencedirect.com/science/article/pii/ +0375947468903953 +227. B. +Gyarmati, +T. +Vertse, +Nucl. +Phys. +A +160(3), +523 +(1971). +DOI +10. +1016/0375-9474(71)90095-9. +URL https://www.sciencedirect.com/science/ +article/pii/0375947471900959 +228. A.I. Baz’, Y.B. Zel’dovich, A.M. Perelomov, Scattering, reactions and decay in non- +relativistic quantum mechanics (Israel Program for Scientific Translation, Jerusalem, +1969) +229. Z. Bacic, J. Simons, J. Phys. Chem. 86(7), 1192 (1982). DOI 10.1021/j100396a027. +URL https://doi.org/10.1021/j100396a027 +230. V.A. Mandelshtam, H.S. Taylor, V. Ryaboy, N. Moiseyev, Phys. Rev. A 50, 2764 +(1994). DOI 10.1103/PhysRevA.50.2764. URL https://link.aps.org/doi/10. +1103/PhysRevA.50.2764 +231. J.K. Koga, M. Murakami, A.V. Arefiev, Y. Nakamiya, S.S. Bulanov, S.V. Bulanov, +Physics Letters A 384(34), 126854 (2020). DOI https://doi.org/10.1016/j.physleta. +2020.126854. +URL https://www.sciencedirect.com/science/article/pii/ +S0375960120307210 +232. O. Klein, Zeitschrift für Physik 53(3), 157 (1929). DOI 10.1007/BF01339716. URL +https://doi.org/10.1007/BF01339716 +233. F. Sauter, Zeitschrift für Physik 69(11), 742 (1931). DOI 10.1007/BF01339461. URL +https://doi.org/10.1007/BF01339461 +234. J. Schwinger, Phys. Rev. 82, 664 (1951). DOI 10.1103/PhysRev.82.664. URL https: +//link.aps.org/doi/10.1103/PhysRev.82.664 +235. A. Hansen, F. Ravndal, Phys. Scr. 23(6), 1036 (1981). DOI 10.1088/0031-8949/23/6/ +002. URL https://doi.org/10.1088/0031-8949/23/6/002 +236. S.M. Wang, W. Nazarewicz, Phys. Rev. Lett. 126, 142501 (2021). +DOI +10.1103/PhysRevLett.126.142501. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.126.142501 +237. H. Rumpf, General Relativity and Gravitation 10(6), 509 (1979). +DOI 10.1007/ +BF00759287. URL https://doi.org/10.1007/BF00759287 +238. H. Rumpf, General Relativity and Gravitation 10(6), 525 (1979). +DOI 10.1007/ +BF00759288. URL https://doi.org/10.1007/BF00759288 + +74 +239. H. Rumpf, General Relativity and Gravitation 10(8), 647 (1979). +DOI 10.1007/ +BF00756901. URL https://doi.org/10.1007/BF00756901 +240. M. Soffel, B. Müller, W. Greiner, Physics Reports 85(2), 51 (1982). +DOI https:// +doi.org/10.1016/0370-1573(82)90129-6. URL https://www.sciencedirect.com/ +science/article/pii/0370157382901296 +241. S.S. Gershtein, Y.B. Zel’dovich, Zh. Eksp. Teor. Fiz. 654-659(57) (1969). +URL +https://www.osti.gov/biblio/4760767 +242. V. Popov, Sov. Phys. JETP 32, 526 (1971) +243. T. Tomoda, Phys. Rev. A 26, 174 (1982). +DOI 10.1103/PhysRevA.26.174. +URL +https://link.aps.org/doi/10.1103/PhysRevA.26.174 +244. U. Müller, T. de Reus, J. Reinhardt, B. Müller, W. Greiner, G. Soff, Phys. Rev. A 37, +1449 (1988). DOI 10.1103/PhysRevA.37.1449. URL https://link.aps.org/doi/ +10.1103/PhysRevA.37.1449 +245. E. Ackad, M. Horbatsch, Phys. Rev. A 78, 062711 (2008). DOI 10.1103/PhysRevA. +78.062711. URL https://link.aps.org/doi/10.1103/PhysRevA.78.062711 +246. V. Kuleshov, V. Mur, A. Fedotov, Y.E. Lozovik, Journal of Experimental and Theoret- +ical Physics 125(6), 1144 (2017) +247. P. Pickl, D. Dürr, EPL (Europhysics Letters) 81(4), 40001 (2008). +DOI 10.1209/ +0295-5075/81/40001. URL https://doi.org/10.1209/0295-5075/81/40001 +248. P. +Pickl, +D. +Dürr, +Communications +in +Mathematical +Physics +282(1), +161 +(2008). +DOI 10.1007/s00220-008-0530-5. +URL https://doi.org/10.1007/ +s00220-008-0530-5 +249. P. Grashin, K. Sveshnikov, Phys. Rev. D 106, 013003 (2022). DOI 10.1103/PhysRevD. +106.013003. URL https://link.aps.org/doi/10.1103/PhysRevD.106.013003 +250. A. Krasnov, K. Sveshnikov, Mod. Phys. Lett. A 37, 2250136 (2022). URL https: +//www.worldscientific.com/doi/abs/10.1142/S021773232250136X +251. A. Gonoskov, T. Blackburn, M. Marklund, S. Bulanov, Rev. Mod. Phys. 94(4), +045001 (2022). +URL https://link.aps.org/doi/10.1103/RevModPhys.94. +045001https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.94. +045001 +252. R. Alkofer, M.B. Hecht, C.D. Roberts, S.M. Schmidt, D.V. Vinnik, Phys. Rev. Lett. +87, 193902 (2001). DOI 10.1103/PhysRevLett.87.193902. URL https://link.aps. +org/doi/10.1103/PhysRevLett.87.193902 +253. L. Klar, N. Szpak, R. Schützhold. Quantum simulation of spontaneous pair creation in +2d optical lattices (2019). DOI 10.48550/ARXIV.1901.09880. URL https://arxiv. +org/abs/1901.09880 +254. A.K. Geim, K.S. Novoselov, Nature Materials 6(3), 183 (2007). +DOI 10.1038/ +nmat1849. URL https://doi.org/10.1038/nmat1849 +255. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. +Phys. 81, 109 (2009). DOI 10.1103/RevModPhys.81.109. URL https://link.aps. +org/doi/10.1103/RevModPhys.81.109 +256. F. Fillion-Gourdeau, S. MacLean, Phys. Rev. B 92, 035401 (2015). DOI 10.1103/ +PhysRevB.92.035401. URL https://link.aps.org/doi/10.1103/PhysRevB.92. +035401 +257. A.I. Berdyugin, N. Xin, H. Gao, S. Slizovskiy, Z. Dong, S. Bhattacharjee, P. Kumar- +avadivel, S. Xu, L.A. Ponomarenko, M. Holwill, D.A. Bandurin, M. Kim, Y. Cao, M.T. +Greenaway, K.S. Novoselov, I.V. Grigorieva, K. Watanabe, T. Taniguchi, V.I. Fal’ko, +L.S. Levitov, R.K. Kumar, A.K. Geim, Science 375(6579), 430 (2022). DOI 10.1126/ +science.abi8627. +URL https://www.science.org/doi/abs/10.1126/science. + +75 +abi8627 +258. H. Tsertos, E. Berdermann, F. Bosch, M. Clemente, S. Huchler, P. Kienle, W. Koenig, +C. Kozhuharov, Zeit. für Phys. A 342(1), 79 (1992). URL https://doi.org/10. +1007/BF01294492https://link.springer.com/article/10.1007/BF01294492 +259. M. Durante, P. Indelicato, B. Jonson, V. Koch, K. Langanke, U.G. Meißner, E. Nappi, +T. Nilsson, T. Stöhlker, E. Widmann, M. Wiescher, Phys. Scr. 94(3), 033001 (2019). +URL http://dx.doi.org/10.1088/1402-4896/aaf93f +260. A. Gumberidze, C. Kozhuharov, R.T. Zhang, S. Trotsenko, Y.S. Kozhedub, +R.D. Du, H. Bois, F. Beyer, K.H. Blumenhagen, C. Brandau, A. Bräuning- +Demian, W. Chen, O. Forstner, B. Gao, T. Gassner, R.E. Grisenti, S. Hag- +mann, P.M. Hillenbrand, P. Indelicato, A. Kumar, M. Lestinsky, Y.A. Litvi- +nov, +N. +Petridis, +D. +Schury, +U. +Spillmann, +C. +Trageser, +M. +Trassinelli, +X. Tu, T. Stöhlker, Journal of Physics: Conference Series 1412(14), 142015 +(2020). +URL http://dx.doi.org/10.1088/1742-6596/1412/14/142015https: +//iopscience.iop.org/article/10.1088/1742-6596/1412/14/142015/pdf +261. P.H. Heenen, J. Skalski, A. Staszczak, D. Vretenar, Nucl. Phys. A 944, 415 (2015). +DOI 10.1016/j.nuclphysa.2015.07.016. +URL http://www.sciencedirect.com/ +science/article/pii/S0375947415001748 +262. Y.S. Kozhedub, I.I. Tupitsyn, V.M. Shabaev, G. Plunien, T. Stöhlker, Journal of +Physics: Conference Series 599, 012037 (2015). +DOI 10.1088/1742-6596/599/1/ +012037. URL https://doi.org/10.1088/1742-6596/599/1/012037 +263. N. Szpak, R. Schützhold, New Journal of Physics 14(3), 035001 (2012) +264. R. Lee, A. Milstein, Physics Letters B 761, 340 (2016) +265. I. Maltsev, V. Shabaev, I. Tupitsyn, Y. Kozhedub, G. Plunien, T. Stöhlker, Nuclear +Instruments and Methods in Physics Research Section B: Beam Interactions with Ma- +terials and Atoms 408, 97 (2017) +266. I.A. Maltsev, V.M. Shabaev, I.I. Tupitsyn, A.I. Bondarev, Y.S. Kozhedub, G. Plunien, +T. Stöhlker, Phys. Rev. A 91, 032708 (2015). DOI 10.1103/PhysRevA.91.032708. +URL https://link.aps.org/doi/10.1103/PhysRevA.91.032708 +267. A.N. Artemyev, U.D. Jentschura, V.G. Serbo, A. Surzhykov, The European Physical +Journal C 72(3), 1935 (2012). DOI 10.1140/epjc/s10052-012-1935-z. URL https: +//doi.org/10.1140/epjc/s10052-012-1935-z +268. I.B. Khriplovich, International Journal of Modern Physics A 31(28n29), 1645035 +(2016). +DOI 10.1142/S0217751X16450354. +URL https://doi.org/10.1142/ +S0217751X16450354 +269. I. Khriplovich, The European Physical Journal Plus 132(2), 1 (2017) +270. R. du Rietz, E. Williams, D.J. Hinde, M. Dasgupta, M. Evers, C.J. Lin, D.H. Luong, +C. Simenel, A. Wakhle, Phys. Rev. C 88, 054618 (2013). DOI 10.1103/PhysRevC.88. +054618. URL https://link.aps.org/doi/10.1103/PhysRevC.88.054618 +271. C. Simenel, K. Godbey, A.S. Umar, Phys. Rev. Lett. 124, 212504 (2020). +DOI +10.1103/PhysRevLett.124.212504. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.124.212504 +272. I.A. Maltsev, V.M. Shabaev, R.V. Popov, Y.S. Kozhedub, G. Plunien, X. Ma, +T. Stöhlker, D.A. Tumakov, Phys. Rev. Lett. 123, 113401 (2019). +DOI +10.1103/PhysRevLett.123.113401. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.123.113401 +273. H. Nakatsuji, H. Nakashima, Phys. Rev. Lett. 95, 050407 (2005). +DOI +10.1103/PhysRevLett.95.050407. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.95.050407 + +76 +274. H. Nakatsuji, Acc. Chem. Res. 45 (2012). +URL https://doi.org/10.1021/ +ar200340j +275. A. Savin, P. Schwerdtfeger, H. Preuss, H. Silberbach, H. Stoll, Chemical Physics Let- +ters 98(3), 226 (1983). DOI https://doi.org/10.1016/0009-2614(83)87155-3. URL +https://www.sciencedirect.com/science/article/pii/0009261483871553 +276. E. Engel, International Journal of Quantum Chemistry 56(4), 217 (1995). DOI https: +//doi.org/10.1002/qua.560560405. URL https://onlinelibrary.wiley.com/doi/ +abs/10.1002/qua.560560405 +277. A. Teale, T. Helgaker, A. Savin, C. Adamo, B. Aradi, A. Arbuznikov, P. Ayers, +E.J. Baerends, V. Barone, P. Calaminici, et al., ChemRxiv (2022). URL 10.26434/ +chemrxiv-2022-13j2v +278. J. Mann, J. Waber, Atomic Data and Nuclear Data Tables 5(2), 201 (1973). DOI https: +//doi.org/10.1016/S0092-640X(73)80004-X. +URL https://www.sciencedirect. +com/science/article/pii/S0092640X7380004X +279. P. Jönsson, G. Gaigalas, J. Biero´n, C.F. Fischer, I. Grant, Computer Physics Commu- +nications 184(9), 2197 (2013). DOI https://doi.org/10.1016/j.cpc.2013.02.016. URL +https://www.sciencedirect.com/science/article/pii/S0010465513000738 +280. C. Froese Fischer, G. Gaigalas, P. Jönsson, J. Biero´n, Computer Physics Commu- +nications 237, 184 (2019). +DOI https://doi.org/10.1016/j.cpc.2018.10.032. +URL +https://www.sciencedirect.com/science/article/pii/S0010465518303928 +281. I. Grant, H. Quiney, Atoms 10(4) (2022). DOI 10.3390/atoms10040108. URL https: +//www.mdpi.com/2218-2004/10/4/108 +282. K. Hirao, Recent Advances in Multireference Methods (WORLD SCIENTIFIC, 1999). +DOI 10.1142/4030. URL https://www.worldscientific.com/doi/abs/10.1142/ +4030 +283. T. Fleig, Chemical Physics 395, 2 (2012). DOI https://doi.org/10.1016/j.chemphys. +2011.06.032. +URL https://www.sciencedirect.com/science/article/pii/ +S0301010411002710. +Recent Advances and Applications of Relativistic Quantum +Chemistry +284. R.R. Li, M.R. Hoffmann, Advances in Quantum Chemistry 81, 105 (2020) +285. I.P. Grant, Advances in Physics 19(82), 747 (1970). URL http://dx.doi.org/10. +1080/00018737000101191 +286. J. Desclaux, J. Dolbeault, M. Esteban, P. Indelicato, E. Séré, in Computational Chem- +istry, Handbook of Numerical Analysis, vol. X (Elsevier, 2003), p. 1032 +287. S.G. Karshenboim, Physics Reports 422(1), 1 (2005). DOI https://doi.org/10.1016/j. +physrep.2005.08.008. URL https://www.sciencedirect.com/science/article/ +pii/S0370157305003637 +288. F.W. Knollmann, A.N. Patel, S.C. Doret, Phys. Rev. A 100(2), 022514 (2019). URL +https://link.aps.org/doi/10.1103/PhysRevA.100.022514 +289. I. Counts, J. Hur, D.P.L.A. Craik, H. Jeon, C. Leung, J.C. Berengut, A. Geddes, +A. Kawasaki, W. Jhe, V. Vuleti´c, Phys. Rev. Lett. 125(12), 123002 (2020). +URL +https://link.aps.org/doi/10.1103/PhysRevLett.125.123002 +290. N. Figueroa, J. Berengut, V. Dzuba, V. Flambaum, D. Budker, D. Antypas, Phys. +Rev. Lett. 128(7), 073001 (2022). +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.128.073001 +291. J.C. Berengut, D. Budker, C. Delaunay, V.V. Flambaum, C. Frugiuele, E. Fuchs, +C. Grojean, R. Harnik, R. Ozeri, G. Perez, Y. Soreq, Phys. Rev. Lett. 120(9), 091801 +(2018). URL https://link.aps.org/doi/10.1103/PhysRevLett.120.091801 + +77 +292. J.C. Berengut, C. Delaunay, A. Geddes, Y. Soreq, Phys. Rev. Res. 2(4), 043444 (2020). +URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.043444 +293. R.A. Müller, V.A. Yerokhin, A.N. Artemyev, A. Surzhykov, Phys. Rev. A 104(2), +L020802 (2021). +URL https://link.aps.org/doi/10.1103/PhysRevA.104. +L020802 +294. S.O. Allehabi, V.A. Dzuba, V.V. Flambaum, A.V. Afanasjev, Phys. Rev. A 103(3), +L030801 (2021). +URL https://link.aps.org/doi/10.1103/PhysRevA.103. +L030801 +295. P. Munro-Laylim, V. Dzuba, V. Flambaum, Phys. Rev. A 105(4), 042814 (2022). URL +https://link.aps.org/doi/10.1103/PhysRevA.105.042814 +296. J. Hur, D.P.A. Craik, I. Counts, E. Knyazev, L. Caldwell, C. Leung, S. Pandey, J.C. +Berengut, A. Geddes, W. Nazarewicz, P.G. Reinhard, A. Kawasaki, H. Jeon, W. Jhe, +V. Vuleti´c, Phys. Rev. Lett. 128(16), 163201 (2022). URL https://link.aps.org/ +doi/10.1103/PhysRevLett.128.163201 +297. C. Froese Fischer, Hartree–Fock method for atoms. A numerical approach (John Wiley +and Sons, Inc., New York, 1977) +298. D.R. Hartree, The calculation of atomic structures (Wiley, New York, 1957) +299. W.R. Johnson, S.A. Blundell, J. Sapirstein, Phys. Rev. A 37, 307 (1988). DOI 10.1103/ +PhysRevA.37.307. URL https://link.aps.org/doi/10.1103/PhysRevA.37.307 +300. J. Sapirstein, W. Johnson, J. Phys. B: At. Mol. Opt. Phys. 29(22), 5213 (1996). URL +https://doi.org/10.1088/0953-4075/29/22/005 +301. C.F. +Fischer, +O. +Zatsarinny, +Comp. +Phys. +Commun. +180(6), +879 +(2009). +URL +http://www.sciencedirect.com/science/article/B6TJ5-4V47CHB-1/2/ +c1e0df07b77f34787c41131fc451a85c +302. E. Kahl, J. Berengut, Computer Physics Communications 238, 232 (2019). +DOI +https://doi.org/10.1016/j.cpc.2018.12.014. URL https://www.sciencedirect.com/ +science/article/pii/S0010465518304302 +303. W.H.E. Schwarz, P. Schwerdtfeger, J.G. Snijders, E.J. Baerends, J. Phys. B 23, 3225 +(1990). URL http://stacks.iop.org/0953-4075/23/i=19/a=010 +304. W.H.E. Schwarz, E.M. van Wezenbeek, E.J. Baerends, J.G. Snijders, J. Phys. B 22, +1515 (1989). URL http://stacks.iop.org/0953-4075/22/i=10/a=008 +305. W. Johnson, S. Blundell, J. Sapirstein, Phys. Rev. A 37(2), 307 (1988). URL https: +//doi.org/10.1103/PhysRevA.37.307 +306. W. Johnson, S. Blundell, J. Sapirstein, Phys. Rev. A 37(8), 2764 (1988). URL https: +//doi.org/10.1103/PhysRevA.37.2764 +307. C.T. Chantler, Journal of Physical and Chemical Reference Data 29(4), 597 (2000). +DOI 10.1063/1.1321055. URL https://doi.org/10.1063/1.1321055 +308. C.T. Chantler, J.A. Lowe, I.P. Grant, Phys. Rev. A 82, 052505 (2010). DOI 10.1103/ +PhysRevA.82.052505. URL https://link.aps.org/doi/10.1103/PhysRevA.82. +052505 +309. H. Tatewaki, T. Koga, Y. Mochizuki, Chemical Physics Letters 375(3), 399 +(2003). DOI https://doi.org/10.1016/S0009-2614(03)00873-X. URL https://www. +sciencedirect.com/science/article/pii/S000926140300873X +310. H. Tatewaki, Y. Watanabe, The Journal of Chemical Physics 121(10), 4528 (2004). +DOI 10.1063/1.1779213. URL https://doi.org/10.1063/1.1779213 +311. L.G.M. de Macedo, A.C. Borin, A.B. da Silva, Atomic Data and Nuclear Data Tables +93(6), 931 (2007). DOI https://doi.org/10.1016/j.adt.2007.06.007. URL https:// +www.sciencedirect.com/science/article/pii/S0092640X07000435 + +78 +312. T.Q. Teodoro, A.B.F. da Silva, R.L.A. Haiduke, Journal of Chemical Theory and Com- +putation 10(9), 3800 (2014). DOI 10.1021/ct500518n. URL https://doi.org/10. +1021/ct500518n +313. A. Almoukhalalati, S. Knecht, H.J.A. Jensen, K.G. Dyall, T. Saue, The Journal of +Chemical Physics 145(7), 074104 (2016). DOI 10.1063/1.4959452. URL https: +//doi.org/10.1063/1.4959452 +314. M.H. Mittleman, Phys. Rev. A 24, 1167 (1981). DOI 10.1103/PhysRevA.24.1167. +URL https://link.aps.org/doi/10.1103/PhysRevA.24.1167 +315. H.M. Quiney, I.P. Grant, S. Wilson, Phys. Scr. 36(3), 460 (1987). +DOI 10.1088/ +0031-8949/36/3/013. URL https://doi.org/10.1088/0031-8949/36/3/013 +316. H.M. Quiney, I.P. Grant, S. Wilson, Journal of Physics B: Atomic and Molecu- +lar Physics 20(7), 1413 (1987). +DOI 10.1088/0022-3700/20/7/010. +URL https: +//doi.org/10.1088/0022-3700/20/7/010 +317. R. Jáuregui, C.F. Bunge, E. Ley-Koo, Phys. Rev. A 55, 1781 (1997). +DOI 10. +1103/PhysRevA.55.1781. URL https://link.aps.org/doi/10.1103/PhysRevA. +55.1781 +318. E. Lindroth, Phys. Scr. 36(3), 485 (1987). DOI 10.1088/0031-8949/36/3/018. URL +https://doi.org/10.1088/0031-8949/36/3/018 +319. O. Gorceix, P. Indelicato, J.P. Desclaux, Journal of Physics B: Atomic and Molecular +Physics 20(4), 639 (1987). DOI 10.1088/0022-3700/20/4/006. URL https://doi. +org/10.1088/0022-3700/20/4/006 +320. F.E. Jorge, A.B.F. da Silva, The Journal of Chemical Physics 105(13), 5503 (1996). +DOI 10.1063/1.472390. URL https://doi.org/10.1063/1.472390 +321. C. Thierfelder, P. Schwerdtfeger, Phys. Rev. A 82, 062503 (2010). +DOI 10.1103/ +PhysRevA.82.062503. +URL http://link.aps.org/doi/10.1103/PhysRevA.82. +062503 +322. K. Huang, Y.K. Kim, K. Cheng, J. Desclaux, Phys. Rev. Lett. 48(18), 1245 +(1982). +DOI 10.1103/PhysRevLett.48.1245. +URL https://journals.aps.org/ +prl/abstract/10.1103/PhysRevLett.48.1245 +323. P. Indelicato, E. Lindroth, J. Desclaux, Phys. Rev. Lett. 94(1), 013002 (2005). DOI +10.1103/PhysRevLett.94.013002. URL https://doi.org/10.1103/PhysRevLett. +94.013002 +324. Y. Kim, F. Parente, J. Marques, P. Indelicato, J. Desclaux, Phys. Rev. A 58(3), +R1885 (1998). DOI 10.1103/PhysRevA.58.1885. URL https://doi.org/10.1103/ +PhysRevA.58.1885 +325. W.E. Lamb, R.C. Retherford, Phys. Rev. 72(3), 241 (1947). URL http://link.aps. +org/doi/10.1103/PhysRev.72.241 +326. P. Kusch, H. Foley, Phys. Rev. 74(3), 250 (1948) +327. H. Bethe, Phys. Rev. 72(4), 339 (1947). +URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.72.339 +328. R. Feynman, Phys. Rev. 76(6), 769 (1949). URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.76.769 +329. J. Schwinger, Phys. Rev. 74(10), 1439 (1948). URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.74.1439 +330. J. Schwinger, Phys. Rev. 75(4), 651 (1949). URL http://link.aps.org/doi/10. +1103/PhysRev.75.651 +331. J. Schwinger, Phys. Rev. 76(6), 790 (1949). URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.76.790 + +79 +332. T. Tati, S. itirô Tomonaga, Progr. Theoret. Phys. (Kyoto) 3(4), 391 (1948). +URL +https://doi.org/10.1143/ptp/3.4.391 +333. E.H. Wichmann, N.M. Kroll, Phys. Rev. 101, 843 (1956). DOI 10.1103/PhysRev.101. +843. URL https://link.aps.org/doi/10.1103/PhysRev.101.843 +334. G. Brown, J. Langer, G. Schaefer, Proceedings of the Royal Society of London Series +A 251, 92 (1959). +URL https://royalsocietypublishing.org/doi/10.1098/ +rspa.1959.0092 +335. G. Brown, D. Mayers, Proceedings of the Royal Society of London Series A 251, 105 +(1959). URL https://royalsocietypublishing.org/doi/10.1098/rspa.1959. +0093 +336. P. Mohr, Annals of Physics 88(1), 26 (1974). +URL https://doi.org/10.1016/ +0003-4916(74)90398-4 +337. P.J. Mohr, Annals of Physics 88(1), 52 (1974). URL https://doi.org/10.1016/ +0003-4916(74)90399-6 +338. K. Cheng, W. Johnson, Canadian Journal of Physics 14(6), 1943 (1976). URL https: +//doi.org/10.1139/p07-106 +339. G. Soff, P. Schlüter, B. Müller, W. Greiner, Phys. Rev. Lett. 48(21), 1465 +(1982). +URL http://prl.aps.org/abstract/PRL/v48/i21/p1465_1http:// +link.aps.org/doi/10.1103/PhysRevLett.48.1465 +340. A.V. Malyshev, D.A. Glazov, V.M. Shabaev, I.I. Tupitsyn, V.A. Yerokhin, V.A. Za- +ytsev, Phys. Rev. A 106(1), 012806 (2022). URL https://link.aps.org/doi/10. +1103/PhysRevA.106.012806 +341. S. Blundell, P. Mohr, W. Johnson, J. Sapirstein, Phys. Rev. A 48(4), 2615 (1993). URL +http://journals.aps.org/pra/abstract/10.1103/PhysRevA.48.2615 +342. P.J. Mohr, J. Sapirstein, Phys. Rev. A 62(5), 052501 (2000). URL https://journals. +aps.org/pra/abstract/10.1103/PhysRevA.62.052501 +343. M.I. Eides, H. Grotch, V.A. Shelyuto, Physics Reports 342(2-3), 63 (2001). URL +https://doi.org/10.1016/S0370-1573(00)00077-6 +344. E. Salpeter, Phys. Rev. 87(2), 328 (1952) +345. N. Nakanishi, Phys. Rev. 138, B1182 (1965). DOI 10.1103/PhysRev.138.B1182. URL +https://link.aps.org/doi/10.1103/PhysRev.138.B1182 +346. N. Nakanishi, Phys. Rev. 139, B1401 (1965). DOI 10.1103/PhysRev.139.B1401. URL +https://link.aps.org/doi/10.1103/PhysRev.139.B1401 +347. H. Sazdjian, Journal of Mathematical Physics 28(11), 2618 (1987). DOI 10.1063/1. +527755. URL https://doi.org/10.1063/1.527755 +348. G. Ramalho, A. Arriaga, M.T. Peña, Phys. Rev. C 65, 034008 (2002). DOI 10.1103/ +PhysRevC.65.034008. URL https://link.aps.org/doi/10.1103/PhysRevC.65. +034008 +349. P. Indelicato, P.J. Mohr, in Handbook of Relativistic Quantum Chemistry, ed. by +W. Liu, C. van Wüllen, P. Indelicato, J. Autschbach, J. Li, Springer Reference +(Springer-Verlag, Berlin Heidelberg, 2017), p. 231 +350. U.D. Jentschura, G.S.A. Adkins, Quantum Electrodynamics: Atoms, Lasers and Grav- +ity. Quantum Electrodynamics: Atoms, Lasers and Gravity (World Scientific, 2021). +URL https://www.worldscientific.com/doi/abs/10.1142/12722 +351. V.M. +Shabaev, +Physics +Reports +356(3), +119 +(2002). +URL +http: +//www.sciencedirect.com/science/article/B6TVP-44CMSTR-1/2/ +f487ad322a16e9b027362334c7823059 +352. A.N. Artemyev, in Handbook of Relativistic Quantum Chemistry (Springer Berlin Hei- +delberg, Berlin, Heidelberg, 2017), p. 287 + +80 +353. A.N. Artemyev, in Handbook of Relativistic Quantum Chemistry (Springer Berlin Hei- +delberg, Berlin, Heidelberg, 2017), p. 243 +354. J. Sapirstein, Phys. Scr. T46, 52 (1993). +URL https://doi.org/10.1088/ +0031-8949/1993/T46/006 +355. I. +Lindgren, +Molecular +Physics +98(16), +1159 +(2000). +URL +http: +//www.informaworld.com/smpp/content~db=all~content=a713831352~frm= +titlelink +356. I. Lindgren, B. Åsén, S. Salomonson, A.M. Mårtensson-Pendrill, Phys. Rev. A 64(6), +062505 (2001). URL https://doi.org/10.1103/PhysRevA.64.062505 +357. I. Lindgren, P. Indelicato, in Handbook of Relativistic Quantum Chemistry (Springer +Berlin Heidelberg, Berlin, Heidelberg, 2017), p. 313 +358. W.H. Furry, Phys. Rev. 81(1), 115 (1951). +URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.81.115 +359. W.H. Furry, J.R. Oppenheimer, Phys. Rev. 45, 245 (1934). DOI 10.1103/PhysRev.45. +245. URL https://link.aps.org/doi/10.1103/PhysRev.45.245 +360. F. Dyson, Phys. Rev. 75(11), 1736 (1949). URL http://journals.aps.org/pr/ +abstract/10.1103/PhysRev.75.1736 +361. M. Gell-Mann, F. Low, Phys. Rev. 84(2), 350 (1951). URL https://journals.aps. +org/pr/abstract/10.1103/PhysRev.84.350 +362. A.L. Fetter, J.D. Walecka, Quantum theory of many-particle systems. International +Series in Pure and Apllied Physics (Mc Graw-Hill, New-York, 1971) +363. J. Sucher, Phys. Rev. 107, 1448 (1957) +364. P. Mohr, in Physics of Highly-Ionized Atoms, Nato Asi Series, vol. 201 (Plenum, New +York, 1989), pp. 111–141 +365. P. Mohr, in Handbook of Atomic Physics (American Institute of Physics, New-York, +1996), p. 125 +366. P. Indelicato, V.M. Shabaev, A.V. Volotka, Phys. Rev. A 69(6), 062506 (2004). URL +http://link.aps.org/abstract/PRA/v69/e062506 +367. O.Y. Andreev, L.N. Labzowsky, G. Plunien, uuml, nter, Phys. Rev. A 79(3), 032515 +(2009). URL http://link.aps.org/doi/10.1103/PhysRevA.79.032515 +368. L.N. Labzowsky, A.V. Shonin, D.A. Solovyev, J. Phys. B: At. Mol. Opt. Phys. 38(3), +265 (2005). URL https://doi.org/10.1088/0953-4075/38/3/010 +369. O.Y. Andreev, L.N. Labzowsky, G. Plunien, D.A. Solovyev, Physics Reports +455(4-6), 135 (2008). URL http://www.sciencedirect.com/science/article/ +B6TVP-4R0KSY7-2/1/4a50b89b9f9f68754542deb9133eb105 +370. I. Lindgren, S. Salomonson, B. Åsén, Physics Reports 389(4), 161 (2004). +URL +https://doi.org/10.1016/j.physrep.2003.09.004 +371. I. Lindgren, S. Salomonson, D. Hedendahl, Phys. Rev. A 73(6), 062502 (2006). URL +http://link.aps.org/abstract/PRA/v73/e062502 +372. I. Lindgren, S. Salomonson, D. Hedendahl, J. Phys. B: At. Mol. Opt. Phys. 2011, +723574 (2011). URL http://dx.doi.org/10.1155/2011/723574 +373. J. Holmberg, S. Salomonson, I. Lindgren, Phys. Rev. A 92(1), 012509 (2015). URL +http://link.aps.org/doi/10.1103/PhysRevA.92.012509 +374. P.J. Mohr, B.N. Taylor, D.B. Newell, Rev. Mod. Phys. 84(4), 1527 (2012). URL http: +//link.aps.org/doi/10.1103/RevModPhys.84.1527 +375. E. Tiesinga, P.J. Mohr, D.B. Newell, B.N. Taylor, Rev. Mod. Phys. 93(2), 025010 +(2021). URL https://link.aps.org/doi/10.1103/RevModPhys.93.025010 +376. P. Mohr, Phys. Rev. A 46(7), 4421 (1992). +URL https://doi.org/10.1103/ +PhysRevA.46.4421 + +81 +377. P. Indelicato, P.J. Mohr, Hyp. Int. 114, 147 (1998). URL http://link.springer. +com/10.1023/A:1012622505367 +378. T. Beier, P.J. Mohr, H. Persson, G. Soff, Phys. Rev. A 58(2), 954 (1998). URL https: +//link.aps.org/doi/10.1103/PhysRevA.58.954 +379. P. Indelicato, U. Jentschura, P. Mohr. Point nucleus self-energy evaluation for high-z +and high-n, 1 ≤ n ≤ 10 levels up to z = 137 (2022) +380. P. Indelicato, P. Mohr. Self-energy evaluation for ns and np1/2 levels 1 ≤ z ≤ 6 with +finite nuclear size correction (2022) +381. P. Mohr, Phys. Rev. A 26(5), 2338 (1982). +URL https://doi.org/10.1103/ +PhysRevA.26.2338 +382. P.J. Mohr, Y.K. Kim, Phys. Rev. A 45(5), 2727 (1992). URL http://link.aps.org/ +doi/10.1103/PhysRevA.45.2727 +383. E.O. Le Bigot, P. Indelicato, P.J. Mohr, Phys. Rev. A 64(5), 052508 (2001). URL +https://link.aps.org/doi/10.1103/PhysRevA.64.052508 +384. G. Soff, P. Mohr, Phys. Rev. A 38(10), 5066 (1988). DOI 10.1103/PhysRevA.38.5066. +URL 10.1103/PhysRevA.38.5066 +385. H. Persson, I. Lindgren, S. Salomonson, P. Sunnergren, Phys. Rev. A 48, 2772 +(1993). DOI 10.1103/PhysRevA.48.2772. URL https://link.aps.org/doi/10. +1103/PhysRevA.48.2772 +386. E.A. Uehling, Phys. Rev. 48, 55 (1935). DOI 10.1103/PhysRev.48.55. URL https: +//link.aps.org/doi/10.1103/PhysRev.48.55 +387. A.M. Frolov, D.M. Wardlaw, The European Physical Journal B 85(10), 348 +(2012). DOI 10.1140/epjb/e2012-30408-4. URL https://doi.org/10.1140/epjb/ +e2012-30408-4 +388. L.W. Fullerton, G.A. Rinker, Phys. Rev. A 13, 1283 (1976). DOI 10.1103/PhysRevA. +13.1283. URL https://link.aps.org/doi/10.1103/PhysRevA.13.1283 +389. S. Klarsfeld, Physics Letters 66B(1), 86 (1977). URL http://dx.doi.org/10.1016/ +0370-2693(77)90620-7 +390. J. Sapirstein, K.T. Cheng, Phys. Rev. A 68, 042111 (2003). DOI 10.1103/PhysRevA. +68.042111. URL https://link.aps.org/doi/10.1103/PhysRevA.68.042111 +391. K. Huang, Phys. Rev. A 14(4), 1311 (1976). URL http://link.aps.org/doi/10. +1103/PhysRevA.14.1311 +392. S.J. Brodsky, P.J. Mohr, in Structure and Collisions of Ions and Atoms, vol. 5 (Springer +Berlin Heidelberg, Berlin, Heidelberg, 1978), pp. 3–67. URL https://doi.org/10. +1007/978-3-642-81210-1_2 +393. E. Borie, G. Rinker, Rev. Mod. Phys. 54(1), 67 (1982). URL http://link.aps.org/ +doi/10.1103/RevModPhys.54.67 +394. N. Paul, G. Bian, T. Azuma, S. Okada, P. Indelicato, Phys. Rev. Lett. 126(17), 173001 +(2021). DOI 10.1103/PhysRevLett.126.173001. URL https://doi.org/10.1103/ +PhysRevLett.126.173001https://hal.archives-ouvertes.fr/hal-03007179 +395. G. Källen, A. Sabry, Mat. Fys. Medd. Dan. Vid. 29, 17 (1955) +396. V. Yerokhin, P. Indelicato, V. Shabaev, Phys. Rev. Lett. 91(7), 073001 (2003). URL +http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.073001 +397. V. Yerokhin, P. Indelicato, V. Shabaev, Eur. Phys. J. D 25(3), 203 (2003). URL https: +//doi.org/10.1140/epjd/e2003-00270-x +398. V.A. Yerokhin, P. Indelicato, V.M. Shabaev, Phys. Rev. Lett. 97(25), 253004 (2006). +URL http://link.aps.org/abstract/PRL/v97/e253004 +399. I. Lindgren, H. Persson, S. Salomonson, V. Karasiev, L. Labzowsky, A. Mitrushenkov, +M. Tokman, J. Phys. B: At. Mol. Opt. Phys. 26, L503 (1993). +URL http:// + +82 +iopscience.iop.org/article/10.1088/0953-4075/26/16/003/meta +400. H. Persson, I. Lindgren, L. Labzowsky, G. Plunien, T. Beier, G. Soff, Phys. Rev. A +54(4), 2805 (1996). +URL https://journals.aps.org/pra/abstract/10.1103/ +PhysRevA.54.2805 +401. H. Persson, S. Salomonson, P. Sunnergren, I. Lindgren, Phys. Rev. Lett. 76(2), 204 +(1996). URL https://journals.aps.org/prl/abstract/10.1103/PhysRevLett. +76.204 +402. P. Indelicato, P.J. Mohr, Phys. Rev. A 63(4), 052507 (2001). +URL https:// +journals.aps.org/pra/abstract/10.1103/PhysRevA.63.052507 +403. G. Breit, Phys. Rev. 34, 553 (1929). DOI 10.1103/PhysRev.34.553. URL https: +//link.aps.org/doi/10.1103/PhysRev.34.553 +404. G. Breit, Phys. Rev. 36, 383 (1930). DOI 10.1103/PhysRev.36.383. URL https: +//link.aps.org/doi/10.1103/PhysRev.36.383 +405. G. Breit, Phys. Rev. 39, 616 (1932). DOI 10.1103/PhysRev.39.616. URL https: +//link.aps.org/doi/10.1103/PhysRev.39.616 +406. J. Mann, W. Johnson, Phys. Rev. A 4(1), 41 (1971). URL https://doi.org/10. +1103/PhysRevA.4.41 +407. J. Hata, I. Grant, J. Phys. B: At. Mol. Opt. Phys. 17, L107 (1984). +URL https: +//doi.org/10.1088/0022-3700/17/5/001 +408. O. Gorceix, P. Indelicato, J.P. Desclaux, J. Phys. B: At. Mol. Opt. Phys. 20(4), 639 +(1987). URL http://dx.doi.org/10.1088/0022-3700/20/4/006 +409. I. Lindgren, J. Phys. B: At. Mol. Opt. Phys. 23(7), 1085 (1990). +URL https:// +iopscience.iop.org/article/10.1088/0953-4075/23/7/006 +410. I. Lindgren, H. Persson, S. Salomonson, L. Labzowsky, Phys. Rev. A 51(2), 1167 +(1995). URL https://doi.org/10.1103/PhysRevA.51.1167 +411. B. Åsén, S. Salomonson, I. Lindgren, Phys. Rev. A 65(3), 032516 (16) (2002). URL +http://doi.org/10.1103/PhysRevA.65.032516 +412. P. Pyykkö", L.B. Zhao, Journal of Physics B: Atomic, Molecular and Optical Physics +36(8), 1469 (2003). DOI 10.1088/0953-4075/36/8/302. URL https://doi.org/10. +1088/0953-4075/36/8/302 +413. K. Dyall, I. Grant, C. Johnson, F. Parpia, E. Plummer, Comp. Phys. Commun. 55, 425 +(1989). +DOI 10.1016/0010-4655(89)90136-7. +URL https://doi.org/10.1016/ +0010-4655(89)90136-7 +414. T. Welton, Phys. Rev. 74(9), 1157 (1948). +DOI 10.1103/PhysRev.74.1157. +URL +https://journals.aps.org/pr/abstract/10.1103/PhysRev.74.1157 +415. J.S.M. Ginges, J.C. Berengut, Phys. Rev. A 93, 052509 (2016). +DOI 10.1103/ +PhysRevA.93.052509. URL https://link.aps.org/doi/10.1103/PhysRevA.93. +052509 +416. K.G. Dyall, The Journal of Chemical Physics 139(2), 021103 (2013). DOI 10.1063/1. +4813483. URL https://doi.org/10.1063/1.4813483 +417. V. Shabaev, I. Tupitsyn, V. Yerokhin, Comput. Phys. Commun. 189, 175 (2015). DOI +https://doi.org/10.1016/j.cpc.2014.12.002. URL http://www.sciencedirect.com/ +science/article/pii/S0010465514004081 +418. T. Hangele, M. Dolg, M. Hanrath, X. Cao, P. Schwerdtfeger, J. Chem. Phys. 136, +214105 (2012). +DOI 10.1063/1.4723805. +URL https://doi.org/10.1063/1. +4723805 +419. T. Hangele, M. Dolg, P. Schwerdtfeger, J. Chem. Phys. 138, 174113 (2013). DOI +10.1063/1.4803148. URL https://doi.org/10.1063/1.4803148 + +83 +420. A. Kramida, Y. Ralchenko, J. Reader, N.A. Team. NIST Atomic Spectra Database +(Version 5.10) (2022). DOI 10.18434/T4W30F. URL https://physics.nist.gov/ +asd +421. K. Andersson, B.O. Roos, Chemical Physics Letters 191(6), 507 (1992). DOI https:// +doi.org/10.1016/0009-2614(92)85581-T. URL https://www.sciencedirect.com/ +science/article/pii/000926149285581T +422. J. Biero´n, G. Gaigalas, E. Gaidamauskas, S. Fritzsche, P. Indelicato, P. Jönsson, Phys. +Rev. A 80, 012513 (2009). DOI 10.1103/PhysRevA.80.012513. URL https://link. +aps.org/doi/10.1103/PhysRevA.80.012513 +423. V.A. Dzuba, V.V. Flambaum, M.G. Kozlov, Phys. Rev. A 54, 3948 (1996). DOI 10. +1103/PhysRevA.54.3948. URL https://link.aps.org/doi/10.1103/PhysRevA. +54.3948 +424. S. Blundell, D. Guo, W. Johnson, J. Sapirstein, Atomic Data and Nuclear Data Tables +37(1), 103 (1987). URL https://doi.org/10.1016/0092-640X(87)90006-4 +425. S. Blundell, W. Johnson, J. Sapirstein, Phys. Rev. A 41(3), 1698 (1990). URL http: +//journals.aps.org/pra/abstract/10.1103/PhysRevA.41.1698 +426. S. Salomonson, H. Warston, I. Lindgren, Phys. Rev. Lett. 76, 3092 (1996). +DOI 10.1103/PhysRevLett.76.3092. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.76.3092 +427. F. Coester, H. Kümmel, Nuclear Physics 17, 477 (1960). +DOI https://doi.org/10. +1016/0029-5582(60)90140-1. +URL https://www.sciencedirect.com/science/ +article/pii/0029558260901401 +428. J. ˇCížek, J. Paldus, Phys. Scr. 21(3-4), 251 (1980). DOI 10.1088/0031-8949/21/3-4/ +006. URL https://doi.org/10.1088/0031-8949/21/3-4/006 +429. R.J. Bartlett, Theoretica chimica acta 80(2), 71 (1991). DOI 10.1007/BF01119614. +URL https://doi.org/10.1007/BF01119614 +430. H.G. Kümmel, International Journal of Modern Physics B 17(28), 5311 (2003) +431. R.J. Bartlett, M. Musiał, Rev. Mod. Phys. 79, 291 (2007). DOI 10.1103/RevModPhys. +79.291. URL https://link.aps.org/doi/10.1103/RevModPhys.79.291 +432. I. Shavitt, R.J. Bartlett, Many-body methods in chemistry and physics: MBPT and +coupled-cluster theory (Cambridge University Press, 2009) +433. R.J. Bartlett, Wiley Interdisciplinary Reviews: Computational Molecular Science 2(1), +126 (2012) +434. S.N. Datta, Computational and Theoretical Chemistry 1166, 112574 (2019). DOI +https://doi.org/10.1016/j.comptc.2019.112574. URL https://www.sciencedirect. +com/science/article/pii/S2210271X19302695 +435. J. Liu, L. Cheng, WIREs Computational Molecular Science 11(6), e1536 (2021). DOI +https://doi.org/10.1002/wcms.1536. URL https://wires.onlinelibrary.wiley. +com/doi/abs/10.1002/wcms.1536 +436. R.K. Chaudhuri, S.K. Chattopadhyay, Many-body methods for atoms and molecules +(CRC Press, 2017) +437. U. Kaldor, Theoretica chimica acta 80(6), 427 (1991). DOI 10.1007/BF01119664. +URL https://doi.org/10.1007/BF01119664 +438. L. Visscher, E. Eliav, U. Kaldor, J. Chem. Phys. 115(21), 9720 (2001). URL https: +//doi.org/10.1063/1.1415746 +439. E. Eliav, U. Kaldor, in Relativistic methods for chemists (Springer, 2010), pp. 279–349 +440. E. Eliav, S. Fritzsche, U. Kaldor, Nucl. Phys. A 944, 518 (2015). DOI 10.1016/j. +nuclphysa.2015.06.017 + +84 +441. A.V. Oleynichenko, A. Zaitsevskii, L.V. Skripnikov, E. Eliav, Symmetry 12(7) (2020). +DOI 10.3390/sym12071101. URL https://www.mdpi.com/2073-8994/12/7/1101 +442. E. Eliav, U. Kaldor, P. Schwerdtfeger, B.A. Hess, Y. Ishikawa, Phys. Rev. Lett. 73, +3203 (1994). DOI 10.1103/PhysRevLett.73.3203. URL http://link.aps.org/doi/ +10.1103/PhysRevLett.73.3203 +443. U. Kaldor, E. Eliav, in High-Accuracy Calculations for Heavy and Super- +Heavy Elements, Advances in Quantum Chemistry, vol. 31, ed. by J.R. Sabin, +M.C. +Zerner, +E. +Brändas, +S. +Wilson, +J. +Maruani, +Y. +Smeyers, +P. +Grout, +R. McWeeny (Academic Press, 1998), pp. 313–336. +DOI https://doi.org/10. +1016/S0065-3276(08)60194-X. URL https://www.sciencedirect.com/science/ +article/pii/S006532760860194X +444. E. Eliav, M.J. Vilkas, Y. Ishikawa, U. Kaldor, The Journal of Chemical Physics +122(22), 224113 (2005). +DOI 10.1063/1.1929727. +URL https://doi.org/10. +1063/1.1929727 +445. A. Borschevsky, L.F. Pašteka, V. Pershina, E. Eliav, U. Kaldor, Phys. Rev. A 91, +020501 (2015). DOI 10.1103/PhysRevA.91.020501. URL https://link.aps.org/ +doi/10.1103/PhysRevA.91.020501 +446. A. Borschevsky, V. Pershina, E. Eliav, U. Kaldor, Phys. Rev. A 87, 022502 (2013). +DOI 10.1103/PhysRevA.87.022502. URL https://link.aps.org/doi/10.1103/ +PhysRevA.87.022502 +447. S. Pal, Phys. Rev. A 39, 39 (1989). DOI 10.1103/PhysRevA.39.39. URL https: +//link.aps.org/doi/10.1103/PhysRevA.39.39 +448. N. Oliphant, L. Adamowicz, International Reviews in Physical Chemistry 12(2), +339 (1993). DOI 10.1080/01442359309353285. URL https://doi.org/10.1080/ +01442359309353285 +449. P. Piecuch, K. Kowalski, International Journal of Molecular Sciences 3(6), 676 (2002). +DOI 10.3390/i3060676. URL https://www.mdpi.com/1422-0067/3/6/676 +450. B. Jeziorski, Molecular Physics 108(21-23), 3043 (2010). DOI 10.1080/00268976. +2010.524169. URL https://doi.org/10.1080/00268976.2010.524169 +451. E. Eliav, U. Kaldor, in Recent Progress in Coupled Cluster Methods (Springer, 2010), +pp. 113–144 +452. A. Köhn, M. Hanauer, L.A. Mueck, T.C. Jagau, J. Gauss, Wiley Interdisciplinary Re- +views: Computational Molecular Science 3(2), 176 (2013) +453. Y.B. Tang, B.Q. Lou, T.Y. Shi, Phys. Rev. A 96, 022513 (2017). +DOI 10.1103/ +PhysRevA.96.022513. URL https://link.aps.org/doi/10.1103/PhysRevA.96. +022513 +454. S.N. Datta, Computational and Theoretical Chemistry 1180, 112794 (2020). DOI +https://doi.org/10.1016/j.comptc.2020.112794. URL https://www.sciencedirect. +com/science/article/pii/S2210271X20300943 +455. F.A. Evangelista, The Journal of Chemical Physics 149(3), 030901 (2018). DOI 10. +1063/1.5039496. URL https://doi.org/10.1063/1.5039496 +456. V.A. Dzuba, W.R. Johnson, Phys. Rev. A 76, 062510 (2007). DOI 10.1103/PhysRevA. +76.062510. URL https://link.aps.org/doi/10.1103/PhysRevA.76.062510 +457. Y. Cheng, S. Liu, S.B. Zhang, Y.B. Tang, Phys. Rev. A 102, 012824 (2020). +DOI 10.1103/PhysRevA.102.012824. URL https://link.aps.org/doi/10.1103/ +PhysRevA.102.012824 +458. M. Safronova, W. Johnson, in All-Order Methods for Relativistic Atomic Structure +Calculations, Advances In Atomic, Molecular, and Optical Physics, vol. 55, ed. by +E. Arimondo, P.R. Berman, C.C. Lin (Academic Press, 2008), pp. 191–233. DOI https: + +85 +//doi.org/10.1016/S1049-250X(07)55004-4. +URL https://www.sciencedirect. +com/science/article/pii/S1049250X07550044 +459. M.S. Safronova, M.G. Kozlov, W.R. Johnson, D. Jiang, Phys. Rev. A 80, 012516 +(2009). DOI 10.1103/PhysRevA.80.012516. URL https://link.aps.org/doi/10. +1103/PhysRevA.80.012516 +460. H. Gharibnejad, E. Eliav, M.S. Safronova, A. Derevianko, Phys. Rev. A 83, 052502 +(2011). DOI 10.1103/PhysRevA.83.052502. URL https://link.aps.org/doi/10. +1103/PhysRevA.83.052502 +461. M.S. Safronova, U.I. Safronova, C.W. Clark, Phys. Rev. A 90, 032512 (2014). +DOI 10.1103/PhysRevA.90.032512. URL https://link.aps.org/doi/10.1103/ +PhysRevA.90.032512 +462. S.G. Porsev, M.G. Kozlov, M.S. Safronova, I.I. Tupitsyn, Phys. Rev. A 93, 012501 +(2016). DOI 10.1103/PhysRevA.93.012501. URL https://link.aps.org/doi/10. +1103/PhysRevA.93.012501 +463. V.A. Dzuba, J.C. Berengut, C. Harabati, V.V. Flambaum, Phys. Rev. A 95, 012503 +(2017). DOI 10.1103/PhysRevA.95.012503. URL https://link.aps.org/doi/10. +1103/PhysRevA.95.012503 +464. S.A. Blundell, W.R. Johnson, J. Sapirstein, Phys. Rev. A 41, 1698 (1990). DOI 10. +1103/PhysRevA.41.1698. URL https://link.aps.org/doi/10.1103/PhysRevA. +41.1698 +465. J. Sapirstein, K.T. Cheng, Phys. Rev. A 83(1), 012504 (2011). URL http://link. +aps.org/doi/10.1103/PhysRevA.83.012504 +466. A. Derevianko, S.G. Porsev, K. Beloy, Phys. Rev. A 78, 010503 (2008). DOI 10.1103/ +PhysRevA.78.010503. URL https://link.aps.org/doi/10.1103/PhysRevA.78. +010503 +467. J. Sapirstein, K.T. Cheng, Phys. Rev. A 91(6), 062508 (2015). URL http://link. +aps.org/doi/10.1103/PhysRevA.91.062508 +468. S.A. Blundell, W.R. Johnson, J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990). +DOI 10.1103/PhysRevLett.65.1411. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.65.1411 +469. P. Beiersdorfer, Journal of Physics B: Atomic, Molecular and Optical Physics 43(7), +074032 (2010). DOI 10.1088/0953-4075/43/7/074032. URL https://doi.org/10. +1088/0953-4075/43/7/074032 +470. A.V. Volotka, D.A. Glazov, G. Plunien, V.M. Shabaev, Annalen der Physik 525(8- +9), 636 (2013). +DOI https://doi.org/10.1002/andp.201300079. +URL https:// +onlinelibrary.wiley.com/doi/abs/10.1002/andp.201300079 +471. V.M. Shabaev, A.I. Bondarev, D.A. Glazov, M.Y. Kaygorodov, Y.S. Kozhedub, I.A. +Maltsev, A.V. Malyshev, R.V. Popov, I.I. Tupitsyn, N.A. Zubova, Hyperfine Interac- +tions 239(1), 60 (2018). DOI 10.1007/s10751-018-1537-8. URL https://doi.org/ +10.1007/s10751-018-1537-8 +472. D.H. Bross, P. Parmar, K.A. Peterson, The Journal of Chemical Physics 143(18), +184308 (2015). +DOI 10.1063/1.4935375. +URL https://doi.org/10.1063/1. +4935375 +473. A. Coste, R. Avril, P. Blancard, J. Chatelet, D. Lambert, J. Legre, S. Liberman, +J. Pinard, J. Opt. Soc. Am. 72(1), 103 (1982). DOI 10.1364/JOSA.72.000103. URL +http://opg.optica.org/abstract.cfm?URI=josa-72-1-103 +474. M. Dolg, X. Cao, Chemical Reviews 112(1), 403 (2012). DOI 10.1021/cr2001383. +URL https://doi.org/10.1021/cr2001383 + +86 +475. P. Schwerdtfeger, ChemPhysChem 12(17), 3143 (2011). DOI https://doi.org/10.1002/ +cphc.201100387. URL https://chemistry-europe.onlinelibrary.wiley.com/ +doi/abs/10.1002/cphc.201100387 +476. V. Dzuba, V. Flambaum, O. Sushkov, Physics Letters A 140(9), 493 (1989). DOI https: +//doi.org/10.1016/0375-9601(89)90129-1. URL https://www.sciencedirect.com/ +science/article/pii/0375960189901291 +477. V.A. Dzuba, Phys. Rev. A 78, 042502 (2008). DOI 10.1103/PhysRevA.78.042502. +URL https://link.aps.org/doi/10.1103/PhysRevA.78.042502 +478. J. Brandejs, J. Višˇnák, L. Veis, M. Maté, Ö. Legeza, J. Pittner, The Journal of Chemical +Physics 152(17), 174107 (2020). DOI 10.1063/1.5144974. URL https://doi.org/ +10.1063/1.5144974 +479. I. Lindgren, International Journal of Quantum Chemistry 114(18), 1176 (2014). DOI +https://doi.org/10.1002/qua.24629. URL https://onlinelibrary.wiley.com/doi/ +abs/10.1002/qua.24629 +480. I. Lindgren, Relativistic many-body theory: a new field-theoretical approach, vol. 63 +(Springer, 2016) +481. P.F.m.c. Loos, P.M.W. Gill, Phys. Rev. Lett. 105, 113001 (2010). +DOI +10.1103/PhysRevLett.105.113001. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.105.113001 +482. J. Karwowski, J. Styszynski, W.H.E. Schwarz, Journal of Physics B: Atomic, Molec- +ular and Optical Physics 24(23), 4877 (1991). DOI 10.1088/0953-4075/24/23/016. +URL https://dx.doi.org/10.1088/0953-4075/24/23/016 +483. Y. Watanabe, H. Nakano, H. Tatewaki, The Journal of Chemical Physics 126(17), +174105 (2007). +DOI 10.1063/1.2733647. +URL https://doi.org/10.1063/1. +2733647 +484. G. Rodrigues, P. Indelicato, J. Santos, P. Patté, F. Parente, Atomic Data and Nuclear +Data Tables 86(2), 117 (2004). DOI https://doi.org/10.1016/j.adt.2003.11.005. URL +https://www.sciencedirect.com/science/article/pii/S0092640X03000846 +485. A. Borschevsky, E. Eliav, M.J. Vilkas, Y. Ishikawa, U. Kaldor, Phys. Rev. A 75, 042514 +(2007). DOI 10.1103/PhysRevA.75.042514. URL https://link.aps.org/doi/10. +1103/PhysRevA.75.042514 +486. P. Chhetri, D. Ackermann, H. Backe, M. Block, B. Cheal, C. Droese, C.E. Düll- +mann, J. Even, R. Ferrer, F. Giacoppo, S. Götz, F.P. Heßberger, M. Huyse, +O. Kaleja, J. Khuyagbaatar, P. Kunz, M. Laatiaoui, F. Lautenschläger, W. Lauth, +N. Lecesne, L. Lens, E. Minaya Ramirez, A.K. Mistry, S. Raeder, P. Van Duppen, +T. Walther, A. Yakushev, Z. Zhang, Phys. Rev. Lett. 120, 263003 (2018). +DOI +10.1103/PhysRevLett.120.263003. +URL https://link.aps.org/doi/10.1103/ +PhysRevLett.120.263003 +487. C. Thierfelder, P. Schwerdtfeger, Phys. Rev. A 79, 032512 (2009). +DOI 10.1103/ +PhysRevA.79.032512. URL https://link.aps.org/doi/10.1103/PhysRevA.79. +032512 +488. V.A. Dzuba, M.S. Safronova, U.I. Safronova, Phys. Rev. A 90, 012504 (2014). +DOI 10.1103/PhysRevA.90.012504. URL https://link.aps.org/doi/10.1103/ +PhysRevA.90.012504 +489. E. Eliav, U. Kaldor, A. Borschevsky, Encyclopedia of Inorganic and Bioinorganic +Chemistry pp. 1–16 (2011) +490. V.A. Dzuba, Phys. Rev. A 93, 032519 (2016). DOI 10.1103/PhysRevA.93.032519. +URL https://link.aps.org/doi/10.1103/PhysRevA.93.032519 + +87 +491. B.G.C. Lackenby, V.A. Dzuba, V.V. Flambaum, Phys. Rev. A 98, 022518 (2018). +DOI 10.1103/PhysRevA.98.022518. URL https://link.aps.org/doi/10.1103/ +PhysRevA.98.022518 +492. H. Arbely, A. Borschevsky, High accuracy calculations of atomic properties of group +v and group x elements. Ph.D. thesis, Van Swinderen Institute for Particle Physics and +Gravity (VSI), University of Groningen (2018) +493. M. Kaygorodov, D. Usov, E. Eliav, Y. Kozhedub, A. Malyshev, A. Oleynichenko, +V. Shabaev, L. Skripnikov, A. Titov, I. Tupitsyn, et al., Physical Review A 105(6), +062805 (2022) +494. V. Pershina, A. Borschevsky, E. Eliav, U. Kaldor, J. Chem. Phys. 128, 024707 (2008). +DOI 10.1063/1.2814242. URL https://doi.org/10.1063/1.2814242 +495. Y. Guo, A. Borschevsky, E. Eliav, L.F. Pasteka, Journal of Physics B: Atomic, Molec- +ular and Optical Physics (2022). URL http://iopscience.iop.org/article/10. +1088/1361-6455/ac761f +496. E. Eliav, U. Kaldor, Y. Ishikawa, M. Seth, P. Pyykkö, Phys. Rev. A 53, 3926 +(1996). DOI 10.1103/PhysRevA.53.3926. URL https://link.aps.org/doi/10. +1103/PhysRevA.53.3926 +497. V. Pershina, A. Borschevsky, E. Eliav, U. Kaldor, J. Phys. Chem. A 112, 13712 (2008). +DOI 10.1021/jp8061306. URL https://doi.org/10.1021/jp8061306 +498. A. Borschevsky, V. Pershina, E. Eliav, U. Kaldor, Chem. Phys. Lett. 480, 49 +(2009). DOI 10.1016/j.cplett.2009.08.059. URL http://www.sciencedirect.com/ +science/article/pii/S0009261409010562 +499. V.A. Dzuba, V.V. Flambaum, Hyperfine Interactions 237(1), 160 (2016). DOI 10.1007/ +s10751-016-1365-7. URL https://doi.org/10.1007/s10751-016-1365-7 +500. R.F. de Farias, Chemical Physics Letters 667, 1 (2017). DOI https://doi.org/10.1016/ +j.cplett.2016.11.023. URL https://www.sciencedirect.com/science/article/ +pii/S0009261416309058 +501. M.Y. Kaygorodov, L.V. Skripnikov, I.I. Tupitsyn, E. Eliav, Y.S. Kozhedub, A.V. Maly- +shev, A.V. Oleynichenko, V.M. Shabaev, A.V. Titov, A.V. Zaitsevskii, Phys. Rev. A +104, 012819 (2021). DOI 10.1103/PhysRevA.104.012819. URL https://link.aps. +org/doi/10.1103/PhysRevA.104.012819 +502. Y. Guo, L.F. Pašteka, E. Eliav, A. Borschevsky, in New Electron Correlation Meth- +ods and their Applications, and Use of Atomic Orbitals with Exponential Asymptotes, +Advances in Quantum Chemistry, vol. 83, ed. by M. Musial, P.E. Hoggan (Academic +Press, 2021), pp. 107–123. +DOI https://doi.org/10.1016/bs.aiq.2021.05.007. +URL +https://www.sciencedirect.com/science/article/pii/S0065327621000095 +503. P. Jerabek, B. Schuetrumpf, P. Schwerdtfeger, W. Nazarewicz, Phys. Rev. Lett. 120, +053001 (2018). DOI 10.1103/PhysRevLett.120.053001. URL https://link.aps. +org/doi/10.1103/PhysRevLett.120.053001 +504. Y. Guo, L.F. Pašteka, E. Eliav, A. Borschevsky. Ionization potentials and electron +affinity of oganesson (2021) +505. A. Landau, E. Eliav, Y. Ishikawa, U. Kaldor, The Journal of Chemical Physics 115(6), +2389 (2001). DOI 10.1063/1.1386413. URL https://doi.org/10.1063/1.1386413 +506. A. Borschevsky, V. Pershina, E. Eliav, U. Kaldor, J. Chem. Phys. 138(12), 124302 +(2013). DOI 10.1063/1.4795433. URL https://doi.org/10.1063/1.4795433 +507. R.S. Mulliken, The Journal of Chemical Physics 2(11), 782 (1934). DOI 10.1063/1. +1749394. URL https://doi.org/10.1063/1.1749394 +508. E. Eliav, U. Kaldor, Y. Ishikawa, P. Pyykkö, Phys. Rev. Lett. 77, 5350 (1996). +DOI 10.1103/PhysRevLett.77.5350. +URL https://link.aps.org/doi/10.1103/ + +88 +PhysRevLett.77.5350 +509. N. Gaston, P. Schwerdtfeger, W. Nazarewicz, Phys. Rev. A 66, 062505 (2002). +DOI 10.1103/PhysRevA.66.062505. URL https://link.aps.org/doi/10.1103/ +PhysRevA.66.062505 +510. C. Thierfelder, P. Schwerdtfeger, F.P. Heßberger, S. Hofmann, The European Physical +Journal A 36(2), 227 (2008). DOI 10.1140/epja/i2008-10584-7. URL https://doi. +org/10.1140/epja/i2008-10584-7 +511. W. Bambynek, H. Behrens, M.H. Chen, B. Crasemann, M.L. Fitzpatrick, K.W.D. +Ledingham, H. Genz, M. Mutterer, R.L. Intemann, Rev. Mod. Phys. 49, 77 (1977). +DOI 10.1103/RevModPhys.49.77. +URL https://link.aps.org/doi/10.1103/ +RevModPhys.49.77 +512. K. Pachucki, U.D. Jentschura, M. Pfützner, Phys. Rev. C 75, 055502 (2007). +DOI 10.1103/PhysRevC.75.055502. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.75.055502 +513. V. Pershina, A. Borschevsky, E. Eliav, U. Kaldor, J. Chem. Phys. 129, 144106 (2008). +DOI 10.1063/1.2988318. URL https://doi.org/10.1063/1.2988318 +514. V. Pershina, J. Anton, T. Jacob, J. Chem. Phys. 131, 084713 (2009). DOI 10.1063/1. +3212449. URL https://doi.org/10.1063/1.3212449 +515. A. Türler, V. Pershina, Chem. Rev. 113, 1237 (2013). DOI 10.1021/cr3002438. URL +http://pubs.acs.org/doi/abs/10.1021/cr3002438 +516. V. Pershina, J. Phys. Chem. C 120, 20232 (2016). DOI 10.1021/acs.jpcc.6b07834. +URL https://doi.org/10.1021/acs.jpcc.6b07834 +517. V. Pershina, Inorg. Chem. 57(7), 3948 (2018). DOI 10.1021/acs.inorgchem.8b00101. +URL https://doi.org/10.1021/acs.inorgchem.8b00101 +518. L. Trombach, S. Ehlert, S. Grimme, P. Schwerdtfeger, J.M. Mewes, Phys. Chem. +Chem. Phys. 21, 18048 (2019). DOI 10.1039/C9CP02455G. URL http://dx.doi. +org/10.1039/C9CP02455G +519. K. Chandrakumar, T.K. Ghanty, S.K. Ghosh, The Journal of Physical Chemistry A +108(32), 6661 (2004). URL https://doi.org/10.1021/jp960276m +520. P. Schwerdtfeger, J.K. Nagle, Molecular Physics 117(9-12), 1200 (2019). DOI 10. +1080/00268976.2018.1535143. URL https://doi.org/10.1080/00268976.2018. +1535143 +521. K.D. Bonin, V.V. Kresin, Electric-dipole polarizabilities of atoms, molecules, and +clusters (World Scientific, 1997) +522. A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92(9), 5397 (1990). DOI 10.1063/1. +458517 +523. B. Silvi, A. Savin, Nature 371(6499), 683 (1994). DOI 10.1038/371683a0. URL +https://doi.org/10.1038/371683a0 +524. A. Savin, R. Nesper, S. Wengert, T.F. Fässler, Angew. Chem. Int. Ed. 36(17), +1808 (1997). +DOI https://doi.org/10.1002/anie.199718081. +URL https:// +onlinelibrary.wiley.com/doi/abs/10.1002/anie.199718081 +525. P.G. Reinhard, J.A. Maruhn, A.S. Umar, V.E. Oberacker, Phys. Rev. C 83, 034312 +(2011). DOI 10.1103/PhysRevC.83.034312. URL https://link.aps.org/doi/10. +1103/PhysRevC.83.034312 +526. C.L. Zhang, B. Schuetrumpf, W. Nazarewicz, Phys. Rev. C 94, 064323 (2016). +DOI 10.1103/PhysRevC.94.064323. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.94.064323 +527. T. Li, M.Z. Chen, C.L. Zhang, W. Nazarewicz, M. Kortelainen, Phys. Rev. C 102, +044305 (2020). DOI 10.1103/PhysRevC.102.044305. URL https://link.aps.org/ + +89 +doi/10.1103/PhysRevC.102.044305 +528. P. Fuentealba, E. Chamorro, J.C. Santos, in Theoretical and computational chemistry, +vol. 19 (Elsevier, 2007), pp. 57–85 +529. E. Florez, O.R. Smits, J.M. Mewes, P. Jerabek, P. Schwerdtfeger, The Journal of +Chemical Physics 157(6), 064304 (2022). DOI 10.1063/5.0097642. URL https: +//doi.org/10.1063/5.0097642 +530. J.M. Mewes, O.R. Smits, G. Kresse, P. Schwerdtfeger, Angew. Chem. Int. Ed. 58(50), +17964 (2019). +DOI 10.1002/anie.201906966. +URL https://onlinelibrary. +wiley.com/doi/abs/10.1002/anie.201906966 +531. J.M. Mewes, P. Schwerdtfeger, Angewandte Chemie International Edition 60(14), +7703 (2021). +DOI https://doi.org/10.1002/anie.202100486. +URL https:// +onlinelibrary.wiley.com/doi/abs/10.1002/anie.202100486 +532. J.M. Mewes, P. Jerabek, O.R. Smits, P. Schwerdtfeger, Angew. Chem. Int. Ed. 58(40), +14260 (2019). +DOI 10.1002/anie.201908327. +URL https://onlinelibrary. +wiley.com/doi/abs/10.1002/anie.201908327 +533. O.R. Smits, J.M. Mewes, P. Jerabek, P. Schwerdtfeger, Angewandte Chemie Inter- +national Edition 59(52), 23636 (2020). DOI https://doi.org/10.1002/anie.202011976. +URL https://onlinelibrary.wiley.com/doi/abs/10.1002/anie.202011976 +534. W.M. Haynes, D.R. Lide, T.J. Bruno, CRC Handbook of Chemistry and Physics (CRC +press, Boca Raton, 2016) +535. B. Fricke, G. Soff, Atomic Data and Nuclear Data Tables 19(1), 83 (1977). DOI https:// +doi.org/10.1016/0092-640X(77)90010-9. URL https://www.sciencedirect.com/ +science/article/pii/0092640X77900109 +536. D.C. Hoffman, D.M. Lee, V. Pershina, in The chemistry of the actinide and transac- +tinide elements (Springer, 2008), pp. 1652–1752 +537. V. Pershina, in The Chemistry of Superheavy Elements, ed. by M. Schädel, +D. Shaughnessy (Springer Berlin Heidelberg, Berlin, Heidelberg, 2014), pp. 135– +239. +DOI 10.1007/978-3-642-37466-1\_3. +URL https://doi.org/10.1007/ +978-3-642-37466-1_3 +538. V. Pershina, Radiochim. Acta 107(9-11), 833 (2019). URL https://doi.org/10. +1515/ract-2018-3098 +539. P. Pyykkö, in EPJ Web of Conferences, vol. 131 (EDP sciences, 2016), vol. 131. URL +https://doi.org/10.1051/epjconf/201613101001 +540. C.S. Cao, H.S. Hu, J. Li, W.H.E. Schwarz, Pure. Appl. Chem. 91(12), 1969 (2019). +DOI 0.1515/pac-2019-0901. URL https://doi.org/10.1515/pac-2019-0901 +541. P. Pyykkö, J.P. Desclaux, Acc. Chem. Res. 12, 276 (1979). DOI 10.1021/ar50140a002. +URL http://dx.doi.org/10.1021/ar50140a002 +542. P. Pyykkö, Chem. Rev. 88(3), 563 (1988). DOI 10.1021/cr00085a006. URL http: +//pubs.acs.org/doi/abs/10.1021/cr00085a006 +543. P. Pyykkö, Annual Rev. Phys. Chem. 63, 45 (2012). URL https://doi.org/10. +1146/annurev-physchem-032511-143755 +544. N.C. Pyper, Philosophical Transactions of the Royal Society A: Mathematical, +Physical and Engineering Sciences 378(2180), 20190305 (2020). +DOI 10.1098/ +rsta.2019.0305. URL https://royalsocietypublishing.org/doi/abs/10.1098/ +rsta.2019.0305 +545. E. Scerri, Educ. Chem 50(6), 24 (2013) +546. E.R. Scerri, W. Parsons, in Mendeleev to Oganesson: A multidiciplinary perspective +on the periodic table (Oxford University Press, Oxford, 2018), pp. 140–151 + +90 +547. B. Fricke, in Recent impact of physics on inorganic chemistry (Springer, 1975), pp. +89–144 +548. K. Umemoto, S. Saito, Journal of the Physical Society of Japan 65(10), 3175 (1996). +DOI 10.1143/JPSJ.65.3175. URL https://doi.org/10.1143/JPSJ.65.3175 +549. J.T. Waber, D.T. Cromer, D. Liberman, The Journal of Chemical Physics 51(2), 664 +(1969). DOI 10.1063/1.1672054. URL https://doi.org/10.1063/1.1672054 +550. J.B. Mann, J.T. Waber, J. Chem. Phys. 53, 2397 (1970). DOI 10.1063/1.1674338. +URL https://doi.org/10.1063/1.1674338 +551. J.P. Dognon, P. Pyykkö, Angew. Chem. Int. Ed. 56, 10132 (2017). DOI 10.1002/anie. +201701609. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/anie. +201701609 +552. R. Eichler, Radiochim. Acta 107(9-11), 865 (2019). +URL https://doi.org/10. +1515/ract-2018-3080 +553. P.J. Karol, R.C. Barber, B.M. Sherrill, E. Vardaci, T. Yamazaki, Pure and Applied +Chemistry 88(1-2), 155 (2016). DOI doi:10.1515/pac-2015-0501. URL https:// +doi.org/10.1515/pac-2015-0501 +554. W. Nazarewicz, M. Bender, S. ´Cwiok, P. Heenen, A. Kruppa, P.G. Reinhard, T. Vertse, +Nucl. Phys. A 701, 165 (2002). DOI 10.1016/S0375-9474(01)01567-6. URL http: +//www.sciencedirect.com/science/article/pii/S0375947401015676 +555. G.T. Seaborg, J.L. Bloom, Scientific American 220(4), 56 (1969). URL https:// +www.jstor.org/stable/24926334 +556. Y.T. Oganessian, K. Rykaczewski, Phys. Today 68, 32 (2015). DOI 10.1063/PT.3. +2880. URL https://doi.org/10.1063/PT.3.2880 +557. C.E. +Düllmann, +M. +Block, +Sci. +Am. +318, +48 +(2018). +URL +https://www.scientificamerican.com/article/ +the-quest-for-superheavy-elements-and-the-island-of-stability/ +558. K. Chapman, Philosophical Transactions of the Royal Society A 378(2180), 20190535 +(2020). URL http://doi.org/10.1098/rsta.2019.0535 +559. A. Staszczak, A. Baran, W. Nazarewicz, Phys. Rev. C 87, 024320 (2013). +DOI 10.1103/PhysRevC.87.024320. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.87.024320 +560. A. Baran, M. Kowal, P.G. Reinhard, L. Robledo, A. Staszczak, M. Warda, Nucl. +Phys. A 944, 442 (2015). +DOI 10.1016/j.nuclphysa.2015.06.002. +URL http: +//www.sciencedirect.com/science/article/pii/S037594741500130X +561. S.A. Giuliani, G. Martínez-Pinedo, L.M. Robledo, Phys. Rev. C 97, 034323 (2018). +DOI 10.1103/PhysRevC.97.034323. +URL https://link.aps.org/doi/10.1103/ +PhysRevC.97.034323 +562. J. Khuyagbaatar, A. Yakushev, C.E. Düllmann, D. Ackermann, L.L. Andersson, +M. Asai, M. Block, R.A. Boll, H. Brand, D.M. Cox, M. Dasgupta, X. Derkx, +A. Di Nitto, K. Eberhardt, J. Even, M. Evers, C. Fahlander, U. Forsberg, J.M. +Gates, N. Gharibyan, P. Golubev, K.E. Gregorich, J.H. Hamilton, W. Hartmann, R.D. +Herzberg, F.P. Heßberger, D.J. Hinde, J. Hoffmann, R. Hollinger, A. Hübner, E. Jäger, +B. Kindler, J.V. Kratz, J. Krier, N. Kurz, M. Laatiaoui, S. Lahiri, R. Lang, B. Lom- +mel, M. Maiti, K. Miernik, S. Minami, A.K. Mistry, C. Mokry, H. Nitsche, J.P. +Omtvedt, G.K. Pang, P. Papadakis, D. Renisch, J.B. Roberto, D. Rudolph, J. Runke, +K.P. Rykaczewski, L.G. Sarmiento, M. Schädel, B. Schausten, A. Semchenkov, D.A. +Shaughnessy, P. Steinegger, J. Steiner, E.E. Tereshatov, P. Thörle-Pospiech, K. Tin- +schert, T. Torres De Heidenreich, N. Trautmann, A. Türler, J. Uusitalo, M. We- +grzecki, N. Wiehl, S.M. Van Cleve, V. Yakusheva, Phys. Rev. C 102, 064602 (2020). + +91 +DOI 10.1103/PhysRevC.102.064602. URL https://link.aps.org/doi/10.1103/ +PhysRevC.102.064602 +563. Y.T. Oganessian, V.K. Utyonkov, N.D. Kovrizhnykh, F.S. Abdullin, S.N. Dmitriev, +D. Ibadullayev, M.G. Itkis, D.A. Kuznetsov, O.V. Petrushkin, A.V. Podshibiakin, A.N. +Polyakov, A.G. Popeko, R.N. Sagaidak, L. Schlattauer, I.V. Shirokovski, V.D. Shubin, +M.V. Shumeiko, D.I. Solovyev, Y.S. Tsyganov, A.A. Voinov, V.G. Subbotin, A.Y. Bo- +drov, A.V. Sabel’nikov, A.V. Khalkin, V.B. Zlokazov, K.P. Rykaczewski, T.T. King, +J.B. Roberto, N.T. Brewer, R.K. Grzywacz, Z.G. Gan, Z.Y. Zhang, M.H. Huang, H.B. +Yang, Phys. Rev. C 106, L031301 (2022). +DOI 10.1103/PhysRevC.106.L031301. +URL https://link.aps.org/doi/10.1103/PhysRevC.106.L031301 +564. M.J. Esteban, Comptes Rendus. Physique 21(2), 177 (2020). DOI 10.5802/crphys.16 +565. R. de la Madrid, European journal of physics 26(2), 287 (2005). +URL https:// +iopscience.iop.org/article/10.1088/0143-0807/26/2/008 +566. N. Arrizabalaga, J. Duoandikoetxea, L. Vega, Journal of Mathematical Physics 54(4), +041504 (2013). +DOI 10.1063/1.4798804. +URL https://doi.org/10.1063/1. +4798804 +567. B.L. Voronov, D.M. Gitman, I.V. Tyutin, Theoretical and Mathematical Physics +150(1), 34 (2007). DOI 10.1007/s11232-007-0004-5. URL https://doi.org/10. +1007/s11232-007-0004-5 +568. T. Kato, Perturbation theory for linear operators, vol. 132 (Springer Science & Busi- +ness Media, 2013) +569. K. Maurin, Lecture Notes, ICTP, Trieste, preprint IC/66/12 (1966) +570. A. Böhm, The Rigged Hilbert Space and Quantum Mechanics: Lectures in Mathemat- +ical Physics at the University of Texas at Austin (Springer-Verlag, 1978) +571. A. Bohm, S. Maxson, M. Loewe, M. Gadella, Physica A: Statistical Me- +chanics and its Applications 236(3), 485 (1997). +DOI https://doi.org/10. +1016/S0378-4371(96)00284-1. URL https://www.sciencedirect.com/science/ +article/pii/S0378437196002841 +572. A.M. Perelomov, Y.B. Zeldovich, Quantum Mechanics, Selected Topics (World Scien- +tific, Singapore, 1998) +573. M. Gadella, F. Gómez, International Journal of Theoretical Physics 42(10), 2225 +(2003). +DOI 10.1023/B:IJTP.0000005956.11617.e9. +URL https://doi.org/10. +1023/B:IJTP.0000005956.11617.e9 +574. J.P. Antoine, A. Bohm, S. Wickramasekara, in Compendium of Quantum Physics +(Springer, 2009), pp. 651–660 +575. J.P. Antoine, Entropy 23(1) (2021). DOI 10.3390/e23010124. URL https://www. +mdpi.com/1099-4300/23/1/124 +576. J.E. Roberts, Journal of Mathematical Physics 7(6), 1097 (1966). DOI 10.1063/1. +1705001. URL https://doi.org/10.1063/1.1705001 +577. J.E. Roberts, Communications in Mathematical Physics 3(2), 98 (1966). DOI 10.1007/ +BF01645448. URL https://doi.org/10.1007/BF01645448 +578. A. Bohm, in Boulder Lectures in Theoretical Physics IX A: Mathematical Methods of +Theoretical Physics (Wiley, New York, NY, USA, 1967) +579. J. Antoine, Journal of Mathematical Physics 10(1), 53 (1969). +DOI 10.1063/1. +1664761. URL https://doi.org/10.1063/1.1664761 +580. J. Antoine, Journal of Mathematical Physics 10(12), 2276 (1969). DOI 10.1063/1. +1664834. URL https://doi.org/10.1063/1.1664834 +581. C. Trapani, S. Triolo, F. Tschinke, Journal of Fourier Analysis and Applications 25(4), +2109 (2019). DOI 10.1007/s00041-018-09659-5. URL https://doi.org/10.1007/ + +92 +s00041-018-09659-5 +582. I.M. Gel’fand, N.Y. Vilenkin, Generalized functions: applications of harmonic analy- +sis, vol. 4 (Academic Press, 2014) +583. P. Blanchard, E. Brüning, in Mathematical Methods in Physics (Springer, 2015), pp. +439–453 +584. P.A.M. Dirac, The principles of quantum mechanics. 27 (Oxford university press, +1981) +585. N.N. Bogolubov, A. Logunov, I. Todorov, Reading, Mass (1975) +586. I. Antoniou, M. Gadella, I. Prigogine, G.P. Pronko, Journal of Mathematical Physics +39(6), 2995 (1998). DOI 10.1063/1.532235. URL https://doi.org/10.1063/1. +532235 +587. E. Celeghini, M. Gadella, M.A. del Olmo, Axioms 8(3) (2019). +DOI 10.3390/ +axioms8030089. URL https://www.mdpi.com/2075-1680/8/3/89 +588. R. de la Madrid, Journal of Mathematical Physics 53(10), 102113 (2012). DOI 10. +1063/1.4758925. URL https://doi.org/10.1063/1.4758925 + diff --git a/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/load_file.txt b/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f439568cd80d189ef5bd80ebe632adf8aa7315f4 --- /dev/null +++ b/o9E0T4oBgHgl3EQfqgFN/content/tmp_files/load_file.txt @@ -0,0 +1,11435 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf,len=11434 +page_content='Pushing the Limits of the Periodic Table – A Review on Atomic Relativistic Electronic Structure Theory and Calculations for the Superheavy Elements∗ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Smitsa,1, P.' metadata={'source': 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F-75005 Paris, France 3Facility for Rare Isotope Beams and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA In memoriam to two of the pioneers in this field, Jean-Paul Desclaux (Grenoble) and Sigurd Hofmann (Darmstadt) Received: date / Accepted: date Contents 1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='2 Hartree-Fock-Bogoliubov equation analogy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='4 Calculation of QED corrections .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='1 One-electron radiative corrections .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='2 Two-electron radiative and non-radiative corrections .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 57 8 Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 58 9 Appendix A: The Self-Adjointness of the Dirac-Coulomb Hamiltonian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 58 10 Appendix B: The Rigged Hilbert Space Formalism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 60 Abstract We review the progress in atomic structure theory with a focus on superheavy ele- ments and the aim to predict their ground state configuration and element’s placement in the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' To understand the electronic structure and correlations in the regime of large atomic numbers, it is important to correctly solve the Dirac equation in strong Coulomb fields, and also to take into account quantum electrodynamic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' We specifically fo- cus on the fundamental difficulties encountered when dealing with the many-particle Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' We further discuss the possibility for future many-electron atomic structure calcu- lations going beyond the critical nuclear charge Zcrit ≈ 170, where levels such as the 1s shell dive into the negative energy continuum (Enκ < −mec2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The nature of the resulting Gamow states within a rigged Hilbert space formalism is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 1 Introduction The periodic table (PT) of the elements, introduced by Dmitri Mendeleev and Lothar Meyer, is based on the Pauli and Aufbau (building-up) principle [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Arguably, the PT is the most important and useful tool concerning the electronic structure of atoms and molecules [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Chemical and physical similarities between the elements within a group or period obtained from their measurable properties is often hailed as a building block of the PT, but these patterns also follow from the underlying electronic shell structure of the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Despite many controversies concerning the PT, for example, the starting and ending points of the f-block elements, the placement of the lightest elements hydrogen and helium, observed anomalies in chemical behavior or even the shape and visual representation [4, 6–8], it is still going strong after 150 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Furthermore, with the nuclear synthesis of the 7p block elements up to oganesson with nuclear charge Z = 118 [9, 10], the full 7th period of the PT is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Hence, what remains to be solved is how the PT can successfully be extended both theoretically and experimentally into the superheavy element region beyond Z = 118 [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' A progress in this direction has been made by placing the unknown elements up to nuclear charge Z = 172 into the Periodic Table [16, 17], see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The existence and properties of new superheavy elements beyond oganesson depends on both nuclear and electronic structure properties [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' There are, however, a number of open questions and major challenges to both electronic and nuclear structure theory concerning the accurate prediction of physical and chemical properties of the superheavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' * Here we define the starting point of the superheavy element region at the transactinides, Z ≥ 103 3 For example, to correctly place an element into the PT and predict its basic properties, one should gain knowledge of its atomic shell structure, such as ground and excited electronic states and underlying dominant configurations [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' In the case of dense spectra, which are prominent in open-shell systems as well as in the superheavy element region where high principal quantum number and angular momentum states are occupied, detailed knowledge of low-lying excited electronic states are required within a window of a few eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' This is often a very challenging task as both relativistic and electron correlation effects play a major role requiring sophisticated multi-reference methods at the relativistic Dirac-Coulomb-Breit level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Currently, the heaviest element for which it is possible to compare theory and experiment is lawrencium (Z = 103) [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Moreover, the Dirac-Coulomb Hamiltonian has its limits in strong Coulomb fields as beyond the critical nuclear charge of Zcrit ≈ 170 for finite-size nuclei, the 1s electron level dives into the negative energy continuum below E = −mec2 [21–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' At the single-particle level of theory, the correct description and interpretation of the resulting resonances can be given in terms of Gamow states [33–37], but how such diving states can correctly and accurately be described within a multi-electron framework, and how the PT can be extended beyond the critical nuclear charge, are open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' At high nuclear charge, the PT is ultimately limited by the nuclear stability, not by its electronic shell structure [18, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' For nuclear structure theory and corresponding pre- dictions of nuclear stability of isotopes see for example Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' [18, 38, 39] and references Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 1 Pyykkö’s periodic table extended to Z = 172 (with permission from PCCP [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='Group ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='132 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='133 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='134 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='135 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='136 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='137 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='5g4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Here we focus solely on the discussion of relativistic electronic structure theory in the superheavy element region [14, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The outline of this Review is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' We first discuss the Dirac equation and its pe- culiarities compared to the non-relativistic Schrödinger equation, specifically for electrons in strong Coulomb fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' We discuss the critical nuclear charge in detail to clarify the region of validity of the Dirac-Coulomb Hamiltonian and discuss how states embedded in the neg- ative energy continuum should be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The process of spontaneous pair creation in a supercritical field is analyzed including most recent references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The importance of quan- tum electrodynamics (QED) effects and how these can be treated in strong Coulomb fields is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The major problem of correctly describing electron correlation for the accurate prediction of electronic spectra in the superheavy element region is addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' We review the current status of electronic structure calculations for the transactinides and discuss the placement of the elements beyond oganesson into the PT based on quantum theoretical pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The literature on this topic is vast [28, 43–45], including a rigorous mathematical treatment of the Dirac equation and its generalizations [29, 46–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 2 The Dirac Equation in Strong Coulomb Fields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='1 The QED Lagrangian Electronic structure theory is based on the QED sector of the Standard Model of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Within the Standard Model, electrons are spin-1/2 Dirac fermions, and their dy- namics is described by the QED Lagrangian density LQED = i¯hc ¯ψ(x)γµ∂µψ(x)−mec2 ¯ψ(x)ψ(x) −1 4FµνFµν −e ¯ψ(x)γµAµ(x)ψ(x), (1) where ψ(x) is the field operator and γµ are the Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The first two terms in (1) are the kinetic and mass terms describing the free electrons with mass me, whereas the third term describes the photon field Aµ = (φ,AAA), corresponding to the electromagnetic scalar and vector potentials (with Fµν = ∂µAν −∂νAµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The last term corresponds to the interaction between electrons and photons, with the elementary charge e acting as the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The interaction picture represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' (1) has been extensively used in quantum field theory and it has been demonstrated to work to astonishingly high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' It would be highly desirable to treat the QED Lagrangian for a many-electron system in an external Coulomb field to avoid divergencies that appear in perturbative treatments [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Such a direct treatment could in principle be performed through lattice gauge theory which is mathematically well defined [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' However, the long-range nature of the Coulomb potential, related to the zero rest-mass of the photon, currently prevents any accurate com- putational treatment using lattice gauge theory in finite boxes [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' Treating the required large boxes is currently computationally too demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' However, progress in this field has recently been made on the nuclear length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' For instance, a combined lattice QCD+QED approach has been used to successfully calculate hadron and meson mass differences, such as the proton-to-neutron mass splitting, and its dependence on both the strong and electro- magnetic coupling constants [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='2 The Many-Electron Dirac-Coulomb-Breit Hamiltonian Atomic physics calculations are performed in the Hamiltonian formalism derived from the Langrangian (1) by a Legendre transformation [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' The resulting first-quantized N- particle Hamiltonian can be written in atomic units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' ¯h = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='e = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content='me = 1) as [45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E0T4oBgHgl3EQfqgFN/content/2301.02553v1.pdf'} +page_content=' 62]: HD = N ∑ k=1 hk + N ∑ k 1, we define the α-Rényi divergence by +Dα(µ∥ν) := +1 +α − 1 log +�� �µ(ω) +ν(ω) +�α +dν(ω) +� +. +(4) +4In the low-dimensional regime d ≤ nϵ2 +dp, the gradient queries used per subproblem improves to +√ +nd +ϵdp +√ +K . +9 + +Dα is quasiconvex in its arguments, i.e. if µ = Eξµξ and ν = Eξνξ (where ξ is a random variable, +and µξ, νξ are distribution families indexed by ξ), then Dα(µ∥ν) ≤ maxξ Dα(µξ∥νξ). +2 +Framework +We now outline our primary technical innovation, a new gradient estimator for stochastic convex +optimization (ReSQue). We define this estimator in Section 2.1 and prove that it satisfies several +local stability properties in a small ball around a “centerpoint” used for its definition. In Section 2.2, +we then give preliminaries on a “ball acceleration” framework developed in [CJJ+20, ACJ+21]. This +framework aggregates solutions to proximal subproblems defined on small (Euclidean) balls, and +uses these subproblem solutions to efficiently solve an optimization problem on a larger domain. +Our algorithms in Sections 3 and 4 instantiate the framework of Section 2.2 with new subproblem +solvers enjoying improved parallelism or privacy, based on our new ReSQue estimator. +2.1 +ReSQue estimators +Throughout we use γρ : Rd → R≥0 to denote the probability density function of N(0, ρ2Id), i.e. +γρ(x) = (2πρ)− d +2 exp(− 1 +2ρ2 ∥x∥2). We first define the Gaussian convolution operation. +Definition 1 (Gaussian convolution). For a function f : Rd → R we denote its convolution with a +Gaussian of covariance ρ2Id by �fρ := f ∗ γρ, i.e. +�fρ(x) := Ey∼N(0,ρ2Id)f(x + y) = +� +y∈Rn f(x − y)γρ(y)dy. +(5) +Three well-known properties of �fρ are that it is differentiable, that if f is L-Lipschitz, so is �fρ for +any ρ, and that | �fρ − f| ≤ Lρ +√ +d pointwise (Lemma 8, [BJL+19]). Next, given a centerpoint ¯x and +a smoothing radius ρ, we define the associated reweighted stochastic query (ReSQue) estimator. +Definition 2 (ReSQue estimator). Let ¯x ∈ Rd and let f : Rd → R be convex. Suppose we have a +gradient estimator g : Rd → Rd satisfying Eg ∈ ∂f. We define the ReSQue estimator of radius ρ as +the random vector +�∇g +¯x �fρ(x) := γρ(x − ¯x − ξ) +γρ(ξ) +g(¯x + ξ) where ξ ∼ N(0, ρ2Id), +where we first sample ξ, and then independently query g at ¯x + ξ. When g is deterministically an +element of ∂f, we drop the superscript and denote the estimator by �∇¯x �fρ. +When g is unbiased for ∂f and enjoys a variance bound, the corresponding ReSQue estimator +is unbiased for the convolved function, and inherits a similar variance bound. +Lemma 1. The estimator in Definition 2 satisfies the following properties, where expectations are +taken over both the randomness in ξ and the randomness in g. +1. Unbiased: E�∇g +¯x �fρ(x) = ∇ �fρ(x). +2. Bounded variance: If E ∥g∥2 ≤ L2 everywhere, and x ∈ B¯x(ρ), then E∥�∇g +¯x �fρ(x)∥2 ≤ 3L2. +10 + +Proof. The first statement follows by expanding the expectation over ξ and g: +Eg +� γρ(x − ¯x − ξ) +γρ(ξ) +g(¯x + ξ)γρ(ξ)dξ = +� γρ(x − ¯x − ξ) +γρ(ξ) +∂f(¯x + ξ)γρ(ξ)dξ += +� +∂f(¯x + ξ)γρ(x − ¯x − ξ)dξ = ∇ �fρ(x). +The last equality used that the integral is a subgradient of �fρ, and �fρ is differentiable. +For the second statement, denote v := x − ¯x for simplicity. Since f is L-Lipschitz, +E∥�∇g +¯x �fρ(x)∥2 = Eg +� (γρ(v − ξ))2 +γρ(ξ) +∥g(¯x + ξ)∥2 dξ +≤ L2(2πρ)− d +2 +� +exp +� +−∥v − ξ∥2 +ρ2 ++ ∥ξ∥2 +2ρ2 +� +dξ. +Next, a standard calculation for Gaussian integrals shows +� +exp +� +2 ⟨v, ξ⟩ − ∥ξ∥2 +2ρ2 +� +dξ = exp +� +∥v∥2 +2ρ2 +� � +exp +� +−∥ξ − v∥2 +2ρ2 +� +dξ = exp +� +∥v∥2 +2ρ2 +� +(2πρ) +d +2 . +(6) +The statement then follows from (6), which yields +� +exp +� +−∥v − ξ∥2 +ρ2 ++ ∥ξ∥2 +2ρ2 +� +dξ = exp +� +−∥v∥2 +ρ2 +� � +exp +� +4 ⟨v, ξ⟩ − ∥ξ∥2 +2ρ2 +� +dξ += (2πρ) +d +2 exp +� +2 ∥v∥2 +ρ2 +� +≤ 3 · (2πρ) +d +2 +(7) +and completes the proof of the second statement. +When the gradient estimator g is deterministically a subgradient of a Lipschitz function, we can +show additional properties about ReSQue. The following lemma will be used in Section 4 both to +obtain higher moment bounds on ReSQue, as well as higher moment bounds on the difference of +ReSQue estimators at nearby points, where the bound scales with the distance between the points. +Lemma 2. If x, x′ ∈ B¯x( ρ +p) for p ≥ 2 then +Eξ∼N(0,ρ2Id) +��γρ(x − ¯x − ξ) +γρ(ξ) +�p� +≤ 2, +Eξ∼N(0,ρ2Id) +����� +γρ(x − ¯x − ξ) − γρ(x′ − ¯x − ξ) +γρ(ξ) +���� +p� +≤ +�24p ∥x − x′∥ +ρ +�p +. +We defer a proof to Appendix A, where a helper calculation (Fact 3) is used to obtain the result. +2.2 +Ball acceleration +We summarize the guarantees of a recent “ball acceleration” framework originally proposed by +[CJJ+20]. For specified parameters 0 < r < R, this framework efficiently aggregates (approximate) +solutions to constrained optimization problems over Euclidean balls of radius r to optimize a function +over a ball of radius R. Here we give an approximation-tolerant variant of the [CJJ+20] algorithm +11 + +in Proposition 1, which was developed by [ACJ+21]. +Before stating the guarantee, we require +the definitions of three types of oracles. +In each of the following definitions, for some function +F : Rd → R, scalars λ, r, and point ¯x ∈ Rd which are clear from context, we will denote +x⋆ +¯x,λ := argminx∈B¯x(r) +� +F(x) + λ +2 ∥x − ¯x∥2 +� +. +(8) +We mention that in the non-private settings of prior work [ACJ+21, CH22] (and under slightly +different oracle access assumptions), it was shown that the implementation of line search oracles +(Definition 3) and stochastic proximal oracles (Definition 5) can be reduced to ball optimization +oracles (Definition 4). Indeed, such a result is summarized in Proposition 2 and used in Section 3 to +obtain our parallel SCO algorithms. To tightly quantify the privacy loss of each oracle for developing +our SCO algorithms in Section 4 (and to implement these oracles under only the function access +afforded by Problem 2), we separate out the requirements of each oracle definition separately. +Definition 3 (Line search oracle). We say Ols is a (∆, λ)-line search oracle for F : Rd → R if +given ¯x ∈ Rd, Ols returns x ∈ Rd with +��x − x⋆ +¯x,λ +�� ≤ ∆. +Definition 4 (Ball optimization oracle). We say Obo is a (φ, λ)-ball optimization oracle for F : +Rd → R if given ¯x ∈ Rd, Obo returns x ∈ Rd with +E +� +F(x) + λ +2 ∥x − ¯x∥2 +� +≤ F(x⋆ +¯x,λ) + λ +2 +��x⋆ +¯x,λ − ¯x +��2 + φ. +Definition 5 (Stochastic proximal oracle). We say Osp is a (∆, σ, λ)-stochastic proximal oracle for +F : Rd → R if given ¯x ∈ Rd, Osp returns x ∈ Rd with +��Ex − x⋆ +¯x,λ +�� ≤ ∆ +λ , E +��x − x⋆ +¯x,λ +��2 ≤ σ2 +λ2 . +Leveraging Definitions 3, 4, and 5, we state a variant of the main result of [ACJ+21]. Roughly +speaking, Proposition 1 states that to optimize a function F over a ball of radius R, it suffices to +query ≈ ( R +r ) +2 +3 oracles which approximately optimize a sufficiently regularized variant of F over a +ball of radius r. We quantify the types of approximate optimization of such regularized functions +in Proposition 1, and defer a detailed discussion of how to derive this statement from [ACJ+21] in +Appendix B, as it is stated slightly differently in the original work.5 +Proposition 1. Let F : Rd → R be L-Lipschitz and convex, and suppose for R ≥ 0 there is +x⋆ ∈ argminxF(x) with x⋆ ∈ B(R). There is an algorithm BallAccel taking parameters r ∈ [0, R] +and ϵopt ∈ (0, LR] with the following guarantee. Define +κ := LR +ϵopt +, K := +�R +r +� 2 +3 +, λ⋆ := ϵoptK2 +R2 +log2 κ. +For a universal constant Cba > 0, BallAccel runs in at most CbaK log κ iterations and produces a +point x such that +EF(x) ≤ F(x⋆) + ϵopt. +Moreover, in each iteration BallAccel requires the following oracle calls (all for F). +5In particular, we use an error tolerance for the ball optimization oracles, which is slightly larger than in [ACJ+21], +following a tighter error analysis given in Proposition 1 of [CH22]. +12 + +1. At most Cba log( Rκ +r ) calls to a ( r +Cba , λ)-line search oracle with values of λ ∈ [ λ⋆ +Cba , CbaL +ϵopt ]. +2. A single call to ( +λr2 +Cba log3 κ, λ)-ball optimization oracle with λ ∈ [ λ⋆ +Cba , CbaL +ϵopt ]. +3. A single call to ( ϵopt +CbaR, ϵopt +√ +K +CbaR , λ)-stochastic proximal oracle with λ ∈ [ λ⋆ +Cba , CbaL +ϵopt ]. +The optimization framework in Proposition 1 is naturally compatible with our ReSQue estima- +tors, whose stability properties are local in the sense that they hold in balls of radius ≈ ρ around +the centerpoint ¯x (see Lemma 2). Conveniently, BallAccel reduces an optimization problem over a +domain of size R to a sequence of approximate optimization problems on potentially much smaller +domains of radius r. In Sections 3 and 4, by instantiating Proposition 1 with r ≈ ρ, we demonstrate +how to use the local stability properties of ReSQue estimators (on smaller balls) to solve constrained +subproblems, and consequently design improved parallel and private algorithms. +Finally, as mentioned previously, in settings where privacy is not a consideration, Proposition +1 of [CH22] gives a direct implementation of all the line search and stochastic proximal oracles +required by Proposition 1 by reducing them to ball optimization oracles. The statement in [CH22] +also assumes access to function evaluations in addition to gradient (estimator) queries; however, it is +straightforward to use geometric aggregation techniques (see Lemma 11) to bypass this requirement. +We give a slight rephrasing of Proposition 1 in [CH22] without the use of function evaluation oracles, +and defer further discussion to Appendix C where we prove the following. +Proposition 2. Let F : Rd → R be L-Lipschitz and convex, and suppose for R ≥ 0 there is +x⋆ ∈ argminxF(x) with x⋆ ∈ B(R). There is an implementation of BallAccel (see Proposition 1) +taking parameters r ∈ [0, R] and ϵopt ∈ (0, LR] with the following guarantee, where we define κ, K, λ⋆ +as in Proposition 1. +For a universal constant Cba > 0, BallAccel runs in at most CbaK log κ +iterations and produces a point x such that EF(x) ≤ F(x⋆) + ϵopt. +1. Each iteration makes at most Cba log2( Rκ +r ) calls to ( λr2 +Cba , λ)-ball optimization oracle with values +of λ ∈ [ λ⋆ +Cba , CbaL +ϵopt ]. +2. For each j ∈ [⌈log2 K+Cba⌉], at most C2 +ba·2−jK log( Rκ +r ) iterations query a ( λr2 +Cba2j ·log−2( Rκ +r ), λ)- +ball optimization oracle for some λ ∈ [ λ⋆ +Cba , CbaL +ϵopt ]. +3 +Parallel stochastic convex optimization +In this section, we present our main results on parallel convex optimization with improved com- +putational depth and total work. We present our main results below in Theorems 1 and 2, after +formally stating our notation and the SCO problem we study in this section. +3.1 +Preliminaries +In this section, we study the following SCO problem, which models access to an objective only +through the stochastic gradient oracle. +Problem 1. Let f : Rd → R be convex. +We assume there exists a stochastic gradient oracle +g : Rd → Rd satisfying for all x ∈ Rd, Eg(x) ∈ ∂f(x), E ∥g(x)∥2 ≤ L2. Our goal is to produce an +ϵopt-approximate minimizer to f constrained to B(R). We define parameter +κ := LR +ϵopt +. +(9) +13 + +When discussing a parallel algorithm which queries a stochastic gradient oracle, in the sense +of Problem 1, we separate its complexity into four parameters. The query depth is the maximum +number of sequential rounds of interaction with the oracle, where queries are submitted in batch. +The total number of queries is the total number of oracle queries used by the algorithm. +The +computational depth and work are the sequential depth and total amount of computational work +done by the algorithm outside of these oracle queries. For simplicity we assume that all d-dimensional +vector operations have a cost of d when discussing computation. +3.2 +Proofs of Theorems 1 and 2 +Theorem 1 (Parallel EpochSGD-based solver). BallAccel (Proposition 2) using parallel EpochSGD +(Algorithm 1) as a ball optimization oracle solves Problem 1 with expected error ϵopt, with +O +� +d +1 +3 κ +2 +3 log3(dκ) +� +query depth and O +� +d +1 +3 κ +2 +3 log3 (dκ) + κ2 log4 (dκ) +� +total queries, +and an additional computational cost of +O +� +d +1 +3 κ +2 +3 log3 (dκ) + κ2 log4 (dκ) +� +depth and O +�� +d +1 +3 κ +2 +3 log3 (dκ) + κ2 log4 (dκ) +� +· d +� +work. +Theorem 2 (Parallel AC-SA-based solver). BallAccel (Proposition 2) using parallel AC-SA (Algo- +rithm 2) as a ball optimization oracle solves Problem 1 with expected error ϵopt, with +O +� +d +1 +3 κ +2 +3 log κ +� +query depth +and O +� +d +1 +3 κ +2 +3 log3 (dκ) + d +1 +4 κ log4 (dκ) + κ2 log4 (dκ) +� +total queries, +and an additional computational cost of +O +� +d +1 +3 κ +2 +3 log3 (dκ) + d +1 +4 κ log4 (dκ) +� +depth +and O +�� +d +1 +3 κ +2 +3 log3 (dκ) + d +1 +4 κ log4 (dκ) + κ2 log4 (dκ) +� +· d +� +work. +The query depth, total number of queries, and total work for both of our results are the same (up +to logarithmic factors). The main difference is that AC-SA attains an improved computational depth +for solving SCO, compared to using EpochSGD. Our results build upon the BallAccel framework +in Section 2.2, combined with careful parallel implementations of the required ball optimization +oracles to achieve improved complexities. +We begin by developing our parallel ball optimization oracles using our ReSQue estimator ma- +chinery from Section 2.1. +First, Proposition 2 reduces Problem 1 to implementation of a ball +optimization oracle. Recall that a ball optimization oracle (Definition 4) requires an approximate +solution x of a regularized subproblem. In particular, for some accuracy parameter φ, and defining +x⋆ +¯x,λ as in (8), we wish to compute a random x ∈ B¯x(r) such that +E +� +�fρ(x) + λ +2 ∥x − ¯x∥2 +� +≤ �fρ(x⋆ +¯x,λ) + λ +2 +��x⋆ +¯x,λ − ¯x +��2 + φ, x ∈ B¯x(r). +Note that such a ball optimization oracle can satisfy the requirements of Proposition 2 with F ← �fρ, +r ← ρ. In particular, Lemma 1 gives a gradient estimator variance bound under the setting r = ρ. +14 + +EpochSGD. +We implement EpochSGD [HK14, ACJ+21], a variant of standard stochastic gradi- +ent descent on regularized objective functions, in parallel using the stochastic ReSQue estimator +constructed in Definition 2. Our main observation is that the gradient queries in Definition 2 can +be implemented in parallel at the beginning of the algorithm. We provide the pseudocode of our +parallel implementation of EpochSGD in Algorithm 1 and state its guarantees in Proposition 3. +Algorithm 1: EpochSGD(f, g, ¯x, r, ρ, λ, φ) +1 Input: f : Rd → R and g : Rd → R satisfying the assumptions of Problem 1, ¯x ∈ Rd, +r, ρ, λ, φ > 0 +2 η1 ← +1 +4λ, T1 ← 16, T ← ⌈ 48L2 +λφ ⌉ +3 Sample ξi ∼ N(0, ρ2Id), i ∈ [2T] independently +4 Query g(¯x + ξi) for all i ∈ [2T] (in parallel) +5 x0 +1 ← ¯x, k ← 1 +6 while � +j∈[k] Tj ≤ T do +7 +x1 +k ← argminx∈B¯x(r) +� +ηkλ +2 ∥x − ¯x∥2 + 1 +2∥x − x0 +k∥2� +8 +for t ∈ [Tk − 1] do +9 +i ← � +j∈[k−1] Tj + t +10 +�∇g +¯x �fρ(xt +k) ← γρ(xt +k−¯x−ξi) +γρ(ξi) +g(¯x + ξi) +11 +xt+1 +k +← argminx∈B¯x(r) +� +ηk⟨�∇g +¯x �fρ(xt +k), x⟩ + ηkλ +2 ∥x − ¯x∥2 + 1 +2∥x − xt +k∥2� +12 +end +13 +x0 +k+1 ← +1 +Tk +� +t∈[Tk] xt +k, Tk+1 ← 2Tk, ηk+1 ← ηk +2 , k ← k + 1 +14 end +15 return x0 +k +Proposition 3 (Proposition 3, [ACJ+21]). Let f, g satisfy the assumptions of Problem 1. When +ρ = r, Algorithm 1 is a (φ, λ)-ball optimization oracle for �fρ which makes O( L2 +φλ) total queries to g +with constant query depth, and an additional computational cost of O( L2 +φλ) depth and work. +AC-SA. +We can also implement AC-SA [GL12], a variant of accelerated gradient descent under +stochastic gradient queries, in parallel using stochastic ReSQue estimators. We provide the pseu- +docode of our parallel implementation of AC-SA in Algorithm 2 and state its guarantees in Lemma 4. +Proposition 4 (Special case of Theorem 1, [GL12]). Let f, g satisfy the assumptions of Problem 1. +When ρ = r, Algorithm 2 is a (φ, λ)-ball optimization oracle for �fρ which makes +O +�� +1 + L +ρλ log +�λr2 +φ +� ++ L2 +λφ +� +total queries +with constant query depth, and an additional computational cost of +O +�� +1 + L +ρλ log +�λr2 +φ +�� +depth and O +�� +1 + L +ρλ log +�λr2 +φ +� ++ L2 +λφ +� +work. +Because the statement of Proposition 4 follows from specific parameter choices in the main result +in [GL12], we defer a more thorough discussion of how to obtain this result to Appendix D. +15 + +Algorithm 2: AC-SA(f, ¯x, r, ρ, λ, φ) +1 Input: f : Rd → R, g : Rd → R satisfying the assumptions of Problem 1, ¯x ∈ Rd, +r, ρ, λ, φ > 0 +2 K ← ⌈log2( λr2 +φ )⌉, T ← ⌈4 +� +L +ρλ + 1⌉, Nk ← +� +48 · 2k · +L2 +λ2r2T +� +for k ∈ [K] +3 Sample ξi ∼ N(0, ρ2Id), i ∈ [N] independently, for N = T · (� +k∈[K] Nk) +4 Query g(¯x + ξi) for all i ∈ [N] (in parallel) +5 xag +0 ← ¯x, x0 ← ¯x +6 for k ∈ [K] do +7 +for t ∈ [T] do +8 +αt ← +2 +t+1, γt ← +4( L +ρ +λ) +t(t+1) +9 +xmd +t +← (1−αt)(λ+γt) +γt+(1−α2 +t )λ xag +t−1 + αt(1−αt)(λ+γt) +γt+(1−α2 +t )λ xt−1 +10 +NT,[k−1] ← T · � +k′∈[k−1] Nk′ +11 +�∇f(xmd +t ) ← +1 +Nk +� +n∈[Nk] +γρ(xmd +t −¯x−ξNT,[k−1]+n) +γρ(ξNT,[k−1]+n) +g(¯x + ξNT,[k−1]+n) +12 +xt ← argminx∈B¯x(r)Ψt(x), where +Ψt(x) := ⟨αt �∇f(xmd +t ) + λ(xmd +t +− ¯x), x − xt⟩ + γt+λ(1−αt) +2 +∥x − xt−1∥2 + λαt +2 ∥x − xmd +t ∥2 +13 +xag +t +← αtxt + (1 − αt)xag +t−1 +14 +end +15 +xag +0 ← xag +T , x0 ← xag +T +16 end +17 Return: xag +T +Main results. +We now use our parallel ball optimization oracles to prove Theorems 1 and 2. +Proofs of Theorems 1 and 2. We use Proposition 2 with r = ρ = +ϵopt +√ +dL on F ← �fρ, which approx- +imates f to additive ϵopt. Rescaling ϵopt by a constant from the guarantee of Proposition 2 gives +the error claim. For the oracle query depths, note that each ball optimization oracle (whether im- +plemented using Algorithm 1 or Algorithm 2) has constant query depth, and at most O(log2(dκ)) +ball optimization oracles are queried per iteration on average. Note that (see Proposition 1) +κ = LR +ϵopt +, K = +�R +r +� 2 +3 += d +1 +3 κ +2 +3 , λ⋆ = ϵoptK2 +R2 +log2 κ = ϵoptd +2 +3 κ +4 +3 +R2 +log2 κ. +For the total oracle queries, computational depth, and work, when implementing each ball +optimization oracle with EpochSGD, we have that for jmax := ⌈log2 K + Cba⌉, these are all +O +� +�K log (dκ) · +� +� � +j∈[jmax] +1 +2j +�L2 · 2j log2(dκ) +λ2⋆r2 +� ++ +� L2 +λ2⋆r2 +� +log2 (dκ) +� +� +� +� += O +� +K log4 (dκ) · L2 +λ2⋆r2 +� += O +� +κ2 log4 (dκ) +� +due to Proposition 3. The additional terms in the theorem statement are due to the number of ball +oracles needed. For the computational depth when implementing each ball optimization oracle with +16 + +AC-SA we have that (due to Proposition 4), it is bounded by +O +� +K log3(dκ) · +� +L +rλ⋆ +log(dκ) +� += O +� +K log4(dκ) · +√κ +K +1 +4 +� += O +� +d +1 +4 κ log4(dκ) +� +. +Finally, for the total oracle queries and work bounds, the bound due to the L2 +λφ term is as was +computed for Theorem 1, and the bound due to the other term is the same as the above display. +4 +Private stochastic convex optimization +We now develop our main result on an improved gradient complexity for private SCO. First, in +Section 4.1, we introduce several variants of differential privacy including a relaxation of Rényi +differential privacy [Mir17], which tolerates a small amount of total variation error. Next, in Sec- +tions 4.2, 4.3, and 4.4, we build several private stochastic optimization subroutines which will be +used in the ball acceleration framework of Proposition 1. Finally, in Sections 4.5 and 4.6, we give +our main results on private ERM and SCO respectively, by leveraging the subroutines we develop. +4.1 +Preliminaries +In this section, we study the following specialization of Problem 1 naturally compatible with pre- +serving privacy with respect to samples, through the formalism of DP (to be defined shortly). +Problem 2. Let P be a distribution over S, and suppose there is a family of functions indexed by +s ∈ S, such that f(·; s) : Rd → R is convex for all s ∈ S. Let D := {si}i∈[n] consist of n i.i.d. draws +from P, and define the empirical risk and population risk by +ferm(x) := 1 +n +� +i∈[n] +f(x; si) and fpop(x) := Es∼Pf(x; s). +We denote fi := f(·; si) for all i ∈ [n], and assume that for all s ∈ S, f(·; s) is L-Lipschitz. We are +given D, and can query subgradients of the “sampled functions” fi. Our goal is to produce an ϵopt +approximate minimizer to fpop constrained to B(R). We again define κ = LR +ϵopt as in (9). +In the “one-pass” setting where we only query each ∂fi a single time, we can treat each ∂fi as a +bounded stochastic gradient of the underlying population risk fpop. We note the related problem of +empirical risk minimization, i.e. optimizing ferm (in the setting of Problem 2), can also be viewed +as a case of Problem 1 where we construct g by querying ∂fi for i ∼unif. [n]. We design (ϵdp, δ)- +DP algorithms for solving Problem 2 which obtain small optimization error for ferm and fpop. To +disambiguate, we will always use ϵopt to denote an optimization error parameter, and ϵdp to denote +a privacy parameter. Our private SCO algorithm will require querying ∂fi multiple times for some +i ∈ [n], and hence incur bias for the population risk gradient. Throughout the rest of the section, +following the notation of Problem 2, we will fix a dataset D ∈ Sn and define the empirical risk ferm +and population risk fpop accordingly. We now move on to our privacy definitions. +We say that two datasets D = {si}i∈[n] ∈ Sn and D′ = {s′ +i}i∈[n] ∈ Sn are neighboring if +|{i | si ̸= s′ +i}| = 1. +We say a mechanism (i.e. a randomized algorithm) M satisfies (ϵdp, δ)- +differential privacy (DP) if, for its output space Ω and all neighboring D, D′, we have for all S ⊆ Ω, +Pr[M(D) ∈ S] ≤ exp(ϵdp) Pr[M(D′) ∈ S] + δ. +(10) +17 + +We extensively use the notion of Rényi differential privacy due to its compatibility with the sub- +sampling arguments we will use, as well as an approximate relaxation of its definition which we +introduce. We say that a mechanism M satisfies (α, ϵ)-Rényi differential privacy (RDP) if for all +neighboring D, D′ ∈ Sn, the α-Rényi divergence (4) satisfies +Dα(M(D)∥M(D′)) ≤ ϵ. +(11) +RDP has several useful properties which we now summarize. +Proposition 5 (Propositions 1, 3, and 7, [Mir17]). RDP has the following properties. +1. (Composition): Let M1 : Sn → Ω satisfy (α, ϵ1)-RDP and M2 : Sn × Ω → Ω′ satisfy (α, ϵ2)- +RDP for any input in Ω. Then the composition of M2 and M1, defined as M2(D, M1(D)) +satisfies (α, ϵ1 + ϵ2)-RDP. +2. (Gaussian mechanism): For µ, µ′ ∈ Rd, Dα(N(µ, σ2Id)∥N(µ′, σ2Id)) ≤ +α +2σ2 ∥µ − µ′∥2. +3. (Standard DP): If M satisfies (α, ϵ)-RDP, then for all δ ∈ (0, 1), M satisfies (ϵ+ +1 +α−1 log 1 +δ, δ)- +DP. +We also use the following definition of approximate Rényi divergence: +Dα,δ(µ∥ν) := +min +DTV(µ′,µ)≤δ,DTV(ν′,ν)≤δ Dα(µ′∥ν′). +(12) +We relax the definition (11) and say that M satisfies (α, ϵ, δ)-RDP if for all neighboring D, +D′ ∈ Sn, recalling definition (12), +Dα,δ(M(D)∥M(D′)) ≤ ϵ. +The following is then immediate from Proposition 5, and our definition of approximate RDP, by +coupling the output distributions with the distributions realizing the minimum (12). +Corollary 1. If M satisfies (α, ϵ, δ)-RDP, then for all δ′ ∈ (0, 1), M satisfies (ϵdp, δ′ + (1 + +exp(ϵdp))δ)-DP for ϵdp := ϵ + +1 +α−1 log 1 +δ′ . +Proof. Let µ, ν be within total variation δ of M(D) and M(D′), such that Dα(µ∥ν) ≤ ϵ and hence +for any event S, +Pr +ω∼µ [ω ∈ S] ≤ exp(ϵdp) Pr +ω∼ν[ω ∈ S] + δ′. +Combining the above with +Pr +ω∼M(D) [ω ∈ S] − δ ≤ Pr +ω∼µ[ω ∈ S], Pr +ω∼ν[ω ∈ S] ≤ +Pr +ω∼M(D′) [ω ∈ S] + δ, +we have +Pr +ω∼M(D)[ω ∈ S] ≤ exp(ϵdp) Pr +ω∼ν[ω ∈ S] + δ′ + δ +≤ exp(ϵdp) +Pr +ω∼M(D′)[ω ∈ S] + δ′ + (1 + exp(ϵdp))δ. +Finally, our approximate RDP notion enjoys a composition property similar to standard RDP. +18 + +Lemma 3. Let M1 : Sn → Ω satisfy (α, ϵ1, δ1)-RDP and M2 : Sn ×Ω → Ω′ satisfy (α, ϵ2, δ2)-RDP +for any input in Ω. Then the composition of M2 and M1, defined as M2(D, M1(D)) satisfies +(α, ϵ1 + ϵ2, δ1 + δ2)-RDP. +Proof. Let D, D′ be neighboring datasets, and let µ, µ′ be distributions within total variation +δ1 of M1(D), M1(D′) realizing the bound Dα(µ∥µ′) ≤ ϵ1. +For any ω ∈ Ω, similarly let νω, +ν′ +ω be the distributions within total variation δ2 of M2(D, ω) and M2(D′, ω) realizing the bound +Dα(νω∥ν′ +ω) ≤ ϵ2. Finally, let P1 be the distribution of ω ∈ Ω according to M1(D), and Q1 to be +the distribution of M1(D′); similarly, let P2,ω, Q2,ω be the distributions of ω′ ∈ Ω′ according to +M2(D, ω) and M2(D′, ω). We first note that by a union bound, +DTV +�� +νω(ω′)µ(ω)dωdω′, +� +P1(ω)P2,ω(ω′)dωdω′ +� +≤ δ1 + δ2, +DTV +�� +ν′ +ω(ω′)µ′(ω)dωdω′, +� +Q1(ω)Q2,ω(ω′)dωdω′ +� +≤ δ1 + δ2. +Finally, by Proposition 1 of [Mir17], we have +Dα +�� +νω(ω′)µ(ω)dωdω′ +����� +� +ν′ +ω(ω′)µ′(ω)dωdω′ +� +≤ ϵ1 + ϵ2. +Combining the above two displays yields the claim. +4.2 +Subsampled smoothed ERM solver: the convex case +We give an ERM algorithm that takes as input a dataset D ∈ Sn, parameters T ∈ N and r, ρ, β > 0, +and a center point ¯x ∈ Rd. Our algorithm is based on a localization approach introduced by [FKT20] +which repeatedly decreases a domain size to bound the error due to adding noise for privacy. In +particular we will obtain an error bound on � +ferm +ρ +with respect to the set B¯x(r), using at most T calls +to the ReSQue estimator in Definition 2 with a deterministic subgradient oracle. Here we recall that +ferm is defined as in Problem 2, and � +ferm +ρ +is correspondingly defined as in Definition 1. Importantly, +our ERM algorithm developed in this section attains RDP bounds improving with the subsampling +parameter T +n when T ≪ n, due to only querying T random samples in our dataset. +We summarize our optimization and privacy guarantees on Algorithm 3 in the following. The +proof follows by combining Lemma 4 (the utility bound) and Lemma 7 (the privacy bound). +Proposition 6. Let x⋆ +¯x ∈ argminx∈B¯x(r) � +ferm +ρ +(x). Algorithm 3 uses at most T gradients and produces +x ∈ B¯x(r) such that, for a universal constant Ccvx, +E +� +� +ferm +ρ +(x) +� +− � +ferm +ρ +(x⋆ +¯x) ≤ CcvxLr +�√ +d +βT + +1 +√ +T +� +. +Moreover, there is a universal constant Cpriv ≥ 1, such that if +T +n ≤ +1 +Cpriv , β2 log2( 1 +δ) ≤ +1 +Cpriv , +δ ∈ (0, 1 +6), and ρ +r ≥ Cpriv log2( log T +δ ), Algorithm 3 satisfies (α, ατ, δ)-RDP for +τ := Cpriv +� +β log +�1 +δ +� +· T +n +�2 +and α ∈ +� +1, +1 +Cprivβ2 log2( 1 +δ) +� +. +19 + +Algorithm 3: Subsampled ReSQued ERM solver, convex case +1 Input: ¯x ∈ Rd, ball radius, convolution radius, and privacy parameter r, ρ, β > 0, dataset +D ∈ Sn, iteration count T ∈ N +2 �T ← 2⌊log2 T⌋, k ← log2 �T, η ← r +L min( 1 +√ +T , β +√ +d), x0 ← ¯x +3 for i ∈ [k] do +4 +Ti ← 2−i �T, ηi ← 4−iη, σi ← Lηi +β +5 +y0 ← xi−1 +6 +for j ∈ [Ti] do +7 +zi,j ∼unif. [n] +8 +yj ← ΠB¯x(r)(yj−1 − ηi �∇¯x �fzi,j +ρ +(yj−1)) ; +▷ PSGD step using ReSQue (See Definition 2) for a +subsampled function. Lemma 5 denotes the random Gaussian sample by ξi,j. +9 +end +10 +¯yi ← 1 +Ti +� +j∈[Ti] yj +11 +xi ← ¯yi + ζi, for ζi ∼ N(0, σ2 +i Id) +12 end +13 return xk +Utility analysis. +We begin by proving a utility guarantee for Algorithm 3, following [FKT20]. +Lemma 4. Let x⋆ +¯x := argminx∈B¯x(r) � +ferm +ρ +(x). We have, for a universal constant Ccvx, +E +� +� +ferm +ρ +(xk) +� +− � +ferm +ρ +(x⋆ +¯x) ≤ CcvxLr +�√ +d +βT + +1 +√ +T +� +. +Proof. Denote F := � +ferm +ρ +, ¯y0 := x⋆ +¯x, and ζ0 := ¯x − x⋆ +¯x, where by assumption ∥ζ0∥ ≤ r. We begin by +observing that in each run of Line 8, by combining the first property in Lemma 1 with the definition +of ferm, we have that E +��∇¯x �fzi,j +ρ +(yj−1) | yj−1 +� +∈ ∂F(yj−1). Moreover, by the second property in +Lemma 1 and the fact that fzi,j is L-Lipschitz, +E +����∇¯x �fzi,j +ρ +(yj−1) +��� +2 +≤ 3L2. +We thus have +E [F(xk)] − F(x⋆ +¯x) = +� +i∈[k] +E[F(¯yi) − F(¯yi−1)] + E [F(xk) − F(¯yk)] +≤ +� +i∈[k] +� +� +E +� +∥xi−1 − ¯yi−1∥2� +2ηiTi ++ 3ηiL2 +2 +� +� + LE [∥xk − ¯yk∥] +≤ 8r2 +ηT + 4 +� +i∈[k−1] +σ2 +i d +ηiTi ++ +� +i∈[k] +3ηiL2 +2 ++ Lσk +√ +d. +(13) +In the second line, we used standard regret guarantees on projected stochastic gradient descent, e.g. +Lemma 7 of [HK14], where we used that all ¯yi ∈ B¯x(r); in the third line, we used +E[∥xk − ¯yk∥] ≤ +� +E +� +∥xk − ¯yk∥2� += +� +E +� +∥ζk∥2� += σk +√ +d +20 + +by Jensen’s inequality. Continuing, we have by our choice of parameters that +σ2 +i +ηiTi ≤ 2−i L2η +2β2 �T , hence +E [F(xk)] − F(x⋆ +¯x) ≤ 8r2 +ηT + 4L2ηd +β2 �T ++ 3ηL2 +2 ++ L2η +√ +d +β +· 1 +�T 2 +≤ +� +8Lr +√ +T ++ 8Lr +√ +d +βT +� ++ 8Lr +√ +d +βT ++ 3Lr +2 +√ +T ++ Lr +√ +T +. +Here we used that 2 �T ≥ T and �T 2 ≥ +√ +T, for all T ∈ N. +Privacy analysis. +We now show that our algorithm satisfies a strong (approximate) RDP guar- +antee. Let D′ = {s′ +i}i∈[n] ∈ Sn be such that D = {si}i∈[n] and D′ are neighboring, and without loss +of generality assume s′ +1 ̸= s1. Define the multiset +I := {zi,j | i ∈ [k], j ∈ [Ti]} +(14) +to contain all sampled indices in [n] throughout Algorithm 3. We begin by giving an (approximate) +RDP guarantee conditioned on the number of times “1” appears in I. The proof of Lemma 5 is +primarily based on providing a potential-based proof of a “drift bound,” i.e. how far away iterates +produced by two neighboring datasets drift apart (coupling all other randomness used). To carry +out this potential proof, we rely on the local stability properties afforded by Lemma 2. +Lemma 5. Define I as in (14) in one call to Algorithm 3. Let I be deterministic (i.e. this statement +is conditioned on the realization of I). Let b be the number of times the index 1 appears in I. Let µ +be the distribution of the output of Algorithm 3 run on D, and µ′ be the distribution when run on D′, +such that D and D′ are neighboring and differ in the first entry, and the only randomness is in the +Gaussian samples used to define ReSQue estimators and on Line 11. Suppose ρ +r ≥ 1728 log2( log T +δ ). +Then we have for any α > 1, +Dα,δ(µ∥µ′) ≤ 1500αβ2b2. +Proof. Throughout this proof we treat I as fixed with b occurrences of the index 1. Let bi be the +number of times 1 appears in Ii := {zi,j | j ∈ [Ti]}, such that � +i∈[k] bi = b. We first analyze the +privacy guarantee of one loop, and then analyze the privacy of the whole algorithm. +We begin by fixing some i ∈ [k], and analyzing the RDP of the ith outer loop in Algorithm 3, +conditioned on the starting point y0. Consider a particular realization of the Ti Gaussian samples +used in implementing Line 8, Ξi := {ξi,j}j∈[Ti], where we let ξi,j ∼ N(0, ρ2Id) denote the Gaussian +sample used to define the update to yj−1. Conditioned on the values of Ii, Ξi, the ith outer loop in +Algorithm 3 (before adding ζi in Line 11) is a deterministic map. For a given realization of Ii and +Ξi, we abuse notation and denote {yj}j∈[Ti] to be the iterates of the ith outer loop in Algorithm 3 +using the dataset D starting at y0, and {y′ +j}j∈[Ti] similarly using D′. Finally, define +Φj := +��yj − y′ +j +��2 , p := +� +5 log +�log T +δ +�� +. +In the following parts of the proof, we will bound for this p the quantity EΦp +Ti, to show that with +high probability it remains small at the end of the loop, regardless of the location of the 1 indices. +Potential growth: iterates with zi,j ̸= 1. We first bound the potential growth in any iteration +j ∈ [Ti] where zi,j ̸= 1. Fix y0, y′ +0 and {ξi,t}t∈[j−1], so that Φj−1 is deterministic. We have (taking +expectations over only ξi,j), +Eξi,jΦp +j ≤ E (Φj−1 + Aj + Bj)p , +(15) +21 + +where +Aj := −2ηiZj +� +∂fzi,j(¯x + ξi,j), yj−1 − y′ +j−1 +� +, +Bj := η2 +i Z2 +j ∥∂fzi,j(¯x + ξi,j)∥2 , +and +Zj := +γρ(yj−1 − ¯x − ξi,j) − γρ(y′ +j−1 − ¯x − ξi,j) +γρ(ξi,j) +. +The inequality in (15) follows from expanding the definition of the update to Φj before projection, +and then using the fact that Euclidean projections onto a convex set only decrease distances. By the +second part of Lemma 2, for all q ∈ [2, p], if +� +Φj−1 ≤ ρ +p (which is always satisfied as +� +Φj−1 ≤ r), +Eξi,jZq +j ≤ +� +24q +� +Φj−1 +ρ +�q +. +By Lipschitzness of fzi,j and Cauchy-Schwarz (on Aj), we thus have +Eξi,j|Aj|q ≤ +�48ηiLqΦj−1 +ρ +�q +for all q ∈ [2, p], +Eξi,jBq +j ≤ +�48ηiLq +ρ +�2q +Φq +j−1 for all q ∈ [1, p]. +(16) +Next, we perform a Taylor expansion of (15), which yields +Eξi,jΦp +j ≤ Φp +j−1 + pΦp−1 +j−1Eξi,j [Aj + Bj] ++ p(p − 1) +� 1 +0 +(1 − t)Eξi,j +� +(Φj−1 + t(Aj + Bj))p−2 (Aj + Bj)2� +dt. +(17) +By monotonicity of convex gradients and the first part of Lemma 1, we have +Eξi,j [Aj] = −2ηi +� +∂ �fzi,j +ρ +(yj−1) − ∂ �fzi,j +ρ +(y′ +j−1), yj−1 − y′ +j−1 +� +≤ 0. +(18) +By applying (16), we have +pΦp−1 +j−1Eξi,jBj ≤ p +�48ηiL +ρ +�2 +Φp +j−1. +(19) +22 + +Next we bound the second-order terms. For any t ∈ [0, 1] we have denoting Cj := Aj + Bj, +Eξi,j +� +(Φj−1 + tCj)p−2 C2 +j +� += +p−2 +� +q=0 +�p − 2 +q +� +Φp−2−q +j−1 +Eξi,j +� +t2+qC2+q +j +� +≤ 4 +p−2 +� +q=0 +2q +�p − 2 +q +� +Φp−2−q +j−1 +Eξi,j +� +|Aj|2+q� ++ 4 +p−2 +� +q=0 +2q +�p − 2 +q +� +Φp−2−q +j−1 +Eξi,j +� +B2+q +j +� +≤ 4Φp +j−1 +�48ηiLp +ρ +�2 p−2 +� +q=0 +2q +�p − 2 +q +� �48ηiLq +ρ +�q ++ 4Φp +j−1 +�48ηiLp +ρ +�2 p−2 +� +q=0 +2q +�p − 2 +q +� �48ηiL(2 + q) +ρ +�2q+2 +≤ 8Φp +j−1 +�48ηiLp +ρ +�2 � +1 + 96ηiLp +ρ +�p−2 +≤ 16Φp +j−1 +�48ηiLp +ρ +�2 +. +(20) +The first inequality used (a + b)p ≤ 2p(ap + bp) for any nonnegative a, b and 0 ≤ t ≤ 1, the second +inequality used (16), and the third and fourth inequalities used +48ηiL(2 + q) +ρ +≤ 1 +2p +for our choices of ηiL ≤ r +4 and ρ. Finally, plugging (18), (19), and (20) into (17), +Eξi,jΦp +j ≤ Φp +j−1 +� +1 + 16p2 +�48ηiLp +ρ +�2� +≤ Φp +j−1 +� +1 + 16p +�48ηiLp +ρ +�2�p +. +Finally, using (ηiL)2 ≤ +r2 +16T ≤ +r2 +16Ti and our assumed bound on r +ρ, which implies 16p +ρ2 (48ηiLp)2 ≤ 1 +Ti , +taking expectations over {ξt}t∈[j−1] yields +EΦp +j ≤ EΦp +j−1 +� +1 + 1 +Ti +�p +when zi,j ̸= 1. +(21) +Potential growth: iterates with zi,j = 1. Next, we handle the case where zi,j = 1. We have that +conditional on fixed values of {ξi,t}t∈[j−1], y0 and y′ +0, +Eξi,jΦp +j ≤ Eξi,j (Φj−1 + Dj + Ej)p +≤ Eξi,j +�� +1 + 1 +bi +� +Φj−1 + 2biEj +�p +, +(22) +where overloading f ← f(·; s1), h ← f(·; s′ +1), +Dj := −2ηi +� +�∇¯x �fρ(yj−1) − �∇¯x�hρ(y′ +j−1), yj−1 − y′ +j−1 +� +, +Ej := η2 +i +����∇¯x �fρ(yj−1) − �∇¯x�hρ(y′ +j−1) +��� +2 +, +23 + +and we use Dj ≤ +1 +bi Φj−1 + biEj by Cauchy-Schwarz and Young’s inequality. Next, convexity of +∥·∥2q implies that +Eq +j ≤ η2q +i 22q−1 +�����∇¯x �fρ(yj−1) +��� +2q ++ +����∇¯x�hρ(y′ +j−1) +��� +2q� +. +Next, we note that since f is Lipschitz, the first part of Lemma 2 implies for all q ≤ p, +E +����∇¯x �fρ(yj−1) +��� +2q +≤ L2qE +��γρ(yj−1 − ¯x − ξ) +γρ(ξ) +�2q� +≤ 2(L)2q, +and a similar calculation holds for h. Here we used our assumed bound on r +ρ to check the requirement +in Lemma 2 is satisfied. By linearity of expectation, we thus have +Eξi,jEq +j ≤ (9ηiL)2q . +(23) +Finally, expanding (22) and plugging in the moment bound (23), +Eξi,jΦp +j ≤ +p +� +q=0 +�p +q +� � +1 + 1 +bi +�q +Φq +j−1(2bi)p−qEξi,j +� +Ep−q +j +� +≤ +p +� +q=0 +�p +q +� � +1 + 1 +bi +�q +Φq +j−1(2bi)p−q(9ηiL)2(p−q) += +�� +1 + 1 +bi +� +Φj−1 + 2bi(9ηiL)2 +�p +. +Taking expectations over {ξi,t}t∈[j−1], and using Fact 4 with Z ← (1+ 1 +bi )Φj−1 and C ← 2bi(9ηiL)2, +EΦp +j ≤ +�� +1 + 1 +bi +� +E +� +Φp +j−1 +� 1 +p + 2bi(9ηiL)2 +�p +, when zi,j = 1. +(24) +One loop privacy. We begin by obtaining a high-probability bound on ΦTi. Define +Wj := E[Φp +j] +1 +p . +By using (21) and (24), we observe +Wj ≤ +� +� +� +� +1 + 1 +Ti +� +Wj−1 +zi,j ̸= 1 +� +1 + 1 +bi +� +Wj−1 + 2bi(9ηiL)2 +zi,j = 1 +. +Hence, regardless of the bi locations of the 1 indices in Ii, we have +WTi ≤ +� +1 + 1 +Ti +�Ti � +1 + 1 +bi +�bi � +2b2 +i (9ηiL)2� +≤ 1200b2 +i (ηiL)2. +Thus, by Markov’s inequality, with probability at least 1 − +δ +log T over the randomness of Ξi = +{ξi,j}j∈[Ti], we have using our choice of p, +��yTi − y′ +Ti +��2 ≤ 1200b2 +i (ηiL)2 · +�log T +δ +� 1 +p +≤ 1500b2 +i (ηiL)2. +(25) +24 + +In the last inequality, we used our choice of p. +Call Ei the event that the sampled Ξi admits +a deterministic map which yields the bound in (25). +By the second part of Proposition 5, the +conditional distribution of the output of the ith outer loop under Ei satisfies (α, 1500β2b2 +i )-RDP, +where we use the value of σi in Line 4 of Algorithm 3. We conclude via Fact 1 with E ← Ei that +the ith outer loop of Algorithm 3 satisfies +� +α, 1500αβ2b2 +i , +δ +log T +� +-RDP. +All loops privacy. By applying composition of RDP (the third part of Proposition 5), for a given +realization of I = ∪i∈[k]Ii with b occurrences of 1, applying composition over the log T outer +iterations (Lemma 3), Algorithm 3 satisfies +� +α, 1500αβ2b2, δ +� +-RDP. +Here, we used � +i∈[k] b2 +i ≤ b2. This is the desired conclusion. +We next apply amplification by subsampling to boost the guarantee of Lemma 5. To do so, we use +the following key Proposition 7, which was proven in [BDRS18]. The use case in [BDRS18] involved +subsampling with replacement and was used in a framework they introduced termed truncated CDP, +but we will not need the framework except through the following powerful fact. +Proposition 7 (Theorem 12, [BDRS18]). Let τ ≤ 1 +3, s ∈ (0, 1 +40). Let P, Q, R be three distributions +over the same probability space, such that for each pair P1, P2 ∈ {P, Q, R}, we have Dα(P1∥P2) ≤ ατ +for all α > 1. Then for all α ∈ (1, 3 +τ ), +Dα(sP + (1 − s)R∥sQ + (1 − s)R) ≤ 13s2ατ. +We also require a straightforward technical fact about binomial distributions. +Lemma 6. Let m, n ∈ N satisfy m +n ≤ 1 +60. Consider the following partition of the elements I ∈ [n]m +with at most b copies of 1: +S0 := {I ∈ [n]m | Ii ̸= 1 for all i ∈ [m]}, +S1 := {I ∈ [n]m | Ii = 1 for between 1 and b many i ∈ [m]}. +Let π0 and π1 be the uniform distributions on S0 and S1 respectively. Then there exists a coupling +Γ(π0, π1) such that for all (I, I′) in the support of Γ, +��� +i | Ii ̸= I′ +i +��� ≤ b. +Proof. Define a probability distribution p on elements of [b] such that +pa := +�m +a +� +(n − 1)m−a +� +a∈[b] +�m +a +� +(n − 1)m−a for all a ∈ [b]. +Clearly, � +a∈[b] pa = 1. Our coupling Γ := Γ(π0, π1) is defined as follows. +1. Draw I ∼ π0 and a ∼ p independently. +2. Let I′ be I with a uniformly random subset of a indices replaced with 1. Return (I, I′). +25 + +This coupling satisfies the requirement, so it suffices to verify it has the correct marginals. This is +immediate for S0 by definition. For I′ ∈ S1, suppose I′ has a occurrences of the index 1. The total +probability I′ is drawn from Γ is then indeed +(n − 1)a +(n − 1)m · pa +�m +a +� = +1 +� +a∈[b] +�m +a +� +(n − 1)m−a = +1 +|S1|. +The first equality follows as the probability we draw I ∼ π0 which agrees with I′ on all the non-1 +locations is (n − 1)a−m, and the probability I′ is drawn given that we selected I is pa · +�m +a +�−1. +Finally, we are ready to state our main privacy guarantee for Algorithm 3. +Lemma 7. There is a universal constant Cpriv ∈ [1, ∞), such that if T +n ≤ +1 +Cpriv , β2 log2( 1 +δ) ≤ +1 +Cpriv , +δ ∈ (0, 1 +6), and ρ +r ≥ Cpriv log2( log T +δ ), Algorithm 3 satisfies (α, ατ, δ)-RDP for +τ := Cpriv +� +β log +�1 +δ +� +· T +n +�2 +, α ∈ +� +1, +1 +Cprivβ2 log2( 1 +δ) +� +. +Proof. Let D, D′ be neighboring, and without loss of generality, suppose they differ in the first +entry. Let Cpriv ≥ 60, and let I be defined as in (14). Let E be the event that I contains at most b +copies of the index 1, where +b := 2 log +�2 +δ +� +. +By a Chernoff bound, E occurs with probability at least 1 − δ +2 over the randomness of I. +We +define P to be the distribution of the output of Algorithm 3 when run on D, conditioned on E and +I containing at least one copy of the index 1 (call this total conditioning event E1, i.e. there are +between 1 and b copies of the index 1). Similarly, we define Q to be the distribution when run on +D′ conditioned on E1, and R to be the distribution conditioned on E ∩ Ec +1 (when run on either D or +D′). We claim that for all P1, P2 ∈ {P, Q, R}, we have +Dα, δ +2 (P1∥P2) ≤ 1500αβ2b2, for all α > 1. +(26) +To see (26) for P1 = P and P2 = Q (or vice versa), we can view P, Q as mixtures of outcomes +conditioned on the realization I. +Then, applying quasiconvexity of Rènyi divergence (over this +mixture), and applying Lemma 5 (with δ ← δ +2), we have the desired claim. To see (26) for the +remaining cases, we first couple the conditional distributions under E1 and E ∩ Ec +1 by their index +sets, according to the coupling in Lemma 6. Then applying quasiconvexity of Rényi divergence +(over this coupling) again yields the claim, where we set m ← �T − 1 ≤ T. Finally, let +s := Pr[E1 | E] = 1 − +� +1 − 1 +n +� �T−1 +Pr[E] +≤ 1 − 1 − 1.1T +n +1 − δ +2 +≤ 1.2T +n +. +Note that conditional on E and the failure event in Lemma 5 not occurring, the distributions of +Algorithm 3 using D and D′ respectively are sP + (1 − s)R and sQ + (1 − s)R. Hence, union +bounding with Ec (see Fact 1), the claim follows from Proposition 7 with τ ← 6000β2 log2( 2 +δ). +26 + +Regularized extension. +We give a slight extension to Algorithm 3 which handles regularization, +and enjoys similar utility and privacy guarantees as stated in Proposition 6. Let +x⋆ +¯x,λ := argminx∈B¯x(r) +� +� +ferm +ρ +(x) + λ +2 ∥x − ¯x∥2 +� +. +(27) +Our extension Algorithm 4 is identical to Algorithm 3, except it requires a regularization parameter +λ, allows for an arbitrary starting point with an expected distance bound (adjusting the step size +accordingly), and takes composite projected steps incorporating the regularization. +Algorithm 4: Subsampled ReSQued ERM solver, regularized case, convex rate +1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, and regularization +parameter r, ρ, β, λ > 0, dataset D ∈ Sn, iteration count T ∈ N, distance bound +r′ ∈ [0, 2r], initial point x0 ∈ B¯x(r) satisfying E∥x0 − x⋆ +¯x,λ∥2 ≤ (r′)2 +2 �T ← 2⌊log2 T⌋, k ← log2 �T, η ← r′ +L min( 1 +√ +T , β +√ +d) +3 for i ∈ [k] do +4 +Ti ← 2−i �T, ηi ← 4−iη, σi ← Lηi +β +5 +y0 ← xi−1 +6 +for j ∈ [Ti] do +7 +zi,j ∼unif. [n] +8 +yj ← argminy∈B¯x(r){⟨ηi �∇¯x �fzi,j +ρ +(yj−1), y⟩ + 1 +2 ∥y − yj−1∥2 + ηiλ +2 ∥y − ¯x∥2} for +zi,j ∼unif. [n] +9 +end +10 +¯yi ← 1 +Ti +� +j∈[Ti] yj +11 +xi ← ¯yi + ζi, for ζi ∼ N(0, σ2 +i Id) +12 end +13 return xk +Corollary 2. Let x⋆ +¯x,λ be defined as in (27). Algorithm 4 uses at most T gradients and produces +x ∈ B¯x(r) such that, for a universal constant Ccvx, +E +� +� +ferm +ρ +(x) + λ +2 ∥x − ¯x∥2 +� +− +� +� +ferm +ρ +(x⋆ +¯x,λ) + λ +2 +��x⋆ +¯x,λ − ¯x +��2 +� +≤ CcvxLr′ +�√ +d +βT + +1 +√ +T +� +. +Moreover, there is a universal constant Cpriv ≥ 1, such that if +T +n ≤ +1 +Cpriv , β2 log2( 1 +δ) ≤ +1 +Cpriv , +δ ∈ (0, 1 +6), and ρ +r ≥ Cpriv log2( log T +δ ), Algorithm 4 satisfies (α, ατ, δ)-RDP for +τ := Cpriv +� +β log +�1 +δ +� +· T +n +�2 +, α ∈ +� +1, +1 +Cprivβ2 log2( 1 +δ) +� +. +Proof. The proof is almost identical to Proposition 6, so we only discuss the differences. Throughout +this proof, for notational convenience, we define +F λ(x) := � +ferm +ρ +(x) + λ +2 ∥x − ¯x∥2 . +Utility. Standard results on composite stochastic mirror descent (e.g. Lemma 12 of [CJST19]) +show the utility bound in (13) still holds with F λ in place of F. In particular each term E[F λ(¯yi) − +27 + +F λ(¯yi−1)] as well as E[F λ(xk)−F λ(¯yk)] enjoys the same bound as its counterpart in (13). The only +other difference is that, defining ζ0 := x0 − x⋆ +¯x,λ in the proof of Lemma 4, we have Eζ2 +0 ≤ (r′)2 in +place of the bound r2, and we appropriately changed η to scale as r′ instead. +Privacy. The subsampling-based reduction from Lemma 7 to Lemma 5 is identical, so we only +discuss how to obtain an analog of Lemma 5 for Algorithm 4. +In each iteration j ∈ [Ti], by +completing the square, we can rewrite Line 8 as +yj ← argminy∈B¯x(r) +� +1 +2 +����y − +� +1 +1 + ηiλyj−1 + +ηiλ +1 + ηiλ ¯x − +ηi +1 + ηiλ +�∇¯x �fzi,j +ρ +(yj−1) +����� +2� +. +Now consider our (conditional) bounds on Eξi,jΦj in (15) and (22). +We claim these still hold +true; before projection, the same arguments used in (15) and (22) still hold (in fact improve by +(1 + ηiλ)2), and projection only decreases distances. Finally, note that the proof of Lemma 5 only +used the choice of step size η through ηL +√ +T ≤ r and used the assumed bound on r +ρ to bound the +drift growth. As we now have ηL +√ +T ≤ r′ ≤ 2r, we adjusted the assumed bound on r +ρ by a factor +of 2. The remainder of the proof of Lemma 5 is identical. +Without loss of generality, Cpriv is the same constant in Proposition 6 and Corollary 2, since we +can set both to be the maximum of the two. The same logic applies to the following Proposition 8 +and Lemma 10 (which will also be parameterized by a Cpriv) so we will not repeat it. Finally, the +following fact about initial error will also be helpful in the following Section 4.3. +Lemma 8. We have +� +ferm +ρ +(¯x) − +� +� +ferm +ρ +(x⋆ +¯x,λ) + λ +2 +��x⋆ +¯x,λ − ¯x +��2 +� +≤ 2L2 +λ . +Proof. By strong convexity and Lipschitzness of � +ferm +ρ +, we have +λ +2 +��x⋆ +¯x,λ − ¯x +��2 ≤ � +ferm +ρ +(¯x) − +� +� +ferm +ρ +(x⋆ +¯x,λ) + λ +2 +��x⋆ +¯x,λ − ¯x +��2 +� +≤ � +ferm +ρ +(¯x) − � +ferm +ρ +(x⋆ +¯x,λ) ≤ L +��x⋆ +¯x,λ − ¯x +�� . +Rearranging gives ∥x⋆ +¯x,λ − ¯x∥ ≤ 2L +λ , which can be plugged in above to yield the conclusion. +We also state a slight extension to Lemma 8 which will be used in Section 4.5. +Lemma 9. Define x⋆ +¯x,x′,λ := argminx∈B¯x(r){ � +ferm +ρ +(x)+ λ +2 ∥x − x′∥2}, where x′ ∈ Rd is not necessarily +in B¯x(r). Let x0 := ΠB¯x(r)(x′). We have +� +� +ferm +ρ +(x0) + λ +2 +��x0 − x′��2 +� +− +� +� +ferm +ρ +(x⋆ +¯x,x′,λ) + λ +2 +��x⋆ +¯x,x′,λ − x′��2 +� +≤ 2L2 +λ . +Proof. The proof is identical to Lemma 8, where we use λ +2 ∥x0 − x′∥2 ≤ λ +2∥x⋆ +¯x,x′,λ − x′∥2. +4.3 +Subsampled smoothed ERM solver: the strongly convex case +We next give an ERM algorithm similar to Algorithm 4, but enjoys an improved optimization rate. +In particular, it again attains RDP bounds improving with the subsampling parameter T +n , and we +obtain error guarantees against x⋆ +¯x,λ defined in (27) at a rate decaying as 1 +T or better. +We now give our analysis of Algorithm 5 below. The proof follows a standard reduction template +from the strongly convex case to the convex case (see e.g. Lemma 4.7 in [KLL21]). +28 + +Algorithm 5: Subsampled ReSQued ERM solver, strongly convex case +1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, and regularization +parameter r, ρ, β, λ > 0, dataset D ∈ Sn, iteration count T ∈ N +2 k ← ⌈log log T⌉, x0 ← ¯x +3 for i ∈ [k] do +4 +βi−1 ← 2 +k−i+1 +2 +β, ri−1 ← min(2r, +� +2Di−1 +λ +) (see (28)), Ti−1 ← 2i−1−kT +5 +xi ← output of Algorithm 4 with inputs (¯x, r, ρ, βi−1, λ, D, Ti−1, ri−1, xi−1) +6 end +7 return xk+1 +Proposition 8. Let x⋆ +¯x,λ be defined as in (27). Algorithm 5 uses at most T gradients and produces +x such that, for a universal constant Csc, +E +� +� +ferm +ρ +(x) + λ +2 ∥x − ¯x∥2 +� +− � +ferm +ρ +(x⋆ +¯x,λ) − λ +2 ∥x⋆ +¯x,λ − ¯x∥2 ≤ CscL2 +λ +� +d +β2T 2 + 1 +T +� +. +Moreover, there is a universal constant Cpriv ≥ 1, such that if T +n ≤ +1 +Cpriv , β2 log2( log log T +δ +) ≤ +1 +Cpriv , +δ ∈ (0, 1 +6), and ρ +r ≥ Cpriv log2( log T +δ ), Algorithm 5 satisfies (α, ατ, δ)-RDP for +τ := Cpriv +� +β log +�log log T +δ +� +· T +n +�2 +, α ∈ +� +1, +1 +Cprivβ2 log2( log log T +δ +) +� +. +Proof. We analyze the utility and privacy separately. +Utility. +Denote for simplicity F λ(x) := � +ferm +ρ +(x) + λ +2∥x − ¯x∥2, F λ +⋆ := F λ(x⋆ +¯x,λ), and ∆i := +E[F λ(xi) − F λ +⋆ ]. Moreover, define for all 0 ≤ i ≤ k, +Ei := 2C2 +cvxL2 +λ +· +� √ +d +βiTi ++ +1 +√Ti +�2 +, Di := 4Ei +2i +� +2L2 +λ +· +1 +4E0 +, +(28) +where we define Tk = T and βk = β. By construction, for all 0 ≤ i ≤ k − 1, Ei+1 = 1 +2Ei, and so +Di+1 +4Ei+1 += +� +Di +4Ei +=⇒ +� +DiEi = Di+1. +(29) +We claim inductively that for all 0 ≤ i ≤ k, ∆i ≤ Di. The base case of the induction follows because +by Lemma 8, we have ∆0 ≤ 2L2 +λ += D0. Next, suppose that the inductive hypothesis is true up to +iteration i. By strong convexity, +E +���xi − x⋆ +¯x,λ +��2� +≤ 2∆i +λ +≤ 2Di +λ , +where we used the inductive hypothesis. Hence, the expected radius upper bound (defined by ri) is +valid for the call to Algorithm 4. Thus, by Corollary 2, +∆i+1 = E +� +F λ(xi+1) − F λ +⋆ +� +≤ CcvxLri +� √ +d +βiTi ++ +1 +√Ti +� +≤ CcvxL +� +2Di +λ +� √ +d +βiTi ++ +1 +√Ti +� += +� +DiEi = Di+1. +29 + +Here we used (29) in the last equation, which completes the induction. Hence, iterating (29) for +k = ⌈log2 log2 T⌉ iterations, where we use E0 ≥ +L2 +2λT so that Dk ≤ 8Ek, we have +∆k ≤ 8Ek ≤ 32C2 +cvxL2 +λ +· +� +d +β2T 2 + 1 +T +� +. +Privacy. The privacy guarantee follows by combining the privacy guarantee in Corollary 2 and +composition of approximate RDP (Lemma 3), where we adjusted the definition of δ by a factor of k. +In particular, we use that the privacy guarantee in each call to Corollary 2 is a geometric sequence +(i.e. β2 +i T 2 +i is doubling), and at the end it is 1 +2β2T 2. +4.4 +Private stochastic proximal estimator +In this section, following the development of [ACJ+21], we give an algorithm which calls Algorithm 5 +with several different iteration counts and returns a (random) point �x which enjoys a substantially +reduced bias for x⋆ +¯x,λ defined in (27) compared to the expected number of gradient queries. +Algorithm 6: Bias-reduced ReSQued stochastic proximal estimator +1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, and regularization +parameter r, ρ, β, λ > 0, dataset D ∈ Sn, iteration count T ∈ N with T ≤ ⌊ +n +2Cpriv ⌋ +2 Tmax ← ⌊ +n +Cpriv ⌋, jmax ← ⌊log2 +Tmax +T +⌋ +3 for k ∈ [jmax] do +4 +Draw J ∼ Geom( 1 +2) +5 +x0 ← output of Algorithm 5 with inputs (¯x, r, ρ, β, λ, D, T) +6 +if J ≤ jmax then +7 +xJ ← output of Algorithm 5 with inputs (¯x, r, ρ, 2− J +2 β, λ, D, 2JT) +8 +xJ−1 ← output of Algorithm 5 with inputs (¯x, r, ρ, 2− J−1 +2 β, λ, D, 2J−1T) +9 +�xk ← x0 + 2J(xJ − xJ−1) +10 +end +11 +else +12 +�xk ← x0 +13 +end +14 end +15 Return: �x ← +1 +jmax +� +k∈[jmax] �xk +Proposition 9. Let x⋆ +¯x,λ be defined as in (27). We have, for a universal constant Cbias: +∥E�x − x⋆ +¯x,λ∥ ≤ Cbias +� +L +λ · +�√ +d +βn + 1 +√n +�� +, +and, for a universal constant Cvar, +E∥�x − x⋆ +¯x,λ∥2 ≤ CvarL2 +λ2 +� +d +β2T 2 + 1 +T +� +. +Proof. We begin by analyzing the output �xk of a single loop k ∈ [jmax]. For J ∼ Geom( 1 +2), we have +Pr[J = j] = 2−j if j ∈ [jmax], and Pr[J = j] = 0 otherwise. We denote xj to be the output of +30 + +Algorithm 3 with privacy parameter 2− j +2 β and gradient bound 2jT. First, +E�xk = Ex0 + +� +j∈[jmax] +Pr[J = j]2j(Exj − Exj−1) = Exjmax. +Since T · 2jmax ≥ Tmax +2 +≥ +n +2Cpriv , applying Jensen’s inequality gives +∥Exjmax − x⋆ +¯x,λ∥ ≤ +� +E∥xjmax − x⋆ +¯x,λ∥2 ≤ +√2CscL +λ +�√ +d +βn + 1 +√n +� +, +where the last inequality follows from Proposition 8 and strong convexity of the regularized function +to convert the function error bound to a distance bound. This implies the first conclusion, our bias +bound. Furthermore, for our variance bound, we have +E∥�xk − E�xk∥2 ≤ E∥�xk − x⋆ +¯x,λ∥2 ≤ 2E∥�xk − x0∥2 + 2E∥x0 − x⋆ +¯x,λ∥2. +By Proposition 8 and strong convexity, E∥x0 − x⋆ +¯x,λ∥2 ≤ CscL2 +λ2 ( +d +β2T 2 + 1 +T ). Next, +E∥�xk − x0∥2 = +� +j∈[jmax] +Pr[J = j]22jE∥xj − xj−1∥2 = +� +j∈[jmax] +2jE∥xj − xj−1∥2. +Note that +E∥xj − xj−1∥2 ≤ 2E∥xj − x⋆ +¯x,λ∥2 + 2E∥xj−1 − x⋆ +¯x,λ∥2 ≤ 2−j · 6CscL2 +λ2 +� +d +β2T 2 + 1 +T +� +, +and hence combining the above bounds yields +E ∥�xk − E�xk∥2 ≤ 14CscjmaxL2 +λ2 +· +� +d +β2T 2 + 1 +T +� +. +Now, averaging jmax independent copies shows that +E +���x − x⋆ +¯x,λ +��2 = ∥�x − E�x∥2 + +��E�x − x⋆ +¯x,λ +��2 +≤ +1 +jmax +· +�14CscjmaxL2 +λ2 +· +� +d +β2T 2 + 1 +T +�� ++ C2 +bias +� +L +λ · +�√ +d +βn + 1 +√n +��2 +, +where we used our earlier bias bound. The conclusion follows by letting Cvar = C2 +bias + 14Csc. +We conclude with a gradient complexity and privacy bound, depending on the sampled J. +Lemma 10. There is a universal constant Cpriv ≥ 1, such that if β2 log2( log log n +δ +) ≤ +1 +Cpriv , δ ∈ (0, 1 +2), +and ρ +r ≥ Cpriv log2( log T +δ ), the following holds. Consider one loop indexed by k ∈ [jmax], and let J be +the result of the Geom( 1 +2) draw. If J ∈ [jmax], loop k of Algorithm 6 uses at most 2J+1T gradients. +Furthermore, the loop satisfies (α, ατ, δ)-RDP for +τ := 2J · Cpriv +� +β log +�log log n +δ +� +· T +n +�2 +, α ∈ +� +�1, +1 +Cprivβ2 log2 � +log log n +δ +� +� +� . +If J ̸∈ [jmax], Algorithm 6 uses at most T gradients, and the loop satisfies (α, ατ, δ)-RDP for +τ := Cpriv +� +β log +�log log n +δ +� +· T +n +�2 +, α ∈ +� +�1, +1 +Cprivβ2 log2 � +log log n +δ +� +� +� . +Proof. This is immediate by Proposition 8, where we applied Lemma 3 and set δ ← δ +3 (taking a +union bound over the at most 3 calls to Algorithm 5, adjusting Cpriv as necessary). +31 + +4.5 +Private ERM solver +In this section, we give our main result on privately solving ERM in the setting of Problem 2, which +will be used in a reduction framework in Section 4.6 to solve the SCO problem as well. Our ERM +algorithm is an instantiation of Proposition 1. We first develop a line search oracle (see Definition 3) +based on the solver of Section 4.3 (Algorithm 5), which succeeds with high probability. To do so, +we leverage the following geometric lemma for aggregating independent runs of our solver. +Lemma 11 (Claim 1, [KLL+22]). There is an algorithm Aggregate which takes as input (S, ∆) ∈ +(Rd)k×R≥0, and returns z ∈ Rd such that ∥z − y∥ ≤ ∆, if for some unknown point y ∈ Rd satisfying +at least 0.51k points x ∈ S, ∥x − y∥ ≤ ∆ +3 . The algorithm runs in time O(dk2). +Algorithm 7: High probability ReSQued ERM solver, strongly convex case +1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, regularization +parameter, and failure probability r, ρ, β, λ, ζ > 0, dataset D ∈ Sn, iteration count T ∈ N +2 k ← 20 log( 1 +ζ ) +3 for i ∈ [k] do +4 +xi ← output of Algorithm 5 with inputs (¯x, r, ρ, β, λ, D, T) +5 end +6 Return: x′ ← Aggregate({xi}i∈[k], 9√2CscL +λ +( +d +β2T 2 + 1 +T ) +1 +2 ) +Proposition 10. Let x⋆ +¯x,λ be defined as in (27). Algorithm 7 uses at most 18T log( 1 +ζ ) gradients and +produces x′ such that with probability at least 1 − ζ, for a universal constant Cls, +∥x′ − x⋆ +¯x,λ∥ ≤ ClsL +λ +· +�√ +d +βT + +1 +√ +T +� +. +Moreover, there exists a universal constant Cpriv ≥ 1 such that T +n ≤ +1 +Cpriv , δ ∈ (0, 1 +6) and ρ +r ≥ +Cpriv log2( 1 +δ log( T +ζ )), Algorithm 7 satisfies (α, ατ, δ)-RDP for +τ := Cpriv log +�1 +ζ +� � +β log +�1 +δ log +�T +ζ +�� +· T +n +�2 +, α ∈ +� +�1, +1 +Cprivβ2 log2 � +1 +δ log +� +T +ζ +�� +� +� . +Proof. For each xi, by Proposition 8, +E +� +� +fermr(xi) + λ +2 ∥xi − ¯x∥2 +� +− � +fermr(x⋆ +¯x,λ) − λ +2 ∥x⋆ +¯x,λ − ¯x∥2 ≤ CscL2 +λ +� +d +β2T 2 + 1 +T +� +. +Further, by strong convexity and Jensen’s inequality we have +E[∥xi − x⋆ +¯x,λ∥] ≤ +√2CscL +λ +� +d +β2T 2 + 1 +T +� 1 +2 +. +Hence, by Markov’s inequality, for each i ∈ [k] we have +Pr +� +∥xi − x⋆ +¯x,λ∥ ≥ 3√2CscL +λ +� +d +β2T 2 + 1 +T +� 1 +2 +� +≤ 1 +3. +32 + +Hence by a Chernoff bound, with probability ≥ 1 − ζ, at least 0.51k points x ∈ {xi}i∈[k] satisfy +∥x − x⋆ +¯x,λ∥ ≤ 3√2CscL +λ +� +d +β2T 2 + 1 +T +� 1 +2 +. +Hence the precondition of Lemma 11 holds, giving the distance guarantee with high probability. +The privacy guarantee follows from Proposition 8 and the composition of approximate RDP, where +we adjusted Cpriv by a constant and the definition of δ by a factor of k. +Now we are ready to prove our main result on private ERM. +Theorem 3 (Private ERM). In the setting of Problem 2, let ϵdp ∈ (0, 1) and δ ∈ (0, 1 +6). There is +an (ϵdp, δ)-DP algorithm which takes as input D and outputs �x ∈ B(R) such that +E +� +ferm(�x) − min +x∈B(R) ferm(x) +� +≤ O +� +�LR · +� +� 1 +√n + +� +d log 1 +δ log1.5( n +δ ) log n +nϵdp +� +� +� +� . +Moreover, with probability at least 1 − δ, the algorithm queries at most +O +� +log6 �n +δ +� � +min +� +n, +n2ϵ2 +dp +d +� ++ min +� +(nd) +2 +3 +ϵdp +, n +4 +3 ϵ +1 +3 +dp +��� +gradients. +Proof. Throughout this proof, set for a sufficiently large constant C, +ϵopt := CLR +� +� 1 +√n + +� +d log 1 +δ log1.5( n +δ ) log n +nϵdp +� +� , κ := LR +ϵopt +, +ρ := ϵopt +L +√ +d +, r := +ρ +√ +C log2( n +δ ) +, α := 4 log 2 +δ +ϵdp +, β := +ϵdp +C log( n +δ ) +� +log 1 +δ +. +(30) +Note that for the given parameter settings, for sufficiently large C, we have +κ ≤ 1 +C min +� +�√n, +nϵdp +� +d log 1 +δ log1.5( n +δ ) log n +� +� , R +r ≤ n log2 +�log n +δ +� +. +(31) +Our algorithm proceeds as follows. We follow the framework of Proposition 1, and instantiate the +necessary oracles as follows for CbaK log κ iterations. +1. We use Algorithm 7 with r, ρ, β defined in (30), and +T1 := +√ +C +� +κ +√ +d +√ +Kβ log2 κ ++ +κ2 +K log3 κ log n +δ +� +, ζ := +1 +κCbaK log κ, +(32) +as a ( r +Cba , λ)-line search oracle Ols. +2. We use Algorithm 5 with r, ρ, β defined in (30), and +T2 := +√ +C +� +κ +√ +d +√ +Kβ√log κ ++ +κ2 +K log κ +� +, +(33) +as a ( +λr2 +Cba log3 κ, λ)-ball optimization oracle Obo. +33 + +3. We use Algorithm 6 with r, ρ, β defined in (30), and +T3 := +√ +C +� +κ +√ +d +√ +Kβ ++ κ2 +K +� +(34) +as a ( ϵopt +CbaR, ϵopt +√ +K +CbaR , λ)-stochastic proximal oracle Osp. +We split the remainder of the proof into four parts. We first show that the oracle definitions +are indeed met. We then bound the overall optimization error against ferm. Finally, we discuss the +privacy guarantee and the gradient complexity bound. +Oracle correctness. For the line search oracle, by Proposition 10, it suffices to show +ClsL +λ +· +� √ +d +βT1 ++ +1 +√T1 +� +≤ +r +Cba +. +This is satisfied for T1 in (32), since Proposition 1 guarantees λ ≥ ϵoptK2 log2 κ +R2Cba +. Hence, +ClsL +λ +· +√ +d +βT1 +· Cba +r +≤ ClsC2 +ba · +κ +√ +d +β log2 κ · +1 +√ +K +· 1 +T1 +≤ 1 +2, +ClsL +λ +· +1 +√T1 +· Cba +r +≤ ClsC2 +ba · +κ +log2 κ · +1 +√ +K +· +1 +√T1 +≤ 1 +2, +for a sufficiently large C, where we used K1.5 = R +r to simplify. By a union bound, the above holds +with probability at least 1 − ϵopt +LR over all calls to Algorithm 7, since there are at most CbaK log κ +iterations. For the remainder of the proof, let Els be the event that all line search oracles succeed. +For the ball optimization oracle, by Proposition 8, it suffices to show +CscL2 +λ +� +d +β2T 2 +2 ++ 1 +T2 +� +≤ +λr2 +Cba log3 κ. +This is satisfied for our choice of T2 in (33), again with λ ≥ ϵoptK2 log2 κ +R2Cba +. Hence, +CscL2 +λ +· +d +β2T 2 +2 +· Cba log3 κ +λr2 +≤ CscC3 +ba · +κ2d +β2 log κ · 1 +K · 1 +T 2 +2 +≤ 1 +2, +CscL2 +λ +· 1 +T2 +· Cba log3 κ +λr2 +≤ CscC3 +ba · +κ2 +log κ · 1 +K · 1 +T2 +≤ 1 +2, +again for large C. Finally, for the proximal gradient oracle, by Proposition 9, it suffices to show +Cbias +� +L +λ · +�√ +d +βn + 1 +√n +�� +≤ +ϵopt +CbaλR, +CvarL2 +λ2 +� +d +β2T 2 +3 ++ 1 +T3 +� +≤ +ϵ2 +optK +C2 +baλ2R2 . +The first inequality is clear. The second is satisfied for our choice of T3 in (34), which implies +CvarL2 +λ2 +· +d +β2T 2 +3 +· C2 +baλ2R2 +ϵ2 +optK += CvarC2 +ba · κ2d +β2 · 1 +K · 1 +T 2 +3 +≤ 1 +2, +CvarL2 +λ2 +· 1 +T3 +· C2 +baλ2R2 +ϵ2 +optK += CvarC2 +ba · κ2 · 1 +K · 1 +T3 +≤ 1 +2. +34 + +Optimization error. By Proposition 1, the expected optimization error against � +ferm +ρ +is bounded +by ϵopt whenever Els occurs. Otherwise, the optimization error is never larger than LR as long as +we return a point in B(R), since the function is L-Lipschitz. Further, we showed Pr[Els] ≥ 1 − ϵopt +LR , +so the total expected error is bounded by 2ϵopt. Finally, the additive error between � +ferm +ρ +and ferm +is bounded by ρL +√ +d = ϵopt. The conclusion follows by setting the error bound to 3ϵopt. +Privacy. We first claim that each call to Ols, and Obo used by Proposition 1 satisfies +� +α, +ϵdp +6CbaK log κ, +δ +18CbaK log κ +� +-RDP. +We first analyze Ols. The preconditions of Proposition 10 are met, where log( 18CbaK log κ +δ +log( T +ζ )) ≤ +2 log n +δ for our parameter settings. Moreover, our α is in the acceptable range. Finally, by Proposi- +tion 10 it suffices to note +8αCprivβ2T 2 +1 log3 � n +δ +� +n2 +≤ 128CCprivβ2 log3( n +δ ) log 1 +δ +n2ϵdp +· +� +κ2d +Kβ2 log κ + +κ4 +K2 log2 κ +� +≤ +ϵdp +6CbaK log κ, +where the second inequality follows for sufficiently large C due to (31). Next, we analyze the privacy +of Obo. The preconditions of Proposition 8 are met, where log( log log T +δ +) ≤ log n +δ for our parameter +settings, and our α is again acceptable. Finally, by Proposition 8 it suffices to note +αCprivβ2T 2 +2 log2( n +δ ) +n2 +≤ 16CCprivβ2 log2( n +δ ) log 1 +δ +n2ϵdp +· +� +κ2d +Kβ2 log κ + +κ4 +K2 log2 κ +� +≤ +ϵdp +6CbaK log κ, +again for sufficiently large C from (31). Hence, by applying Lemma 3, all of the at most CbaK log κ +calls to Ols and Obo used by the algorithm combined satisfy +� +α, ϵdp +3 , δ +9 +� +-RDP. +Finally, we analyze the privacy of Osp. Let +jmax := +� +log2 +� 1 +T3 +· +� +n +Cpriv +��� +be the truncation parameter in Algorithm 6. The total number of draws from Geom( 1 +2) in Algo- +rithm 6 over the course of the algorithm is CbaK log κ · jmax. It is straightforward to check that the +expected number of draws where J = j for all j ∈ [jmax] is +2−jmaxCbaκ log κ · jmax = Ω +�T3 +n · K log κ · jmax +� +, +which is superconstant. +By Chernoff and a union bound, with probability ≥ 1 − δ +n, there is a +constant C′ such that for all j ∈ [jmax], the number of times we draw J = j is bounded by +2−jC′K log κ log n +δ . +Similarly, the number of times we draw J ̸∈ [jmax] is bounded by C′K log κ log n +δ . This implies by +Lemma 3 that all calls to Osp used by the algorithm combined satisfy +� +α, ϵdp +6 , δ +18 +� +-RDP. +35 + +Here, we summed the privacy loss in Lemma 10 over 0 ≤ J ≤ jmax, which gives +� +0≤j≤jmax +� +2j · αCprivβ2 log2( n +δ )T 2 +3 +n2 +� +· +� +2−jC′K log κ log n +δ +� +≤ (jmax + 1) · 16CC′CprivKβ2 log3( n +δ ) log 1 +δ log κ +n2ϵdp +· +� κ2d +Kβ2 + κ4 +K2 +� +≤ ϵdp +6 , +for sufficiently large C, where we use log κ, jmax ≤ log n, and K ≥ log 1 +δ for our parameter set- +tings. Finally, combining these bounds shows that our whole algorithm satisfies (α, ϵdp +2 , δ +6)-RDP, +and applying Corollary 1, gives the desired privacy guarantee. +Gradient complexity. We have argued that with probability at least 1 − δ, the number of times +we encounter the J = j case of Lemma 10 for all 0 ≤ j ≤ jmax is bounded by 2−jC′K log κ log n +δ . +Under this event, Proposition 10, Proposition 8, and Lemma 10 imply the total gradient complexity +of our algorithm is at most +CbaK log κ · +� +�18T1 log 1 +ζ + T2 + +� +0≤j≤jmax +� +2−jC′ log n +δ +� � +2j+1T3 +� +� +� +≤ 36CbaC′K log n +� +T1 log n + T2 + T3 log n log n +δ +� +, +where we use ζ ≥ n−2, jmax ≤ log n, and κ ≤ n. The conclusion follows from plugging in our +parameter choices from (32), (33), and (34). +Finally, we note that following the strategy of Section 4.3, it is straightforward to extend The- +orem 3 to the strongly convex setting. We state this result as follows. +Corollary 3 (Private regularized ERM). In the setting of Problem 2, let ϵdp ∈ (0, 1), δ ∈ (0, 1 +6), +λ ≥ 0, and x′ ∈ B(R). There is an (ϵdp, δ)-DP algorithm which outputs �x ∈ B(R) such that +E +� +ferm(�x) + λ +2 +��x − x′��2 − min +x∈B(R) +� +ferm(x) + λ +2 +��x − x′��2 +�� +≤ O +� +L2 +λ · +� +1 +n + d log 1 +δ log3( n +δ ) log2 n +n2ϵ2 +dp +�� +. +Moreover, with probability at least 1 − δ, the algorithm queries at most +O +� +log6 �n +δ +� � +min +� +n, +n2ϵ2 +dp +d +� ++ min +� +(nd) +2 +3 +ϵdp +, n +4 +3 ϵ +1 +3 +dp +��� +gradients. +Proof. We first note that similar to Corollary 2 (an extension of Proposition 6), it is straightfor- +ward to extend Theorem 3 to handle both regularization and an improved upper bound on the +distance to the optimum, with the same error rate and privacy guarantees otherwise. The handling +of the improved upper bound on the distance follows because the convergence rate of the [ACJ+21] +algorithm scales proportionally to the distance to the optimum, when it is smaller than R. The +regularization is handled in the same way as Corollary 2, where regularization can only improve +the contraction in the privacy proof. One subtle point is that for the regularized problems, we need +to obtain starting points for Algorithm 5 when the constraint set is B¯x(r), but the regularization +in the objective is centered around a point not in B¯x(r) (in our case, the centerpoint will be a +weighted combination of ¯x and x′). However, by initializing Algorithm 5 at the projection of the +regularization centerpoint, the initial function error guarantee in Lemma 8 still holds (see Lemma 9). +36 + +The reduction from the claimed rate in this corollary statement to the regularized extension of +Theorem 3 then proceeds identically to the proof of Proposition 8, which calls Corollary 2 repeatedly. +4.6 +Private SCO solver +Finally, we give our main result on private SCO in this section. To obtain it, we will combine +Corollary 3 with a generic reduction in [FKT20, KLL21], which uses a private ERM solver as a +black box. The reduction is based on the iterative localization technique proposed by [FKT20] +(which is the same strategy used by Section 4.3), and derived in greater generality by [KLL21]. +Proposition 11 (Modification of Theorem 5.1 in [KLL21]). Suppose there is an (ϵdp, δ)-DP algo- +rithm Aerm with expected excess loss +O +� +L2 +λ · +� +1 +n + d log 1 +δ log3( n +δ ) log2 n +n2ϵ2 +dp +�� +, +using N(n, ϵdp, δ) gradient queries, for some function N, when applied to an L-Lipschitz empirical +risk (with n samples, constrained to B(R) ⊂ Rd) plus a λ-strongly convex regularizer. Then there is +an (ϵdp, δ)-DP algorithm Asco using � +i∈⌈log n⌉ N( n +2i , ϵdp +2i , δ +2i ) gradient queries, with expected excess +population loss +O +� +�LR · +� +� 1 +√n + +� +d log 1 +δ log1.5( n +δ ) log n +nϵdp +� +� +� +� . +Theorem 5.1 in [KLL21] assumes a slightly smaller risk guarantee for Aerm (removing the ex- +traneous log3( n +δ ) log2 n factor), but it is straightforward to see that the proof extends to handle our +larger risk assumption. Combining Proposition 11 and Corollary 3 then gives our main result. +Theorem 4 (Private SCO). In the setting of Problem 2, let ϵdp ∈ (0, 1) and δ ∈ (0, 1 +6). There is +an (ϵdp, δ)-DP algorithm which takes as input D and outputs �x ∈ B(R) such that +E +� +fpop(�x) − min +x∈B(R) fpop(x) +� +≤ O +� +�LR · +� +� 1 +√n + +� +d log 1 +δ log1.5( n +δ ) log n +nϵdp +� +� +� +� . +Moreover, with probability at least 1 − δ, the algorithm queries at most +O +� +log6 �n +δ +� � +min +� +n, +n2ϵ2 +dp +d +� ++ min +� +(nd) +2 +3 +ϵdp +, n +4 +3 ϵ +1 +3 +dp +��� +gradients. +Acknowledgements +We thank Vijaykrishna Gurunathan for helpful conversations on parallel convex optimization that +facilitated initial insights regarding ReSQue. +YC was supported in part by the Israeli Science +Foundation (ISF) grant no. 2486/21 and the Len Blavatnik and the Blavatnik Family foundation. +AS was supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF- +1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship. +37 + +References +[Abo16] +John M. Abowd. The challenge of scientific reproducibility and privacy protection for +statistical agencies. Technical report, Census Scientific Advisory Committee, 2016. +[ACG+16] +Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal +Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the +2016 ACM SIGSAC conference on computer and communications security, 2016. +[ACJ+21] +Hilal Asi, Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford. Stochastic +bias-reduced gradient methods. In Advances in Neural Information Processing Systems, +NeurIPS, 2021. +[AFKT21] +Hilal Asi, Vitaly Feldman, Tomer Koren, and Kunal Talwar. Private stochastic convex +optimization: Optimal rates in l1 geometry. In International Conference on Machine +Learning, ICML, 2021. +[App17] +Differential Privacy Team Apple. Learning with privacy at scale. Technical report, +Apple, 2017. +[BBG18] +Borja Balle, Gilles Barthe, and Marco Gaboardi. Privacy amplification by subsampling: +Tight analyses via couplings and divergences. +In Advances in Neural Information +Processing Systems, NeurIPS, 2018. +[BDRS18] +Mark Bun, Cynthia Dwork, Guy N. Rothblum, and Thomas Steinke. Composable and +versatile privacy via truncated CDP. In Proceedings of the 50th Annual ACM SIGACT +Symposium on Theory of Computing, STOC, 2018. +[BFGT20] +Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, and Kunal Talwar. +Stability of +stochastic gradient descent on nonsmooth convex losses. Advances in Neural Informa- +tion Processing Systems, 33:4381–4391, 2020. +[BFTT19] +Raef Bassily, Vitaly Feldman, Kunal Talwar, and Abhradeep Guha Thakurta. Private +stochastic convex optimization with optimal rates. In Advances in Neural Information +Processing Systems, NeurIPS, 2019. +[BJL+19] +Sébastien Bubeck, Qijia Jiang, Yin Tat Lee, Yuanzhi Li, and Aaron Sidford. Com- +plexity of highly parallel non-smooth convex optimization. +In Advances in Neural +Information Processing Systems, NeurIPS, 2019. +[BM99] +Guy E. Blelloch and Bruce M. Maggs. Parallel algorithms. In Mikhail J. Atallah, editor, +Algorithms and Theory of Computation Handbook, Chapman & Hall/CRC Applied +Algorithms and Data Structures series. CRC Press, 1999. +[Bot12] +Léon Bottou. Stochastic gradient descent tricks. In Grégoire Montavon, Genevieve B. +Orr, and Klaus-Robert Müller, editors, Neural Networks: Tricks of the Trade - Second +Edition, volume 7700 of Lecture Notes in Computer Science, pages 421–436. Springer, +2012. +[BS18] +Eric Balkanski and Yaron Singer. +Parallelization does not accelerate convex op- +timization: Adaptivity lower bounds for non-smooth convex minimization. +arXiv: +1808.03880, 2018. +38 + +[BST14] +Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimiza- +tion: Efficient algorithms and tight error bounds. In IEEE 55th Annual Symposium +on Foundations of Computer Science, FOCS, 2014. +[Bub15] +Sébastien Bubeck. Convex optimization: Algorithms and complexity. Found. Trends +Mach. Learn., 8(3-4):231–357, 2015. +[BV14] +Stephen P. Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge Univer- +sity Press, 2014. +[CH22] +Yair Carmon and Danielle Hausler. Distributionally robust optimization via ball oracle +acceleration. arXiv:2203.13225, 2022. +[CJJ+20] +Yair Carmon, Arun Jambulapati, Qijia Jiang, Yujia Jin, Yin Tat Lee, Aaron Sidford, +and Kevin Tian. Acceleration with a ball optimization oracle. In Advances in Neural +Information Processing Systems, NeurIPS, 2020. +[CJJS21] +Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford. Thinking inside the +ball: Near-optimal minimization of the maximal loss. In Conference on Learning The- +ory, COLT, 2021. +[CJST19] +Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian. Variance reduction for matrix +games. In Advances in Neural Information Processing Systems, NeurIPS, 2019. +[CM08] +Kamalika Chaudhuri and Claire Monteleoni. Privacy-preserving logistic regression. In +Advances in Neural Information Processing Systems, NeurIPS, 2008. +[CMS11] +Kamalika Chaudhuri, Claire Monteleoni, and Anand D Sarwate. Differentially private +empirical risk minimization. Journal of Machine Learning Research, 12(3), 2011. +[DBW12] +John C Duchi, Peter L Bartlett, and Martin J Wainwright. Randomized smoothing +for stochastic optimization. SIAM Journal on Optimization, 22(2):674–701, 2012. +[DG19] +Jelena Diakonikolas and Cristóbal Guzmán. Lower bounds for parallel and randomized +convex optimization. In Conference on Learning Theory, COLT, 2019. +[DGBSX12] Ofer Dekel, Ran Gilad-Bachrach, Ohad Shamir, and Lin Xiao. Optimal distributed +online prediction using mini-batches. Journal of Machine Learning Research, 13(1), +2012. +[DR14] +Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. +Found. Trends Theor. Comput. Sci., 9(3-4):211–407, 2014. +[DRY18] +John Duchi, Feng Ruan, and Chulhee Yun. Minimax bounds on stochastic batched +convex optimization. In Conference On Learning Theory, COLT, 2018. +[Duc18] +John C Duchi. Introductory lectures on stochastic optimization. The Mathematics of +Data, pages 99–186, 2018. +[EPK14] +Úlfar Erlingsson, Vasyl Pihur, and Aleksandra Korolova. Rappor: Randomized aggre- +gatable privacy-preserving ordinal response. In Proceedings of the 2014 ACM SIGSAC +conference on computer and communications security, 2014. +39 + +[FKT20] +Vitaly Feldman, Tomer Koren, and Kunal Talwar. Private stochastic convex optimiza- +tion: optimal rates in linear time. In Proceedings of the 52nd Annual ACM SIGACT +Symposium on Theory of Computing, STOC, 2020. +[FTS17] +Kazuto Fukuchi, Quang Khai Tran, and Jun Sakuma. Differentially private empirical +risk minimization with input perturbation. In International Conference on Discovery +Science, 2017. +[GDG+19] +Alexander Gasnikov, Pavel Dvurechensky, Eduard Gorbunov, Evgeniya Vorontsova, +Daniil Selikhanovych, César A Uribe, Bo Jiang, Haoyue Wang, Shuzhong Zhang, +Sébastien Bubeck, et al. Near optimal methods for minimizing convex functions with +lipschitz p-th derivatives. In Conference on Learning Theory, COLT, 2019. +[GL12] +Saeed Ghadimi and Guanghui Lan. Optimal stochastic approximation algorithms for +strongly convex stochastic composite optimization i: A generic algorithmic framework. +SIAM Journal on Optimization, 22(4):1469–1492, 2012. +[GLL22] +Sivakanth Gopi, Yin Tat Lee, and Daogao Liu. Private convex optimization via expo- +nential mechanism. arXiv:2203.00263, 2022. +[Gol64] +A. A. Goldstein. Convex programming in hilbert space. 70(5):709––710, 1964. +[Haz16] +Elad Hazan. Introduction to online convex optimization. Found. Trends Optim., 2(3- +4):157–325, 2016. +[HK14] +Elad Hazan and Satyen Kale. Beyond the regret minimization barrier: optimal algo- +rithms for stochastic strongly-convex optimization. J. Mach. Learn. Res., 15(1):2489– +2512, 2014. +[INS+19] +Roger Iyengar, Joseph P Near, Dawn Song, Om Thakkar, Abhradeep Thakurta, and +Lun Wang. +Towards practical differentially private convex optimization. +In IEEE +Symposium on Security and Privacy (SP), 2019. +[JLSW20] +Haotian Jiang, Yin Tat Lee, Zhao Song, and Sam Chiu-wai Wong. An improved cutting +plane method for convex optimization, convex-concave games, and its applications. In +Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, +STOC, 2020. +[JT14] +Prateek Jain and Abhradeep Guha Thakurta. +(near) dimension independent risk +bounds for differentially private learning. +In International Conference on Machine +Learning, ICML, 2014. +[KJ16] +Shiva Prasad Kasiviswanathan and Hongxia Jin. Efficient private empirical risk mini- +mization for high-dimensional learning. In International Conference on Machine Learn- +ing, ICML, 2016. +[KLL21] +Janardhan Kulkarni, Yin Tat Lee, and Daogao Liu. Private non-smooth erm and sco in +subquadratic steps. In Advances in Neural Information Processing Systems, NeurIPS, +2021. +[KLL+22] +Jonathan A. Kelner, Jerry Li, Allen Liu, Aaron Sidford, and Kevin Tian. Semi-random +sparse recovery in nearly-linear time. arXiv:2203.04002, 2022. +40 + +[KTE88] +Leonid G. Khachiyan, Sergei Pavlovich Tarasov, and I. I. Erlikh. +The method of +inscribed ellipsoids. Soviet Math. Dokl., 37:226–230, 1988. +[LLH+22] +Xuechen Li, Daogao Liu, Tatsunori Hashimoto, Huseyin A Inan, Janardhan Kulkarni, +Yin Tat Lee, and Abhradeep Guha Thakurta. When does differentially private learning +not suffer in high dimensions? arXiv:2207.00160, 2022. +[Mir17] +Ilya Mironov. Rényi differential privacy. In IEEE 30th Computer Security Foundations +Symposium, CSF, 2017. +[MS13] +Renato DC Monteiro and Benar Fux Svaiter. An accelerated hybrid proximal extra- +gradient method for convex optimization and its implications to second-order methods. +SIAM Journal on Optimization, 23(2):1092–1125, 2013. +[Nem94] +Arkadi Nemirovski. On parallel complexity of nonsmooth convex optimization. Journal +of Complexity, 10(4):451–463, 1994. +[Nes83] +Yu E Nesterov. A method for solving the convex programming problem with conver- +gence rate o(1/k2). In Dokl. Akad. Nauk SSSR,, 1983. +[Nes03] +Yurii Nesterov. +Introductory lectures on convex optimization: A basic course, vol- +ume 87. Springer Science & Business Media, 2003. +[Nes18] +Yurii Nesterov. Lectures on convex optimization, volume 137. Springer, 2018. +[NY83] +Arkadi S. Nemirovski and David B. Yudin. Problem complexity and method efficiency +in optimization. 1983. +[Pol64] +Boris T. Polyak. Some methods of speeding up the convergence of iteration methods. +USSR Computational Mathematics and Mathematical Physics, 4(5):1–17, 1964. +[RBHT12] +Benjamin IP Rubinstein, Peter L Bartlett, Ling Huang, and Nina Taft. Learning in +a large function space: Privacy-preserving mechanisms for svm learning. Journal of +Privacy and Confidentiality, 4(1):65–100, 2012. +[SB14] +Shai Shalev-Shwartz and Shai Ben-David. Understanding Machine Learning - From +Theory to Algorithms. Cambridge University Press, 2014. +[SBB+18] +Kevin Scaman, Francis Bach, Sébastien Bubeck, Laurent Massoulié, and Yin Tat Lee. +Optimal algorithms for non-smooth distributed optimization in networks. In Advances +in Neural Information Processing Systems, NeurIPS, 2018. +[Sha07] +Shai Shalev-Shwartz. +Online learning: Theory, algorithms, and applications. +PhD +thesis, Hebrew University, 2007. +[Smi09] +Adam +Smith. +Differential +privacy +and +the +secrecy +of +the +sample. +https://adamdsmith.wordpress.com/2009/09/02/sample-secrecy/, 2009. +Accessed: +2022-11-06. +[SSTT21] +Shuang Song, Thomas Steinke, Om Thakkar, and Abhradeep Thakurta. Evading the +curse of dimensionality in unconstrained private glms. In International Conference on +Artificial Intelligence and Statistics, AISTATS, 2021. +41 + +[SU15] +Thomas Steinke and Jonathan Ullman. Between pure and approximate differential +privacy. arXiv:1501.06095, 2015. +[Tal22] +Kunal Talwar. Ppml workshop talk: Open questions in differentially private machine +learning. https://machinelearning.apple.com/video/open-questions, 2022. Accessed: +2022-11-06. +[Wan18] +Yu-Xiang Wang. Revisiting differentially private linear regression: optimal and adap- +tive prediction & estimation in unbounded domain. arXiv:1803.02596, 2018. +[WBSS21] +Blake E Woodworth, Brian Bullins, Ohad Shamir, and Nathan Srebro. The min-max +complexity of distributed stochastic convex optimization with intermittent communi- +cation. In Conference on Learning Theory, COLT, 2021. +[ZZMW17] +Jiaqi Zhang, Kai Zheng, Wenlong Mou, and Liwei Wang. Efficient private erm for +smooth objectives. In International Joint Conference on Artificial Intelligence, IJCAI, +2017. +42 + +A +Helper facts +Fact 2. Let p ∈ N. For any integer r such that 0 ≤ r ≤ p − 1, � +0≤q≤p(−1)q�p +q +� +qr = 0. +Proof. We recognize the formula as a scaling of the Stirling number of the second kind with r objects +and p bins, i.e. the number of ways to put r objects into p bins such that each bin has at least one +object. When r < p there are clearly no such ways. +Fact 3. Let p ∈ N be even and p ≥ 2. Let ∥x∥ , ∥y∥ ≤ 1 +p. Then +� +0≤q≤p +(−1)q +�p +q +� +exp +�1 +2 +�� +(p − q)2 − (p − q) +� +∥x∥2 + (q2 − q) ∥y∥2 + 2q(p − q) ⟨x, y⟩ +�� +≤ (12p ∥x − y∥)p. +Proof. Fix some x. Let fx(y) be the left-hand side displayed above, and let +fq +x(y) := exp +�1 +2 +�� +(p − q)2 − (p − q) +� +∥x∥2 + (q2 − q) ∥y∥2 + 2q(p − q) ⟨x, y⟩ +�� +. +We will perform a pth order Taylor expansion of fx around x, where we show that partial derivatives +of order at most p − 1 are all zero at x, and we bound the largest order derivative tensor. +Derivatives of fq +x. Fix some 0 ≤ q ≤ p, and define +Cq := q2 − q, Fq := fq +x(y), vq := (q2 − q)y + q(p − q)x. +(35) +Note that for fixed q, Fq and vq are functions of y, and we defined them such that ∇yvq = CqId, +∇yFq = vqFq. Next, in the following we use � +sym to mean a symmetric sum over all choices of +tensor modes, e.g. � +sym v⊗2 +q +⊗Id means we will choose 2 of the 4 modes where the action is v⊗2 +q . To +gain some intuition for the derivatives of Fq, we begin by evaluating the first few via product rule: +∇fq +x(y) = Fqvq, +∇2fq +x(y) = Fqv⊗2 +q ++ CqFqId, +∇3fq +x(y) = Fqv⊗3 +q ++ CqFq +� +sym +vq ⊗ Id, +∇4fq +x(y) = Fqv⊗4 +q ++ CqFq +� +sym +v⊗2 +q +⊗ Id + 3C2 +q FqId ⊗ Id. +For any fixed 0 ≤ r ≤ p, we claim that the rth derivative tensor has the form +∇rfq +x(y) = Fq +� +� +� +0≤s≤⌊ r +2 ⌋ +Nr,s +� r +2s +� +� +(Cq)s � +sym +v⊗(r−2s) +q +⊗ I⊗s +d +�� +� , +(36) +where the Nr,s are nonnegative coefficients which importantly do not depend on q. To see this we +proceed by induction; the base cases are computed above. Every time we take the derivative of a +“monomial” term of the form Fq(Cq)sv⊗(r−2s) +q +⊗I⊗s +d +via product rule, we will have one term in which +Fq becomes vqFq (and hence we obtain a FqCs +qv⊗(r+1−2s) +q +⊗ I⊗s +d +monomial), and r − 2s many terms +where a vq becomes CqId (and hence we obtain a FqCs+1 +q +v⊗(r−1−2s) +q +⊗ I⊗(s+1) +d +monomial). For fixed +0 ≤ s ≤ ⌊r+1 +2 ⌋, we hence again see that Nr+1,s has no dependence on q. +43 + +Next, note � +0≤s≤⌊ r +2 ⌋ Nr,s has a natural interpretation as the total number of “monomial” terms +of the form Fq(Cq)sv⊗(r−2s) +q +⊗ I⊗s +d +when expanding ∇rfq +x(y). We claim that for all 0 ≤ q ≤ p and +0 ≤ r ≤ p − 1, +� +0≤s≤⌊ r+1 +2 ⌋ Nr+1,s +� +0≤s≤⌊ r +2 ⌋ Nr,s +≤ p. +(37) +To see this, consider taking an additional derivative of (36) with respect to y. Each monomial of +the form Fq(Cq)sv⊗(r−2s) +q +⊗ I⊗s +d +contributes at most r − 2s + 1 ≤ p monomials to the next derivative +tensor via product rule, namely one from Fq and one from each copy of vq. Averaging this bound +over all monomials yields the claim (37), since each contributes at most p. +Taylor expansion at x. Next, we claim that for all 0 ≤ r ≤ p − 1, +∇rfx(x) = 0. +(38) +To see this, we have that ((p − q)2 − (p − q)) + (q2 − q) + 2q(p − q) = p2 − p is independent of q, +and hence all of the Fq are equal to some value F when y = x. Furthermore, when y = x we have +that vq = q(p − 1)x. Now, from the characterization (36) and summing over all q, any monomial of +the form x⊗(r−2s) ⊗ I⊗s +d +has a total coefficient of +FNr,s +� +0≤q≤p +(−1)q +�p +q +� +(Cq)s(q(p − 1))r−2s = FNr,s(p − 1)r−2s � +0≤q≤p +(−1)q +�p +q +� +Cs +qqr−2s. +Since Cq is a quadratic in q, each summand (Cq)sqr−2s is a polynomial of degree at most r ≤ p − 1 +in q, so applying Fact 2 to each monomial yields the claim (38). +Taylor expansion at y. Finally, we will bound the injective tensor norm of ∇pfx(y), where the +injective tensor norm of a degree-p symmetric tensor T is the maximum value of T[v⊗p] over unit +norm v. We proceed by bounding the injective tensor norm of each monomial and then summing. +First, for any 0 ≤ p ≤ q, under our parameter settings it is straightforward to see ∥vq∥ ≤ p +and Fq ≤ 2. Also, for any 0 ≤ s ≤ p +2 we have Cs +q ≤ p2s, and by repeatedly applying (37), we have +� +0≤s≤⌊ p +2 ⌋ Np,s ≤ pp. In other words, each of the monomials of the form Fq(Cq)sv⊗(r−2s) +q +⊗ I⊗s +d +has +injective tensor norm at most 2pp (since each Cq contributes two powers of p, and each vq contributes +one power of p), and there are at most pp such monomials. Hence, by triangle inequality over the +sum of all monomials, +��∇pfq +x(y)[(y − x)⊗p] +�� ≤ 2p2p ∥y − x∥p . +By summing the above over all q (reweighting by (−1)q�p +q +� +), and using that the unsigned coefficients +sum to � +0≤q≤p +�q +p +� += 2p, we have +��∇pfx(y)[(y − x)⊗p] +�� ≤ 4pp2p ∥x − y∥p . +The conclusion follows by a Taylor expansion from x to y of order p, and using pp ≤ 3pp!. +Proof of Lemma 2. For the first claim, +� (γρ(x − ¯x − ξ))p +(γρ(ξ))p−1 +dξ = (2πρ)− d +2 +� +exp +� +− 1 +2ρ2 +� +p ∥x − ¯x∥2 − 2p ⟨x − ¯x, ξ⟩ + ∥ξ∥2�� +dξ += exp +�p2 − p +2ρ2 +∥x − ¯x∥2 +� +≤ 2, +44 + +where the second equality used the calculation in (6), and the inequality used the assumed bound +on ∥x − ¯x∥. We move onto the second claim. First, we prove the statement for all even p ∈ N. +Denote v := x − ¯x and v′ := x′ − ¯x for simplicity. Explicitly expanding the numerator yields that +(2πρ) +d +2 +� (γρ(v − ξ) − γρ(v′ − ξ))p +(γρ(ξ))p−1 +dξ = +� +0≤q≤p +(−1)q +�p +q +� +Sq +where we define +Sq := (2πρ) +d +2 +� (γρ(v − ξ))p−q(γr(v′ − ξ))q +(γρ(ξ))p−1 +dξ += +� +exp +� +− 1 +2ρ2 +� +(p − q) ∥v∥2 + q +��v′��2 − 2(p − q) ⟨v, ξ⟩ − 2q +� +v′, ξ +� ++ ∥ξ∥2�� +dξ += (2πρ) +d +2 exp +� 1 +2ρ2 +�� +(p − q)2 − (p − q) +� +∥v∥2 + (q2 − q) +��v′��2 + 2q(p − q) +� +v, v′��� +. +In the last line, we again used (6) to compute the integral. When p ≥ 2 and is even, a strengthening +of the conclusion then follows from Fact 3 (where we overload x ← v +ρ, y ← v′ +ρ in its application). +In particular, this shows the desired claim where the base of the exponent is 12p +ρ ∥x − x′∥ instead of +24p +ρ ∥x − x′∥. We move to general p ≥ 2. Define the random variable +Z := +���� +γρ(x − ¯x − ξ) − γρ(x′ − ¯x − ξ) +γρ(ξ) +���� . +Recall that we have shown for all even p ≥ 2, +EZp ≤ +�12p ∥x − x′∥ +ρ +�p +. +Now, let p ≥ 2 be sandwiched between the even integers q and q + 2. Hölder’s inequality and the +above inequality (for p ← q and p ← q + 2) demonstrate +EZp ≤ (EZq) +q+2−p +2 +� +EZq+2� p−q +2 +≤ +�12(q + 2) ∥x − x′∥ +ρ +�p +, +where we use q(q + 2 − p) + (q + 2)(p − q) = 2p. The conclusion follows since q + 2 ≤ 2p. +Fact 4. Let Z be a nonnegative scalar random variable, let C ≥ 0 be a fixed scalar, and let p ∈ N +and p ≥ 2. Then +(E [(Z + C)p]) +1 +p ≤ E [Zp] +1 +p + C. +Proof. Denote A := E [Zp] +1 +p . Taking pth powers of both sides, we have the conclusion if +(A + C)p − E [(Z + C)p] ≥ 0 ⇐⇒ +� +q∈[p−1] +�p +q +� +Cp−q (Aq − E [Zq]) ≥ 0. +Here we use that the q = 0 and q = p terms cancel. We conclude since Jensen’s inequality yields +E[Zp] ≥ E[Zq] +p +q =⇒ Aq ≥ E[Zq], for all q ∈ [p − 1]. +45 + +B +Discussion of Proposition 1 +In this section, we discuss how to obtain Proposition 1 from the analysis in [ACJ+21]. We separate +the discussion into four parts, corresponding to the iteration count, the line search oracle parameters, +the ball optimization oracle parameters, and the proximal gradient oracle parameters. We note that +Proposition 2 in [ACJ+21] states that they obtain function error ϵopt with constant probability; +however, examining the proof shows it actually yields an expected error bound. +Iteration count. +The bound CbaK log κ on the number of iterations follows immediately from +the value Kmax stated in Proposition 2 of [ACJ+21], where we set λmin ← λ⋆ and ϵ ← ϵopt. +Line search oracle parameters. +The line search oracle is called in the implementation of Line +2 of Algorithm 4 in [ACJ+21]. Our implementation follows the development of Appendix D.2.3 +in [ACJ+21], which is a restatement of Proposition 2 in [CJJS21]. The bound Cba log( Rκ +r ) on the +number of calls to the oracle is immediate from the statement of Proposition 2. For the oracle +parameter ∆ = +r +Cba , we note that the proof of Proposition 2 of [CJJS21] only requires that we +obtain points at distance O(r) from x⋆ +¯x,λ given a choice of λ, although it is stated as requiring a +function error guarantee. This is evident where the proof applies Lemma 3 of the same paper. +Ball optimization oracle parameters. +The ball optimization oracle is called in the implemen- +tation of Line 5 of Algorithm 4 in [ACJ+21]. In iteration k of the algorithm, the error requirement +is derived through the potential bound in Lemma 5 of [ACJ+21]. More precisely, Lemma 5 shows +that (following their notation), conditioned on all randomness through iteration k, +E +� +Ak+1 (F(xk+1) − F(x⋆)) + ∥vk+1 − x⋆∥2� +− +� +Ak (F(xk) − F(x⋆)) + ∥vk − x⋆∥2� +≤ −1 +6λk+1Ak+1 ∥�xk+1 − yk∥2 + Ak+1φk+1 + a2 +k+1σ2 +k+1 + 2Rak+1δk+1, +where the terms a2 +k+1σ2 +k+1 +2Rak+1δk+1 are handled identically in [ACJ+21] and our Proposition 1 +(see the following discussion). For the remaining two terms, Proposition 4 of [ACJ+21] guarantees +that as long as the method does not terminate, one of the following occurs. +1. ∥�xk+1 − yk∥2 = Ω(r2). +2. λk+1 = O(λ⋆). +In the first case, as long as φk+1 (the error tolerance to the ball optimization oracle) is set to be +λk+1r2 +Cba +for a sufficiently large Cba (which it is smaller than by logarithmic factors), up to constant +factors the potential proof is unaffected. The total contributions to the potential due to all Ak+1φk+1 +losses from the iterations of the second case across the entire algorithm is bounded by +O +� +(K log κ) · R2 +ϵopt +· λ⋆r2 +log3 κ +� += O +� +R2� +. +Here, the first term is the iteration count, the second term is due to an upper bound on Ak+1, and +the third term is bounded since λk+1 = O(λ⋆). The initial potential in the proof of Proposition 2 +of [ACJ+21] is R2, so the final potential is unaffected by more than constant factors. For a more +formal derivation of the same improved error tolerance, we refer the reader to [CH22], Lemma 8. +46 + +Stochastic proximal oracle parameters. +Our stochastic proximal oracle parameters are ex- +actly the settings of δk, σk required by Proposition 2 of [ACJ+21], except we simplified the bound +on σ2 +k = O( ϵ +ak ) (note we use ϵopt in place of ϵ). In particular, following notation of [ACJ+21], we +have +ϵ +ak += ϵ√λk +√Ak += Ω +� +ϵ · +� +λ⋆ · +√ϵ +R +� += Ω +�ϵ2K +R2 log κ +� +. +The first equality used λka2 +k = Ak for the parameter choices of Algorithm 4 in [ACJ+21]. The +second equality used that all λk = Ω(λ⋆) and all Ak = O( R2 +ϵ ) in Algorithm 4 in [ACJ+21], where +we chose λ⋆ = ϵK2 +R2 log2 κ. The final equality plugged in this bound on λ⋆ and simplified. Hence, +obtaining a variance as declared in Proposition 1 suffices to meet the requirement. +C +Discussion of Proposition 2 +In this section, we discuss how to obtain Proposition 2 (which is based on Proposition 1 in [CH22]) +from the analysis in [CH22]. The iteration count discussion is the same as in Appendix B. We sepa- +rate the discussion into parts corresponding to the two requirements in Proposition 2. Throughout, +we will show how to use the analysis in [CH22] to guarantee that with probability at least 1−Ω( 1 +κ), +the algorithm has expected function error O(ϵopt); because the maximum error over B(R) is ≤ LR, +this corresponds to an overall error O(ϵopt), and we may adjust Cba by a constant to compensate. +Per-iteration requirements. +The ball optimization error guarantees are as stated in Proposition +1 of [CH22], except we dropped the function evaluations requirement. To see that this is doable, +note that [CH22] obtains their line search oracle (see Proposition 1) by running O(log( Rκ +r )) ball +optimization oracles to O(λr2) expected error, querying the function value, and applying Markov +to argue at least one will succeed with high probability. We can instead apply a Chernoff bound +with O(log( Rκ +r )) independent runs to argue that with probability O( +1 +Kκ·polylog(Kκ)), the precondi- +tions of the geometric aggregation in Lemma 11 are met with ∆ = O(r), as required by the line +search oracle (see Algorithm 7). Finally, applying a union bound over all iterations implies that the +overall failure probability due to these line search oracles is O( 1 +κ) as required by our earlier argument. +Additional requirements. +The error requirements of the queries which occur every ≈ 2−j iter- +ations are as stated in [CH22]. The only difference is that we state the complexity deterministically +(Proposition 1 of [CH22] implicitly states an expected gradient bound). The stochastic proximal +oracle is implemented as Algorithm 2, [CH22]; it is also adapted with slightly different parame- +ters as Algorithm 6 of this paper. The expected complexity bound is derived by summing over +all j ∈ [⌈log2 K + Cba⌉], the probability j is sampled in each iteration of Algorithm 2 of [CH22]. +For all j a Chernoff bound shows that the number of times in the entire algorithm j is sampled is +O(2−jK log( Rκ +r )) (within a constant of its expectation), with probability 1 − Ω(poly( r +Rκ)). Taking +a union bound over all j shows the failure probability of our complexity bound is O( 1 +κ) as required. +D +Discussion of Proposition 4 +In this section, we discuss how to obtain Proposition 4 using results in [GL12]. We first state the +following helper fact on the smoothness of a convolved function �fρ (see Definition 1). +47 + +Fact 5 (Lemma 8, [BJL+19]). If f : Rd → R is L-Lipschitz, �fρ (see Definition 1) is L +ρ -smooth. +The statement of Proposition 4 then follows from recursively applying Proposition 9 of [GL12] +on the objective Ψ = �fρ + λ +2 ∥· − ¯x∥2, which is λ-strongly convex and ( L +ρ +λ)-smooth, together with +the divergence choice of V (x0, x∗) := 1 +2∥x0 − x∗∥2, which satisfies ν = 1. Our parameter choices +in Algorithm 2 are the same as in [GL12], where we use that our variance bound is 3L2 (Lemma 1). +In particular, denote the iterate xag +T after the kth outer loop by xk. We will inductively assume +that E 1 +2∥xk−1 − x⋆ +¯x,λ∥2 ≤ +r2 +2k−1 (clearly the base case holds). This then implies +E +�λ +2 ∥xk − x⋆ +¯x,λ∥2 +� +≤ E +� +Ψ(xk) − Ψ(x⋆ +¯x,λ) +� +≤ +2( L +ρ + λ)∥xk−1 − x⋆ +¯x,λ∥2 +T(T + 1) ++ +24L2 +λNk(T + 1) ≤ λ +2k r2 +where the second inequality is Proposition 9 in [GL12] (cf. equation (4.21) therein), and the last is +by our choice of T and Nk. Thus, when K > log2( λr2 +φ ) we have EΨ(xag +T ) − Ψ(x⋆ +¯x,λ) ≤ φ as in the +last outer loop k = K. The computational depth follows immediately from computing TK, and the +total oracle queries and computational complexity follow since NK asymptotically dominates: +T · +� +� � +k∈[K] +Nk +� +� = O (TNK + TK) = O +�� +1 + L +ρλ log +�λr2 +φ +� ++ L2 +λφ +� +. +48 + diff --git a/otAyT4oBgHgl3EQflvhP/content/tmp_files/load_file.txt b/otAyT4oBgHgl3EQflvhP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fbb26e6a3aeac2cd1af359f4cf9c5a17d346743 --- /dev/null +++ b/otAyT4oBgHgl3EQflvhP/content/tmp_files/load_file.txt @@ -0,0 +1,1899 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf,len=1898 +page_content='ReSQueing Parallel and Private Stochastic Convex Optimization Yair Carmon∗ Arun Jambulapati† Yujia Jin‡ Yin Tat Lee§ Daogao Liu† Aaron Sidford‡ Kevin Tian§ Abstract We introduce a new tool for stochastic convex optimization (SCO): a Reweighted Stochastic Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Combining ReSQue with recent advances in ball oracle acceleration [CJJ+20, ACJ+21], we develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a SCO objective constrained to the unit ball in Rd, we obtain the following results (up to polylogarithmic factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We give a parallel algorithm obtaining optimization error ϵopt with d1/3ϵ−2/3 opt gradient oracle query depth and d1/3ϵ−2/3 opt + ϵ−2 opt gradient queries in total, assuming access to a bounded-variance stochastic gradient estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For ϵopt ∈ [d−1, d−1/4], our algorithm matches the state-of-the-art oracle depth of [BJL+19] while maintaining the optimal total work of stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We give an (ϵdp, δ)-differentially private algorithm which, given n samples of Lipschitz loss functions, obtains near-optimal optimization error and makes min(n, n2ϵ2 dpd−1) + min(n4/3ϵ1/3 dp , (nd)2/3ϵ−1 dp ) queries to the gradients of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the regime d ≤ nϵ2 dp, where privacy comes at no cost in terms of the optimal loss up to constants, our algo- rithm uses n+(nd)2/3ϵ−1 dp queries and improves recent advancements of [KLL21, AFKT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the moderately low-dimensional setting d ≤ √nϵ3/2 dp , our query complexity is near-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ∗Tel Aviv University, ycarmon@tauex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' †University of Washington, {jmblpati, dgliu}@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ‡Stanford University, {yujiajin, sidford}@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' §Microsoft Research, {yintatlee, tiankevin}@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='00457v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='OC] 1 Jan 2023 Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 Parallelism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 37 References 37 A Helper facts 43 B Discussion of Proposition 1 46 C Discussion of Proposition 2 47 D Discussion of Proposition 4 47 1 Introduction Stochastic convex optimization (SCO) is a foundational problem in optimization theory, machine learning, theoretical computer science, and modern data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Variants of the problem underpin a wide variety of applications in machine learning, statistical inference, operations research, signal processing, and control and systems engineering [Sha07, SB14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, it provides a fertile ground for the design and analysis of scalable optimization algorithms such as the celebrated stochastic gradient descent (SGD), which is ubiquitous in machine learning practice [Bot12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' SGD approximately minimizes a function f : Rd → R by iterating xt+1 ← xt − ηg(xt), where g(xt) is an unbiased estimator to a (sub)gradient of f at iterate xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When f is convex, E ∥g(x)∥2 ≤ 1 for all x and f is minimized at x⋆ in the unit ball, SGD finds an ϵopt-optimal point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x satisfying Ef(x) ≤ f(x⋆) + ϵopt) using O(ϵ−2 opt) stochastic gradient evaluations [Bub15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This complexity is unimprovable without further assumptions [Duc18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' for sufficiently large d, this complexity is optimal even if g is an exact subgradient of f [DG19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Although SGD is widely-used and theoretically optimal in this simple setting, the algorithm in its basic form has natural limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For example, when parallel computational resources are given (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' multiple stochastic gradients can be queried in batch), SGD has suboptimal sequential depth in certain regimes [DBW12, BJL+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Further, standard SGD is not differentially private, and existing private1 SCO algorithms are not as efficient as SGD in terms of gradient evaluation complexity [BST14, BFTT19, FKT20, BFGT20, AFKT21, KLL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Despite substantial advances in both the parallel and private settings, the optimal complexity of each SCO problem remains open (see Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 for more precise definitions of problem settings and the state-of-the-art rates, and Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 for a broader discussion of related work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Though seemingly disparate at first glance, in spirit parallelism and privacy impose similar constraints on effective algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Parallel algorithms must find a way to query the oracle multiple times (possibly at multiple points) without using the oracle’s output at these points to determine where they were queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In other words, they cannot be too reliant on a particular outcome to adaptively choose the next query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Likewise, private algorithms must make optimization progress without over-relying on any individual sample to determine the optimization trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In both cases, oracle queries must be suitably robust to preceding oracle outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In this paper, we provide a new stochastic gradient estimation tool which we call Reweighted Stochastic Query (ReSQue) estimators (defined more precisely in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ReSQue is essentially an efficient parallel method for computing an unbiased estimate of the gradient of a convolution of f with a continuous (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Gaussian) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' These estimators are particularly well-suited for opti- mizing a convolved function over small Euclidean balls, as they enjoy improved stability properties over these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, these local stability properties facilitate tighter control over the stability of SGD-like procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We show that careful applications of ReSQue in conjunction with recent advances in accelerated ball-constrained optimization [CJJ+20, ACJ+21] yield complexity improvements for both parallel and private SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Paper organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 respectively, we formally describe the problems of parallel and private SCO we study, stating our results and contextualizing them in the prior litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We then cover additional related work in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 and, in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4, give an overview of our approach to obtaining these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5, we describe the notation we use throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 we introduce our ReSQue estimator and prove some of its fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1Throughout this paper, when we use the description “private” without further description we always refer to differential privacy [DR14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For formal definitions of differential privacy, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1 In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 we describe our adaptation of the ball acceleration frameworks of [ACJ+21, CH22], reducing SCO to minimizing the objective over small Euclidean balls, subproblems which are suitable for ReSQue-based stochastic gradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, in Sections 3 and 4, we prove our main results for parallel and private SCO (deferring problem statements to Problem 1 and Problem 2), respectively, by providing suitable implementations of our ReSQue ball acceleration framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 Parallelism In Section 3 we consider the following formulation of the SCO problem, simplified for the purposes of the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We assume there is a convex function f : Rd → R which can be queried through a stochastic gradient oracle g, satisfying Eg ∈ ∂f and E ∥g∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We wish to minimize the restriction of f to the unit Euclidean ball to expected additive error ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the standard sequential setting, SGD achieves this goal using roughly ϵ−2 opt queries to g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' as previously mentioned, this complexity is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A generalization of this formulation is restated in Problem 1 with a variance bound L2 and a radius bound R, which are both set to 1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In settings where multiple machines can be queried simultaneously, the parallel complexity of an SCO algorithm is a further important measure for consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In [Nem94], this problem was formalized in the setting of oracle-based convex optimization, where the goal is to develop iterative methods with a number of parallel query batches to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In each batch, the algorithm can submit polynomially many queries to g in parallel, and then perform computations (which do not use g) on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The query depth of a parallel algorithm in the [Nem94] model is the number of parallel rounds used to query g, and was later considered in stochastic algorithms [DBW12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Ideally, a parallel SCO algorithm will also have bounded total queries (the number of overall queries to g), and bounded computational depth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' the parallel depth used by the algorithm outside of oracle queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We discuss these three complexity measures more formally in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the low-accuracy regime ϵopt ≥ d−1/4, recent work [BJL+19] showed that SGD indeed achieves the optimal oracle query depth among parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 Moreover, in the high-accuracy regime ϵopt ≤ d−1, cutting plane methods (CPMs) by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [KTE88] (see [JLSW20] for an updated overview) achieve the state-of-the-art oracle query depth of d, up to logarithmic factors in d, ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the intermediate regime ϵopt ∈ [d−1, d−1/4], [DBW12, BJL+19] designed algorithms with or- acle query depths that improved upon SGD, as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, [BJL+19] obtained an algorithm with query depth �O(d1/3ϵ−2/3 opt ), which they conjectured is optimal for in- termediate ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' However, the total oracle query complexity of [BJL+19] is �O(d4/3ϵ−8/3 opt ), a (fairly large) polynomial factor worse than SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The main result of Section 3 is a pair of improved parallel algorithms in the setting of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Both of our algorithms achieve the “best of both worlds” between the [BJL+19] parallel algorithm and SGD, in that their oracle query depth is bounded by �O(d1/3ϵ−2/3 opt ) (as in [BJL+19]), but their total query complexity matches SGD’s in the regime ϵopt ≤ d−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We note that ϵopt ≤ d−1/4 is the regime where a depth of �O(d1/3ϵ−2/3 opt ) improves upon [DBW12] and SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our guarantees are formally stated in Theorems 1 and 2, and summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our first algorithm (Theorem 1) is based on a batched SGD using our ReSQue estimators, within the “ball acceleration” framework of [ACJ+21] (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By replacing SGD with an accelerated counterpart [GL12], we obtain a further improved computational depth in Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Theorem 2 simultaneously achieves the query depth of [BJL+19], the computational depth of [DBW12], and the total query complexity of SGD in the intermediate regime ϵopt ∈ [d−1, d−1/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2We omit logarithmic factors when discussing parameter regimes throughout the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2 Method g query depth computational depth # g queries SGD [Nes18] ϵ−2 ϵ−2 ϵ−2 [DBW12] d 1 4 ϵ−1 d 1 4 ϵ−1 d 1 4 ϵ−1 + ϵ−2 [BJL+19] d 1 3 ϵ− 2 3 d 4 3 ϵ− 8 3 d 4 3 ϵ− 8 3 CPM [KTE88] d d d BallAccel + EpochSGD (Theorem 1) d 1 3 ϵ− 2 3 d 1 3 ϵ− 2 3 + ϵ−2 d 1 3 ϵ− 2 3 + ϵ−2 BallAccel + AC-SA (Theorem 2) d 1 3 ϵ− 2 3 d 1 3 ϵ− 2 3 + d 1 4 ϵ−1 d 1 3 ϵ− 2 3 + ϵ−2 Table 1: Comparison of parallel SCO results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The complexity of finding a point with expected error ϵ := ϵopt in Problem 1, where L = R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We hide polylogarithmic factors in d and ϵ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 Differential privacy Differential privacy (DP) is a mathematical quantification for privacy risks in algorithms involving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When performing stochastic convex optimization with respect to a sampled dataset from a population, privacy is frequently a natural practical desideratum [BST14, EPK14, Abo16, App17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For example, the practitioner may want to privately learn a linear classifier or estimate a regression model or a statistical parameter from measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In this paper, we obtain improved rates for private SCO in the following model, which is standard in the literature and restated in Problem 2 in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Symmetrically to the previous section, in the introduction, we only discuss the specialization of Problem 2 with L = R = 1, where L is a Lipschitz parameter and R is a domain size bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We assume there is a distribution P over a population S, and we obtain independent samples {si}i∈[n] ∼ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Every element s ∈ S induces a 1-Lipschitz convex function f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s), and the goal of SCO is to approximately optimize the population loss fpop := Es∼P[f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The setting of Problem 2 can be viewed as a specialization of Problem 1 which is more compatible with the notion of DP, discussed in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The cost of achieving approximate DP with privacy loss parameter ϵdp (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 for definitions) has been studied by a long line of work, starting with [BST14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The optimal error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' excess population loss) given n samples scales as (omitting logarithmic factors) 1 √n + √ d nϵdp , (1) with matching lower and upper bounds given by [BST14] and [BFTT19], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The n−1/2 term is achieved (without privacy considerations) by simple one-pass SGD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' treating sample gradients as unbiased for the population loss, and discarding samples after we query their gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, the additional term √ d·(nϵdp)−1 can be viewed as the “cost of privacy” in SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Due to this, the moderately low-dimension regime d ≤ nϵ2 dp is natural, as this is the setting where privacy comes at no asymptotic cost from the perspective of the bound (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, many real-world problems in data analysis have low intrinsic dimension, meaning that the effective number of degrees of freedom in the optimization problem is much smaller than the ambient dimension [SSTT21, LLH+22], which can be captured via a dimension-reducing preprocessing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For these reasons, we primarily focus 3 Method excess fpop loss # gradient queries to samples [BST14] 4√ d log n δ √n + √ d log2 n δ nϵ n2 [BFTT19] 1 √n + � d log 1 δ nϵ n 9 2 [FKT20] 1 √n + � d log 1 δ nϵ n2 [BFGT20] 1 √n + � d log 1 δ nϵ n2 [AFKT21] 1 √n + � d log 1 δ nϵ min � n 3 2 , n2ϵ √ d � [KLL21] 1 √n + � d log 1 δ nϵ min � n 5 4 d 1 8 √ϵ, n 3 2 ϵ d 1 8 � Theorem 4 1 √n + � d log 1 δ log n log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 n δ nϵ min � n, n2ϵ2 d � + min � (nd) 2 3 ϵ , n 4 3 ϵ 1 3 � Table 2: Comparison of private SCO results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The excess loss and gradient complexity of (ϵ := ϵdp, δ)-DP in Problem 2, where L = R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We hide polylogarithmic factors in d, n, δ−1, ϵ−1 in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The optimal loss [BST14, SU15] is achieved by rows 2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' on the moderate-dimensional regime in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' An unfortunate property of private SCO algorithms achieving error (1) is they all query substan- tially more than n sample gradients without additional smoothness assumptions [BST14, BFTT19, FKT20, BFGT20, AFKT21, KLL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For example, analyses of simple perturbed SGD variants re- sult in query bounds of ≈ n2 [BFGT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In fact, [BFGT20] conjectured this quadratic complexity was necessary, which was disproven by [AFKT21, KLL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The problem of obtaining the optimal error (1) using n gradient queries has been repeatedly highlighted as an important open problem by the private optimization community, as discussed in [BFGT20, AFKT21, KLL21, ACJ+21] as well as the recent research overview [Tal22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The main result of Section 4 is a new private SCO algorithm, which achieves the error bound (1) up to logarithmic factors with an improved gradient query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our guarantee is formally stated in Theorem 4 and summarized in Table 2 and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Omitting logarithmic factors, our gradient query complexity is min � n, n2ϵ2 dp d � + min � (nd) 2 3 ϵdp , n 4 3 ϵ 1 3 dp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the regime d ≤ nϵ2 dp, our query complexity is bounded by n+(nd)2/3ϵ−1 dp , and when d ≤ √nϵ3/2 dp is sufficiently small, the above bound is linear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our result improves upon the prior state-of-the-art by polynomial factors whenever d ≪ n4/3 (omitting ϵdp dependencies for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Table 2 and Figure 1, we summarize our private SCO result and compare its query complexity with the prior art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the regime d ≈ n, ϵdp = Θ(1), the state-of-the-art bound [KLL21] is ≈ n11/8 gradient queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' on the other hand, Theorem 4 uses ≈ n4/3 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When d ≪ √n, the previous best algorithm [KLL21] still uses ≈ n21/16 queries, whereas our bound improves to near-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4 α : dimension d ∝ nα β : gradient complexity ∝ nβ Result in [KLL21] Result in [AFKT21] Our result Figure 1: Comparison among our gradient complexity and previous results in [AFKT21, KLL21] for the non-trivial regime d ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We omit dependencies on ϵdp (treated as Θ(1) in this figure) and logarithmic terms for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 Related work Stochastic convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Convex optimization is a fundamental task with numerous applications in computer science, operations research, and statistics [BV14, Bub15, Nes18], and has been the focus of extensive research over the past several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The primary setting of interest in this paper is non-smooth (Lipschitz) stochastic convex optimization in private and parallel computational models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Previously, [Gol64] gave a gradient method that used O(ϵ−2) gradient queries to compute a point achieving ϵ error for Lipschitz convex minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This rate which was shown to be optimal in an information-theoretic sense in [NY83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The stochastic gradient descent method extends [Gol64] to tolerate randomized, unbiased gradient oracles with bounded second moment: this yields algorithms for Problem 1 and Problem 2 (when privacy is not a consideration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Since the first proposal of accelerated (momentum-based) methods [Pol64, Nes83, Nes03], acceleration has become a central topic in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This work builds on the seminal Monteiro-Svaiter acceleration technique [MS13] and its higher-order variants [GDG+19, BJL+19], More specifically, our work follows recent developments in accelerated ball optimization [CJJ+20, CJJS21, ACJ+21], which can be viewed as a limiting case of high-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our algorithms directly leverage error-robust variants of this framework developed by [ACJ+21, CH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Parallel SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Recently, parallel optimization has received increasing interest in the context of large-scale machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Speeding up SGD by averaging stochastic gradients across mini- batches is extremely common in practice, and optimal in certain distributed optimization set- tings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DGBSX12, DRY18, WBSS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Related to the setting we study are the distributed optimization methods proposed in [SBB+18], which also leverage convolution-based randomized smoothing and apply to both stochastic and deterministic gradient-based methods (but do not focus on parallel depth in the sense of [Nem94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, lower bounds against the oracle query depth of parallel SCO algorithms in the setting we consider have been an active area of study, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Nem94, BS18, DG19, BJL+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Both the private stochastic convex optimization problem (DP-SCO) and the pri- vate empirical risk minimization problem (DP-ERM) are well-studied by the DP community [CM08, 5 RBHT12, CMS11, JT14, BST14, KJ16, FTS17, ZZMW17, Wan18, INS+19, BFTT19, FKT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, [BST14] shows that the exponential mechanism and noisy stochastic gradient de- scent achieve the optimal loss for DP-ERM for (ϵdp, 0)-DP and (ϵdp, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In follow-up works, [BFTT19, FKT20] show that one can achieve the optimal loss for DP-SCO as well, by a suitable modification of noisy stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' However, these algorithms suffer from large (at least quadratic in n) gradient complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Under an additional assumption that the loss functions are sufficiently smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' have Lipschitz gradient), [FKT20] remedies this issue by obtaining op- timal loss and optimal gradient complexity under differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In a different modification of Problem 2’s setting (where sample function access is modeled through value oracle queries instead of subgradients), [GLL22] designs an exponential mechanism-based method that uses the optimal value oracle complexity to obtain the optimal SCO loss for non-smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Most directly related to our approach are the recent works [KLL21] and [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Both pro- pose methods improving upon the quadratic gradient complexity achieved by noisy SGD, by using variants of smoothing via Gaussian convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The former proposes an algorithm that uses noisy accelerated gradient descent for private SCO with subquadratic gradient complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The latter suggests a ball acceleration framework to solve private SCO with linear gradient queries, under a hypothetical algorithm to estimate subproblem solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our work can be viewed as a formalization of the connection between ball acceleration strategies and private SCO as suggested in [ACJ+21], by way of ReSQue estimators, which we use to obtain improved query complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4 Our approach Here we give an overview of our approach towards obtaining the results outlined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 and Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Both sets of results build off a common framework based on a new stochastic gradient estimation tool we introduce and call Reweighted Stochastic Query (ReSQue) estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our new tool is naturally compatible with ball-constrained optimization frameworks, where an optimization problem is localized to a sequence of constrained subproblems (solved to sufficient accuracy), whose solutions are then stitched together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We exploit this synergy, as well as the local stability properties of our ReSQue estimators, to design SCO algorithms with improved parallelism or privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We separate the discussion of our approach into three parts regarding our optimization framework, and our instantiations of it in parallel and private settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ReSQue estimators and ball acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The convolution of a function of interest f : Rd → R with a Gaussian density γρ (with covariance ρ2Id), which we denote by �fρ, is a well-studied tool for SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In a line of work building upon [DBW12] and including [Haz16, BJL+19, KLL21], SCO algorithms capitalized on smoothness properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' high-order derivative bounds) of the convolved function �fρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In this work, we introduce a new tool that capitalizes upon a different property: the fact that the Gaussian density is locally stable in a small ball around its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Given a reference point ¯x and a query point x, our proposed estimator for ∇ �fρ(x) is draw ξ ∼ N(0, ρ2Id), and output estimate γρ(x − ¯x − ξ) γρ(ξ) g(¯x + ξ), (2) where g(z) is an unbiased estimate for a subgradient of f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', Eg(z) ∈ ∂f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' That is, to estimate the gradient of �fρ, we simply reweight (stochastic) gradients of f that were queried at random perturbations of reference point ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This reweighted stochastic query (ReSQue) estimator is unbiased for ∇ �fρ(x), regardless of ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' However, when ∥x − ¯x∥ ≪ ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x is contained in a small ball around ¯x, the reweighting factor γρ(x−¯x−ξ) γρ(ξ) is likely to be close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' As a result, when g is bounded and 6 x is near ¯x, the estimator (2) enjoys regularity properties such as moment bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Crucially, the stochastic gradient queries performed by ReSQue (at points of the form ¯x + ξ) do not depend on the point x at which we eventually estimate the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We develop this theory in Section 2, but mention one additional property here, which can be thought of as a “relative smoothness” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We show that when ∥x − x′∥ is sufficiently smaller than ρ, the difference of estimators of the form (2) has many bounded moments, where bounds scale as a function of ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When we couple a sequence of stochastic gradient updates by the randomness used in defining (2), we can use this property to bound how far sequences drift apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, initially nearby points are likely to stay close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We exploit this property when analyzing the stability of private stochastic gradient descent algorithms later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To effectively use these local stability properties of (2), we combine them with an optimization framework called ball-constrained optimization [CJJ+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' It is motivated by the question: given parameters 0 < r < R, and an oracle which minimizes f : Rd in a ball of radius r around an input point, how many oracles must we query to optimize f in a ball of larger radius R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' It is not hard to show that simply iterating calls to the oracle gives a good solution in roughly R r queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In recent work, [CJJ+20] demonstrated that the optimal number of calls scales (up to logarithmic factors) as ( R r )2/3, and [ACJ+21] gave an approximation-tolerant variant of the [CJJ+20] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We refer to these algorithms as ball acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Roughly, [ACJ+21] shows that running stochastic gradient methods on ≈ ( R r )2/3 subproblems constrained to balls of radius r obtains total gradient query complexity comparable to directly running SGD on the global function of domain radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Importantly, in many structured cases, we have dramatically more freedom in solving these subproblems, compared to the original optimization problem, since we are only required to optimize over a small radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' One natural form of complexity gain from ball acceleration is when there is a much cheaper gradient estimator, which is only locally defined, compared to a global estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This was the original motivation for combining ball acceleration with stochastic gradient methods in [CJJS21], which exploited local smoothness of the softmax function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' the form of our ReSQue estimator (2) is motivated by the [CJJS21] estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In this work, we show that using ReSQue with reference point ¯x significantly improves the parallel and private complexity of minimizing the convolution �fρ inside a ball of radius r ≈ ρ centered at ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Parallel subproblem solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A key property of the ReSQue estimator (2) is that its estimate of ∇ �fρ(x) is a scalar reweighting of g(¯x + ξ), where ξ ∼ N(0, ρ2Id) and ¯x is a fixed reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, in each ball subproblem (assuming r = ρ), we can make all the stochastic gradient queries in parallel, and use the resulting pool of vectors to perform standard (ball-constrained) stochastic optimization using ReSQue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Thus, we solve each ball subproblem with a single parallel stochastic gradient query, and — using ball acceleration — minimize �fρ with query depth of roughly ρ−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To ensure that �fρ is a uniform ϵopt-approximation of the original f, we must set ρ to be roughly ϵopt/ √ d, leading to the claimed d1/3ϵ−2/3 opt depth bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Furthermore, the ball acceleration frame- work guarantees that we require no more than roughly ρ−2/3+ϵ−2 opt stochastic gradient computations throughout the optimization, yielding the claimed total query bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' However, the computational depth of the algorithm described thus far is roughly ϵ−2 opt, which is no better than SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Sec- tion 3 we combine our approach with the randomized smoothing algorithm of [DBW12] by using an accelerated mini-batched method [GL12] for the ball-constrained stochastic optimization, leading to improved computational depth as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our parallel SCO results use the ReSQue/ball acceleration technique in a simpler manner than our private SCO results described next and in Section 4, so we chose to present them first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 7 Private subproblem solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To motivate our improved private SCO solvers, we make the following connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, it is straightforward to show that the convolved function �fρ is 1 ρ- smooth whenever the underlying function f is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Further, recently [FKT20] obtained a linear gradient query complexity for SCO, under the stronger assumption that each sample function (see Problem 2) is ≲ √n-smooth (for L = R = 1 in Problem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This bound is satisfied by the result of Gaussian convolution with radius 1 √n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' however, two difficulties arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, to preserve the function value approximately up to ϵopt, we must take a Gaussian convolution of radius ρ ≈ ϵopt √ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For ϵopt in (1), this is much smaller than 1 √n in many regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Second, we cannot access the exact gradients of the convolved sampled functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, it is natural to ask: is there a way to simulate the smoothness of the convolved function, under stochastic query access?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taking a step back, the primary way in which [FKT20] used the smoothness assumption was through the fact that gradient steps on a sufficiently smooth function are contractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This obser- vation is formalized as follows: if x′ ← x − η∇f(x) and y′ ← y − η∇f(y), when f is O( 1 η)-smooth, then ∥x′ − y′∥ ≤ ∥x − y∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' As alluded to earlier, we show that ReSQue estimators (2) allow us to simulate this contractivity up to polylogarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We show that by coupling the randomness ξ in the estimator (2), the drift growth in two-point sequences updated with (2) is predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We give a careful potential-based argument (see Lemma 5) to bound higher moments of our drift after a sequence of updates using ReSQue estimators, when they are used in an SGD subroutine over a ball of radius ≪ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This allows for the use of “iterative localization” strategies introduced by [FKT20], based on iterate perturbation via the Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have not yet dealt with the fact that while this “smoothness simulation” strategy allows us to privately solve one constrained ball subproblem, we still need to solve K ≈ ( 1 r)2/3 ball subproblems to optimize our original function, where r ≪ ρ is the radius of each subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we rely on arguments based on amplification by subsampling, a common strategy in the private SCO literature [ACG+16, BBG18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We set our privacy budget for each ball subproblem to be approximately (ϵdp, δ) (our final overall budget), before subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We then use solvers by suitably combining the [FKT20] framework and our estimator (2) to solve these ball subproblems using ≈ n · K−1/2 gradient queries each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, our algorithm obtains the desired query complexity: ≈ n √ K ���� gradient queries per subproblem K ���� number of subproblems = n √ K, and privacy: ≈ ϵdp ���� privacy budget per subproblem 1 √ K ���� subsampling √ K ���� advanced composition = ϵdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (3) Here we used the standard technique of advanced composition (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2, [DR14]) to bound the privacy loss over K consecutive ball subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let us briefly derive the resulting complexity bound and explain the bottleneck for improving it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, the ball radius r must be set to ≈ ρ (the smoothing parameter) for our ReSQue estimators to be well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, we have to set ρ ≈ ϵopt √ d , otherwise the effect of the convolution begins to dominate the optimization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For ϵopt ≈ 1 √n + √ d(nϵdp)−1 (see (1)), this results in 1 r ≈ min( √ nd, nϵdp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, K ≈ ( 1 r)2/3 is known to be essentially tight for ball acceleration with R = 1 [CJJ+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the subproblem accuracies required by the [ACJ+21] ball acceleration framework,3 known lower bounds on private empirical risk minimization imply that 3These subproblem accuracy requirements cannot be lowered in general, because combined they recover the optimal gradient complexities of SGD over the entire problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 8 ≈ n √ K gradients are necessary for each subproblem to preserve a privacy budget of ϵdp [BST14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' As subsampling requires the privacy loss before amplification to already be small (see discussion in [Smi09, BBG18]), all of these parameter choices are optimized, leading to a gradient complexity of n √ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For our lower bound on 1 r, this scales as ≈ min(n4/3, (nd)2/3) as we derive in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4 To go beyond the strategies we employ, it is natural to look towards other privacy amplification arguments (for aggregating ball subproblems) beyond subsampling, which we defer to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our final algorithm is analyzed through the machinery of Rényi differential privacy (RDP) [Mir17], which allows for more fine-grained control on the effects of composition and subsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We modify the standard RDP machinery in two main ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We define an approximate relaxation and control the failure probability of our relaxation using high moment bounds on our drift (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We also provide an analysis of amplification under subsampling with replacement by modifying the truncated CDP (concentrated DP) tools introduced by [BDRS18], who analyzed subsampling without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Sampling with replacement is crucial in order to guarantee that our ReSQue estimators are unbiased for the empirical risks we minimize when employing a known reduction [FKT20, KLL21] from private SCO to private regularized empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 Notation Throughout �O hides polylogarithmic factors in problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For n ∈ N, we let [n] := {i | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For x ∈ Rd we let ∥x∥ denote the Euclidean norm of x, and let Bx(r) := {x′ ∈ Rd | ∥x′ − x∥ ≤ r} denote a Euclidean ball of radius r centered at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' when x is unspecified we take it to be the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', B(r) := {x′ ∈ Rd | ∥x′∥ ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We let N(µ, Σ) denote a multivariate Gaussian distribution with mean µ ∈ Rd and covariance Σ ∈ Rd×d, and Id is the identity matrix in Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For K ⊆ Rd, we define the Euclidean projection onto K by ΠK(x) := argminx′∈K ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For p ∈ [0, 1], we let Geom(p) denote the geometric distribution with parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say a function f : Rd → R is L-Lipschitz if for all x, x′ ∈ Rd we have |f(x) − f(x′)| ≤ L ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say f is λ-strongly convex if for all x, x′ ∈ Rd and t ∈ [0, 1] we have f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) − λt(1 − t) 2 ��x − x′��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We denote the subdifferential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', set of all subgradients) of a convex function f : Rd → R at x ∈ Rd by ∂f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Overloading notation, when clear from the context we will write ∂f(x) to denote an arbitrary subgradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let µ, ν be two probability densities µ, ν on the same probability space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We let DTV(µ, ν) := 1 2 � |µ(ω) − ν(ω)|dω denote the total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The following fact is straightforward to see and will be frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let E be any event that occurs with probability at least 1 − δ under the density µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then DTV(µ, µ | E) ≤ δ, where µ | E denotes the conditional distribution of µ under E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For two densities µ, ν, we say that a joint distribution Γ(µ, ν) over the product space of outcomes is a coupling of µ, ν if for (x, x′) ∼ Γ(µ, ν), the marginals of x and x′ are µ and ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When µ is absolutely continuous with respect to ν, and α > 1, we define the α-Rényi divergence by Dα(µ∥ν) := 1 α − 1 log �� �µ(ω) ν(ω) �α dν(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (4) 4In the low-dimensional regime d ≤ nϵ2 dp, the gradient queries used per subproblem improves to √ nd ϵdp √ K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 9 Dα is quasiconvex in its arguments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' if µ = Eξµξ and ν = Eξνξ (where ξ is a random variable, and µξ, νξ are distribution families indexed by ξ), then Dα(µ∥ν) ≤ maxξ Dα(µξ∥νξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2 Framework We now outline our primary technical innovation, a new gradient estimator for stochastic convex optimization (ReSQue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We define this estimator in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 and prove that it satisfies several local stability properties in a small ball around a “centerpoint” used for its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2, we then give preliminaries on a “ball acceleration” framework developed in [CJJ+20, ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This framework aggregates solutions to proximal subproblems defined on small (Euclidean) balls, and uses these subproblem solutions to efficiently solve an optimization problem on a larger domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our algorithms in Sections 3 and 4 instantiate the framework of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 with new subproblem solvers enjoying improved parallelism or privacy, based on our new ReSQue estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 ReSQue estimators Throughout we use γρ : Rd → R≥0 to denote the probability density function of N(0, ρ2Id), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' γρ(x) = (2πρ)− d 2 exp(− 1 2ρ2 ∥x∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first define the Gaussian convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Definition 1 (Gaussian convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a function f : Rd → R we denote its convolution with a Gaussian of covariance ρ2Id by �fρ := f ∗ γρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' �fρ(x) := Ey∼N(0,ρ2Id)f(x + y) = � y∈Rn f(x − y)γρ(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (5) Three well-known properties of �fρ are that it is differentiable, that if f is L-Lipschitz, so is �fρ for any ρ, and that | �fρ − f| ≤ Lρ √ d pointwise (Lemma 8, [BJL+19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, given a centerpoint ¯x and a smoothing radius ρ, we define the associated reweighted stochastic query (ReSQue) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Definition 2 (ReSQue estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let ¯x ∈ Rd and let f : Rd → R be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Suppose we have a gradient estimator g : Rd → Rd satisfying Eg ∈ ∂f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We define the ReSQue estimator of radius ρ as the random vector �∇g ¯x �fρ(x) := γρ(x − ¯x − ξ) γρ(ξ) g(¯x + ξ) where ξ ∼ N(0, ρ2Id), where we first sample ξ, and then independently query g at ¯x + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When g is deterministically an element of ∂f, we drop the superscript and denote the estimator by �∇¯x �fρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When g is unbiased for ∂f and enjoys a variance bound, the corresponding ReSQue estimator is unbiased for the convolved function, and inherits a similar variance bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The estimator in Definition 2 satisfies the following properties, where expectations are taken over both the randomness in ξ and the randomness in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Unbiased: E�∇g ¯x �fρ(x) = ∇ �fρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Bounded variance: If E ∥g∥2 ≤ L2 everywhere, and x ∈ B¯x(ρ), then E∥�∇g ¯x �fρ(x)∥2 ≤ 3L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The first statement follows by expanding the expectation over ξ and g: Eg � γρ(x − ¯x − ξ) γρ(ξ) g(¯x + ξ)γρ(ξ)dξ = � γρ(x − ¯x − ξ) γρ(ξ) ∂f(¯x + ξ)γρ(ξ)dξ = � ∂f(¯x + ξ)γρ(x − ¯x − ξ)dξ = ∇ �fρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The last equality used that the integral is a subgradient of �fρ, and �fρ is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the second statement, denote v := x − ¯x for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Since f is L-Lipschitz, E∥�∇g ¯x �fρ(x)∥2 = Eg � (γρ(v − ξ))2 γρ(ξ) ∥g(¯x + ξ)∥2 dξ ≤ L2(2πρ)− d 2 � exp � −∥v − ξ∥2 ρ2 + ∥ξ∥2 2ρ2 � dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, a standard calculation for Gaussian integrals shows � exp � 2 ⟨v, ξ⟩ − ∥ξ∥2 2ρ2 � dξ = exp � ∥v∥2 2ρ2 � � exp � −∥ξ − v∥2 2ρ2 � dξ = exp � ∥v∥2 2ρ2 � (2πρ) d 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (6) The statement then follows from (6), which yields � exp � −∥v − ξ∥2 ρ2 + ∥ξ∥2 2ρ2 � dξ = exp � −∥v∥2 ρ2 � � exp � 4 ⟨v, ξ⟩ − ∥ξ∥2 2ρ2 � dξ = (2πρ) d 2 exp � 2 ∥v∥2 ρ2 � ≤ 3 · (2πρ) d 2 (7) and completes the proof of the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When the gradient estimator g is deterministically a subgradient of a Lipschitz function, we can show additional properties about ReSQue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The following lemma will be used in Section 4 both to obtain higher moment bounds on ReSQue, as well as higher moment bounds on the difference of ReSQue estimators at nearby points, where the bound scales with the distance between the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' If x, x′ ∈ B¯x( ρ p) for p ≥ 2 then Eξ∼N(0,ρ2Id) ��γρ(x − ¯x − ξ) γρ(ξ) �p� ≤ 2, Eξ∼N(0,ρ2Id) ����� γρ(x − ¯x − ξ) − γρ(x′ − ¯x − ξ) γρ(ξ) ���� p� ≤ �24p ∥x − x′∥ ρ �p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We defer a proof to Appendix A, where a helper calculation (Fact 3) is used to obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 Ball acceleration We summarize the guarantees of a recent “ball acceleration” framework originally proposed by [CJJ+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For specified parameters 0 < r < R, this framework efficiently aggregates (approximate) solutions to constrained optimization problems over Euclidean balls of radius r to optimize a function over a ball of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we give an approximation-tolerant variant of the [CJJ+20] algorithm 11 in Proposition 1, which was developed by [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Before stating the guarantee, we require the definitions of three types of oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In each of the following definitions, for some function F : Rd → R, scalars λ, r, and point ¯x ∈ Rd which are clear from context, we will denote x⋆ ¯x,λ := argminx∈B¯x(r) � F(x) + λ 2 ∥x − ¯x∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (8) We mention that in the non-private settings of prior work [ACJ+21, CH22] (and under slightly different oracle access assumptions), it was shown that the implementation of line search oracles (Definition 3) and stochastic proximal oracles (Definition 5) can be reduced to ball optimization oracles (Definition 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Indeed, such a result is summarized in Proposition 2 and used in Section 3 to obtain our parallel SCO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To tightly quantify the privacy loss of each oracle for developing our SCO algorithms in Section 4 (and to implement these oracles under only the function access afforded by Problem 2), we separate out the requirements of each oracle definition separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Definition 3 (Line search oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say Ols is a (∆, λ)-line search oracle for F : Rd → R if given ¯x ∈ Rd, Ols returns x ∈ Rd with ��x − x⋆ ¯x,λ �� ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Definition 4 (Ball optimization oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say Obo is a (φ, λ)-ball optimization oracle for F : Rd → R if given ¯x ∈ Rd, Obo returns x ∈ Rd with E � F(x) + λ 2 ∥x − ¯x∥2 � ≤ F(x⋆ ¯x,λ) + λ 2 ��x⋆ ¯x,λ − ¯x ��2 + φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Definition 5 (Stochastic proximal oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say Osp is a (∆, σ, λ)-stochastic proximal oracle for F : Rd → R if given ¯x ∈ Rd, Osp returns x ∈ Rd with ��Ex − x⋆ ¯x,λ �� ≤ ∆ λ , E ��x − x⋆ ¯x,λ ��2 ≤ σ2 λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Leveraging Definitions 3, 4, and 5, we state a variant of the main result of [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Roughly speaking, Proposition 1 states that to optimize a function F over a ball of radius R, it suffices to query ≈ ( R r ) 2 3 oracles which approximately optimize a sufficiently regularized variant of F over a ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We quantify the types of approximate optimization of such regularized functions in Proposition 1, and defer a detailed discussion of how to derive this statement from [ACJ+21] in Appendix B, as it is stated slightly differently in the original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let F : Rd → R be L-Lipschitz and convex, and suppose for R ≥ 0 there is x⋆ ∈ argminxF(x) with x⋆ ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an algorithm BallAccel taking parameters r ∈ [0, R] and ϵopt ∈ (0, LR] with the following guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define κ := LR ϵopt , K := �R r � 2 3 , λ⋆ := ϵoptK2 R2 log2 κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a universal constant Cba > 0, BallAccel runs in at most CbaK log κ iterations and produces a point x such that EF(x) ≤ F(x⋆) + ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, in each iteration BallAccel requires the following oracle calls (all for F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 5In particular, we use an error tolerance for the ball optimization oracles, which is slightly larger than in [ACJ+21], following a tighter error analysis given in Proposition 1 of [CH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' At most Cba log( Rκ r ) calls to a ( r Cba , λ)-line search oracle with values of λ ∈ [ λ⋆ Cba , CbaL ϵopt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A single call to ( λr2 Cba log3 κ, λ)-ball optimization oracle with λ ∈ [ λ⋆ Cba , CbaL ϵopt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A single call to ( ϵopt CbaR, ϵopt √ K CbaR , λ)-stochastic proximal oracle with λ ∈ [ λ⋆ Cba , CbaL ϵopt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The optimization framework in Proposition 1 is naturally compatible with our ReSQue estima- tors, whose stability properties are local in the sense that they hold in balls of radius ≈ ρ around the centerpoint ¯x (see Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Conveniently, BallAccel reduces an optimization problem over a domain of size R to a sequence of approximate optimization problems on potentially much smaller domains of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Sections 3 and 4, by instantiating Proposition 1 with r ≈ ρ, we demonstrate how to use the local stability properties of ReSQue estimators (on smaller balls) to solve constrained subproblems, and consequently design improved parallel and private algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, as mentioned previously, in settings where privacy is not a consideration, Proposition 1 of [CH22] gives a direct implementation of all the line search and stochastic proximal oracles required by Proposition 1 by reducing them to ball optimization oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The statement in [CH22] also assumes access to function evaluations in addition to gradient (estimator) queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' however, it is straightforward to use geometric aggregation techniques (see Lemma 11) to bypass this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We give a slight rephrasing of Proposition 1 in [CH22] without the use of function evaluation oracles, and defer further discussion to Appendix C where we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let F : Rd → R be L-Lipschitz and convex, and suppose for R ≥ 0 there is x⋆ ∈ argminxF(x) with x⋆ ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an implementation of BallAccel (see Proposition 1) taking parameters r ∈ [0, R] and ϵopt ∈ (0, LR] with the following guarantee, where we define κ, K, λ⋆ as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a universal constant Cba > 0, BallAccel runs in at most CbaK log κ iterations and produces a point x such that EF(x) ≤ F(x⋆) + ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Each iteration makes at most Cba log2( Rκ r ) calls to ( λr2 Cba , λ)-ball optimization oracle with values of λ ∈ [ λ⋆ Cba , CbaL ϵopt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For each j ∈ [⌈log2 K+Cba⌉], at most C2 ba·2−jK log( Rκ r ) iterations query a ( λr2 Cba2j ·log−2( Rκ r ), λ)- ball optimization oracle for some λ ∈ [ λ⋆ Cba , CbaL ϵopt ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 3 Parallel stochastic convex optimization In this section, we present our main results on parallel convex optimization with improved com- putational depth and total work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We present our main results below in Theorems 1 and 2, after formally stating our notation and the SCO problem we study in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 Preliminaries In this section, we study the following SCO problem, which models access to an objective only through the stochastic gradient oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let f : Rd → R be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We assume there exists a stochastic gradient oracle g : Rd → Rd satisfying for all x ∈ Rd, Eg(x) ∈ ∂f(x), E ∥g(x)∥2 ≤ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our goal is to produce an ϵopt-approximate minimizer to f constrained to B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We define parameter κ := LR ϵopt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (9) 13 When discussing a parallel algorithm which queries a stochastic gradient oracle, in the sense of Problem 1, we separate its complexity into four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The query depth is the maximum number of sequential rounds of interaction with the oracle, where queries are submitted in batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The total number of queries is the total number of oracle queries used by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The computational depth and work are the sequential depth and total amount of computational work done by the algorithm outside of these oracle queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For simplicity we assume that all d-dimensional vector operations have a cost of d when discussing computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 Proofs of Theorems 1 and 2 Theorem 1 (Parallel EpochSGD-based solver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' BallAccel (Proposition 2) using parallel EpochSGD (Algorithm 1) as a ball optimization oracle solves Problem 1 with expected error ϵopt, with O � d 1 3 κ 2 3 log3(dκ) � query depth and O � d 1 3 κ 2 3 log3 (dκ) + κ2 log4 (dκ) � total queries, and an additional computational cost of O � d 1 3 κ 2 3 log3 (dκ) + κ2 log4 (dκ) � depth and O �� d 1 3 κ 2 3 log3 (dκ) + κ2 log4 (dκ) � d � work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Theorem 2 (Parallel AC-SA-based solver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' BallAccel (Proposition 2) using parallel AC-SA (Algo- rithm 2) as a ball optimization oracle solves Problem 1 with expected error ϵopt, with O � d 1 3 κ 2 3 log κ � query depth and O � d 1 3 κ 2 3 log3 (dκ) + d 1 4 κ log4 (dκ) + κ2 log4 (dκ) � total queries, and an additional computational cost of O � d 1 3 κ 2 3 log3 (dκ) + d 1 4 κ log4 (dκ) � depth and O �� d 1 3 κ 2 3 log3 (dκ) + d 1 4 κ log4 (dκ) + κ2 log4 (dκ) � d � work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The query depth, total number of queries, and total work for both of our results are the same (up to logarithmic factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The main difference is that AC-SA attains an improved computational depth for solving SCO, compared to using EpochSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our results build upon the BallAccel framework in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2, combined with careful parallel implementations of the required ball optimization oracles to achieve improved complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by developing our parallel ball optimization oracles using our ReSQue estimator ma- chinery from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, Proposition 2 reduces Problem 1 to implementation of a ball optimization oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Recall that a ball optimization oracle (Definition 4) requires an approximate solution x of a regularized subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, for some accuracy parameter φ, and defining x⋆ ¯x,λ as in (8), we wish to compute a random x ∈ B¯x(r) such that E � �fρ(x) + λ 2 ∥x − ¯x∥2 � ≤ �fρ(x⋆ ¯x,λ) + λ 2 ��x⋆ ¯x,λ − ¯x ��2 + φ, x ∈ B¯x(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Note that such a ball optimization oracle can satisfy the requirements of Proposition 2 with F ← �fρ, r ← ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, Lemma 1 gives a gradient estimator variance bound under the setting r = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 14 EpochSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We implement EpochSGD [HK14, ACJ+21], a variant of standard stochastic gradi- ent descent on regularized objective functions, in parallel using the stochastic ReSQue estimator constructed in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our main observation is that the gradient queries in Definition 2 can be implemented in parallel at the beginning of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We provide the pseudocode of our parallel implementation of EpochSGD in Algorithm 1 and state its guarantees in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 1: EpochSGD(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' φ) 1 Input: f : Rd → R and g : Rd → R satisfying the assumptions of Problem 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ¯x ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' φ > 0 2 η1 ← 1 4λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' T1 ← 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' T ← ⌈ 48L2 λφ ⌉ 3 Sample ξi ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ2Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' i ∈ [2T] independently 4 Query g(¯x + ξi) for all i ∈ [2T] (in parallel) 5 x0 1 ← ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' k ← 1 6 while � j∈[k] Tj ≤ T do 7 x1 k ← argminx∈B¯x(r) � ηkλ 2 ∥x − ¯x∥2 + 1 2∥x − x0 k∥2� 8 for t ∈ [Tk − 1] do 9 i ← � j∈[k−1] Tj + t 10 �∇g ¯x �fρ(xt k) ← γρ(xt k−¯x−ξi) γρ(ξi) g(¯x + ξi) 11 xt+1 k ← argminx∈B¯x(r) � ηk⟨�∇g ¯x �fρ(xt k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x⟩ + ηkλ 2 ∥x − ¯x∥2 + 1 2∥x − xt k∥2� 12 end 13 x0 k+1 ← 1 Tk � t∈[Tk] xt k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Tk+1 ← 2Tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ηk+1 ← ηk 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' k ← k + 1 14 end 15 return x0 k Proposition 3 (Proposition 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [ACJ+21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let f, g satisfy the assumptions of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When ρ = r, Algorithm 1 is a (φ, λ)-ball optimization oracle for �fρ which makes O( L2 φλ) total queries to g with constant query depth, and an additional computational cost of O( L2 φλ) depth and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' AC-SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We can also implement AC-SA [GL12], a variant of accelerated gradient descent under stochastic gradient queries, in parallel using stochastic ReSQue estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We provide the pseu- docode of our parallel implementation of AC-SA in Algorithm 2 and state its guarantees in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 4 (Special case of Theorem 1, [GL12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let f, g satisfy the assumptions of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When ρ = r, Algorithm 2 is a (φ, λ)-ball optimization oracle for �fρ which makes O �� 1 + L ρλ log �λr2 φ � + L2 λφ � total queries with constant query depth, and an additional computational cost of O �� 1 + L ρλ log �λr2 φ �� depth and O �� 1 + L ρλ log �λr2 φ � + L2 λφ � work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Because the statement of Proposition 4 follows from specific parameter choices in the main result in [GL12], we defer a more thorough discussion of how to obtain this result to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 15 Algorithm 2: AC-SA(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' φ) 1 Input: f : Rd → R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' g : Rd → R satisfying the assumptions of Problem 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ¯x ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' φ > 0 2 K ← ⌈log2( λr2 φ )⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' T ← ⌈4 � L ρλ + 1⌉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Nk ← � 48 · 2k · L2 λ2r2T � for k ∈ [K] 3 Sample ξi ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ2Id),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' i ∈ [N] independently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' for N = T · (� k∈[K] Nk) 4 Query g(¯x + ξi) for all i ∈ [N] (in parallel) 5 xag 0 ← ¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x0 ← ¯x 6 for k ∈ [K] do 7 for t ∈ [T] do 8 αt ← 2 t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' γt ← 4( L ρ +λ) t(t+1) 9 xmd t ← (1−αt)(λ+γt) γt+(1−α2 t )λ xag t−1 + αt(1−αt)(λ+γt) γt+(1−α2 t )λ xt−1 10 NT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='[k−1] ← T · � k′∈[k−1] Nk′ 11 �∇f(xmd t ) ← 1 Nk � n∈[Nk] γρ(xmd t −¯x−ξNT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='[k−1]+n) γρ(ξNT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='[k−1]+n) g(¯x + ξNT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='[k−1]+n) 12 xt ← argminx∈B¯x(r)Ψt(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' where Ψt(x) := ⟨αt �∇f(xmd t ) + λ(xmd t − ¯x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x − xt⟩ + γt+λ(1−αt) 2 ∥x − xt−1∥2 + λαt 2 ∥x − xmd t ∥2 13 xag t ← αtxt + (1 − αt)xag t−1 14 end 15 xag 0 ← xag T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' x0 ← xag T 16 end 17 Return: xag T Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We now use our parallel ball optimization oracles to prove Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proofs of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We use Proposition 2 with r = ρ = ϵopt √ dL on F ← �fρ, which approx- imates f to additive ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Rescaling ϵopt by a constant from the guarantee of Proposition 2 gives the error claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the oracle query depths, note that each ball optimization oracle (whether im- plemented using Algorithm 1 or Algorithm 2) has constant query depth, and at most O(log2(dκ)) ball optimization oracles are queried per iteration on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Note that (see Proposition 1) κ = LR ϵopt , K = �R r � 2 3 = d 1 3 κ 2 3 , λ⋆ = ϵoptK2 R2 log2 κ = ϵoptd 2 3 κ 4 3 R2 log2 κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the total oracle queries, computational depth, and work, when implementing each ball optimization oracle with EpochSGD, we have that for jmax := ⌈log2 K + Cba⌉, these are all O � �K log (dκ) · � � � j∈[jmax] 1 2j �L2 · 2j log2(dκ) λ2⋆r2 � + � L2 λ2⋆r2 � log2 (dκ) � � � � = O � K log4 (dκ) · L2 λ2⋆r2 � = O � κ2 log4 (dκ) � due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The additional terms in the theorem statement are due to the number of ball oracles needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the computational depth when implementing each ball optimization oracle with 16 AC-SA we have that (due to Proposition 4), it is bounded by O � K log3(dκ) · � L rλ⋆ log(dκ) � = O � K log4(dκ) · √κ K 1 4 � = O � d 1 4 κ log4(dκ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, for the total oracle queries and work bounds, the bound due to the L2 λφ term is as was computed for Theorem 1, and the bound due to the other term is the same as the above display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4 Private stochastic convex optimization We now develop our main result on an improved gradient complexity for private SCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1, we introduce several variants of differential privacy including a relaxation of Rényi differential privacy [Mir17], which tolerates a small amount of total variation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, in Sec- tions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4, we build several private stochastic optimization subroutines which will be used in the ball acceleration framework of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='6, we give our main results on private ERM and SCO respectively, by leveraging the subroutines we develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 Preliminaries In this section, we study the following specialization of Problem 1 naturally compatible with pre- serving privacy with respect to samples, through the formalism of DP (to be defined shortly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let P be a distribution over S, and suppose there is a family of functions indexed by s ∈ S, such that f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s) : Rd → R is convex for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let D := {si}i∈[n] consist of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' draws from P, and define the empirical risk and population risk by ferm(x) := 1 n � i∈[n] f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' si) and fpop(x) := Es∼Pf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We denote fi := f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' si) for all i ∈ [n], and assume that for all s ∈ S, f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s) is L-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We are given D, and can query subgradients of the “sampled functions” fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our goal is to produce an ϵopt approximate minimizer to fpop constrained to B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We again define κ = LR ϵopt as in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the “one-pass” setting where we only query each ∂fi a single time, we can treat each ∂fi as a bounded stochastic gradient of the underlying population risk fpop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We note the related problem of empirical risk minimization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' optimizing ferm (in the setting of Problem 2), can also be viewed as a case of Problem 1 where we construct g by querying ∂fi for i ∼unif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We design (ϵdp, δ)- DP algorithms for solving Problem 2 which obtain small optimization error for ferm and fpop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To disambiguate, we will always use ϵopt to denote an optimization error parameter, and ϵdp to denote a privacy parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our private SCO algorithm will require querying ∂fi multiple times for some i ∈ [n], and hence incur bias for the population risk gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Throughout the rest of the section, following the notation of Problem 2, we will fix a dataset D ∈ Sn and define the empirical risk ferm and population risk fpop accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We now move on to our privacy definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say that two datasets D = {si}i∈[n] ∈ Sn and D′ = {s′ i}i∈[n] ∈ Sn are neighboring if |{i | si ̸= s′ i}| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say a mechanism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' a randomized algorithm) M satisfies (ϵdp, δ)- differential privacy (DP) if, for its output space Ω and all neighboring D, D′, we have for all S ⊆ Ω, Pr[M(D) ∈ S] ≤ exp(ϵdp) Pr[M(D′) ∈ S] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (10) 17 We extensively use the notion of Rényi differential privacy due to its compatibility with the sub- sampling arguments we will use, as well as an approximate relaxation of its definition which we introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We say that a mechanism M satisfies (α, ϵ)-Rényi differential privacy (RDP) if for all neighboring D, D′ ∈ Sn, the α-Rényi divergence (4) satisfies Dα(M(D)∥M(D′)) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (11) RDP has several useful properties which we now summarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 5 (Propositions 1, 3, and 7, [Mir17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' RDP has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (Composition): Let M1 : Sn → Ω satisfy (α, ϵ1)-RDP and M2 : Sn × Ω → Ω′ satisfy (α, ϵ2)- RDP for any input in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then the composition of M2 and M1, defined as M2(D, M1(D)) satisfies (α, ϵ1 + ϵ2)-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (Gaussian mechanism): For µ, µ′ ∈ Rd, Dα(N(µ, σ2Id)∥N(µ′, σ2Id)) ≤ α 2σ2 ∥µ − µ′∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (Standard DP): If M satisfies (α, ϵ)-RDP, then for all δ ∈ (0, 1), M satisfies (ϵ+ 1 α−1 log 1 δ, δ)- DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We also use the following definition of approximate Rényi divergence: Dα,δ(µ∥ν) := min DTV(µ′,µ)≤δ,DTV(ν′,ν)≤δ Dα(µ′∥ν′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (12) We relax the definition (11) and say that M satisfies (α, ϵ, δ)-RDP if for all neighboring D, D′ ∈ Sn, recalling definition (12), Dα,δ(M(D)∥M(D′)) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The following is then immediate from Proposition 5, and our definition of approximate RDP, by coupling the output distributions with the distributions realizing the minimum (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' If M satisfies (α, ϵ, δ)-RDP, then for all δ′ ∈ (0, 1), M satisfies (ϵdp, δ′ + (1 + exp(ϵdp))δ)-DP for ϵdp := ϵ + 1 α−1 log 1 δ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let µ, ν be within total variation δ of M(D) and M(D′), such that Dα(µ∥ν) ≤ ϵ and hence for any event S, Pr ω∼µ [ω ∈ S] ≤ exp(ϵdp) Pr ω∼ν[ω ∈ S] + δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Combining the above with Pr ω∼M(D) [ω ∈ S] − δ ≤ Pr ω∼µ[ω ∈ S], Pr ω∼ν[ω ∈ S] ≤ Pr ω∼M(D′) [ω ∈ S] + δ, we have Pr ω∼M(D)[ω ∈ S] ≤ exp(ϵdp) Pr ω∼ν[ω ∈ S] + δ′ + δ ≤ exp(ϵdp) Pr ω∼M(D′)[ω ∈ S] + δ′ + (1 + exp(ϵdp))δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, our approximate RDP notion enjoys a composition property similar to standard RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 18 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let M1 : Sn → Ω satisfy (α, ϵ1, δ1)-RDP and M2 : Sn ×Ω → Ω′ satisfy (α, ϵ2, δ2)-RDP for any input in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then the composition of M2 and M1, defined as M2(D, M1(D)) satisfies (α, ϵ1 + ϵ2, δ1 + δ2)-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let D, D′ be neighboring datasets, and let µ, µ′ be distributions within total variation δ1 of M1(D), M1(D′) realizing the bound Dα(µ∥µ′) ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For any ω ∈ Ω, similarly let νω, ν′ ω be the distributions within total variation δ2 of M2(D, ω) and M2(D′, ω) realizing the bound Dα(νω∥ν′ ω) ≤ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, let P1 be the distribution of ω ∈ Ω according to M1(D), and Q1 to be the distribution of M1(D′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' similarly, let P2,ω, Q2,ω be the distributions of ω′ ∈ Ω′ according to M2(D, ω) and M2(D′, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first note that by a union bound, DTV �� νω(ω′)µ(ω)dωdω′, � P1(ω)P2,ω(ω′)dωdω′ � ≤ δ1 + δ2, DTV �� ν′ ω(ω′)µ′(ω)dωdω′, � Q1(ω)Q2,ω(ω′)dωdω′ � ≤ δ1 + δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, by Proposition 1 of [Mir17], we have Dα �� νω(ω′)µ(ω)dωdω′ ����� � ν′ ω(ω′)µ′(ω)dωdω′ � ≤ ϵ1 + ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Combining the above two displays yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2 Subsampled smoothed ERM solver: the convex case We give an ERM algorithm that takes as input a dataset D ∈ Sn, parameters T ∈ N and r, ρ, β > 0, and a center point ¯x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our algorithm is based on a localization approach introduced by [FKT20] which repeatedly decreases a domain size to bound the error due to adding noise for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular we will obtain an error bound on � ferm ρ with respect to the set B¯x(r), using at most T calls to the ReSQue estimator in Definition 2 with a deterministic subgradient oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we recall that ferm is defined as in Problem 2, and � ferm ρ is correspondingly defined as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Importantly, our ERM algorithm developed in this section attains RDP bounds improving with the subsampling parameter T n when T ≪ n, due to only querying T random samples in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We summarize our optimization and privacy guarantees on Algorithm 3 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The proof follows by combining Lemma 4 (the utility bound) and Lemma 7 (the privacy bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x ∈ argminx∈B¯x(r) � ferm ρ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 3 uses at most T gradients and produces x ∈ B¯x(r) such that, for a universal constant Ccvx, E � � ferm ρ (x) � − � ferm ρ (x⋆ ¯x) ≤ CcvxLr �√ d βT + 1 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, there is a universal constant Cpriv ≥ 1, such that if T n ≤ 1 Cpriv , β2 log2( 1 δ) ≤ 1 Cpriv , δ ∈ (0, 1 6), and ρ r ≥ Cpriv log2( log T δ ), Algorithm 3 satisfies (α, ατ, δ)-RDP for τ := Cpriv � β log �1 δ � T n �2 and α ∈ � 1, 1 Cprivβ2 log2( 1 δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 19 Algorithm 3: Subsampled ReSQued ERM solver, convex case 1 Input: ¯x ∈ Rd, ball radius, convolution radius, and privacy parameter r, ρ, β > 0, dataset D ∈ Sn, iteration count T ∈ N 2 �T ← 2⌊log2 T⌋, k ← log2 �T, η ← r L min( 1 √ T , β √ d), x0 ← ¯x 3 for i ∈ [k] do 4 Ti ← 2−i �T, ηi ← 4−iη, σi ← Lηi β 5 y0 ← xi−1 6 for j ∈ [Ti] do 7 zi,j ∼unif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [n] 8 yj ← ΠB¯x(r)(yj−1 − ηi �∇¯x �fzi,j ρ (yj−1)) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ▷ PSGD step using ReSQue (See Definition 2) for a subsampled function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 5 denotes the random Gaussian sample by ξi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 9 end 10 ¯yi ← 1 Ti � j∈[Ti] yj 11 xi ← ¯yi + ζi, for ζi ∼ N(0, σ2 i Id) 12 end 13 return xk Utility analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by proving a utility guarantee for Algorithm 3, following [FKT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x := argminx∈B¯x(r) � ferm ρ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have, for a universal constant Ccvx, E � � ferm ρ (xk) � − � ferm ρ (x⋆ ¯x) ≤ CcvxLr �√ d βT + 1 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Denote F := � ferm ρ , ¯y0 := x⋆ ¯x, and ζ0 := ¯x − x⋆ ¯x, where by assumption ∥ζ0∥ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by observing that in each run of Line 8, by combining the first property in Lemma 1 with the definition of ferm, we have that E ��∇¯x �fzi,j ρ (yj−1) | yj−1 � ∈ ∂F(yj−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, by the second property in Lemma 1 and the fact that fzi,j is L-Lipschitz, E ����∇¯x �fzi,j ρ (yj−1) ��� 2 ≤ 3L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We thus have E [F(xk)] − F(x⋆ ¯x) = � i∈[k] E[F(¯yi) − F(¯yi−1)] + E [F(xk) − F(¯yk)] ≤ � i∈[k] � � E � ∥xi−1 − ¯yi−1∥2� 2ηiTi + 3ηiL2 2 � � + LE [∥xk − ¯yk∥] ≤ 8r2 ηT + 4 � i∈[k−1] σ2 i d ηiTi + � i∈[k] 3ηiL2 2 + Lσk √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (13) In the second line, we used standard regret guarantees on projected stochastic gradient descent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 7 of [HK14], where we used that all ¯yi ∈ B¯x(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' in the third line, we used E[∥xk − ¯yk∥] ≤ � E � ∥xk − ¯yk∥2� = � E � ∥ζk∥2� = σk √ d 20 by Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Continuing, we have by our choice of parameters that σ2 i ηiTi ≤ 2−i L2η 2β2 �T , hence E [F(xk)] − F(x⋆ ¯x) ≤ 8r2 ηT + 4L2ηd β2 �T + 3ηL2 2 + L2η √ d β 1 �T 2 ≤ � 8Lr √ T + 8Lr √ d βT � + 8Lr √ d βT + 3Lr 2 √ T + Lr √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we used that 2 �T ≥ T and �T 2 ≥ √ T, for all T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We now show that our algorithm satisfies a strong (approximate) RDP guar- antee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let D′ = {s′ i}i∈[n] ∈ Sn be such that D = {si}i∈[n] and D′ are neighboring, and without loss of generality assume s′ 1 ̸= s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define the multiset I := {zi,j | i ∈ [k], j ∈ [Ti]} (14) to contain all sampled indices in [n] throughout Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by giving an (approximate) RDP guarantee conditioned on the number of times “1” appears in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The proof of Lemma 5 is primarily based on providing a potential-based proof of a “drift bound,” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' how far away iterates produced by two neighboring datasets drift apart (coupling all other randomness used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To carry out this potential proof, we rely on the local stability properties afforded by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define I as in (14) in one call to Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let I be deterministic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' this statement is conditioned on the realization of I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let b be the number of times the index 1 appears in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let µ be the distribution of the output of Algorithm 3 run on D, and µ′ be the distribution when run on D′, such that D and D′ are neighboring and differ in the first entry, and the only randomness is in the Gaussian samples used to define ReSQue estimators and on Line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Suppose ρ r ≥ 1728 log2( log T δ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then we have for any α > 1, Dα,δ(µ∥µ′) ≤ 1500αβ2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Throughout this proof we treat I as fixed with b occurrences of the index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let bi be the number of times 1 appears in Ii := {zi,j | j ∈ [Ti]}, such that � i∈[k] bi = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first analyze the privacy guarantee of one loop, and then analyze the privacy of the whole algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by fixing some i ∈ [k], and analyzing the RDP of the ith outer loop in Algorithm 3, conditioned on the starting point y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Consider a particular realization of the Ti Gaussian samples used in implementing Line 8, Ξi := {ξi,j}j∈[Ti], where we let ξi,j ∼ N(0, ρ2Id) denote the Gaussian sample used to define the update to yj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Conditioned on the values of Ii, Ξi, the ith outer loop in Algorithm 3 (before adding ζi in Line 11) is a deterministic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a given realization of Ii and Ξi, we abuse notation and denote {yj}j∈[Ti] to be the iterates of the ith outer loop in Algorithm 3 using the dataset D starting at y0, and {y′ j}j∈[Ti] similarly using D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, define Φj := ��yj − y′ j ��2 , p := � 5 log �log T δ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the following parts of the proof, we will bound for this p the quantity EΦp Ti, to show that with high probability it remains small at the end of the loop, regardless of the location of the 1 indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Potential growth: iterates with zi,j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first bound the potential growth in any iteration j ∈ [Ti] where zi,j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fix y0, y′ 0 and {ξi,t}t∈[j−1], so that Φj−1 is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have (taking expectations over only ξi,j), Eξi,jΦp j ≤ E (Φj−1 + Aj + Bj)p , (15) 21 where Aj := −2ηiZj � ∂fzi,j(¯x + ξi,j), yj−1 − y′ j−1 � , Bj := η2 i Z2 j ∥∂fzi,j(¯x + ξi,j)∥2 , and Zj := γρ(yj−1 − ¯x − ξi,j) − γρ(y′ j−1 − ¯x − ξi,j) γρ(ξi,j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The inequality in (15) follows from expanding the definition of the update to Φj before projection, and then using the fact that Euclidean projections onto a convex set only decrease distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By the second part of Lemma 2, for all q ∈ [2, p], if � Φj−1 ≤ ρ p (which is always satisfied as � Φj−1 ≤ r), Eξi,jZq j ≤ � 24q � Φj−1 ρ �q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By Lipschitzness of fzi,j and Cauchy-Schwarz (on Aj), we thus have Eξi,j|Aj|q ≤ �48ηiLqΦj−1 ρ �q for all q ∈ [2, p], Eξi,jBq j ≤ �48ηiLq ρ �2q Φq j−1 for all q ∈ [1, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (16) Next, we perform a Taylor expansion of (15), which yields Eξi,jΦp j ≤ Φp j−1 + pΦp−1 j−1Eξi,j [Aj + Bj] + p(p − 1) � 1 0 (1 − t)Eξi,j � (Φj−1 + t(Aj + Bj))p−2 (Aj + Bj)2� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (17) By monotonicity of convex gradients and the first part of Lemma 1, we have Eξi,j [Aj] = −2ηi � ∂ �fzi,j ρ (yj−1) − ∂ �fzi,j ρ (y′ j−1), yj−1 − y′ j−1 � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (18) By applying (16), we have pΦp−1 j−1Eξi,jBj ≤ p �48ηiL ρ �2 Φp j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (19) 22 Next we bound the second-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For any t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1] we have denoting Cj := Aj + Bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Eξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='j � (Φj−1 + tCj)p−2 C2 j � = p−2 � q=0 �p − 2 q � Φp−2−q j−1 Eξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='j � t2+qC2+q j � ≤ 4 p−2 � q=0 2q �p − 2 q � Φp−2−q j−1 Eξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='j � |Aj|2+q� + 4 p−2 � q=0 2q �p − 2 q � Φp−2−q j−1 Eξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='j � B2+q j � ≤ 4Φp j−1 �48ηiLp ρ �2 p−2 � q=0 2q �p − 2 q � �48ηiLq ρ �q + 4Φp j−1 �48ηiLp ρ �2 p−2 � q=0 2q �p − 2 q � �48ηiL(2 + q) ρ �2q+2 ≤ 8Φp j−1 �48ηiLp ρ �2 � 1 + 96ηiLp ρ �p−2 ≤ 16Φp j−1 �48ηiLp ρ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (20) The first inequality used (a + b)p ≤ 2p(ap + bp) for any nonnegative a, b and 0 ≤ t ≤ 1, the second inequality used (16), and the third and fourth inequalities used 48ηiL(2 + q) ρ ≤ 1 2p for our choices of ηiL ≤ r 4 and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, plugging (18), (19), and (20) into (17), Eξi,jΦp j ≤ Φp j−1 � 1 + 16p2 �48ηiLp ρ �2� ≤ Φp j−1 � 1 + 16p �48ηiLp ρ �2�p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, using (ηiL)2 ≤ r2 16T ≤ r2 16Ti and our assumed bound on r ρ, which implies 16p ρ2 (48ηiLp)2 ≤ 1 Ti , taking expectations over {ξt}t∈[j−1] yields EΦp j ≤ EΦp j−1 � 1 + 1 Ti �p when zi,j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (21) Potential growth: iterates with zi,j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, we handle the case where zi,j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have that conditional on fixed values of {ξi,t}t∈[j−1], y0 and y′ 0, Eξi,jΦp j ≤ Eξi,j (Φj−1 + Dj + Ej)p ≤ Eξi,j �� 1 + 1 bi � Φj−1 + 2biEj �p , (22) where overloading f ← f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s1), h ← f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' s′ 1), Dj := −2ηi � �∇¯x �fρ(yj−1) − �∇¯x�hρ(y′ j−1), yj−1 − y′ j−1 � , Ej := η2 i ����∇¯x �fρ(yj−1) − �∇¯x�hρ(y′ j−1) ��� 2 , 23 and we use Dj ≤ 1 bi Φj−1 + biEj by Cauchy-Schwarz and Young’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, convexity of ∥·∥2q implies that Eq j ≤ η2q i 22q−1 �����∇¯x �fρ(yj−1) ��� 2q + ����∇¯x�hρ(y′ j−1) ��� 2q� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, we note that since f is Lipschitz, the first part of Lemma 2 implies for all q ≤ p, E ����∇¯x �fρ(yj−1) ��� 2q ≤ L2qE ��γρ(yj−1 − ¯x − ξ) γρ(ξ) �2q� ≤ 2(L)2q, and a similar calculation holds for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we used our assumed bound on r ρ to check the requirement in Lemma 2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By linearity of expectation, we thus have Eξi,jEq j ≤ (9ηiL)2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (23) Finally, expanding (22) and plugging in the moment bound (23), Eξi,jΦp j ≤ p � q=0 �p q � � 1 + 1 bi �q Φq j−1(2bi)p−qEξi,j � Ep−q j � ≤ p � q=0 �p q � � 1 + 1 bi �q Φq j−1(2bi)p−q(9ηiL)2(p−q) = �� 1 + 1 bi � Φj−1 + 2bi(9ηiL)2 �p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taking expectations over {ξi,t}t∈[j−1], and using Fact 4 with Z ← (1+ 1 bi )Φj−1 and C ← 2bi(9ηiL)2, EΦp j ≤ �� 1 + 1 bi � E � Φp j−1 � 1 p + 2bi(9ηiL)2 �p , when zi,j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (24) One loop privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by obtaining a high-probability bound on ΦTi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define Wj := E[Φp j] 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By using (21) and (24), we observe Wj ≤ � � � � 1 + 1 Ti � Wj−1 zi,j ̸= 1 � 1 + 1 bi � Wj−1 + 2bi(9ηiL)2 zi,j = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, regardless of the bi locations of the 1 indices in Ii, we have WTi ≤ � 1 + 1 Ti �Ti � 1 + 1 bi �bi � 2b2 i (9ηiL)2� ≤ 1200b2 i (ηiL)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Thus, by Markov’s inequality, with probability at least 1 − δ log T over the randomness of Ξi = {ξi,j}j∈[Ti], we have using our choice of p, ��yTi − y′ Ti ��2 ≤ 1200b2 i (ηiL)2 · �log T δ � 1 p ≤ 1500b2 i (ηiL)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (25) 24 In the last inequality, we used our choice of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Call Ei the event that the sampled Ξi admits a deterministic map which yields the bound in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By the second part of Proposition 5, the conditional distribution of the output of the ith outer loop under Ei satisfies (α, 1500β2b2 i )-RDP, where we use the value of σi in Line 4 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We conclude via Fact 1 with E ← Ei that the ith outer loop of Algorithm 3 satisfies � α, 1500αβ2b2 i , δ log T � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' All loops privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By applying composition of RDP (the third part of Proposition 5), for a given realization of I = ∪i∈[k]Ii with b occurrences of 1, applying composition over the log T outer iterations (Lemma 3), Algorithm 3 satisfies � α, 1500αβ2b2, δ � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here, we used � i∈[k] b2 i ≤ b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We next apply amplification by subsampling to boost the guarantee of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To do so, we use the following key Proposition 7, which was proven in [BDRS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The use case in [BDRS18] involved subsampling with replacement and was used in a framework they introduced termed truncated CDP, but we will not need the framework except through the following powerful fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 7 (Theorem 12, [BDRS18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let τ ≤ 1 3, s ∈ (0, 1 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let P, Q, R be three distributions over the same probability space, such that for each pair P1, P2 ∈ {P, Q, R}, we have Dα(P1∥P2) ≤ ατ for all α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then for all α ∈ (1, 3 τ ), Dα(sP + (1 − s)R∥sQ + (1 − s)R) ≤ 13s2ατ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We also require a straightforward technical fact about binomial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let m, n ∈ N satisfy m n ≤ 1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Consider the following partition of the elements I ∈ [n]m with at most b copies of 1: S0 := {I ∈ [n]m | Ii ̸= 1 for all i ∈ [m]}, S1 := {I ∈ [n]m | Ii = 1 for between 1 and b many i ∈ [m]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let π0 and π1 be the uniform distributions on S0 and S1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then there exists a coupling Γ(π0, π1) such that for all (I, I′) in the support of Γ, ��� i | Ii ̸= I′ i ��� ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define a probability distribution p on elements of [b] such that pa := �m a � (n − 1)m−a � a∈[b] �m a � (n − 1)m−a for all a ∈ [b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Clearly, � a∈[b] pa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our coupling Γ := Γ(π0, π1) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Draw I ∼ π0 and a ∼ p independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let I′ be I with a uniformly random subset of a indices replaced with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Return (I, I′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 25 This coupling satisfies the requirement, so it suffices to verify it has the correct marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is immediate for S0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For I′ ∈ S1, suppose I′ has a occurrences of the index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The total probability I′ is drawn from Γ is then indeed (n − 1)a (n − 1)m · pa �m a � = 1 � a∈[b] �m a � (n − 1)m−a = 1 |S1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The first equality follows as the probability we draw I ∼ π0 which agrees with I′ on all the non-1 locations is (n − 1)a−m, and the probability I′ is drawn given that we selected I is pa · �m a �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, we are ready to state our main privacy guarantee for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is a universal constant Cpriv ∈ [1, ∞), such that if T n ≤ 1 Cpriv , β2 log2( 1 δ) ≤ 1 Cpriv , δ ∈ (0, 1 6), and ρ r ≥ Cpriv log2( log T δ ), Algorithm 3 satisfies (α, ατ, δ)-RDP for τ := Cpriv � β log �1 δ � T n �2 , α ∈ � 1, 1 Cprivβ2 log2( 1 δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let D, D′ be neighboring, and without loss of generality, suppose they differ in the first entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let Cpriv ≥ 60, and let I be defined as in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let E be the event that I contains at most b copies of the index 1, where b := 2 log �2 δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By a Chernoff bound, E occurs with probability at least 1 − δ 2 over the randomness of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We define P to be the distribution of the output of Algorithm 3 when run on D, conditioned on E and I containing at least one copy of the index 1 (call this total conditioning event E1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' there are between 1 and b copies of the index 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Similarly, we define Q to be the distribution when run on D′ conditioned on E1, and R to be the distribution conditioned on E ∩ Ec 1 (when run on either D or D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We claim that for all P1, P2 ∈ {P, Q, R}, we have Dα, δ 2 (P1∥P2) ≤ 1500αβ2b2, for all α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (26) To see (26) for P1 = P and P2 = Q (or vice versa), we can view P, Q as mixtures of outcomes conditioned on the realization I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then, applying quasiconvexity of Rènyi divergence (over this mixture), and applying Lemma 5 (with δ ← δ 2), we have the desired claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To see (26) for the remaining cases, we first couple the conditional distributions under E1 and E ∩ Ec 1 by their index sets, according to the coupling in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then applying quasiconvexity of Rényi divergence (over this coupling) again yields the claim, where we set m ← �T − 1 ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, let s := Pr[E1 | E] = 1 − � 1 − 1 n � �T−1 Pr[E] ≤ 1 − 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1T n 1 − δ 2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2T n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Note that conditional on E and the failure event in Lemma 5 not occurring, the distributions of Algorithm 3 using D and D′ respectively are sP + (1 − s)R and sQ + (1 − s)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, union bounding with Ec (see Fact 1), the claim follows from Proposition 7 with τ ← 6000β2 log2( 2 δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 26 Regularized extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We give a slight extension to Algorithm 3 which handles regularization, and enjoys similar utility and privacy guarantees as stated in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x,λ := argminx∈B¯x(r) � � ferm ρ (x) + λ 2 ∥x − ¯x∥2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (27) Our extension Algorithm 4 is identical to Algorithm 3, except it requires a regularization parameter λ, allows for an arbitrary starting point with an expected distance bound (adjusting the step size accordingly), and takes composite projected steps incorporating the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 4: Subsampled ReSQued ERM solver, regularized case, convex rate 1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, and regularization parameter r, ρ, β, λ > 0, dataset D ∈ Sn, iteration count T ∈ N, distance bound r′ ∈ [0, 2r], initial point x0 ∈ B¯x(r) satisfying E∥x0 − x⋆ ¯x,λ∥2 ≤ (r′)2 2 �T ← 2⌊log2 T⌋, k ← log2 �T, η ← r′ L min( 1 √ T , β √ d) 3 for i ∈ [k] do 4 Ti ← 2−i �T, ηi ← 4−iη, σi ← Lηi β 5 y0 ← xi−1 6 for j ∈ [Ti] do 7 zi,j ∼unif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [n] 8 yj ← argminy∈B¯x(r){⟨ηi �∇¯x �fzi,j ρ (yj−1), y⟩ + 1 2 ∥y − yj−1∥2 + ηiλ 2 ∥y − ¯x∥2} for zi,j ∼unif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [n] 9 end 10 ¯yi ← 1 Ti � j∈[Ti] yj 11 xi ← ¯yi + ζi, for ζi ∼ N(0, σ2 i Id) 12 end 13 return xk Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x,λ be defined as in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 4 uses at most T gradients and produces x ∈ B¯x(r) such that, for a universal constant Ccvx, E � � ferm ρ (x) + λ 2 ∥x − ¯x∥2 � − � � ferm ρ (x⋆ ¯x,λ) + λ 2 ��x⋆ ¯x,λ − ¯x ��2 � ≤ CcvxLr′ �√ d βT + 1 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, there is a universal constant Cpriv ≥ 1, such that if T n ≤ 1 Cpriv , β2 log2( 1 δ) ≤ 1 Cpriv , δ ∈ (0, 1 6), and ρ r ≥ Cpriv log2( log T δ ), Algorithm 4 satisfies (α, ατ, δ)-RDP for τ := Cpriv � β log �1 δ � T n �2 , α ∈ � 1, 1 Cprivβ2 log2( 1 δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The proof is almost identical to Proposition 6, so we only discuss the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Throughout this proof, for notational convenience, we define F λ(x) := � ferm ρ (x) + λ 2 ∥x − ¯x∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Standard results on composite stochastic mirror descent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 12 of [CJST19]) show the utility bound in (13) still holds with F λ in place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular each term E[F λ(¯yi) − 27 F λ(¯yi−1)] as well as E[F λ(xk)−F λ(¯yk)] enjoys the same bound as its counterpart in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The only other difference is that, defining ζ0 := x0 − x⋆ ¯x,λ in the proof of Lemma 4, we have Eζ2 0 ≤ (r′)2 in place of the bound r2, and we appropriately changed η to scale as r′ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The subsampling-based reduction from Lemma 7 to Lemma 5 is identical, so we only discuss how to obtain an analog of Lemma 5 for Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In each iteration j ∈ [Ti], by completing the square, we can rewrite Line 8 as yj ← argminy∈B¯x(r) � 1 2 ����y − � 1 1 + ηiλyj−1 + ηiλ 1 + ηiλ ¯x − ηi 1 + ηiλ �∇¯x �fzi,j ρ (yj−1) ����� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Now consider our (conditional) bounds on Eξi,jΦj in (15) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We claim these still hold true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' before projection, the same arguments used in (15) and (22) still hold (in fact improve by (1 + ηiλ)2), and projection only decreases distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, note that the proof of Lemma 5 only used the choice of step size η through ηL √ T ≤ r and used the assumed bound on r ρ to bound the drift growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' As we now have ηL √ T ≤ r′ ≤ 2r, we adjusted the assumed bound on r ρ by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The remainder of the proof of Lemma 5 is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Without loss of generality, Cpriv is the same constant in Proposition 6 and Corollary 2, since we can set both to be the maximum of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The same logic applies to the following Proposition 8 and Lemma 10 (which will also be parameterized by a Cpriv) so we will not repeat it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, the following fact about initial error will also be helpful in the following Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have � ferm ρ (¯x) − � � ferm ρ (x⋆ ¯x,λ) + λ 2 ��x⋆ ¯x,λ − ¯x ��2 � ≤ 2L2 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By strong convexity and Lipschitzness of � ferm ρ , we have λ 2 ��x⋆ ¯x,λ − ¯x ��2 ≤ � ferm ρ (¯x) − � � ferm ρ (x⋆ ¯x,λ) + λ 2 ��x⋆ ¯x,λ − ¯x ��2 � ≤ � ferm ρ (¯x) − � ferm ρ (x⋆ ¯x,λ) ≤ L ��x⋆ ¯x,λ − ¯x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Rearranging gives ∥x⋆ ¯x,λ − ¯x∥ ≤ 2L λ , which can be plugged in above to yield the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We also state a slight extension to Lemma 8 which will be used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define x⋆ ¯x,x′,λ := argminx∈B¯x(r){ � ferm ρ (x)+ λ 2 ∥x − x′∥2}, where x′ ∈ Rd is not necessarily in B¯x(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x0 := ΠB¯x(r)(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have � � ferm ρ (x0) + λ 2 ��x0 − x′��2 � − � � ferm ρ (x⋆ ¯x,x′,λ) + λ 2 ��x⋆ ¯x,x′,λ − x′��2 � ≤ 2L2 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The proof is identical to Lemma 8, where we use λ 2 ∥x0 − x′∥2 ≤ λ 2∥x⋆ ¯x,x′,λ − x′∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 Subsampled smoothed ERM solver: the strongly convex case We next give an ERM algorithm similar to Algorithm 4, but enjoys an improved optimization rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, it again attains RDP bounds improving with the subsampling parameter T n , and we obtain error guarantees against x⋆ ¯x,λ defined in (27) at a rate decaying as 1 T or better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We now give our analysis of Algorithm 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The proof follows a standard reduction template from the strongly convex case to the convex case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='7 in [KLL21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 28 Algorithm 5: Subsampled ReSQued ERM solver, strongly convex case 1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, and regularization parameter r, ρ, β, λ > 0, dataset D ∈ Sn, iteration count T ∈ N 2 k ← ⌈log log T⌉, x0 ← ¯x 3 for i ∈ [k] do 4 βi−1 ← 2 k−i+1 2 β, ri−1 ← min(2r, � 2Di−1 λ ) (see (28)), Ti−1 ← 2i−1−kT 5 xi ← output of Algorithm 4 with inputs (¯x, r, ρ, βi−1, λ, D, Ti−1, ri−1, xi−1) 6 end 7 return xk+1 Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x,λ be defined as in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 5 uses at most T gradients and produces x such that, for a universal constant Csc, E � � ferm ρ (x) + λ 2 ∥x − ¯x∥2 � − � ferm ρ (x⋆ ¯x,λ) − λ 2 ∥x⋆ ¯x,λ − ¯x∥2 ≤ CscL2 λ � d β2T 2 + 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, there is a universal constant Cpriv ≥ 1, such that if T n ≤ 1 Cpriv , β2 log2( log log T δ ) ≤ 1 Cpriv , δ ∈ (0, 1 6), and ρ r ≥ Cpriv log2( log T δ ), Algorithm 5 satisfies (α, ατ, δ)-RDP for τ := Cpriv � β log �log log T δ � T n �2 , α ∈ � 1, 1 Cprivβ2 log2( log log T δ ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We analyze the utility and privacy separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Denote for simplicity F λ(x) := � ferm ρ (x) + λ 2∥x − ¯x∥2, F λ ⋆ := F λ(x⋆ ¯x,λ), and ∆i := E[F λ(xi) − F λ ⋆ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, define for all 0 ≤ i ≤ k, Ei := 2C2 cvxL2 λ � √ d βiTi + 1 √Ti �2 , Di := 4Ei 2i � 2L2 λ 1 4E0 , (28) where we define Tk = T and βk = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By construction, for all 0 ≤ i ≤ k − 1, Ei+1 = 1 2Ei, and so Di+1 4Ei+1 = � Di 4Ei =⇒ � DiEi = Di+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (29) We claim inductively that for all 0 ≤ i ≤ k, ∆i ≤ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The base case of the induction follows because by Lemma 8, we have ∆0 ≤ 2L2 λ = D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, suppose that the inductive hypothesis is true up to iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By strong convexity, E ���xi − x⋆ ¯x,λ ��2� ≤ 2∆i λ ≤ 2Di λ , where we used the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, the expected radius upper bound (defined by ri) is valid for the call to Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Thus, by Corollary 2, ∆i+1 = E � F λ(xi+1) − F λ ⋆ � ≤ CcvxLri � √ d βiTi + 1 √Ti � ≤ CcvxL � 2Di λ � √ d βiTi + 1 √Ti � = � DiEi = Di+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 29 Here we used (29) in the last equation, which completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, iterating (29) for k = ⌈log2 log2 T⌉ iterations, where we use E0 ≥ L2 2λT so that Dk ≤ 8Ek, we have ∆k ≤ 8Ek ≤ 32C2 cvxL2 λ � d β2T 2 + 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The privacy guarantee follows by combining the privacy guarantee in Corollary 2 and composition of approximate RDP (Lemma 3), where we adjusted the definition of δ by a factor of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, we use that the privacy guarantee in each call to Corollary 2 is a geometric sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' β2 i T 2 i is doubling), and at the end it is 1 2β2T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='4 Private stochastic proximal estimator In this section, following the development of [ACJ+21], we give an algorithm which calls Algorithm 5 with several different iteration counts and returns a (random) point �x which enjoys a substantially reduced bias for x⋆ ¯x,λ defined in (27) compared to the expected number of gradient queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 6: Bias-reduced ReSQued stochastic proximal estimator 1 Input: ¯x ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ball radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' convolution radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' privacy parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' and regularization parameter r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' dataset D ∈ Sn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' iteration count T ∈ N with T ≤ ⌊ n 2Cpriv ⌋ 2 Tmax ← ⌊ n Cpriv ⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' jmax ← ⌊log2 Tmax T ⌋ 3 for k ∈ [jmax] do 4 Draw J ∼ Geom( 1 2) 5 x0 ← output of Algorithm 5 with inputs (¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' T) 6 if J ≤ jmax then 7 xJ ← output of Algorithm 5 with inputs (¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2− J 2 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2JT) 8 xJ−1 ← output of Algorithm 5 with inputs (¯x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2− J−1 2 β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2J−1T) 9 �xk ← x0 + 2J(xJ − xJ−1) 10 end 11 else 12 �xk ← x0 13 end 14 end 15 Return: �x ← 1 jmax � k∈[jmax] �xk Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x,λ be defined as in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have, for a universal constant Cbias: ∥E�x − x⋆ ¯x,λ∥ ≤ Cbias � L λ · �√ d βn + 1 √n �� , and, for a universal constant Cvar, E∥�x − x⋆ ¯x,λ∥2 ≤ CvarL2 λ2 � d β2T 2 + 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We begin by analyzing the output �xk of a single loop k ∈ [jmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For J ∼ Geom( 1 2), we have Pr[J = j] = 2−j if j ∈ [jmax], and Pr[J = j] = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We denote xj to be the output of 30 Algorithm 3 with privacy parameter 2− j 2 β and gradient bound 2jT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, E�xk = Ex0 + � j∈[jmax] Pr[J = j]2j(Exj − Exj−1) = Exjmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Since T · 2jmax ≥ Tmax 2 ≥ n 2Cpriv , applying Jensen’s inequality gives ∥Exjmax − x⋆ ¯x,λ∥ ≤ � E∥xjmax − x⋆ ¯x,λ∥2 ≤ √2CscL λ �√ d βn + 1 √n � , where the last inequality follows from Proposition 8 and strong convexity of the regularized function to convert the function error bound to a distance bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This implies the first conclusion, our bias bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Furthermore, for our variance bound, we have E∥�xk − E�xk∥2 ≤ E∥�xk − x⋆ ¯x,λ∥2 ≤ 2E∥�xk − x0∥2 + 2E∥x0 − x⋆ ¯x,λ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By Proposition 8 and strong convexity, E∥x0 − x⋆ ¯x,λ∥2 ≤ CscL2 λ2 ( d β2T 2 + 1 T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, E∥�xk − x0∥2 = � j∈[jmax] Pr[J = j]22jE∥xj − xj−1∥2 = � j∈[jmax] 2jE∥xj − xj−1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Note that E∥xj − xj−1∥2 ≤ 2E∥xj − x⋆ ¯x,λ∥2 + 2E∥xj−1 − x⋆ ¯x,λ∥2 ≤ 2−j · 6CscL2 λ2 � d β2T 2 + 1 T � , and hence combining the above bounds yields E ∥�xk − E�xk∥2 ≤ 14CscjmaxL2 λ2 � d β2T 2 + 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Now, averaging jmax independent copies shows that E ���x − x⋆ ¯x,λ ��2 = ∥�x − E�x∥2 + ��E�x − x⋆ ¯x,λ ��2 ≤ 1 jmax �14CscjmaxL2 λ2 � d β2T 2 + 1 T �� + C2 bias � L λ · �√ d βn + 1 √n ��2 , where we used our earlier bias bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The conclusion follows by letting Cvar = C2 bias + 14Csc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We conclude with a gradient complexity and privacy bound, depending on the sampled J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is a universal constant Cpriv ≥ 1, such that if β2 log2( log log n δ ) ≤ 1 Cpriv , δ ∈ (0, 1 2), and ρ r ≥ Cpriv log2( log T δ ), the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Consider one loop indexed by k ∈ [jmax], and let J be the result of the Geom( 1 2) draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' If J ∈ [jmax], loop k of Algorithm 6 uses at most 2J+1T gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Furthermore, the loop satisfies (α, ατ, δ)-RDP for τ := 2J · Cpriv � β log �log log n δ � T n �2 , α ∈ � �1, 1 Cprivβ2 log2 � log log n δ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' If J ̸∈ [jmax], Algorithm 6 uses at most T gradients, and the loop satisfies (α, ατ, δ)-RDP for τ := Cpriv � β log �log log n δ � T n �2 , α ∈ � �1, 1 Cprivβ2 log2 � log log n δ � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is immediate by Proposition 8, where we applied Lemma 3 and set δ ← δ 3 (taking a union bound over the at most 3 calls to Algorithm 5, adjusting Cpriv as necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 Private ERM solver In this section, we give our main result on privately solving ERM in the setting of Problem 2, which will be used in a reduction framework in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='6 to solve the SCO problem as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our ERM algorithm is an instantiation of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first develop a line search oracle (see Definition 3) based on the solver of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 (Algorithm 5), which succeeds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To do so, we leverage the following geometric lemma for aggregating independent runs of our solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lemma 11 (Claim 1, [KLL+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an algorithm Aggregate which takes as input (S, ∆) ∈ (Rd)k×R≥0, and returns z ∈ Rd such that ∥z − y∥ ≤ ∆, if for some unknown point y ∈ Rd satisfying at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='51k points x ∈ S, ∥x − y∥ ≤ ∆ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The algorithm runs in time O(dk2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 7: High probability ReSQued ERM solver, strongly convex case 1 Input: ¯x ∈ Rd, ball radius, convolution radius, privacy parameter, regularization parameter, and failure probability r, ρ, β, λ, ζ > 0, dataset D ∈ Sn, iteration count T ∈ N 2 k ← 20 log( 1 ζ ) 3 for i ∈ [k] do 4 xi ← output of Algorithm 5 with inputs (¯x, r, ρ, β, λ, D, T) 5 end 6 Return: x′ ← Aggregate({xi}i∈[k], 9√2CscL λ ( d β2T 2 + 1 T ) 1 2 ) Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let x⋆ ¯x,λ be defined as in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Algorithm 7 uses at most 18T log( 1 ζ ) gradients and produces x′ such that with probability at least 1 − ζ, for a universal constant Cls, ∥x′ − x⋆ ¯x,λ∥ ≤ ClsL λ �√ d βT + 1 √ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, there exists a universal constant Cpriv ≥ 1 such that T n ≤ 1 Cpriv , δ ∈ (0, 1 6) and ρ r ≥ Cpriv log2( 1 δ log( T ζ )), Algorithm 7 satisfies (α, ατ, δ)-RDP for τ := Cpriv log �1 ζ � � β log �1 δ log �T ζ �� T n �2 , α ∈ � �1, 1 Cprivβ2 log2 � 1 δ log � T ζ �� � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For each xi, by Proposition 8, E � � fermr(xi) + λ 2 ∥xi − ¯x∥2 � − � fermr(x⋆ ¯x,λ) − λ 2 ∥x⋆ ¯x,λ − ¯x∥2 ≤ CscL2 λ � d β2T 2 + 1 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Further, by strong convexity and Jensen’s inequality we have E[∥xi − x⋆ ¯x,λ∥] ≤ √2CscL λ � d β2T 2 + 1 T � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, by Markov’s inequality, for each i ∈ [k] we have Pr � ∥xi − x⋆ ¯x,λ∥ ≥ 3√2CscL λ � d β2T 2 + 1 T � 1 2 � ≤ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 32 Hence by a Chernoff bound, with probability ≥ 1 − ζ, at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='51k points x ∈ {xi}i∈[k] satisfy ∥x − x⋆ ¯x,λ∥ ≤ 3√2CscL λ � d β2T 2 + 1 T � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence the precondition of Lemma 11 holds, giving the distance guarantee with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The privacy guarantee follows from Proposition 8 and the composition of approximate RDP, where we adjusted Cpriv by a constant and the definition of δ by a factor of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Now we are ready to prove our main result on private ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Theorem 3 (Private ERM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the setting of Problem 2, let ϵdp ∈ (0, 1) and δ ∈ (0, 1 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an (ϵdp, δ)-DP algorithm which takes as input D and outputs �x ∈ B(R) such that E � ferm(�x) − min x∈B(R) ferm(x) � ≤ O � �LR · � � 1 √n + � d log 1 δ log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5( n δ ) log n nϵdp � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, with probability at least 1 − δ, the algorithm queries at most O � log6 �n δ � � min � n, n2ϵ2 dp d � + min � (nd) 2 3 ϵdp , n 4 3 ϵ 1 3 dp ��� gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Throughout this proof, set for a sufficiently large constant C, ϵopt := CLR � � 1 √n + � d log 1 δ log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5( n δ ) log n nϵdp � � , κ := LR ϵopt , ρ := ϵopt L √ d , r := ρ √ C log2( n δ ) , α := 4 log 2 δ ϵdp , β := ϵdp C log( n δ ) � log 1 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (30) Note that for the given parameter settings, for sufficiently large C, we have κ ≤ 1 C min � �√n, nϵdp � d log 1 δ log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5( n δ ) log n � � , R r ≤ n log2 �log n δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (31) Our algorithm proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We follow the framework of Proposition 1, and instantiate the necessary oracles as follows for CbaK log κ iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We use Algorithm 7 with r, ρ, β defined in (30), and T1 := √ C � κ √ d √ Kβ log2 κ + κ2 K log3 κ log n δ � , ζ := 1 κCbaK log κ, (32) as a ( r Cba , λ)-line search oracle Ols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We use Algorithm 5 with r, ρ, β defined in (30), and T2 := √ C � κ √ d √ Kβ√log κ + κ2 K log κ � , (33) as a ( λr2 Cba log3 κ, λ)-ball optimization oracle Obo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We use Algorithm 6 with r, ρ, β defined in (30), and T3 := √ C � κ √ d √ Kβ + κ2 K � (34) as a ( ϵopt CbaR, ϵopt √ K CbaR , λ)-stochastic proximal oracle Osp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We split the remainder of the proof into four parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first show that the oracle definitions are indeed met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We then bound the overall optimization error against ferm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, we discuss the privacy guarantee and the gradient complexity bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Oracle correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the line search oracle, by Proposition 10, it suffices to show ClsL λ � √ d βT1 + 1 √T1 � ≤ r Cba .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is satisfied for T1 in (32), since Proposition 1 guarantees λ ≥ ϵoptK2 log2 κ R2Cba .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, ClsL λ √ d βT1 Cba r ≤ ClsC2 ba · κ √ d β log2 κ · 1 √ K 1 T1 ≤ 1 2, ClsL λ 1 √T1 Cba r ≤ ClsC2 ba · κ log2 κ · 1 √ K 1 √T1 ≤ 1 2, for a sufficiently large C, where we used K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5 = R r to simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By a union bound, the above holds with probability at least 1 − ϵopt LR over all calls to Algorithm 7, since there are at most CbaK log κ iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the remainder of the proof, let Els be the event that all line search oracles succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the ball optimization oracle, by Proposition 8, it suffices to show CscL2 λ � d β2T 2 2 + 1 T2 � ≤ λr2 Cba log3 κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is satisfied for our choice of T2 in (33), again with λ ≥ ϵoptK2 log2 κ R2Cba .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, CscL2 λ d β2T 2 2 Cba log3 κ λr2 ≤ CscC3 ba · κ2d β2 log κ · 1 K · 1 T 2 2 ≤ 1 2, CscL2 λ 1 T2 Cba log3 κ λr2 ≤ CscC3 ba · κ2 log κ · 1 K · 1 T2 ≤ 1 2, again for large C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, for the proximal gradient oracle, by Proposition 9, it suffices to show Cbias � L λ · �√ d βn + 1 √n �� ≤ ϵopt CbaλR, CvarL2 λ2 � d β2T 2 3 + 1 T3 � ≤ ϵ2 optK C2 baλ2R2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The first inequality is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The second is satisfied for our choice of T3 in (34), which implies CvarL2 λ2 d β2T 2 3 C2 baλ2R2 ϵ2 optK = CvarC2 ba · κ2d β2 · 1 K · 1 T 2 3 ≤ 1 2, CvarL2 λ2 1 T3 C2 baλ2R2 ϵ2 optK = CvarC2 ba · κ2 · 1 K · 1 T3 ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 34 Optimization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By Proposition 1, the expected optimization error against � ferm ρ is bounded by ϵopt whenever Els occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Otherwise, the optimization error is never larger than LR as long as we return a point in B(R), since the function is L-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Further, we showed Pr[Els] ≥ 1 − ϵopt LR , so the total expected error is bounded by 2ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, the additive error between � ferm ρ and ferm is bounded by ρL √ d = ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The conclusion follows by setting the error bound to 3ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first claim that each call to Ols, and Obo used by Proposition 1 satisfies � α, ϵdp 6CbaK log κ, δ 18CbaK log κ � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first analyze Ols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The preconditions of Proposition 10 are met, where log( 18CbaK log κ δ log( T ζ )) ≤ 2 log n δ for our parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, our α is in the acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, by Proposi- tion 10 it suffices to note 8αCprivβ2T 2 1 log3 � n δ � n2 ≤ 128CCprivβ2 log3( n δ ) log 1 δ n2ϵdp � κ2d Kβ2 log κ + κ4 K2 log2 κ � ≤ ϵdp 6CbaK log κ, where the second inequality follows for sufficiently large C due to (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, we analyze the privacy of Obo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The preconditions of Proposition 8 are met, where log( log log T δ ) ≤ log n δ for our parameter settings, and our α is again acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, by Proposition 8 it suffices to note αCprivβ2T 2 2 log2( n δ ) n2 ≤ 16CCprivβ2 log2( n δ ) log 1 δ n2ϵdp � κ2d Kβ2 log κ + κ4 K2 log2 κ � ≤ ϵdp 6CbaK log κ, again for sufficiently large C from (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, by applying Lemma 3, all of the at most CbaK log κ calls to Ols and Obo used by the algorithm combined satisfy � α, ϵdp 3 , δ 9 � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, we analyze the privacy of Osp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let jmax := � log2 � 1 T3 � n Cpriv ��� be the truncation parameter in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The total number of draws from Geom( 1 2) in Algo- rithm 6 over the course of the algorithm is CbaK log κ · jmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' It is straightforward to check that the expected number of draws where J = j for all j ∈ [jmax] is 2−jmaxCbaκ log κ · jmax = Ω �T3 n · K log κ · jmax � , which is superconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By Chernoff and a union bound, with probability ≥ 1 − δ n, there is a constant C′ such that for all j ∈ [jmax], the number of times we draw J = j is bounded by 2−jC′K log κ log n δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Similarly, the number of times we draw J ̸∈ [jmax] is bounded by C′K log κ log n δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This implies by Lemma 3 that all calls to Osp used by the algorithm combined satisfy � α, ϵdp 6 , δ 18 � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 35 Here, we summed the privacy loss in Lemma 10 over 0 ≤ J ≤ jmax, which gives � 0≤j≤jmax � 2j · αCprivβ2 log2( n δ )T 2 3 n2 � � 2−jC′K log κ log n δ � ≤ (jmax + 1) · 16CC′CprivKβ2 log3( n δ ) log 1 δ log κ n2ϵdp � κ2d Kβ2 + κ4 K2 � ≤ ϵdp 6 , for sufficiently large C, where we use log κ, jmax ≤ log n, and K ≥ log 1 δ for our parameter set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, combining these bounds shows that our whole algorithm satisfies (α, ϵdp 2 , δ 6)-RDP, and applying Corollary 1, gives the desired privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Gradient complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We have argued that with probability at least 1 − δ, the number of times we encounter the J = j case of Lemma 10 for all 0 ≤ j ≤ jmax is bounded by 2−jC′K log κ log n δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Under this event, Proposition 10, Proposition 8, and Lemma 10 imply the total gradient complexity of our algorithm is at most CbaK log κ · � �18T1 log 1 ζ + T2 + � 0≤j≤jmax � 2−jC′ log n δ � � 2j+1T3 � � � ≤ 36CbaC′K log n � T1 log n + T2 + T3 log n log n δ � , where we use ζ ≥ n−2, jmax ≤ log n, and κ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The conclusion follows from plugging in our parameter choices from (32), (33), and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, we note that following the strategy of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3, it is straightforward to extend The- orem 3 to the strongly convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We state this result as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Corollary 3 (Private regularized ERM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the setting of Problem 2, let ϵdp ∈ (0, 1), δ ∈ (0, 1 6), λ ≥ 0, and x′ ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an (ϵdp, δ)-DP algorithm which outputs �x ∈ B(R) such that E � ferm(�x) + λ 2 ��x − x′��2 − min x∈B(R) � ferm(x) + λ 2 ��x − x′��2 �� ≤ O � L2 λ · � 1 n + d log 1 δ log3( n δ ) log2 n n2ϵ2 dp �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, with probability at least 1 − δ, the algorithm queries at most O � log6 �n δ � � min � n, n2ϵ2 dp d � + min � (nd) 2 3 ϵdp , n 4 3 ϵ 1 3 dp ��� gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first note that similar to Corollary 2 (an extension of Proposition 6), it is straightfor- ward to extend Theorem 3 to handle both regularization and an improved upper bound on the distance to the optimum, with the same error rate and privacy guarantees otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The handling of the improved upper bound on the distance follows because the convergence rate of the [ACJ+21] algorithm scales proportionally to the distance to the optimum, when it is smaller than R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The regularization is handled in the same way as Corollary 2, where regularization can only improve the contraction in the privacy proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' One subtle point is that for the regularized problems, we need to obtain starting points for Algorithm 5 when the constraint set is B¯x(r), but the regularization in the objective is centered around a point not in B¯x(r) (in our case, the centerpoint will be a weighted combination of ¯x and x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' However, by initializing Algorithm 5 at the projection of the regularization centerpoint, the initial function error guarantee in Lemma 8 still holds (see Lemma 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 36 The reduction from the claimed rate in this corollary statement to the regularized extension of Theorem 3 then proceeds identically to the proof of Proposition 8, which calls Corollary 2 repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='6 Private SCO solver Finally, we give our main result on private SCO in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To obtain it, we will combine Corollary 3 with a generic reduction in [FKT20, KLL21], which uses a private ERM solver as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The reduction is based on the iterative localization technique proposed by [FKT20] (which is the same strategy used by Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3), and derived in greater generality by [KLL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proposition 11 (Modification of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 in [KLL21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Suppose there is an (ϵdp, δ)-DP algo- rithm Aerm with expected excess loss O � L2 λ · � 1 n + d log 1 δ log3( n δ ) log2 n n2ϵ2 dp �� , using N(n, ϵdp, δ) gradient queries, for some function N, when applied to an L-Lipschitz empirical risk (with n samples, constrained to B(R) ⊂ Rd) plus a λ-strongly convex regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then there is an (ϵdp, δ)-DP algorithm Asco using � i∈⌈log n⌉ N( n 2i , ϵdp 2i , δ 2i ) gradient queries, with expected excess population loss O � �LR · � � 1 √n + � d log 1 δ log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5( n δ ) log n nϵdp � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='1 in [KLL21] assumes a slightly smaller risk guarantee for Aerm (removing the ex- traneous log3( n δ ) log2 n factor), but it is straightforward to see that the proof extends to handle our larger risk assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Combining Proposition 11 and Corollary 3 then gives our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Theorem 4 (Private SCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the setting of Problem 2, let ϵdp ∈ (0, 1) and δ ∈ (0, 1 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' There is an (ϵdp, δ)-DP algorithm which takes as input D and outputs �x ∈ B(R) such that E � fpop(�x) − min x∈B(R) fpop(x) � ≤ O � �LR · � � 1 √n + � d log 1 δ log1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='5( n δ ) log n nϵdp � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Moreover, with probability at least 1 − δ, the algorithm queries at most O � log6 �n δ � � min � n, n2ϵ2 dp d � + min � (nd) 2 3 ϵdp , n 4 3 ϵ 1 3 dp ��� gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Acknowledgements We thank Vijaykrishna Gurunathan for helpful conversations on parallel convex optimization that facilitated initial insights regarding ReSQue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' YC was supported in part by the Israeli Science Foundation (ISF) grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2486/21 and the Len Blavatnik and the Blavatnik Family foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' AS was supported in part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF- 1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 37 References [Abo16] John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Abowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The challenge of scientific reproducibility and privacy protection for statistical agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Technical report, Census Scientific Advisory Committee, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [ACG+16] Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Deep learning with differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [ACJ+21] Hilal Asi, Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Stochastic bias-reduced gradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [AFKT21] Hilal Asi, Vitaly Feldman, Tomer Koren, and Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private stochastic convex optimization: Optimal rates in l1 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Conference on Machine Learning, ICML, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [App17] Differential Privacy Team Apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Learning with privacy at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Technical report, Apple, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BBG18] Borja Balle, Gilles Barthe, and Marco Gaboardi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy amplification by subsampling: Tight analyses via couplings and divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BDRS18] Mark Bun, Cynthia Dwork, Guy N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Rothblum, and Thomas Steinke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Composable and versatile privacy via truncated CDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BFGT20] Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, and Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Stability of stochastic gradient descent on nonsmooth convex losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems, 33:4381–4391, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BFTT19] Raef Bassily, Vitaly Feldman, Kunal Talwar, and Abhradeep Guha Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private stochastic convex optimization with optimal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BJL+19] Sébastien Bubeck, Qijia Jiang, Yin Tat Lee, Yuanzhi Li, and Aaron Sidford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Com- plexity of highly parallel non-smooth convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BM99] Guy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Blelloch and Bruce M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Maggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Parallel algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Mikhail J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Atallah, editor, Algorithms and Theory of Computation Handbook, Chapman & Hall/CRC Applied Algorithms and Data Structures series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' CRC Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Bot12] Léon Bottou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Stochastic gradient descent tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Grégoire Montavon, Genevieve B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Orr, and Klaus-Robert Müller, editors, Neural Networks: Tricks of the Trade - Second Edition, volume 7700 of Lecture Notes in Computer Science, pages 421–436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BS18] Eric Balkanski and Yaron Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Parallelization does not accelerate convex op- timization: Adaptivity lower bounds for non-smooth convex minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv: 1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='03880, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 38 [BST14] Raef Bassily, Adam Smith, and Abhradeep Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private empirical risk minimiza- tion: Efficient algorithms and tight error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In IEEE 55th Annual Symposium on Foundations of Computer Science, FOCS, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Bub15] Sébastien Bubeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Convex optimization: Algorithms and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Trends Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', 8(3-4):231–357, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [BV14] Stephen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Boyd and Lieven Vandenberghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Convex Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Cambridge Univer- sity Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CH22] Yair Carmon and Danielle Hausler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Distributionally robust optimization via ball oracle acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='13225, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CJJ+20] Yair Carmon, Arun Jambulapati, Qijia Jiang, Yujia Jin, Yin Tat Lee, Aaron Sidford, and Kevin Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Acceleration with a ball optimization oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CJJS21] Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Thinking inside the ball: Near-optimal minimization of the maximal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Conference on Learning The- ory, COLT, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CJST19] Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Variance reduction for matrix games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CM08] Kamalika Chaudhuri and Claire Monteleoni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Privacy-preserving logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [CMS11] Kamalika Chaudhuri, Claire Monteleoni, and Anand D Sarwate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Differentially private empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Journal of Machine Learning Research, 12(3), 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DBW12] John C Duchi, Peter L Bartlett, and Martin J Wainwright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Randomized smoothing for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' SIAM Journal on Optimization, 22(2):674–701, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DG19] Jelena Diakonikolas and Cristóbal Guzmán.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lower bounds for parallel and randomized convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Conference on Learning Theory, COLT, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DGBSX12] Ofer Dekel, Ran Gilad-Bachrach, Ohad Shamir, and Lin Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Optimal distributed online prediction using mini-batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Journal of Machine Learning Research, 13(1), 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DR14] Cynthia Dwork and Aaron Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The algorithmic foundations of differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Trends Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', 9(3-4):211–407, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [DRY18] John Duchi, Feng Ruan, and Chulhee Yun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Minimax bounds on stochastic batched convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Conference On Learning Theory, COLT, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Duc18] John C Duchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Introductory lectures on stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The Mathematics of Data, pages 99–186, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [EPK14] Úlfar Erlingsson, Vasyl Pihur, and Aleksandra Korolova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Rappor: Randomized aggre- gatable privacy-preserving ordinal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Proceedings of the 2014 ACM SIGSAC conference on computer and communications security, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 39 [FKT20] Vitaly Feldman, Tomer Koren, and Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private stochastic convex optimiza- tion: optimal rates in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [FTS17] Kazuto Fukuchi, Quang Khai Tran, and Jun Sakuma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Differentially private empirical risk minimization with input perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Conference on Discovery Science, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [GDG+19] Alexander Gasnikov, Pavel Dvurechensky, Eduard Gorbunov, Evgeniya Vorontsova, Daniil Selikhanovych, César A Uribe, Bo Jiang, Haoyue Wang, Shuzhong Zhang, Sébastien Bubeck, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Near optimal methods for minimizing convex functions with lipschitz p-th derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Conference on Learning Theory, COLT, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [GL12] Saeed Ghadimi and Guanghui Lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i: A generic algorithmic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' SIAM Journal on Optimization, 22(4):1469–1492, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [GLL22] Sivakanth Gopi, Yin Tat Lee, and Daogao Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private convex optimization via expo- nential mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='00263, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Gol64] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Goldstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Convex programming in hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 70(5):709––710, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Haz16] Elad Hazan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Introduction to online convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Trends Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', 2(3- 4):157–325, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [HK14] Elad Hazan and Satyen Kale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Beyond the regret minimization barrier: optimal algo- rithms for stochastic strongly-convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', 15(1):2489– 2512, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [INS+19] Roger Iyengar, Joseph P Near, Dawn Song, Om Thakkar, Abhradeep Thakurta, and Lun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Towards practical differentially private convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In IEEE Symposium on Security and Privacy (SP), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [JLSW20] Haotian Jiang, Yin Tat Lee, Zhao Song, and Sam Chiu-wai Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' An improved cutting plane method for convex optimization, convex-concave games, and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [JT14] Prateek Jain and Abhradeep Guha Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (near) dimension independent risk bounds for differentially private learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Conference on Machine Learning, ICML, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [KJ16] Shiva Prasad Kasiviswanathan and Hongxia Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Efficient private empirical risk mini- mization for high-dimensional learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Conference on Machine Learn- ing, ICML, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [KLL21] Janardhan Kulkarni, Yin Tat Lee, and Daogao Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Private non-smooth erm and sco in subquadratic steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [KLL+22] Jonathan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Kelner, Jerry Li, Allen Liu, Aaron Sidford, and Kevin Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Semi-random sparse recovery in nearly-linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='04002, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 40 [KTE88] Leonid G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Khachiyan, Sergei Pavlovich Tarasov, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Erlikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The method of inscribed ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Soviet Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=', 37:226–230, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [LLH+22] Xuechen Li, Daogao Liu, Tatsunori Hashimoto, Huseyin A Inan, Janardhan Kulkarni, Yin Tat Lee, and Abhradeep Guha Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When does differentially private learning not suffer in high dimensions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='00160, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Mir17] Ilya Mironov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Rényi differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In IEEE 30th Computer Security Foundations Symposium, CSF, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [MS13] Renato DC Monteiro and Benar Fux Svaiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' An accelerated hybrid proximal extra- gradient method for convex optimization and its implications to second-order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' SIAM Journal on Optimization, 23(2):1092–1125, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Nem94] Arkadi Nemirovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' On parallel complexity of nonsmooth convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Journal of Complexity, 10(4):451–463, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Nes83] Yu E Nesterov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' A method for solving the convex programming problem with conver- gence rate o(1/k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Nauk SSSR,, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Nes03] Yurii Nesterov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Introductory lectures on convex optimization: A basic course, vol- ume 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Springer Science & Business Media, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Nes18] Yurii Nesterov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Lectures on convex optimization, volume 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [NY83] Arkadi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Nemirovski and David B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Yudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Problem complexity and method efficiency in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Pol64] Boris T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Polyak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Some methods of speeding up the convergence of iteration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' USSR Computational Mathematics and Mathematical Physics, 4(5):1–17, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [RBHT12] Benjamin IP Rubinstein, Peter L Bartlett, Ling Huang, and Nina Taft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Learning in a large function space: Privacy-preserving mechanisms for svm learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Journal of Privacy and Confidentiality, 4(1):65–100, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [SB14] Shai Shalev-Shwartz and Shai Ben-David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Understanding Machine Learning - From Theory to Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Cambridge University Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [SBB+18] Kevin Scaman, Francis Bach, Sébastien Bubeck, Laurent Massoulié, and Yin Tat Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Optimal algorithms for non-smooth distributed optimization in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, NeurIPS, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Sha07] Shai Shalev-Shwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Online learning: Theory, algorithms, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' PhD thesis, Hebrew University, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Smi09] Adam Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Differential privacy and the secrecy of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' https://adamdsmith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='com/2009/09/02/sample-secrecy/, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Accessed: 2022-11-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [SSTT21] Shuang Song, Thomas Steinke, Om Thakkar, and Abhradeep Thakurta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Evading the curse of dimensionality in unconstrained private glms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Conference on Artificial Intelligence and Statistics, AISTATS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 41 [SU15] Thomas Steinke and Jonathan Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Between pure and approximate differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='06095, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Tal22] Kunal Talwar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Ppml workshop talk: Open questions in differentially private machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' https://machinelearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='com/video/open-questions, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Accessed: 2022-11-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [Wan18] Yu-Xiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Revisiting differentially private linear regression: optimal and adap- tive prediction & estimation in unbounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='02596, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [WBSS21] Blake E Woodworth, Brian Bullins, Ohad Shamir, and Nathan Srebro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The min-max complexity of distributed stochastic convex optimization with intermittent communi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In Conference on Learning Theory, COLT, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' [ZZMW17] Jiaqi Zhang, Kai Zheng, Wenlong Mou, and Liwei Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Efficient private erm for smooth objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In International Joint Conference on Artificial Intelligence, IJCAI, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 42 A Helper facts Fact 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For any integer r such that 0 ≤ r ≤ p − 1, � 0≤q≤p(−1)q�p q � qr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We recognize the formula as a scaling of the Stirling number of the second kind with r objects and p bins, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' the number of ways to put r objects into p bins such that each bin has at least one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When r < p there are clearly no such ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let p ∈ N be even and p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let ∥x∥ , ∥y∥ ≤ 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then � 0≤q≤p (−1)q �p q � exp �1 2 �� (p − q)2 − (p − q) � ∥x∥2 + (q2 − q) ∥y∥2 + 2q(p − q) ⟨x, y⟩ �� ≤ (12p ∥x − y∥)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fix some x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let fx(y) be the left-hand side displayed above, and let fq x(y) := exp �1 2 �� (p − q)2 − (p − q) � ∥x∥2 + (q2 − q) ∥y∥2 + 2q(p − q) ⟨x, y⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We will perform a pth order Taylor expansion of fx around x, where we show that partial derivatives of order at most p − 1 are all zero at x, and we bound the largest order derivative tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Derivatives of fq x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fix some 0 ≤ q ≤ p, and define Cq := q2 − q, Fq := fq x(y), vq := (q2 − q)y + q(p − q)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (35) Note that for fixed q, Fq and vq are functions of y, and we defined them such that ∇yvq = CqId, ∇yFq = vqFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, in the following we use � sym to mean a symmetric sum over all choices of tensor modes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' � sym v⊗2 q ⊗Id means we will choose 2 of the 4 modes where the action is v⊗2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To gain some intuition for the derivatives of Fq, we begin by evaluating the first few via product rule: ∇fq x(y) = Fqvq, ∇2fq x(y) = Fqv⊗2 q + CqFqId, ∇3fq x(y) = Fqv⊗3 q + CqFq � sym vq ⊗ Id, ∇4fq x(y) = Fqv⊗4 q + CqFq � sym v⊗2 q ⊗ Id + 3C2 q FqId ⊗ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For any fixed 0 ≤ r ≤ p, we claim that the rth derivative tensor has the form ∇rfq x(y) = Fq � � � 0≤s≤⌊ r 2 ⌋ Nr,s � r 2s � � (Cq)s � sym v⊗(r−2s) q ⊗ I⊗s d �� � , (36) where the Nr,s are nonnegative coefficients which importantly do not depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To see this we proceed by induction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' the base cases are computed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Every time we take the derivative of a “monomial” term of the form Fq(Cq)sv⊗(r−2s) q ⊗I⊗s d via product rule, we will have one term in which Fq becomes vqFq (and hence we obtain a FqCs qv⊗(r+1−2s) q ⊗ I⊗s d monomial), and r − 2s many terms where a vq becomes CqId (and hence we obtain a FqCs+1 q v⊗(r−1−2s) q ⊗ I⊗(s+1) d monomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For fixed 0 ≤ s ≤ ⌊r+1 2 ⌋, we hence again see that Nr+1,s has no dependence on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 43 Next, note � 0≤s≤⌊ r 2 ⌋ Nr,s has a natural interpretation as the total number of “monomial” terms of the form Fq(Cq)sv⊗(r−2s) q ⊗ I⊗s d when expanding ∇rfq x(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We claim that for all 0 ≤ q ≤ p and 0 ≤ r ≤ p − 1, � 0≤s≤⌊ r+1 2 ⌋ Nr+1,s � 0≤s≤⌊ r 2 ⌋ Nr,s ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (37) To see this, consider taking an additional derivative of (36) with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Each monomial of the form Fq(Cq)sv⊗(r−2s) q ⊗ I⊗s d contributes at most r − 2s + 1 ≤ p monomials to the next derivative tensor via product rule, namely one from Fq and one from each copy of vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Averaging this bound over all monomials yields the claim (37), since each contributes at most p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taylor expansion at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Next, we claim that for all 0 ≤ r ≤ p − 1, ∇rfx(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' (38) To see this, we have that ((p − q)2 − (p − q)) + (q2 − q) + 2q(p − q) = p2 − p is independent of q, and hence all of the Fq are equal to some value F when y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Furthermore, when y = x we have that vq = q(p − 1)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Now, from the characterization (36) and summing over all q, any monomial of the form x⊗(r−2s) ⊗ I⊗s d has a total coefficient of FNr,s � 0≤q≤p (−1)q �p q � (Cq)s(q(p − 1))r−2s = FNr,s(p − 1)r−2s � 0≤q≤p (−1)q �p q � Cs qqr−2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Since Cq is a quadratic in q, each summand (Cq)sqr−2s is a polynomial of degree at most r ≤ p − 1 in q, so applying Fact 2 to each monomial yields the claim (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taylor expansion at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, we will bound the injective tensor norm of ∇pfx(y), where the injective tensor norm of a degree-p symmetric tensor T is the maximum value of T[v⊗p] over unit norm v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We proceed by bounding the injective tensor norm of each monomial and then summing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, for any 0 ≤ p ≤ q, under our parameter settings it is straightforward to see ∥vq∥ ≤ p and Fq ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Also, for any 0 ≤ s ≤ p 2 we have Cs q ≤ p2s, and by repeatedly applying (37), we have � 0≤s≤⌊ p 2 ⌋ Np,s ≤ pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In other words, each of the monomials of the form Fq(Cq)sv⊗(r−2s) q ⊗ I⊗s d has injective tensor norm at most 2pp (since each Cq contributes two powers of p, and each vq contributes one power of p), and there are at most pp such monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, by triangle inequality over the sum of all monomials, ��∇pfq x(y)[(y − x)⊗p] �� ≤ 2p2p ∥y − x∥p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' By summing the above over all q (reweighting by (−1)q�p q � ), and using that the unsigned coefficients sum to � 0≤q≤p �q p � = 2p, we have ��∇pfx(y)[(y − x)⊗p] �� ≤ 4pp2p ∥x − y∥p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The conclusion follows by a Taylor expansion from x to y of order p, and using pp ≤ 3pp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='. Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the first claim, � (γρ(x − ¯x − ξ))p (γρ(ξ))p−1 dξ = (2πρ)− d 2 � exp � − 1 2ρ2 � p ∥x − ¯x∥2 − 2p ⟨x − ¯x, ξ⟩ + ∥ξ∥2�� dξ = exp �p2 − p 2ρ2 ∥x − ¯x∥2 � ≤ 2, 44 where the second equality used the calculation in (6), and the inequality used the assumed bound on ∥x − ¯x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We move onto the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' First, we prove the statement for all even p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Denote v := x − ¯x and v′ := x′ − ¯x for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Explicitly expanding the numerator yields that (2πρ) d 2 � (γρ(v − ξ) − γρ(v′ − ξ))p (γρ(ξ))p−1 dξ = � 0≤q≤p (−1)q �p q � Sq where we define Sq := (2πρ) d 2 � (γρ(v − ξ))p−q(γr(v′ − ξ))q (γρ(ξ))p−1 dξ = � exp � − 1 2ρ2 � (p − q) ∥v∥2 + q ��v′��2 − 2(p − q) ⟨v, ξ⟩ − 2q � v′, ξ � + ∥ξ∥2�� dξ = (2πρ) d 2 exp � 1 2ρ2 �� (p − q)2 − (p − q) � ∥v∥2 + (q2 − q) ��v′��2 + 2q(p − q) � v, v′��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the last line, we again used (6) to compute the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' When p ≥ 2 and is even, a strengthening of the conclusion then follows from Fact 3 (where we overload x ← v ρ, y ← v′ ρ in its application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, this shows the desired claim where the base of the exponent is 12p ρ ∥x − x′∥ instead of 24p ρ ∥x − x′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We move to general p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Define the random variable Z := ���� γρ(x − ¯x − ξ) − γρ(x′ − ¯x − ξ) γρ(ξ) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Recall that we have shown for all even p ≥ 2, EZp ≤ �12p ∥x − x′∥ ρ �p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Now, let p ≥ 2 be sandwiched between the even integers q and q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hölder’s inequality and the above inequality (for p ← q and p ← q + 2) demonstrate EZp ≤ (EZq) q+2−p 2 � EZq+2� p−q 2 ≤ �12(q + 2) ∥x − x′∥ ρ �p , where we use q(q + 2 − p) + (q + 2)(p − q) = 2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The conclusion follows since q + 2 ≤ 2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Let Z be a nonnegative scalar random variable, let C ≥ 0 be a fixed scalar, and let p ∈ N and p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Then (E [(Z + C)p]) 1 p ≤ E [Zp] 1 p + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Denote A := E [Zp] 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taking pth powers of both sides, we have the conclusion if (A + C)p − E [(Z + C)p] ≥ 0 ⇐⇒ � q∈[p−1] �p q � Cp−q (Aq − E [Zq]) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here we use that the q = 0 and q = p terms cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We conclude since Jensen’s inequality yields E[Zp] ≥ E[Zq] p q =⇒ Aq ≥ E[Zq], for all q ∈ [p − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 45 B Discussion of Proposition 1 In this section, we discuss how to obtain Proposition 1 from the analysis in [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We separate the discussion into four parts, corresponding to the iteration count, the line search oracle parameters, the ball optimization oracle parameters, and the proximal gradient oracle parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We note that Proposition 2 in [ACJ+21] states that they obtain function error ϵopt with constant probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' however, examining the proof shows it actually yields an expected error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The bound CbaK log κ on the number of iterations follows immediately from the value Kmax stated in Proposition 2 of [ACJ+21], where we set λmin ← λ⋆ and ϵ ← ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Line search oracle parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The line search oracle is called in the implementation of Line 2 of Algorithm 4 in [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our implementation follows the development of Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='3 in [ACJ+21], which is a restatement of Proposition 2 in [CJJS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The bound Cba log( Rκ r ) on the number of calls to the oracle is immediate from the statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the oracle parameter ∆ = r Cba , we note that the proof of Proposition 2 of [CJJS21] only requires that we obtain points at distance O(r) from x⋆ ¯x,λ given a choice of λ, although it is stated as requiring a function error guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This is evident where the proof applies Lemma 3 of the same paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Ball optimization oracle parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The ball optimization oracle is called in the implemen- tation of Line 5 of Algorithm 4 in [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In iteration k of the algorithm, the error requirement is derived through the potential bound in Lemma 5 of [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' More precisely, Lemma 5 shows that (following their notation), conditioned on all randomness through iteration k, E � Ak+1 (F(xk+1) − F(x⋆)) + ∥vk+1 − x⋆∥2� − � Ak (F(xk) − F(x⋆)) + ∥vk − x⋆∥2� ≤ −1 6λk+1Ak+1 ∥�xk+1 − yk∥2 + Ak+1φk+1 + a2 k+1σ2 k+1 + 2Rak+1δk+1, where the terms a2 k+1σ2 k+1 +2Rak+1δk+1 are handled identically in [ACJ+21] and our Proposition 1 (see the following discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For the remaining two terms, Proposition 4 of [ACJ+21] guarantees that as long as the method does not terminate, one of the following occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' ∥�xk+1 − yk∥2 = Ω(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' λk+1 = O(λ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In the first case, as long as φk+1 (the error tolerance to the ball optimization oracle) is set to be λk+1r2 Cba for a sufficiently large Cba (which it is smaller than by logarithmic factors), up to constant factors the potential proof is unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The total contributions to the potential due to all Ak+1φk+1 losses from the iterations of the second case across the entire algorithm is bounded by O � (K log κ) · R2 ϵopt λ⋆r2 log3 κ � = O � R2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Here, the first term is the iteration count, the second term is due to an upper bound on Ak+1, and the third term is bounded since λk+1 = O(λ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The initial potential in the proof of Proposition 2 of [ACJ+21] is R2, so the final potential is unaffected by more than constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For a more formal derivation of the same improved error tolerance, we refer the reader to [CH22], Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 46 Stochastic proximal oracle parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our stochastic proximal oracle parameters are ex- actly the settings of δk, σk required by Proposition 2 of [ACJ+21], except we simplified the bound on σ2 k = O( ϵ ak ) (note we use ϵopt in place of ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, following notation of [ACJ+21], we have ϵ ak = ϵ√λk √Ak = Ω � ϵ · � λ⋆ · √ϵ R � = Ω �ϵ2K R2 log κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The first equality used λka2 k = Ak for the parameter choices of Algorithm 4 in [ACJ+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The second equality used that all λk = Ω(λ⋆) and all Ak = O( R2 ϵ ) in Algorithm 4 in [ACJ+21], where we chose λ⋆ = ϵK2 R2 log2 κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The final equality plugged in this bound on λ⋆ and simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Hence, obtaining a variance as declared in Proposition 1 suffices to meet the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' C Discussion of Proposition 2 In this section, we discuss how to obtain Proposition 2 (which is based on Proposition 1 in [CH22]) from the analysis in [CH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The iteration count discussion is the same as in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We sepa- rate the discussion into parts corresponding to the two requirements in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Throughout, we will show how to use the analysis in [CH22] to guarantee that with probability at least 1−Ω( 1 κ), the algorithm has expected function error O(ϵopt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' because the maximum error over B(R) is ≤ LR, this corresponds to an overall error O(ϵopt), and we may adjust Cba by a constant to compensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Per-iteration requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The ball optimization error guarantees are as stated in Proposition 1 of [CH22], except we dropped the function evaluations requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' To see that this is doable, note that [CH22] obtains their line search oracle (see Proposition 1) by running O(log( Rκ r )) ball optimization oracles to O(λr2) expected error, querying the function value, and applying Markov to argue at least one will succeed with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We can instead apply a Chernoff bound with O(log( Rκ r )) independent runs to argue that with probability O( 1 Kκ·polylog(Kκ)), the precondi- tions of the geometric aggregation in Lemma 11 are met with ∆ = O(r), as required by the line search oracle (see Algorithm 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Finally, applying a union bound over all iterations implies that the overall failure probability due to these line search oracles is O( 1 κ) as required by our earlier argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Additional requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The error requirements of the queries which occur every ≈ 2−j iter- ations are as stated in [CH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The only difference is that we state the complexity deterministically (Proposition 1 of [CH22] implicitly states an expected gradient bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The stochastic proximal oracle is implemented as Algorithm 2, [CH22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' it is also adapted with slightly different parame- ters as Algorithm 6 of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The expected complexity bound is derived by summing over all j ∈ [⌈log2 K + Cba⌉], the probability j is sampled in each iteration of Algorithm 2 of [CH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' For all j a Chernoff bound shows that the number of times in the entire algorithm j is sampled is O(2−jK log( Rκ r )) (within a constant of its expectation), with probability 1 − Ω(poly( r Rκ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Taking a union bound over all j shows the failure probability of our complexity bound is O( 1 κ) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' D Discussion of Proposition 4 In this section, we discuss how to obtain Proposition 4 using results in [GL12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We first state the following helper fact on the smoothness of a convolved function �fρ (see Definition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 47 Fact 5 (Lemma 8, [BJL+19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' If f : Rd → R is L-Lipschitz, �fρ (see Definition 1) is L ρ -smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The statement of Proposition 4 then follows from recursively applying Proposition 9 of [GL12] on the objective Ψ = �fρ + λ 2 ∥· − ¯x∥2, which is λ-strongly convex and ( L ρ +λ)-smooth, together with the divergence choice of V (x0, x∗) := 1 2∥x0 − x∗∥2, which satisfies ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Our parameter choices in Algorithm 2 are the same as in [GL12], where we use that our variance bound is 3L2 (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' In particular, denote the iterate xag T after the kth outer loop by xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' We will inductively assume that E 1 2∥xk−1 − x⋆ ¯x,λ∥2 ≤ r2 2k−1 (clearly the base case holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' This then implies E �λ 2 ∥xk − x⋆ ¯x,λ∥2 � ≤ E � Ψ(xk) − Ψ(x⋆ ¯x,λ) � ≤ 2( L ρ + λ)∥xk−1 − x⋆ ¯x,λ∥2 T(T + 1) + 24L2 λNk(T + 1) ≤ λ 2k r2 where the second inequality is Proposition 9 in [GL12] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content='21) therein), and the last is by our choice of T and Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' Thus, when K > log2( λr2 φ ) we have EΨ(xag T ) − Ψ(x⋆ ¯x,λ) ≤ φ as in the last outer loop k = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' The computational depth follows immediately from computing TK, and the total oracle queries and computational complexity follow since NK asymptotically dominates: T · � � � k∈[K] Nk � � = O (TNK + TK) = O �� 1 + L ρλ log �λr2 φ � + L2 λφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} +page_content=' 48' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAyT4oBgHgl3EQflvhP/content/2301.00457v1.pdf'} diff --git a/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/2301.01639v1.pdf.txt b/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/2301.01639v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..de155156cdec3a2276c9a60853deaee67b4d8bf2 --- /dev/null +++ b/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/2301.01639v1.pdf.txt @@ -0,0 +1,1482 @@ +arXiv:2301.01639v1 [math.PR] 4 Jan 2023 +On Lamperti transformation and +characterisations of discrete random fields +Marko Voutilainen∗, Lauri Viitasaari†, Pauliina Ilmonen‡ +January 5, 2023 +Abstract +In this article we characterise discrete time stationary fields by differ- +ence equations involving stationary increment fields and self-similar +fields. +This gives connections between stationary fields, stationary +increment fields and, through Lamperti transformation, self-similar +fields. Our contribution is a natural generalisation of recently proved +results covering the case of stationary processes. +AMS 2010 Mathematics Subject Classification: 60G60, 60G10, 60G18 +Keywords: random fields, stationary fields, self-similar fields, Lamperti transfor- +mation, fractional Ornstein-Uhlenbeck fields +1 +Introduction +Stationary processes X = (Xt)t∈T have numerous applications in many dif- +ferent fields, and they are a topic of active research. Similarly, self-similar +processes and stationary increment processes have many applications in var- +ious disciplines of science. For details on self-similar processes, we refer to +the monograph [4] and the references therein. +All of these three classes are intimately connected. Indeed, it was already +observed by Lamperti in [8] that there exists a one-to-one correspondence +between stationary processes and self-similar processes. Later on, this con- +nection was used in [13] to obtain relation between stationary processes and +∗Turku School of Economics, Department of Accounting and Finance, FI-20014 Uni- +versity of Turku, Finland +†Uppsala University, Department of Mathematics, Box 480, 751 06 Uppsala, Sweden +‡Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 11100, +FI-00076 Aalto, Finland +1 + +stationary increment processes in continuous time, i.e. +T = R, through +Langevin equation +dXt = −θXtdt + dGt, +(1) +where θ > 0 is a parameter, X is stationary, and G has stationary increments +(along with certain other properties). Most notably, this gives rise to the well- +known Ornstein-Uhlenbeck process when one plugs in G = W, the Brownian +motion. Connection (1) was later extended for discrete time processes X, i.e. +T = Z, in [15], where the authors proved discrete analogue +∆Xt = −θXt−1 + ∆Gt +(2) +of (1), and studied the estimation of the unknown parameter θ. A vector- +valued version was later provided in [16] and [14], covering both continuous +and discrete time cases together with estimation of the unknown parameter +matrix. +Similarly to stationary processes, stationary fields form an important sub- +class of random objects. In this case, X = (Xt)t∈T with T = RN in contin- +uous time or T = ZN in discrete time (naturally, T can be a more general +parameter space). Also, self-similarity and stationarity of the increments are +wanted features in many applications. However, while the notion of station- +arity is essentially unchanged in the context of random fields, the notion of +self-similarity and stationarity of the increments become more complicated +when t is multidimensional. For notion of self-similarity for fields, one typi- +cally introduces componentwise self-similarity and considers H-self-similarity +with H as an N-dimensional vector of componentwise self-similarity indices, +see e.g. [11, 5]. In [5] a version of the Lamperti theorem was proved (in +continuous time) for fields, providing a connection between stationary fields +and self-similar fields. For other notions of self-similarity for fields, see for +example [3] and [1]. +The notion of stationary increments for fields is even more complicated +due to the fact that the definition of increment is not obvious. +One ap- +proach is to consider rectangular increments, where increments are taken +over N-dimensional rectangulars. Gaussian self-similar fields and Gaussian +rectangular increment fields were studied, again in continuous time, e.g. in +[10] and [9]. +In this article we extend the Characterisation (2) provided in [15] to +discrete time fields, providing a connection between stationary fields, self- +similar fields, and stationary increment fields. More precisely, we provide a +characterisation of the type (see Theorem 2.20) +Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG. +2 + +Here ˆΘ is a vector of parameters, and ˆX− +t +is a vector of ”previous value” +consisting of previous values in different coordinate directions. Our notion +of increment ∆tG corresponds to the notion of stationary rectangular incre- +ments of [10], cf. Remark 2.9. As such and exactly as in [13, 15] in the case +of processes, we obtain correspondence between stationary fields, stationary +rectangular increment fields, and self-similar fields. +The rest of the article is organised as follows. In Section 2 we introduce +and prove our main results. We introduce our notation and main definitions +in Section 2.1, while our main results and their proofs are presented in Section +2.2. In Section 2.3 we briefly illustrate how our characterisation can be used +to construct discrete time fractional Ornstein-Uhlenbeck fields, extending +notions of (generalized) Ornstein-Uhlenbeck processes of [13]. We end the +paper with conclusions, Section 3, describing future directions, in particular +to cover continuous time parameter space and statistical inference. +2 +Connections between stationary, self-similar, +and stationary increment fields +2.1 +Preliminaries and notations +We begin with by introducing some definitions and notations. +Definition 2.1 (Stationarity). A random field X = (Xt)t∈ZN is stationary +if +(Xt+s)t∈ZN +law += (Xt)t∈ZN +for every s ∈ ZN in the sense of finite dimensional distributions. +Definition 2.2 (Self-similarity). Let Y = (Yet)t∈ZN = (Yet1,...,etN )t∈ZN be +a random field. In addition, let Θ = (θ1, . . . , θN) ∈ (0, ∞)N be a positive +multi-index. If +(Yet+s)t∈ZN +law += (e⟨s,Θ⟩Yet)t∈ZN +for every s ∈ ZN, where ⟨s, Θ⟩ is the standard inner product of vectors, then +Y is a Θ-self-similar random field. +Remark 2.3. The exponential terms in Definition 2.2 are introduced in or- +der to take into account the discrete nature of the field. Definition 2.2 is +analogous to the classical definition in continuous time. See e.g. [8] for the +definition in the one parameter continuous case and [13] for the definition in +the one parameter discrete case. +3 + +The following definition provides a notion of Lamperti transformation in +our setting. +Definition 2.4 (Lamperti). Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N. The Lamperti +transformation LΘ and its inverse L−1 +Θ for discrete random fields are defined +by +(LΘX)et = e⟨t,Θ⟩Xt, +t ∈ ZN, +(L−1 +Θ Y )t = e−⟨t,Θ⟩Yet, +t ∈ ZN. +Remark 2.5. The formulae in Definitions 2.2 and 2.4 differ slightly from +the standard forms in continuous settings. +In the case of random fields, +the definition of component-wise self-similarity and the corresponding Lam- +perti transformation together with a one-to-one correspondence between self- +similar and stationary fields was presented in [5]. In comparison, our defini- +tions are obtained via change of variables from the standard ones, ensuring +that we stay within our discrete parameter set when applying the transforma- +tion. +Definition 2.6 (Increments). The square increment ∆tX of a field (Xt)t∈ZN +at a point t = (t1, . . . , tN) ∈ ZN is given by +∆tX = +� +(i1,...,iN)∈{0,1}N +(−1) +�N +l=1 ilXt1−i1,...,tN−iN. +(3) +Example 2.7. In the particular case N = 2, we have +∆tX = Xt1,t2 + Xt1−1,t2−1 − Xt1−1,t2 − Xt1,t2−1. +Definition 2.8 (Stationary increment field). A field X = (Xt)t∈ZN has sta- +tionary increments if the increment field (∆tX)t∈ZN is stationary. That is +(∆t+sX)t∈ZN +law += (∆tX)t∈ZN +for every s ∈ ZN in the sense of finite dimensional distributions. +Remark 2.9. The authors in [10] introduced a continuous time analogous +notion of strictly stationary rectangular increments by assuming stationar- +ity of the increments over arbitrary rectangular increments. In comparison, +in our definition, we consider stationary increments over unit rectangulars. +However, due to the discrete nature of our index space t ∈ ZN, one can +show that our definition is equivalent to assuming stationarity over arbitrary +(discrete) rectangular increments. +4 + +We also need a notion of previous value that is not so straightforward in +a multi-parameter setting. +Definition 2.10 (Previous value). The previous value of the field X = +(Xt)t∈ZN at a point t = (t1, . . . , tN) ∈ ZN is given by +X− +t = Xt − ∆tX = +� +(i1,...,iN)∈{0,1}N +(i1,...,iN)̸=0 +(−1)1+�N +l=1 ilXt1−i1,...,tN−iN. +Remark 2.11. Note that in our definition of the previous value, we take +into account the terms in (3) that have a smaller index in at least one of the +coordinate directions. Indeed, in the two-dimensional case, we have +∆tX = Xt1,t2 + Xt1−1,t2−1 − Xt1−1,t2 − Xt1,t2−1 +while the previous value is given by +X− +t = Xt1−1,t2 + Xt1,t2−1 − Xt1−1,t2−1. +Definition 2.12 (Inner product ⟨ˆΘ, ˆX− +t ⟩). Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N +and X = (Xt)t∈ZN be a field. We define vectors ˆΘ and ˆX− +t of length 2N − 1 +having elements of the forms +(−1)1+�N +l=1 ile−⟨i,Θ⟩ +and +Xt1−i1,...,tN−iN, +respectively, where i = (i1, . . . , iN) ∈ {0, 1}N, i ̸= 0. Then the inner product +of the vectors is +⟨ˆΘ, ˆX− +t ⟩ = +� +(i1,...,iN)∈{0,1}N +i̸=0 +(−1)1+�N +l=1 ile−⟨i,Θ⟩Xt1−i1,...,tN−iN. +Example 2.13. In the two-dimensional case we obtain that, for Θ = (θ1, θ2) ∈ +(0, ∞)2 and X = (Xt)t∈Z2, the vectors ˆΘ and ˆX− +t are given as +ˆΘ = (e−θ1, e−θ2, −e−θ1−θ2) +and +ˆX− +t = + + +Xt1−1,t2 +Xt1,t2−1 +Xt1−1,t2−1 + + +and we have +⟨ˆΘ, ˆX− +t ⟩ = e−θ1Xt1−1,t2 + e−θ2Xt1,t2−1 − e−θ1−θ2Xt1−1,t2−1. +(4) +Remark 2.14. The vectors ˆΘ and ˆX− +t +are unique up to permutations of +their elements. +5 + +Definition 2.15 (Class GΘ). Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N. Let G = +(Gt)t∈ZN be a stationary increment field with Gt = 0 for all t such that +�N +i=1 ti ∈ {0, −1, . . . , −N + 1}. If +lim +M1→∞ . . . +lim +MN→∞ +t1 +� +j1=−M1 +· · · +tN +� +jN=−MN +e +�N +l=1 jlθl∆(j1,...,jN)G +(5) +converges in probability defining an almost surely finite random variable for +every t = (t1, . . . , tN) ∈ ZN, then G ∈ GΘ. +Remark 2.16. It turns out that the order of the limits in (5) is irrelevant, +and any permutation of the order leads to the convergence towards the same +limiting random variable that turns out to be the stationary field Xt, cf. proof +of Theorem 2.28. +Remark 2.17. Condition Gt = 0 for all t such that �N +i=1 ti ∈ {0, −1, . . . , −N+ +1} is rather peculiar and it essentially means that G has to vanish along dis- +crete points of certain N − 1-dimensional planes. However, this condition is +not required a priori for the characterisation, but turns out to hold true and +is also required to obtain uniqueness of the representation, cf. Theorem 2.20 +and Remark 2.29. +Remark 2.18. Note that if G ∈ L1 (or ∆G ∈ L1), then G ∈ GΘ for every +Θ. Indeed, this can be seen from +E| +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθl∆(j1,...,jN)G| +≤ +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθlE|∆(j1,...,jN)G| += +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθlE|∆(1,...,1)G| +≤ +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθl +� +(i1,...,iN)∈{0,1}N +E|G1−i1,...,1−iN|. +Example 2.19. In the two-dimensional case, we have G ∈ GΘ for Θ = +(θ1, θ2) ∈ (0, ∞)2 provided that G has stationary increments, Gt,−t = Gt,−t−1 = +0 and +lim +M1→∞ lim +M2→∞ +t1 +� +j1=−M1 +t2 +� +j2=−M2 +ej1θ1ej2θ2∆(j1,j2)G +6 + +exists as an almost surely finite random variable. That is, G has stationary +increments and Gt1,t2 is set to zero on lines t1 = −t2 and t1 = −t2 − 1. The +existence of such fields follow as a by-product of our main results. +2.2 +AR(1) type characterisation of stationary fields +Our main result is the following characterisation that is a natural extension +of the one-dimensional case presented in [15]. +Theorem 2.20. Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N. A field X = (Xt)t∈ZN is +stationary if and only if the following conditions are satisfied. +(i) +lim +m→−∞ emθjXt1,...,tj−1,m,tj+1,...,tN +P +−→ 0 +for every j ∈ {1, . . . , N} and t1, . . . , tj−1, tj+1, . . . , tN ∈ Z. +(ii) There exists G = (Gt)t∈ZN ∈ GΘ such that +Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG +for every t ∈ ZN, +(6) +where ⟨ˆΘ, ˆX− +t ⟩ is given by Definition 2.12. +Moreover, for a given Θ, the stationary increment field G ∈ GΘ in (6) is +unique. +As a direct corollary we obtain the following version in a two-dimensional +case. +Corollary 2.21. Let Θ = (θ1, θ2) ∈ (0, ∞)2. A field X = (Xt)t∈Z2 is sta- +tionary if and only if the following conditions are satisfied. +(i) +lim +m→−∞ emθ2Xt1,m +P +−→ 0 +and +lim +m→−∞ emθ1Xm,t2 +P +−→ 0 +for every t1 and t2. +(ii) There exists G = (Gt)t∈Z2 ∈ GΘ such that +Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG +for every t ∈ Z2, +(7) +where ⟨ˆΘ, ˆX− +t ⟩ is given by (4). +The stationary increment field G ∈ GΘ in (7) is unique. +7 + +The proof of Theorem 2.20 is split into a series of lemmas and auxiliary +theorems. We begin with the following result that is a version of Lamperti +theorem. +Theorem 2.22. If X = (Xt)t∈ZN is stationary, then (LΘX)et is Θ-self- +similar. +Conversely, if Y = (Yet)t∈ZN is Θ-self-similar, then (L−1 +Θ Y )t is +stationary. +Proof. First, assume that X is stationary. Set Yet = (LΘX)et and let n ∈ N. +Now +(Yet1+s, . . . , Yetn+s) = (e⟨t1+s,Θ⟩Xt1+s, . . . e⟨tn+s,Θ⟩Xtn+s) +law += (e⟨s,Θ⟩e⟨t1,Θ⟩Xt1, . . . , e⟨s,Θ⟩e⟨tn,Θ⟩Xtn) += (e⟨s,Θ⟩Yet1, . . . , e⟨s,Θ⟩Yetn), +proving the first part of the claim. Next, assume that Y is Θ-self-similar. +Set Xt = (L−1 +Θ Y )t and let n ∈ N. Now +(Xt1+s, . . . , Xtn+s) = (e−⟨t1+s,Θ⟩Yet1+s, . . . , e−⟨tn+s,Θ⟩Yetn+s) +law += (e−⟨t1,Θ⟩Yet1, . . . , e−⟨tn,Θ⟩Yetn) += (Xt1, . . . , Xtn), +completing the proof. +The following lemma provides one of our key observations. +Lemma 2.23. Let (Yet)t∈ZN be Θ-self-similar. Set +∆tY = +� +(i1,...,iN)∈{0,1}N +(−1) +�N +l=1 ilYet1−i1,...,etN −iN . +For �N +l=1 tl ≥ 1, we set +Gt = +t1 +� +k1=1−t2−···−tN +t2 +� +k2=1−k1−t3−···−tN +· · · +tN +� +kN=1−k1−···−kN−1 +e−⟨k,Θ⟩∆kY, +and, for �N +l=1 tl ≤ 0, we set +Gt = (−1)N +−t2−···−tN−N+1 +� +k1=t1+1 +−k1−t3−···−tN−N+2 +� +k2=t2+1 +· · · +−k1−···−kN−1 +� +kN=tN+1 +e−⟨k,Θ⟩∆kY. +Here ⟨k, Θ⟩ is the standard inner product and sums of the type �s1 +s2 with +s1 < s2 are interpreted as empty sums. Now +8 + +(i) Gt = 0 for all t such that �N +l=1 tl ∈ {0, −1, . . . , −N + 1}, +(ii) ∆tG = e−⟨t,Θ⟩∆tY for every t ∈ ZN, +(iii) G = (Gt)t∈ZN is a stationary increment field. +Remark 2.24. It turns out that G defined as above satisfies G ∈ GΘ, see +also Lemma 2.31 below. +Example 2.25. In the two-dimensional case, for Θ-self-similar (Yet)t∈Z2, we +denote +∆tY = Yet1,et2 − Yet1−1,et2 − Yet1,et2−1 + Yet1−1,et2−1. +The field G = (Gt)t∈Z2 defined as +Gt1,t2 = +��t1 +k1=1−t2 +�t2 +k2=1−k1 e−⟨(k1,k2),Θ⟩∆kY, +t1 + t2 ≥ 1 +�−t2−1 +k1=t1+1 +�−k1 +k2=t2+1 e−⟨(k1,k2),Θ⟩∆kY, +t1 + t2 ≤ 0, +belongs to the class GΘ. Here sums of the type �s +s+1 are interpreted as empty +sums. +The proof of Lemma 2.23 is based on the following additional lemmas +that we prove first. The first one provides an auxiliary result on sums of +binomial coefficients. Although the result is quite elementary, we provide a +proof for the reader’s convenience. +Lemma 2.26. We have the following identities: +M−1 +2 +� +m=0 +�M +2m +� += 2M−1, +M−1 +2 +� +m=0 +� +M +2m + 1 +� += 2M−1, +when M ≥ 1 is odd. +M +2 +� +m=0 +�M +2m +� += 2M−1, +M +2 −1 +� +m=0 +� +M +2m + 1 +� += 2M−1, +when M ≥ 2 is even. +Proof. In the odd case +M−1 +2 +� +m=0 +�M +2m +� += +�M +0 +� ++. . .+ +� +M +M − 1 +� +and +M−1 +2 +� +m=0 +� +M +2m + 1 +� += +�M +1 +� ++. . .+ +�M +M +� +. +These sums are equal since +�M +k +� += +� M +M−k +� +. In addition, +M−1 +2 +� +m=0 +�M +2m +� ++ +M−1 +2 +� +m=0 +� +M +2m + 1 +� += 2M +9 + +completing the proof of the first case. For the even case, we obtain +M +2 −1 +� +m=0 +� +M +2m + 1 +� += +�M +1 +� ++ . . . + +� +M +M − 1 +� += +�M − 1 +0 +� ++ +�M − 1 +1 +� ++ · · · + +�M − 1 +M − 2 +� ++ +�M − 1 +M − 1 +� += 2M−1. +Observing that +M +2 +� +m=0 +�M +2m +� ++ +M +2 −1 +� +m=0 +� +M +2m + 1 +� += 2M +completes the proof. +The following lemma sheds light on how the field G of Lemma 2.23 is +constructed from a self-similar field Y . +Lemma 2.27. +(i) Let �N +l=1 tl ≥ 1. Then a term e−⟨j,Θ⟩∆jY belongs to the +sum defining Gt in Lemma 2.23 if and only if +jl ≤ tl +for every l and +N +� +l=1 +jl ≥ 1. +(ii) Let �N +l=1 tl ≤ −N. Then a term e−⟨j,Θ⟩∆jY belongs to the sum defining +Gt in Lemma 2.23 if and only if +jl ≥ tl + 1 +for every l and +N +� +l=1 +jl ≤ 0. +Proof. Item (i): By the upper bounds in the sum defining Gt, it is clear that +we have jl ≤ tl for all l. By the lower bound of the inmost summation, we +obtain jN ≥ 1 − j1 − · · · − jN−1. That is, �N +l=1 jl ≥ 1. We also observe that +the other lower bounds of the sum defining Gt yield conditions +jN−h ≥ 1 − +N−h−1 +� +l=1 +jl − +N +� +l=N−h+1 +tl +for every h ∈ {0, . . . , N − 1}. +These conditions are satisfied since +N−h +� +l=1 +jl + +N +� +N−h+1 +tl ≥ +N +� +l=1 +jl ≥ 1. +10 + +This completes the proof of the first item. +Item (ii): By the lower bounds in the sum defining Gt, it is clear that we +have jl ≥ tl + 1 for all l. By the upper bound of the inmost summation, we +obtain also that jN ≤ −j1 −· · ·−jN−1. That is, �N +l=1 jl ≤ 0. We also observe +that the other upper bounds of the sum defining Gt yield conditions +jN−h ≤ − +N−h−1 +� +l=1 +jl − +N +� +l=N−h+1 +tl − N + (N − h) +for every h ∈ {0, . . . , N − 1}. +These conditions are satisfied since +N−h +� +l=1 +jl + +N +� +N−h+1 +tl ≤ +N−h +� +l=1 +jl + +N +� +N−h+1 +(jl − 1) = +N +� +l=1 +jl − h ≤ −h. +This completes the proof of the second item, and thus the whole proof is +completed. +Proof of Lemma 2.23. Item (i): Let �N +l=1 tl ∈ {0, −1, . . . , −N + 1}. Then +N +� +l=2 +tl ∈ {−t1, −1 − t1, . . . , −N + 1 − t1} +and for the upper bound of the first summation in the definition of Gt it +holds that +− +N +� +l=2 +tl − N + 1 ∈ {t1 − N + 1, t1 − N + 2, . . . , t1}. +Hence, Gt is given by an empty sum. +Item (ii): Recall that Gt = 0 for all t such that �N +i=1 ti ∈ {0, −1, . . . , −N+1}. +We treat the case �N +l=1 tl ≥ 1 first. Let M be such that �N +l=1 tl −M = 1. +Then +∆tG = +� +(i1,...,iN)∈{0,1}N +�N +l=1 il≤M +(−1) +�N +l=1 ilGt1−i1,...,tN−iN. +(8) +By Lemma 2.27, ∆tG consists of terms e−⟨j,Θ⟩∆jY with jl ≤ tl for every l +and �N +l=1 jl ≥ 1. Let m be the number of indices l for which jl = tl. +Assume that m < N. By Lemma 2.27, e−⟨j,Θ⟩∆jY belongs to summands of +11 + +(8) that satisfy jl ≤ tl − il for every l. That is, m of the indices il are zero +while the remaining N − m indices may be zeros or ones. In addition, +1 ≤ +N +� +l=1 +jl ≤ +N +� +l=1 +tl − (N − m) +giving +N − m ≤ +N +� +l=1 +tl − 1 = M. +(9) +Now if N − m is odd, then by Lemma 2.26 and (9), the number of terms +e−⟨j,Θ⟩∆jY in (8) with a positive sign is +�N − m +0 +� ++ . . . + +� +N − m +min{N − m, M} − 1 +� += +�N − m +0 +� ++ . . . + +� +N − m +N − m − 1 +� += 2N−m−1. +Thus, terms e−⟨j,Θ⟩∆jY cancel out in (8). +Similarly, if N−m is even, then the number of terms e−jΘ∆jY with a positive +sign is +�N − m +0 +� ++ · · · + +� +N − m +min{N − m, M} +� += +�N − m +0 +� ++ · · · + +�N − m +N − m +� +. +Again, by Lemma 2.26, terms e−⟨j,Θ⟩∆jY cancel out in (8). +If m = N, we have that e−⟨j,Θ⟩∆jY = e−⟨t,Θ⟩∆tY belongs only to the sum- +mand of (8) with i = 0. Hence we have shown that +∆tG = e−⟨t,Θ⟩∆tY +for every t such that +N +� +l=1 +tl ≥ 1. +This proves the claim for the case �N +l=1 tl ≥ 1. +Assume next that �N +l=1 tl ≤ 0 and let M be such that �N +l=1 tl−M = −N. +Then +∆tG = +� +(i1,...,iN)∈{0,1}N +�N +l=1 il≥M +(−1) +�N +l=1 ilGt1−i1,...,tN−iN. +(10) +By Lemma 2.27, ∆tG consists of terms e−⟨j,Θ⟩∆jY with jl ≥ tl for every l and +�N +l=1 jl ≤ 0. As before, let m be the number of indices l for which jl = tl. If +m < N, then, by Lemma 2.27, e−⟨j,Θ⟩∆jY belongs to summands of (10) that +satisfy jl ≥ tl − il + 1 for every l. That is, m of the indices il are equal to +one while the remaining N − m indices may be zeros or ones. In addition, +0 ≥ +N +� +l=1 +jl ≥ +N +� +l=1 +tl + (N − m) = M − m +12 + +giving m ≥ M and N − m ≤ N − M. If N − m is odd, by Lemma 2.26, the +number of terms e−⟨j,Θ⟩∆jY in (10) with the sign (−1)N+m is equal to +�N − m +0 +� ++ · · · + +� +N − m +N − m − 1 +� += 2N−m−1. +Thus, terms cancel out in (10). Similarly, terms cancel out when N − m is +even. Finally, for the case m = N we observe that e−⟨j,Θ⟩∆jY = e−⟨t,Θ⟩∆tY +belongs only to the summand of (10) with i = 1. The corresponding sign is +(−1)N(−1)N = 1. +To conclude, we have shown that +∆tG = e−⟨t,Θ⟩∆tY +for every t such that +N +� +l=1 +tl ≤ 0. +This completes the proof of item (ii). +Item (iii): For s = (s1, . . . , sN) ∈ ZN, +∆t+sG = e−⟨t+s,Θ⟩∆t+sY += e−⟨t,Θ⟩e−⟨s,Θ⟩ +� +(i1,...,iN)∈{0,1}N +(−1) +�N +l=1 ilYet1+s1−i1,...,etN +sN −iN +law += e−⟨t,Θ⟩ +� +(i1,...,iN)∈{0,1}N +(−1) +�N +l=1 ilYet1−i1,...,etN −iN += e−⟨t,Θ⟩∆tY = ∆tG. +Treating multidimensional distribution similarly completes the proof of item +(iii). +We are now ready to prove three results, Theorem 2.28, Theorem 2.30, +and Lemma 2.32, that give us the main result of this article, Theorem 2.20. +Theorem 2.28. Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N and let X = (Xt)t∈ZN be a +random field. If for some G = (Gt)t∈ZN ∈ GΘ it holds that +Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG +for every t ∈ ZN, +(11) +and +lim +m→−∞ emθjXt1,...,tj−1,m,tj+1,...,tN +P +−→ 0 +for every j ∈ {1, . . . , N} and t1, . . . , tj−1, tj+1, . . . , tN ∈ Z, then X = (Xt)t∈ZN +is stationary. +13 + +Proof. Denote Zt = ∆tG and i = (i1, . . . , iN). From (11) and Definition 2.12, +we get +Xt = +� +(i1,...,iN)∈{0,1}N +i̸=0 +(−1)1+�N +l=1 ile−⟨i,Θ⟩Xt1−i1,...,tN−iN + Zt, +which gives +Xt − +� +(i1,...,iN−1)∈{0,1}N−1 +i̸=0 +(−1)1+�N−1 +l=1 ile−⟨(i1,...,iN−1,0),Θ⟩Xt1−i1,...,tN−1−iN−1,tN += +� +(i1,...,iN−1)∈{0,1}N−1 +(−1) +�N−1 +l=1 ile−⟨(i1,...,iN−1,0),Θ⟩Xt1−i1,...,tN−1−iN−1,tN += +� +(i1,...,iN−1)∈{0,1}N−1 +(−1) +�N−1 +l=1 ile−⟨(i1,...,iN−1,1),Θ⟩Xt1−i1,...,tN−1−iN−1,tN−1 + Zt. +(12) +Set +Yt1,...,tN−1(tN) = +� +(i1,...,iN−1)∈{0,1}N−1 +(−1) +�N−1 +l=1 ile−⟨(i1,...,iN−1,0),Θ⟩Xt1−i1,...,tN−1−iN−1,tN. +Then, by iterating the recursive Equation (12), we get +Yt1,...,tN−1(tN) =e−θNYt1,...,tN−1(tN − 1) + Zt +=e−(n+1)θNYt1,...,tN−1(tN − n − 1) + +n +� +jN=0 +e−jNθNZt1,...,tN−1,tN−jN +=e−(n+1)θNYt1,...,tN−1(tN − n − 1) + e−tNθN +tN +� +jN=tN−n +ejNθNZt1,...,tN−1,jN, +for every n ∈ N. Above +e−(n+1)θNYt1,...,tN−1(tN − n − 1) = e−tN θNe(tN −n−1)θNYt1,...,tN−1(tN − n − 1) += e−tN θNemθNYt1,...,tN−1(m) +by the change of variable m = tN − n − 1. Furthermore +emθNYt1,...,tN−1(m) +=emθN +� +(i1,...,iN−1)∈{0,1}N−1 +(−1) +�N−1 +l=1 ile−⟨(i1,...,iN−1,0),Θ⟩Xt1−i1,...,tN−1−iN−1,m, +14 + +which, by the assumptions, converges to zero in probability as m → −∞. +Hence +Yt1,...,tN−1(tN) += +� +(i1,...,iN−1)∈{0,1}N−1 +(−1) +�N−1 +l=1 ile−⟨(i1,...,iN−1,0),Θ⟩Xt1−i1,...,tN−1−iN−1,tN += e−tNθN +tN +� +jN=−∞ +ejNθNZt1,...,tN−1,jN =: Q(N) +t +. +In the following summations, let (i1, . . . , iN−k−1) ∈ {0, 1}N−k−1 and (i1, . . . , iN−k−2) ∈ +{0, 1}N−k−2. We proceed by induction and assume that for some k ∈ N∪{0} +it holds that +� +(i1,...,iN−k−1) +(−1) +�N−k−1 +l=1 +ile−⟨(i1,...,iN−k−1,0,...,0),Θ⟩Xt1−i1,...,tN−k−1−iN−k−1,tN−k,...,tN +=e− �k +l=0 tN−lθN−l +tN−k +� +jN−k=−∞ +· · · +tN +� +jN=−∞ +e +�k +l=0 jN−lθN−lZt1,...,tN−k−1,jN−k,...,jN =: Q(N−k) +t +. +(13) +Now +� +(i1,...,iN−k−2) +� +(−1) +�N−k−2 +l=1 +ile−⟨(i1,...,iN−k−2,0,...,0),Θ⟩ +· Xt1−i1,...,tN−k−2−iN−k−2,tN−k−1,...,tN +� += − +� +(i1,...,iN−k−2) +� +(−1)1+�N−k−2 +l=1 +ile−⟨(i1,...,iN−k−2,1,0,...,0),Θ⟩ +· Xt1−i1,...,tN−k−2−iN−k−2,tN−k−1−1,...,tN +� ++ Q(N−k) +t += +� +(i1,...,iN−k−2) +� +(−1) +�N−k−2 +l=1 +ile−⟨(i1,...,iN−k−2,1,0,...,0),Θ⟩ +· Xt1−i1,...,tN−k−2−iN−k−2,tN−k−1−1,...,tN +� ++ Q(N−k) +t +. +(14) +Let t∗ = (t1, . . . , tN−k−2, tN−k, . . . , tN) and define +Yt∗(tN−k−1) += +� +(i1,...,iN−k−2) +(−1) +�N−k−2 +l=1 +ile−⟨(i1,...,iN−k−2,0,...,0),Θ⟩Xt1−i1,...,tN−k−2−iN−k−2,tN−k−1,...,tN. +15 + +Equation (14) gives +Yt∗(tN−k−1) =e−θN−k−1Yt∗(tN−k−1 − 1) + Q(N−k) +t +=e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) ++ +n +� +jN−k−1=0 +e−jN−k−1θN−k−1Q(N−k) +t1,...,tN−k−2,tN−k−1−jN−k−1,tN−k,...,tN +=e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) ++ e−tN−k−1θN−k−1 +tN−k−1 +� +jN−k−1=tN−k−1−n +ejN−k−1θN−k−1Q(N−k) +t1,...,tN−k−2,jN−k−1,tN−k,...,tN +for every n ∈ N. Above +e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) =e−tN−k−1θN−k−1e(tN−k−1−n−1)θN−k−1Yt∗(tN−k−1 − n − 1) +=e−tN−k−1θN−k−1emθN−k−1Yt∗(m) +by the change of variable m = tN−k−1 − n − 1. As before, the expression +converges to zero in probability as m → ∞. Hence, we obtain that +Yt∗(tN−k−1) += +� +(i1,...,iN−k−2) +� +(−1) +�N−k−2 +l=1 +ile−⟨(i1,...,iN−k−2,0,...,0),Θ⟩ +· Xt1−i1,...,tN−k−2−iN−k−2,tN−k−1,...,tN +� +=e−tN−k−1θN−k−1 +tN−k−1 +� +jN−k−1=−∞ +ejN−k−1θN−k−1Q(N−k) +t1,...,tN−k−2,jN−k−1,tN−k,...,tN +=e−tN−k−1θN−k−1 +� tN−k−1 +� +jN−k−1=−∞ +ejN−k−1θN−k−1e− �k +l=0 tN−lθN−l +· +tN−k +� +jN−k=−∞ +· · · +tN +� +jN=−∞ +e +�k +l=0 jN−lθN−lZt1,...,tN−k−2,jN−k−1,...,jN +� +=e− �k+1 +l=0 tN−lθN−l +tN−k−1 +� +jN−k−1=−∞ +· · · +tN +� +jN=−∞ +e +�k+1 +l=0 jN−lθN−lZt1,...,tN−k−2,jN−k−1,...,jN, +which proves the induction step. Therefore choosing k = N −2 in (13) yields +� +i1∈{0,1} +(−1)i1e−⟨(i1,0,...,0),Θ⟩Xt1−i1,t2,...,tN = Xt − e−θ1Xt1−1,t2,...,tN += e− �N−2 +l=0 tN−lθN−l +t2 +� +j2=−∞ +· · · +tN +� +jN=−∞ +e +�N−2 +l=0 jN−lθN−lZt1,j2,...,jN =: Q(2) +t +16 + +and we obtain a recursive equation +Xt = e−θ1Xt1−1,t1,...,tN + Q(2) +t . +By repeating the earlier procedure once more, we obtain that +Xt =e−t1θ1 +t1 +� +j1=−∞ +ej1θ1Q(2) +j1,t2,...,tN +=e−t1θ1 +t1 +� +j1=−∞ +ej1θ1e− �N−2 +l=0 tN−lθN−l +t2 +� +j2=−∞ +· · · +tN +� +jN=−∞ +e +�N−2 +l=0 jN−lθN−lZj1,j2,...,jN +=e− �N +l=1 tlθl +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθlZj1,j2,...,jN +=e− �N +l=1 tlθl +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθl∆j1,j2,...,jNG, +which, since G ∈ GΘ, defines an almost surely finite random variable. Hence +it remains to prove that X is stationary. To this end, we show that the one +dimensional distributions of X are stationary. The proof extends straightfor- +wardly to multidimensional distributions. Let s = (s1, . . . , sN) ∈ ZN. Note +that +Xt = +∞ +� +j1=0 +· · · +∞ +� +jN=0 +e− �N +l=1 jlθl∆t1−j1,...,tN−jNG. +Since G is a stationary increment field, we have that +M1 +� +j1=0 +· · · +MN +� +jN=0 +e− �N +l=1 jlθl∆t1+s1−j1,...,tN+sN−jNG +law += +M1 +� +j1=0 +· · · +MN +� +jN=0 +e− �N +l=1 jlθl∆t1−j1,...,tN−jNG +for every M1, . . . , MN ∈ N. Moreover, since G ∈ GΘ, the iterated limits of +both sides converge and hence, the limits are equal in distribution. This gives +Xt+s = +∞ +� +j1=0 +· · · +∞ +� +jN=0 +e− �N +l=1 jlθl∆t1+s1−j1,...,tN+sN−jNG +law += +∞ +� +j1=0 +· · · +∞ +� +jN=0 +e− �N +l=1 jlθl∆t1−j1,...,tN−jNG = Xt +and thus the proof is completed. +17 + +Remark 2.29. The property Gt = 0 for all t such that �N +i=1 ti ∈ {0, −1, . . . , −N+ +1} of the class GΘ is not utilized in the proof of Theorem 2.28. However, in +order to obtain uniqueness in the characterising Equation (11), we need to +pose this additional assumption, see Lemma 2.32. +Theorem 2.30. Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N. +Assume that the field +X = (Xt)t∈ZN is stationary. Then there exists G = (Gt)t∈ZN ∈ GΘ such that +Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG +for every t ∈ ZN. +Proof. Applying Lamperti transformation gives +∆tX = Xt − X− +t = e−⟨t,Θ⟩Yet − X− +t = e−⟨t,Θ⟩(Yet − ∆tY ) − X− +t + e−⟨t,Θ⟩∆tY. +Hence +Xt = e−⟨t,Θ⟩(Yet − ∆tY ) + e−⟨t,Θ⟩∆tY, +where +Yet − ∆tY = +� +(i1,...,iN)∈{0,1}N ,i̸=0 +(−1)1+�N +l=1 ilYet1−i1,...,etN −iN . +Furthermore +e−⟨t,Θ⟩(Yet − ∆tY ) = +� +(i1,...,iN)∈{0,1}N ,i̸=0 +(−1)1+�N +l=1 ile−⟨t−i,Θ⟩e−⟨i,Θ⟩Yet1−i1,...,etN −iN += +� +(i1,...,iN)∈{0,1}N ,i̸=0 +(−1)1+�N +l=1 ile−⟨i,Θ⟩Xt1−i1,...,tN−iN = ⟨ˆΘ, ˆX− +t ⟩, +where the last equality follows by Lamperti transformation. Let G be the +field defined in Lemma 2.23. Then Xt = ⟨ˆΘ, ˆX− +t ⟩ + ∆tG. Moreover, the +proof of Theorem 2.28 and the property +lim +m→−∞ emθiXt1,...,ti−1,m,ti+1,...,tN +P +−→ 0, +for every i ∈ {1, . . . , N} and t1, . . . , ti−1, ti+1, . . . , tN ∈ Z, give us the repre- +sentation +Xt = e− �N +l=1 tlθl +t1 +� +j1=−∞ +· · · +tN +� +jN=−∞ +e +�N +l=1 jlθl∆(j1,j2,...,jN)G. +(15) +Hence, G ∈ GΘ. +As a by-product, we observe the following result. +18 + +Lemma 2.31. Let G be defined as in Lemma 2.23. Then G ∈ GΘ. +Proof. By Lemma 2.23, it suffices to show that the limit (5) exists. Now +since Y is Θ-self-similar, there exists a stationary X such that Y = LΘX. +Following the lines of the proof of Theorem 2.30, we obtain Representation +(15), where G is defined as in Lemma 2.23. Hence G ∈ GΘ by the proof of +Theorem 2.30. +Lemma 2.32. Let Θ = (θ1, . . . , θN) ∈ (0, ∞)N be fixed. Then the stationary +increment field G ∈ GΘ in Theorems 2.28 and 2.30 is unique. +Proof. Assume that G, G′ ∈ GΘ satisfy the recursive Equation (11) of Theo- +rems 2.28 and 2.30. Then, ∆tG = ∆tG′ for all t ∈ ZN. It remains to show +that this implies Gt = G′ +t for all t, which we do by induction. +Since Gt = G′ +t = 0 for all t such that �N +l=1 tl ∈ {0, −1, . . . , −N + 1}, we +may set an induction assumption that there exists s ∈ Z such that Gt = G′ +t +for all t such that �N +l=1 tl ∈ {s, s − 1, . . . , s − N + 1}. In order to complete +the proof, it remains to show that Gt = G′ +t under conditions �N +l=1 tl = s + 1 +and �N +l=1 tl = s − N. Let �N +l=1 tl = s + 1. We now have +∆tG = +� +(i1,...,iN)∈{0,1}N +(−1) +�N +l=1 ilGt1−i1,...,tN−iN += Gt + +� +(i1,...,iN)∈{0,1}N +i̸=0 +(−1) +�N +l=1 ilGt1−i1,...,tN−iN, +where for the indices of G it holds that +N +� +l=1 +(tl − il) ∈ {s, s − 1, . . . , s − N + 1}. +Therefore, by the induction assumption, Gt = G′ +t for all t such that �N +l=1 tl = +s + 1. We proceed proving that Gt = G′ +t for all t such that �N +l=1 tl = s − N. +Let �N +l=1 tl = s − N. Now +(−1)NGt1,...,tN + +� +(i1,...,iN)∈{0,1}N +i̸=1 +(−1) +�N +l=1 ilGt1+1−i1,...,tN+1−iN += (−1)NG′ +t1,...,tN + +� +(i1,...,iN)∈{0,1}N +i̸=1 +(−1) +�N +l=1 ilG′ +t1+1−i1,...,tN+1−iN, +19 + +where +N +� +l=1 +(tl + 1 − il) ∈ {s, s − 1, . . . , s − N + 1}. +Hence, by the induction assumption, Gt = G′ +t for all t such that �N +l=1 tl = +s − N. This completes the proof. +The proof of our main theorem, Theorem 2.20, now follows by combining +Theorems 2.28 and 2.30 together with Lemma 2.32. +2.3 +Fractional Ornstein-Uhlenbeck fields on ZN +We illustrate our approach by constructing a stationary fractional Ornstein- +Uhlenbeck field on Z2. For this purpose, let Gt1,t2 be a discrete fractional +Gaussian field on (t1, t2) ∈ N2. A discrete fractional Gaussian field can be +constructed from its continuous time analogue by embedding to the space N2. +That is, let ˜X be a self-similar Gaussian field having stationary rectangular +increments such that E ˜X2(1, 1) = 1. Existence of such fields is studied in +[10]. We can now embed ˜X into N2 by considering values of ˜X at points +(t1, t2) ∈ N2. Now Definition 2.15 of GΘ allows us to extend G into Z2 by +setting Gt,−t = Gt,−t−1 = 0 and requiring that G has stationary increments +in the sense of Definition 2.8. As G is Gaussian, in view of Remark 2.18, +it follows that our field G = (Gt)t∈Z2 belongs to GΘ for any Θ. This allows +us to define a discrete time stationary fractional Ornstein-Uhlenbeck field of +the first kind by Representation (6). On the other hand, using self-similarity +of ˜X, we can define a stationary field through Lamperti transformation and +obtain, in view of Theorem 2.22, stationary fractional Ornstein-Uhlenbeck +field of the second kind corresponding to a different noise G ∈ GΘ. Obviously, +this approach can be used to define fractional Ornstein-Uhlenbeck fields in +arbitrary dimensions N ≥ 2. +3 +Conclusions +In this article we have provided a characterisation of discrete stationary fields +in terms of fields having stationary rectangular increments. Our characterisa- +tion is analogous to the one-parameter case provided in [15]. As an example, +we have extended the notion of fractional Ornstein-Uhlenbeck processes, in- +troduced in [2] and [7], to the multi-parameter case. +Natural further prospects of our study are two folded. Firstly, it would be +interesting to generalise the characterisation into the continuous parameter +space t ∈ RN. In this case however, the increments ∆tG are replaced with +20 + +differentials, and the notion of differential dtG is more subtle when t ∈ RN. +This is a topic of a forthcoming article. Secondly, parameter estimation is of +paramount importance in statistics, which in our setting would correspond to +the estimation of the parameter Θ. This has been a topic of active research +in the context of fractional Ornstein-Uhlenbeck processes, see e.g. [6, 12] +and the references therein. +For a more general setting in the context of +multivariate time series, parameter estimation is studied in [16]. We believe +that ideas and methods presented in [16] could be useful in our context as +well. +References +[1] Hermine Bierm´e, Mark M Meerschaert, and Hans-Peter Scheffler. Oper- +ator scaling stable random fields. Stoch. Process. their Appl., 117(3):312– +332, 2007. +[2] P. Cheridito, H. Kawaguchi, and M. Maejima. +Fractional Ornstein- +Uhlenbeck processes. Electron. J. Probab., 8(3):1–14, 2003. +[3] Marianne Clausel. Gaussian fields satisfying simultaneous operator scal- +ing relations. In Recent Developments in Fractals and Related Fields, +pages 327–341. Springer, 2010. +[4] Paul Embrechts. Selfsimilar processes. In Princeton Series in Applied +Mathematics. Princeton University Press, 2002. +[5] Marc G Genton, Olivier Perrin, and Murad S Taqqu. Self-similarity and +Lamperti transformation for random fields. Stoch. Models, 23(3):397– +411, 2007. +[6] Y. Hu and D. Nualart. Parameter estimation for fractional Ornstein- +Uhlenbeck processes. Stat. Probab. Lett., 80(11–12):1030–1038, 2010. +[7] T. Kaarakka and P. Salminen. Fractional Ornstein-Uhlenbeck processes. +Commun. Stoch. Anal., 5(1):121–133, 2011. +[8] J.W. Lamperti. Semi-stable processes. Trans. Amer. Math. Soc., 104:62– +78, 1962. +[9] Vitalii Makogin and Yuliya Mishura. Example of a Gaussian self-similar +field with stationary rectangular increments that is not a fractional +Brownian sheet. Stoch. Anal. Appl., 33(3):413–428, 2015. +21 + +[10] Vitalii Makogin and Yuliya Mishura. Gaussian multi-self-similar random +fields with distinct stationary properties of their rectangular increments. +Stoch. Models, 35(4):391–428, 2019. +[11] Gennady Samorodnitsky, Murad S Taqqu, and RW Linde. Stable non- +Gaussian random processes: stochastic models with infinite variance. +Bull. London Math. Soc., 28(134):554–555, 1996. +[12] T. Sottinen and L. Viitasaari. Parameter estimation for the Langevin +equation with stationary-increment Gaussian noise. +Stat. Inference +Stoch. Process., 21(3):569–601, 2018. +[13] Lauri Viitasaari. Representation of stationary and stationary increment +processes via Langevin equation and self-similar processes. Stat. Prob. +Lett., 115:45–53, 2016. +[14] Marko Voutilainen. Modeling and estimation of multivariate discrete +and continuous time stationary processes. Front. Appl. Math. Stat., 6, +2020. +[15] Marko Voutilainen, Lauri Viitasaari, and Pauliina Ilmonen. On model +fitting and estimation of strictly stationary processes. Mod. Stoch.: The- +ory Appl., 4(4):381–406, 2017. +[16] Marko Voutilainen, Lauri Viitasaari, Pauliina Ilmonen, Soledad Torres, +and Ciprian Tudor. Vector-valued generalized Ornstein–Uhlenbeck pro- +cesses: Properties and parameter estimation. Scand. J. Stat., 49(3):992– +1022, 2022. +22 + diff --git a/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/load_file.txt b/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..745b51598af6bac5e7d88f589c5b4d10b5d862dd --- /dev/null +++ b/qNAzT4oBgHgl3EQfq_3c/content/tmp_files/load_file.txt @@ -0,0 +1,948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf,len=947 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='01639v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='PR] 4 Jan 2023 On Lamperti transformation and characterisations of discrete random fields Marko Voutilainen∗, Lauri Viitasaari†, Pauliina Ilmonen‡ January 5, 2023 Abstract In this article we characterise discrete time stationary fields by differ- ence equations involving stationary increment fields and self-similar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This gives connections between stationary fields, stationary increment fields and, through Lamperti transformation, self-similar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Our contribution is a natural generalisation of recently proved results covering the case of stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' AMS 2010 Mathematics Subject Classification: 60G60, 60G10, 60G18 Keywords: random fields, stationary fields, self-similar fields, Lamperti transfor- mation, fractional Ornstein-Uhlenbeck fields 1 Introduction Stationary processes X = (Xt)t∈T have numerous applications in many dif- ferent fields, and they are a topic of active research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Similarly, self-similar processes and stationary increment processes have many applications in var- ious disciplines of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For details on self-similar processes, we refer to the monograph [4] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' All of these three classes are intimately connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Indeed, it was already observed by Lamperti in [8] that there exists a one-to-one correspondence between stationary processes and self-similar processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Later on, this con- nection was used in [13] to obtain relation between stationary processes and ∗Turku School of Economics, Department of Accounting and Finance, FI-20014 Uni- versity of Turku, Finland †Uppsala University, Department of Mathematics, Box 480, 751 06 Uppsala, Sweden ‡Aalto University, Department of Mathematics and Systems Analysis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Box 11100, FI-00076 Aalto, Finland 1 stationary increment processes in continuous time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' T = R, through Langevin equation dXt = −θXtdt + dGt, (1) where θ > 0 is a parameter, X is stationary, and G has stationary increments (along with certain other properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Most notably, this gives rise to the well- known Ornstein-Uhlenbeck process when one plugs in G = W, the Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Connection (1) was later extended for discrete time processes X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' T = Z, in [15], where the authors proved discrete analogue ∆Xt = −θXt−1 + ∆Gt (2) of (1), and studied the estimation of the unknown parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A vector- valued version was later provided in [16] and [14], covering both continuous and discrete time cases together with estimation of the unknown parameter matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Similarly to stationary processes, stationary fields form an important sub- class of random objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In this case, X = (Xt)t∈T with T = RN in contin- uous time or T = ZN in discrete time (naturally, T can be a more general parameter space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Also, self-similarity and stationarity of the increments are wanted features in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' However, while the notion of station- arity is essentially unchanged in the context of random fields, the notion of self-similarity and stationarity of the increments become more complicated when t is multidimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For notion of self-similarity for fields, one typi- cally introduces componentwise self-similarity and considers H-self-similarity with H as an N-dimensional vector of componentwise self-similarity indices, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [11, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In [5] a version of the Lamperti theorem was proved (in continuous time) for fields, providing a connection between stationary fields and self-similar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For other notions of self-similarity for fields, see for example [3] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The notion of stationary increments for fields is even more complicated due to the fact that the definition of increment is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' One ap- proach is to consider rectangular increments, where increments are taken over N-dimensional rectangulars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Gaussian self-similar fields and Gaussian rectangular increment fields were studied, again in continuous time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' in [10] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In this article we extend the Characterisation (2) provided in [15] to discrete time fields, providing a connection between stationary fields, self- similar fields, and stationary increment fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' More precisely, we provide a characterisation of the type (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20) Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 2 Here ˆΘ is a vector of parameters, and ˆX− t is a vector of ”previous value” consisting of previous values in different coordinate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Our notion of increment ∆tG corresponds to the notion of stationary rectangular incre- ments of [10], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As such and exactly as in [13, 15] in the case of processes, we obtain correspondence between stationary fields, stationary rectangular increment fields, and self-similar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The rest of the article is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In Section 2 we introduce and prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We introduce our notation and main definitions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='1, while our main results and their proofs are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='3 we briefly illustrate how our characterisation can be used to construct discrete time fractional Ornstein-Uhlenbeck fields, extending notions of (generalized) Ornstein-Uhlenbeck processes of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We end the paper with conclusions, Section 3, describing future directions, in particular to cover continuous time parameter space and statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 2 Connections between stationary, self-similar, and stationary increment fields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='1 Preliminaries and notations We begin with by introducing some definitions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='1 (Stationarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A random field X = (Xt)t∈ZN is stationary if (Xt+s)t∈ZN law = (Xt)t∈ZN for every s ∈ ZN in the sense of finite dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2 (Self-similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Y = (Yet)t∈ZN = (Yet1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN )t∈ZN be a random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In addition, let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N be a positive multi-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If (Yet+s)t∈ZN law = (e⟨s,Θ⟩Yet)t∈ZN for every s ∈ ZN, where ⟨s, Θ⟩ is the standard inner product of vectors, then Y is a Θ-self-similar random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The exponential terms in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2 are introduced in or- der to take into account the discrete nature of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2 is analogous to the classical definition in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [8] for the definition in the one parameter continuous case and [13] for the definition in the one parameter discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 3 The following definition provides a notion of Lamperti transformation in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='4 (Lamperti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The Lamperti transformation LΘ and its inverse L−1 Θ for discrete random fields are defined by (LΘX)et = e⟨t,Θ⟩Xt, t ∈ ZN, (L−1 Θ Y )t = e−⟨t,Θ⟩Yet, t ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The formulae in Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='4 differ slightly from the standard forms in continuous settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the case of random fields, the definition of component-wise self-similarity and the corresponding Lam- perti transformation together with a one-to-one correspondence between self- similar and stationary fields was presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In comparison, our defini- tions are obtained via change of variables from the standard ones, ensuring that we stay within our discrete parameter set when applying the transforma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='6 (Increments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The square increment ∆tX of a field (Xt)t∈ZN at a point t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN) ∈ ZN is given by ∆tX = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (−1) �N l=1 ilXt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (3) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the particular case N = 2, we have ∆tX = Xt1,t2 + Xt1−1,t2−1 − Xt1−1,t2 − Xt1,t2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='8 (Stationary increment field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A field X = (Xt)t∈ZN has sta- tionary increments if the increment field (∆tX)t∈ZN is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is (∆t+sX)t∈ZN law = (∆tX)t∈ZN for every s ∈ ZN in the sense of finite dimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The authors in [10] introduced a continuous time analogous notion of strictly stationary rectangular increments by assuming stationar- ity of the increments over arbitrary rectangular increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In comparison, in our definition, we consider stationary increments over unit rectangulars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' However, due to the discrete nature of our index space t ∈ ZN, one can show that our definition is equivalent to assuming stationarity over arbitrary (discrete) rectangular increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 4 We also need a notion of previous value that is not so straightforward in a multi-parameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='10 (Previous value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The previous value of the field X = (Xt)t∈ZN at a point t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN) ∈ ZN is given by X− t = Xt − ∆tX = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)̸=0 (−1)1+�N l=1 ilXt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Note that in our definition of the previous value, we take into account the terms in (3) that have a smaller index in at least one of the coordinate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Indeed, in the two-dimensional case, we have ∆tX = Xt1,t2 + Xt1−1,t2−1 − Xt1−1,t2 − Xt1,t2−1 while the previous value is given by X− t = Xt1−1,t2 + Xt1,t2−1 − Xt1−1,t2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='12 (Inner product ⟨ˆΘ, ˆX− t ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N and X = (Xt)t∈ZN be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We define vectors ˆΘ and ˆX− t of length 2N − 1 having elements of the forms (−1)1+�N l=1 ile−⟨i,Θ⟩ and Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN, respectively, where i = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , iN) ∈ {0, 1}N, i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then the inner product of the vectors is ⟨ˆΘ, ˆX− t ⟩ = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N i̸=0 (−1)1+�N l=1 ile−⟨i,Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the two-dimensional case we obtain that, for Θ = (θ1, θ2) ∈ (0, ∞)2 and X = (Xt)t∈Z2, the vectors ˆΘ and ˆX− t are given as ˆΘ = (e−θ1, e−θ2, −e−θ1−θ2) and ˆX− t = \uf8eb \uf8ed Xt1−1,t2 Xt1,t2−1 Xt1−1,t2−1 \uf8f6 \uf8f8 and we have ⟨ˆΘ, ˆX− t ⟩ = e−θ1Xt1−1,t2 + e−θ2Xt1,t2−1 − e−θ1−θ2Xt1−1,t2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (4) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The vectors ˆΘ and ˆX− t are unique up to permutations of their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='15 (Class GΘ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let G = (Gt)t∈ZN be a stationary increment field with Gt = 0 for all t such that �N i=1 ti ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If lim M1→∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' lim MN→∞ t1 � j1=−M1 · · tN � jN=−MN e �N l=1 jlθl∆(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN)G (5) converges in probability defining an almost surely finite random variable for every t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN) ∈ ZN, then G ∈ GΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' It turns out that the order of the limits in (5) is irrelevant, and any permutation of the order leads to the convergence towards the same limiting random variable that turns out to be the stationary field Xt, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Condition Gt = 0 for all t such that �N i=1 ti ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N+ 1} is rather peculiar and it essentially means that G has to vanish along dis- crete points of certain N − 1-dimensional planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' However, this condition is not required a priori for the characterisation, but turns out to hold true and is also required to obtain uniqueness of the representation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Note that if G ∈ L1 (or ∆G ∈ L1), then G ∈ GΘ for every Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Indeed, this can be seen from E| t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθl∆(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN)G| ≤ t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθlE|∆(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN)G| = t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθlE|∆(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',1)G| ≤ t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθl � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N E|G1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',1−iN|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the two-dimensional case, we have G ∈ GΘ for Θ = (θ1, θ2) ∈ (0, ∞)2 provided that G has stationary increments, Gt,−t = Gt,−t−1 = 0 and lim M1→∞ lim M2→∞ t1 � j1=−M1 t2 � j2=−M2 ej1θ1ej2θ2∆(j1,j2)G 6 exists as an almost surely finite random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, G has stationary increments and Gt1,t2 is set to zero on lines t1 = −t2 and t1 = −t2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The existence of such fields follow as a by-product of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='2 AR(1) type characterisation of stationary fields Our main result is the following characterisation that is a natural extension of the one-dimensional case presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A field X = (Xt)t∈ZN is stationary if and only if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (i) lim m→−∞ emθjXt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tj−1,m,tj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN P −→ 0 for every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , N} and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tj−1, tj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (ii) There exists G = (Gt)t∈ZN ∈ GΘ such that Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG for every t ∈ ZN, (6) where ⟨ˆΘ, ˆX− t ⟩ is given by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Moreover, for a given Θ, the stationary increment field G ∈ GΘ in (6) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As a direct corollary we obtain the following version in a two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, θ2) ∈ (0, ∞)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A field X = (Xt)t∈Z2 is sta- tionary if and only if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (i) lim m→−∞ emθ2Xt1,m P −→ 0 and lim m→−∞ emθ1Xm,t2 P −→ 0 for every t1 and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (ii) There exists G = (Gt)t∈Z2 ∈ GΘ such that Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG for every t ∈ Z2, (7) where ⟨ˆΘ, ˆX− t ⟩ is given by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The stationary increment field G ∈ GΘ in (7) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 7 The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20 is split into a series of lemmas and auxiliary theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We begin with the following result that is a version of Lamperti theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If X = (Xt)t∈ZN is stationary, then (LΘX)et is Θ-self- similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Conversely, if Y = (Yet)t∈ZN is Θ-self-similar, then (L−1 Θ Y )t is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' First, assume that X is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Set Yet = (LΘX)et and let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now (Yet1+s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , Yetn+s) = (e⟨t1+s,Θ⟩Xt1+s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' e⟨tn+s,Θ⟩Xtn+s) law = (e⟨s,Θ⟩e⟨t1,Θ⟩Xt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , e⟨s,Θ⟩e⟨tn,Θ⟩Xtn) = (e⟨s,Θ⟩Yet1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , e⟨s,Θ⟩Yetn), proving the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Next, assume that Y is Θ-self-similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Set Xt = (L−1 Θ Y )t and let n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now (Xt1+s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , Xtn+s) = (e−⟨t1+s,Θ⟩Yet1+s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , e−⟨tn+s,Θ⟩Yetn+s) law = (e−⟨t1,Θ⟩Yet1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , e−⟨tn,Θ⟩Yetn) = (Xt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , Xtn), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The following lemma provides one of our key observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let (Yet)t∈ZN be Θ-self-similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Set ∆tY = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (−1) �N l=1 ilYet1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN −iN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For �N l=1 tl ≥ 1, we set Gt = t1 � k1=1−t2−···−tN t2 � k2=1−k1−t3−···−tN · · tN � kN=1−k1−···−kN−1 e−⟨k,Θ⟩∆kY, and, for �N l=1 tl ≤ 0, we set Gt = (−1)N −t2−···−tN−N+1 � k1=t1+1 −k1−t3−···−tN−N+2 � k2=t2+1 · · −k1−···−kN−1 � kN=tN+1 e−⟨k,Θ⟩∆kY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Here ⟨k, Θ⟩ is the standard inner product and sums of the type �s1 s2 with s1 < s2 are interpreted as empty sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now 8 (i) Gt = 0 for all t such that �N l=1 tl ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N + 1}, (ii) ∆tG = e−⟨t,Θ⟩∆tY for every t ∈ ZN, (iii) G = (Gt)t∈ZN is a stationary increment field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' It turns out that G defined as above satisfies G ∈ GΘ, see also Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='31 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the two-dimensional case, for Θ-self-similar (Yet)t∈Z2, we denote ∆tY = Yet1,et2 − Yet1−1,et2 − Yet1,et2−1 + Yet1−1,et2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The field G = (Gt)t∈Z2 defined as Gt1,t2 = ��t1 k1=1−t2 �t2 k2=1−k1 e−⟨(k1,k2),Θ⟩∆kY, t1 + t2 ≥ 1 �−t2−1 k1=t1+1 �−k1 k2=t2+1 e−⟨(k1,k2),Θ⟩∆kY, t1 + t2 ≤ 0, belongs to the class GΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Here sums of the type �s s+1 are interpreted as empty sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23 is based on the following additional lemmas that we prove first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The first one provides an auxiliary result on sums of binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Although the result is quite elementary, we provide a proof for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We have the following identities: M−1 2 � m=0 �M 2m � = 2M−1, M−1 2 � m=0 � M 2m + 1 � = 2M−1, when M ≥ 1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' M 2 � m=0 �M 2m � = 2M−1, M 2 −1 � m=0 � M 2m + 1 � = 2M−1, when M ≥ 2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the odd case M−1 2 � m=0 �M 2m � = �M 0 � +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='+ � M M − 1 � and M−1 2 � m=0 � M 2m + 1 � = �M 1 � +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='+ �M M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' These sums are equal since �M k � = � M M−k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In addition, M−1 2 � m=0 �M 2m � + M−1 2 � m=0 � M 2m + 1 � = 2M 9 completing the proof of the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For the even case, we obtain M 2 −1 � m=0 � M 2m + 1 � = �M 1 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' + � M M − 1 � = �M − 1 0 � + �M − 1 1 � + · · · + �M − 1 M − 2 � + �M − 1 M − 1 � = 2M−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Observing that M 2 � m=0 �M 2m � + M 2 −1 � m=0 � M 2m + 1 � = 2M completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The following lemma sheds light on how the field G of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23 is constructed from a self-similar field Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (i) Let �N l=1 tl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then a term e−⟨j,Θ⟩∆jY belongs to the sum defining Gt in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23 if and only if jl ≤ tl for every l and N � l=1 jl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (ii) Let �N l=1 tl ≤ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then a term e−⟨j,Θ⟩∆jY belongs to the sum defining Gt in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23 if and only if jl ≥ tl + 1 for every l and N � l=1 jl ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Item (i): By the upper bounds in the sum defining Gt, it is clear that we have jl ≤ tl for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' By the lower bound of the inmost summation, we obtain jN ≥ 1 − j1 − · · · − jN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, �N l=1 jl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We also observe that the other lower bounds of the sum defining Gt yield conditions jN−h ≥ 1 − N−h−1 � l=1 jl − N � l=N−h+1 tl for every h ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' These conditions are satisfied since N−h � l=1 jl + N � N−h+1 tl ≥ N � l=1 jl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 10 This completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Item (ii): By the lower bounds in the sum defining Gt, it is clear that we have jl ≥ tl + 1 for all l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' By the upper bound of the inmost summation, we obtain also that jN ≤ −j1 −· · ·−jN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, �N l=1 jl ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We also observe that the other upper bounds of the sum defining Gt yield conditions jN−h ≤ − N−h−1 � l=1 jl − N � l=N−h+1 tl − N + (N − h) for every h ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , N − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' These conditions are satisfied since N−h � l=1 jl + N � N−h+1 tl ≤ N−h � l=1 jl + N � N−h+1 (jl − 1) = N � l=1 jl − h ≤ −h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This completes the proof of the second item, and thus the whole proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Item (i): Let �N l=1 tl ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then N � l=2 tl ∈ {−t1, −1 − t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N + 1 − t1} and for the upper bound of the first summation in the definition of Gt it holds that − N � l=2 tl − N + 1 ∈ {t1 − N + 1, t1 − N + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , t1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence, Gt is given by an empty sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Item (ii): Recall that Gt = 0 for all t such that �N i=1 ti ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We treat the case �N l=1 tl ≥ 1 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let M be such that �N l=1 tl −M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then ∆tG = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N �N l=1 il≤M (−1) �N l=1 ilGt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (8) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='27, ∆tG consists of terms e−⟨j,Θ⟩∆jY with jl ≤ tl for every l and �N l=1 jl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let m be the number of indices l for which jl = tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Assume that m < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='27, e−⟨j,Θ⟩∆jY belongs to summands of 11 (8) that satisfy jl ≤ tl − il for every l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, m of the indices il are zero while the remaining N − m indices may be zeros or ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In addition, 1 ≤ N � l=1 jl ≤ N � l=1 tl − (N − m) giving N − m ≤ N � l=1 tl − 1 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (9) Now if N − m is odd, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='26 and (9), the number of terms e−⟨j,Θ⟩∆jY in (8) with a positive sign is �N − m 0 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' + � N − m min{N − m, M} − 1 � = �N − m 0 � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' + � N − m N − m − 1 � = 2N−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Thus, terms e−⟨j,Θ⟩∆jY cancel out in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Similarly, if N−m is even, then the number of terms e−jΘ∆jY with a positive sign is �N − m 0 � + · · · + � N − m min{N − m, M} � = �N − m 0 � + · · · + �N − m N − m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Again, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='26, terms e−⟨j,Θ⟩∆jY cancel out in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If m = N, we have that e−⟨j,Θ⟩∆jY = e−⟨t,Θ⟩∆tY belongs only to the sum- mand of (8) with i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence we have shown that ∆tG = e−⟨t,Θ⟩∆tY for every t such that N � l=1 tl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This proves the claim for the case �N l=1 tl ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Assume next that �N l=1 tl ≤ 0 and let M be such that �N l=1 tl−M = −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then ∆tG = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N �N l=1 il≥M (−1) �N l=1 ilGt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (10) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='27, ∆tG consists of terms e−⟨j,Θ⟩∆jY with jl ≥ tl for every l and �N l=1 jl ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As before, let m be the number of indices l for which jl = tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If m < N, then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='27, e−⟨j,Θ⟩∆jY belongs to summands of (10) that satisfy jl ≥ tl − il + 1 for every l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, m of the indices il are equal to one while the remaining N − m indices may be zeros or ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In addition, 0 ≥ N � l=1 jl ≥ N � l=1 tl + (N − m) = M − m 12 giving m ≥ M and N − m ≤ N − M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If N − m is odd, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='26, the number of terms e−⟨j,Θ⟩∆jY in (10) with the sign (−1)N+m is equal to �N − m 0 � + · · · + � N − m N − m − 1 � = 2N−m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Thus, terms cancel out in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Similarly, terms cancel out when N − m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Finally, for the case m = N we observe that e−⟨j,Θ⟩∆jY = e−⟨t,Θ⟩∆tY belongs only to the summand of (10) with i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The corresponding sign is (−1)N(−1)N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' To conclude, we have shown that ∆tG = e−⟨t,Θ⟩∆tY for every t such that N � l=1 tl ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This completes the proof of item (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Item (iii): For s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , sN) ∈ ZN, ∆t+sG = e−⟨t+s,Θ⟩∆t+sY = e−⟨t,Θ⟩e−⟨s,Θ⟩ � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (−1) �N l=1 ilYet1+s1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN +sN −iN law = e−⟨t,Θ⟩ � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (−1) �N l=1 ilYet1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN −iN = e−⟨t,Θ⟩∆tY = ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Treating multidimensional distribution similarly completes the proof of item (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We are now ready to prove three results, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='32, that give us the main result of this article, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N and let X = (Xt)t∈ZN be a random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' If for some G = (Gt)t∈ZN ∈ GΘ it holds that Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG for every t ∈ ZN, (11) and lim m→−∞ emθjXt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tj−1,m,tj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN P −→ 0 for every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , N} and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tj−1, tj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN ∈ Z, then X = (Xt)t∈ZN is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Denote Zt = ∆tG and i = (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , iN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' From (11) and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='12, we get Xt = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N i̸=0 (−1)1+�N l=1 ile−⟨i,Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN + Zt, which gives Xt − � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 i̸=0 (−1)1+�N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,tN = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 (−1) �N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,tN = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 (−1) �N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,1),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,tN−1 + Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (12) Set Yt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN) = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 (−1) �N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then, by iterating the recursive Equation (12), we get Yt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN) =e−θNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN − 1) + Zt =e−(n+1)θNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN − n − 1) + n � jN=0 e−jNθNZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1,tN−jN =e−(n+1)θNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN − n − 1) + e−tNθN tN � jN=tN−n ejNθNZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1,jN, for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Above e−(n+1)θNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN − n − 1) = e−tN θNe(tN −n−1)θNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN − n − 1) = e−tN θNemθNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(m) by the change of variable m = tN − n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Furthermore emθNYt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(m) =emθN � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 (−1) �N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,m, 14 which, by the assumptions, converges to zero in probability as m → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence Yt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1(tN) = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1)∈{0,1}N−1 (−1) �N−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−1,0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1−iN−1,tN = e−tNθN tN � jN=−∞ ejNθNZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−1,jN =: Q(N) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In the following summations, let (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , iN−k−1) ∈ {0, 1}N−k−1 and (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , iN−k−2) ∈ {0, 1}N−k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We proceed by induction and assume that for some k ∈ N∪{0} it holds that � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−1) (−1) �N−k−1 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−1−iN−k−1,tN−k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN =e− �k l=0 tN−lθN−l tN−k � jN−k=−∞ · · tN � jN=−∞ e �k l=0 jN−lθN−lZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−1,jN−k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN =: Q(N−k) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (13) Now � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2) � (−1) �N−k−2 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩ Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2−iN−k−2,tN−k−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN � = − � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2) � (−1)1+�N−k−2 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩ Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2−iN−k−2,tN−k−1−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN � + Q(N−k) t = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2) � (−1) �N−k−2 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩ Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2−iN−k−2,tN−k−1−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN � + Q(N−k) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (14) Let t∗ = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN−k−2, tN−k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN) and define Yt∗(tN−k−1) = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2) (−1) �N−k−2 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2−iN−k−2,tN−k−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 15 Equation (14) gives Yt∗(tN−k−1) =e−θN−k−1Yt∗(tN−k−1 − 1) + Q(N−k) t =e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) + n � jN−k−1=0 e−jN−k−1θN−k−1Q(N−k) t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2,tN−k−1−jN−k−1,tN−k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN =e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) + e−tN−k−1θN−k−1 tN−k−1 � jN−k−1=tN−k−1−n ejN−k−1θN−k−1Q(N−k) t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2,jN−k−1,tN−k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Above e−(n+1)θN−k−1Yt∗(tN−k−1 − n − 1) =e−tN−k−1θN−k−1e(tN−k−1−n−1)θN−k−1Yt∗(tN−k−1 − n − 1) =e−tN−k−1θN−k−1emθN−k−1Yt∗(m) by the change of variable m = tN−k−1 − n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As before, the expression converges to zero in probability as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence, we obtain that Yt∗(tN−k−1) = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2) � (−1) �N−k−2 l=1 ile−⟨(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN−k−2,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩ Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2−iN−k−2,tN−k−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN � =e−tN−k−1θN−k−1 tN−k−1 � jN−k−1=−∞ ejN−k−1θN−k−1Q(N−k) t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2,jN−k−1,tN−k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN =e−tN−k−1θN−k−1 � tN−k−1 � jN−k−1=−∞ ejN−k−1θN−k−1e− �k l=0 tN−lθN−l tN−k � jN−k=−∞ · · tN � jN=−∞ e �k l=0 jN−lθN−lZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2,jN−k−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN � =e− �k+1 l=0 tN−lθN−l tN−k−1 � jN−k−1=−∞ · · tN � jN=−∞ e �k+1 l=0 jN−lθN−lZt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−k−2,jN−k−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN, which proves the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Therefore choosing k = N −2 in (13) yields � i1∈{0,1} (−1)i1e−⟨(i1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',0),Θ⟩Xt1−i1,t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN = Xt − e−θ1Xt1−1,t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN = e− �N−2 l=0 tN−lθN−l t2 � j2=−∞ · · tN � jN=−∞ e �N−2 l=0 jN−lθN−lZt1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN =: Q(2) t 16 and we obtain a recursive equation Xt = e−θ1Xt1−1,t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN + Q(2) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' By repeating the earlier procedure once more, we obtain that Xt =e−t1θ1 t1 � j1=−∞ ej1θ1Q(2) j1,t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN =e−t1θ1 t1 � j1=−∞ ej1θ1e− �N−2 l=0 tN−lθN−l t2 � j2=−∞ · · tN � jN=−∞ e �N−2 l=0 jN−lθN−lZj1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN =e− �N l=1 tlθl t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθlZj1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN =e− �N l=1 tlθl t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθl∆j1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jNG, which, since G ∈ GΘ, defines an almost surely finite random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence it remains to prove that X is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' To this end, we show that the one dimensional distributions of X are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The proof extends straightfor- wardly to multidimensional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , sN) ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Note that Xt = ∞ � j1=0 · · ∞ � jN=0 e− �N l=1 jlθl∆t1−j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−jNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Since G is a stationary increment field, we have that M1 � j1=0 · · MN � jN=0 e− �N l=1 jlθl∆t1+s1−j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN+sN−jNG law = M1 � j1=0 · · MN � jN=0 e− �N l=1 jlθl∆t1−j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−jNG for every M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , MN ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Moreover, since G ∈ GΘ, the iterated limits of both sides converge and hence, the limits are equal in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This gives Xt+s = ∞ � j1=0 · · ∞ � jN=0 e− �N l=1 jlθl∆t1+s1−j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN+sN−jNG law = ∞ � j1=0 · · ∞ � jN=0 e− �N l=1 jlθl∆t1−j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−jNG = Xt and thus the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 17 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The property Gt = 0 for all t such that �N i=1 ti ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N+ 1} of the class GΘ is not utilized in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' However, in order to obtain uniqueness in the characterising Equation (11), we need to pose this additional assumption, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Assume that the field X = (Xt)t∈ZN is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then there exists G = (Gt)t∈ZN ∈ GΘ such that Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG for every t ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Applying Lamperti transformation gives ∆tX = Xt − X− t = e−⟨t,Θ⟩Yet − X− t = e−⟨t,Θ⟩(Yet − ∆tY ) − X− t + e−⟨t,Θ⟩∆tY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence Xt = e−⟨t,Θ⟩(Yet − ∆tY ) + e−⟨t,Θ⟩∆tY, where Yet − ∆tY = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N ,i̸=0 (−1)1+�N l=1 ilYet1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN −iN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Furthermore e−⟨t,Θ⟩(Yet − ∆tY ) = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N ,i̸=0 (−1)1+�N l=1 ile−⟨t−i,Θ⟩e−⟨i,Θ⟩Yet1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',etN −iN = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N ,i̸=0 (−1)1+�N l=1 ile−⟨i,Θ⟩Xt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN = ⟨ˆΘ, ˆX− t ⟩, where the last equality follows by Lamperti transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let G be the field defined in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then Xt = ⟨ˆΘ, ˆX− t ⟩ + ∆tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Moreover, the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28 and the property lim m→−∞ emθiXt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',ti−1,m,ti+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN P −→ 0, for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , N} and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , ti−1, ti+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , tN ∈ Z, give us the repre- sentation Xt = e− �N l=1 tlθl t1 � j1=−∞ · · tN � jN=−∞ e �N l=1 jlθl∆(j1,j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',jN)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' (15) Hence, G ∈ GΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As a by-product, we observe the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 18 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let G be defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then G ∈ GΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23, it suffices to show that the limit (5) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now since Y is Θ-self-similar, there exists a stationary X such that Y = LΘX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Following the lines of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30, we obtain Representation (15), where G is defined as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence G ∈ GΘ by the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let Θ = (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , θN) ∈ (0, ∞)N be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then the stationary increment field G ∈ GΘ in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Assume that G, G′ ∈ GΘ satisfy the recursive Equation (11) of Theo- rems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Then, ∆tG = ∆tG′ for all t ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' It remains to show that this implies Gt = G′ t for all t, which we do by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Since Gt = G′ t = 0 for all t such that �N l=1 tl ∈ {0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , −N + 1}, we may set an induction assumption that there exists s ∈ Z such that Gt = G′ t for all t such that �N l=1 tl ∈ {s, s − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , s − N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In order to complete the proof, it remains to show that Gt = G′ t under conditions �N l=1 tl = s + 1 and �N l=1 tl = s − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let �N l=1 tl = s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We now have ∆tG = � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N (−1) �N l=1 ilGt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN = Gt + � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N i̸=0 (−1) �N l=1 ilGt1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN−iN, where for the indices of G it holds that N � l=1 (tl − il) ∈ {s, s − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , s − N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Therefore, by the induction assumption, Gt = G′ t for all t such that �N l=1 tl = s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We proceed proving that Gt = G′ t for all t such that �N l=1 tl = s − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Let �N l=1 tl = s − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now (−1)NGt1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN + � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N i̸=1 (−1) �N l=1 ilGt1+1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN+1−iN = (−1)NG′ t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN + � (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',iN)∈{0,1}N i̸=1 (−1) �N l=1 ilG′ t1+1−i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=',tN+1−iN, 19 where N � l=1 (tl + 1 − il) ∈ {s, s − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' , s − N + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hence, by the induction assumption, Gt = G′ t for all t such that �N l=1 tl = s − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' The proof of our main theorem, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='20, now follows by combining Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='28 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='30 together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='3 Fractional Ornstein-Uhlenbeck fields on ZN We illustrate our approach by constructing a stationary fractional Ornstein- Uhlenbeck field on Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For this purpose, let Gt1,t2 be a discrete fractional Gaussian field on (t1, t2) ∈ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' A discrete fractional Gaussian field can be constructed from its continuous time analogue by embedding to the space N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' That is, let ˜X be a self-similar Gaussian field having stationary rectangular increments such that E ˜X2(1, 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Existence of such fields is studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We can now embed ˜X into N2 by considering values of ˜X at points (t1, t2) ∈ N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Now Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='15 of GΘ allows us to extend G into Z2 by setting Gt,−t = Gt,−t−1 = 0 and requiring that G has stationary increments in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As G is Gaussian, in view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='18, it follows that our field G = (Gt)t∈Z2 belongs to GΘ for any Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This allows us to define a discrete time stationary fractional Ornstein-Uhlenbeck field of the first kind by Representation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' On the other hand, using self-similarity of ˜X, we can define a stationary field through Lamperti transformation and obtain, in view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='22, stationary fractional Ornstein-Uhlenbeck field of the second kind corresponding to a different noise G ∈ GΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Obviously, this approach can be used to define fractional Ornstein-Uhlenbeck fields in arbitrary dimensions N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 3 Conclusions In this article we have provided a characterisation of discrete stationary fields in terms of fields having stationary rectangular increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Our characterisa- tion is analogous to the one-parameter case provided in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' As an example, we have extended the notion of fractional Ornstein-Uhlenbeck processes, in- troduced in [2] and [7], to the multi-parameter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Natural further prospects of our study are two folded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Firstly, it would be interesting to generalise the characterisation into the continuous parameter space t ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In this case however, the increments ∆tG are replaced with 20 differentials, and the notion of differential dtG is more subtle when t ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This is a topic of a forthcoming article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Secondly, parameter estimation is of paramount importance in statistics, which in our setting would correspond to the estimation of the parameter Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' This has been a topic of active research in the context of fractional Ornstein-Uhlenbeck processes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [6, 12] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' For a more general setting in the context of multivariate time series, parameter estimation is studied in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' We believe that ideas and methods presented in [16] could be useful in our context as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' References [1] Hermine Bierm´e, Mark M Meerschaert, and Hans-Peter Scheffler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Oper- ator scaling stable random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' their Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 117(3):312– 332, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Cheridito, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Kawaguchi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Maejima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Fractional Ornstein- Uhlenbeck processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 8(3):1–14, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [3] Marianne Clausel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Gaussian fields satisfying simultaneous operator scal- ing relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In Recent Developments in Fractals and Related Fields, pages 327–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [4] Paul Embrechts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Selfsimilar processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' In Princeton Series in Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Princeton University Press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [5] Marc G Genton, Olivier Perrin, and Murad S Taqqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Self-similarity and Lamperti transformation for random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Models, 23(3):397– 411, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Hu and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Nualart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Parameter estimation for fractional Ornstein- Uhlenbeck processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 80(11–12):1030–1038, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Kaarakka and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Salminen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Fractional Ornstein-Uhlenbeck processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 5(1):121–133, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lamperti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Semi-stable processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 104:62– 78, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [9] Vitalii Makogin and Yuliya Mishura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 33(3):413–428, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 21 [10] Vitalii Makogin and Yuliya Mishura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Models, 35(4):391–428, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [11] Gennady Samorodnitsky, Murad S Taqqu, and RW Linde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stable non- Gaussian random processes: stochastic models with infinite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 28(134):554–555, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Sottinen and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Viitasaari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Parameter estimation for the Langevin equation with stationary-increment Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Inference Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 21(3):569–601, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [13] Lauri Viitasaari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Representation of stationary and stationary increment processes via Langevin equation and self-similar processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 115:45–53, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [14] Marko Voutilainen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Modeling and estimation of multivariate discrete and continuous time stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [15] Marko Voutilainen, Lauri Viitasaari, and Pauliina Ilmonen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' On model fitting and estimation of strictly stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' : The- ory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 4(4):381–406, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' [16] Marko Voutilainen, Lauri Viitasaari, Pauliina Ilmonen, Soledad Torres, and Ciprian Tudor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Vector-valued generalized Ornstein–Uhlenbeck pro- cesses: Properties and parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=', 49(3):992– 1022, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNAzT4oBgHgl3EQfq_3c/content/2301.01639v1.pdf'} diff --git a/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/2301.11731v1.pdf.txt b/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/2301.11731v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3edda983cc7e914f7f9f242b9242e12fc533d5b0 --- /dev/null +++ b/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/2301.11731v1.pdf.txt @@ -0,0 +1,4761 @@ +Excited states, symmetry breaking, and unphysical solutions in state-specific +CASSCF theory +Antoine Marie1, 2 and Hugh G. A. Burton1, a) +1)Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.K. +2)Current address: Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, Toulouse, +France +(Dated: 30 January 2023) +State-specific electronic structure theory provides a route towards +balanced excited-state wave functions by exploiting higher-energy +stationary points of the electronic energy. Multiconfigurational +wave function approximations can describe both closed- and open- +shell excited states and avoid the issues associated with state- +averaged approaches. +We investigate the existence of higher- +energy solutions in complete active space self-consistent field +(CASSCF) theory and characterise their topological properties. +We demonstrate that state-specific approximations can provide +accurate higher-energy excited states in H2 (6-31G) with more +compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary +points, demonstrating that they arise from redundant orbitals when the active space is too large, or symmetry breaking +when the active space is too small. Furthermore, we investigate the conical intersection in CH2 (6-31G) and the avoided +crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave +quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting +the advantages and challenges of practical state-specific calculations. +I. +Introduction +Electronic excited states are fundamentally higher-energy so- +lutions to the time-independent Schrödinger equation. “State- +specific” representations can be identified using higher-energy +stationary points of the electronic energy landscape.1 The ex- +act excited states in full configuration interaction (FCI) corre- +spond to energy saddle points and the number of downhill Hes- +sian eigenvalues increases with each energy level.1–7 Higher- +energy stationary points also exist in non-linear wave function +approximations, but the development of practical state-specific +methods has been hindered by the challenges of non-ground- +state optimisation, the non-linearity of the electronic energy +landscape, and the presence of unphysical solutions. +Instead, the workhorse of modern excited-state electronic +struture theory is linear-response time-dependent density func- +tional theory (LR-TDDFT), which predicts excitation energies +from the response of the ground-state electron density to a weak +external perturbation.8–10 Despite its computational efficiency, +LR-TDDFT inherits the failures of approximate Kohn-Sham +(KS) exchange-correlation functionals, creating large errors +for bond dissociation or open-shell electronic states.11 Further- +more, the ubiquitous adiabatic approximation excludes double +excitations and their associated avoided crossings.10,12 Alter- +native single-reference methods, such as algebraic diagram- +matic construction13,14 (ADC) and equation-of-motion coupled +cluster15,16 (EOM-CC) can provide more accurate excitation +energies at a greater computational cost, but depend strongly +a)Electronic mail: hgaburton@gmail.com +on the quality of the reference determinant. The strong influ- +ence of the ground-state orbitals can also create an unbalanced +description of charge transfer and Rydberg excitations,17,18 +where significant electronic relaxation can occur.8–10,19–21 +These challenges have encouraged researchers to revisit ex- +cited state-specific approximations. For higher-energy SCF +calculations (∆SCF), this progress has been catalysed by +the development of new optimisation algorithms that avoid +variational collapse to the ground state, including the maxi- +mum overlap method,22–24 square-gradient optimisation,25,26 +state-targeted energy projection,27 quasi-Newton direct or- +bital optimisation,28–30 and generalised variational principles.31 +Recent calculations have shown that higher-energy Hartree– +Fock (HF) and KS-DFT solutions can accurately describe +charge transfer and double excitations at a low computa- +tional cost.22,26 Beyond SCF approximations, higher-energy +variational or projective coupled-cluster (∆CC) solutions +can provide more accurate double and double-core exci- +tations by incorporating dynamic electron correlation.32–38 +While ∆SCF and ∆CC are successful for double and charge +transfer excitations, these single-reference methods can- +not describe open-shell excited states and statically corre- +lated ground states. +The onset of this failure usually be- +comes apparent through spin contamination,39,40 spontaneous +symmetry breaking,11,23,39,41–45 and additional unphysical +solutions.32–35,37–39 Furthermore, the solutions of interest can +disappear as the molecular structure changes, creating discon- +tinuous excited-state energy surfaces or gradients.38,39,46–50 +Multiconfigurational SCF (MCSCF) methods,51 particu- +larly the complete-active space self-consistent field (CASSCF) +formulation,52–54 are the state-of-the-art for describing stati- +arXiv:2301.11731v1 [physics.chem-ph] 27 Jan 2023 + +State-Specific +CASSCF2 +cally correlated electronic systems.55 The CASSCF wave func- +tion is a linear expansion of all the configurations that can be +constructed from a set of partially occupied “active orbitals”, +and the energy is optimised with respect to the configuration +interaction (CI) and orbital coefficients simultaneously.53 It +has long been known that higher-energy MCSCF solutions +can represent electronic excited states,56–61 and that multiple +symmetry-broken CASSCF solutions can occur for an inad- +equate active space.62,63 More recently, MCSCF expansions +truncated to single excitations have shown promise for singly +excited charge transfer states,64–68 while state-specific config- +uration interaction with higher degrees of truncation can han- +dle challenging multireference problems, singly-, and doubly- +excited states.69 However, the strong coupling between the or- +bital and CI degrees of freedom makes the optimisation chal- +lenging, and second-order optimisation algorithms are gener- +ally required to reach convergence in practice.70–83 +Extensive research in the 1980s focused on characterising +higher-energy MCSCF solutions. It was originally suggested +that an nth excited state approximation should be the nth state +in the configuration expansion.73 However, this requirement is +often not achieved, resulting in “root flipping”.2,56,75 Further- +more, several stationary points satisfying this condition can +often be identified.3,5,84 The enormous complexity of the mul- +ticonfigurational solution space led Golab et al. to conclude +that “selecting an MCSCF stationary point is a very severe +problem.”3 Instead, the state-averaged (SA) approach is gener- +ally used, where a weighted average energy of the n lowest CI +states constructed from one set of orbitals is optimised.75 While +this approach has become the method of choice for excited- +state CASSCF, it has several disadvantages: discontinuities +can occur on the SA-CASSCF potential energy surface if two +states require orbitals with significantly different character;85 +the number of states is limited by the size of the active space; +large active spaces are required to target high-lying states; and +the Helmann-Feynamn theorem cannot be applied to compute +nuclear gradients because individual SA-CASSCF solutions +are not stationary points of the energy. +Recently, the limitations of SA-CASSCF and the develop- +ment of non-ground-state SCF optimisation algorithms has in- +spired several new investigations into state-specific CASSCF +excited states. In particular, Neuscamman and co-workers have +developed generalized variational principles86,87 and the WΓ +approach inspired by MOM-SCF,88 demonstrating that the is- +sues of root flipping and variational collapse to the ground state +can be successfully avoided. Despite these advances, we still +do not have a complete understanding of the multiple station- +ary points on the SS-CASSCF energy landscape and several +practical questions remain. For example, how many stationary +points are there and how does this change with the active space +or basis set size? Where do unphysical solutions arise, what +are their characteristics, and when does symmetry breaking oc- +cur? And finally, do state-specific excitations behave diabati- +cally or adiabatically as the molecular structure evolves? +Our aim in this work is to answer these questions and estab- +lish a theoretical foundation for practical excited state-specific +calculations. Using second-order optimisation techniques, we +investigate the existence and properties of multiple CASSCF +solutions in typical molecular systems. Our numerical optimi- +sation exploits analytic gradients and second derivatives of the +CASSCF energy, and the relevant differential geometry is sum- +marised below. Using these techniques, we comprehensively +enumerate the multiple CASSCF solutions in H2 (6-31G) and +characterise the resulting unphysical solutions. We find that +state-specific calculations can accurately describe high-lying +excitations with fewer active orbitals than state-averaged for- +malisms, and reveal that multiple solutions can arise from ac- +tive spaces that are too large or too small. We then investigate +the conical intersection in CH2 (6-31G) and the avoided cross- +ing of LiF (6-31G), demonstrating the importance and diffi- +culty of selecting the correct physical solution. +II. +Exploring the multiconfigurational energy landscape +A. +Defining the CASSCF wave function +A multiconfigurational wave function is defined as the linear +combination of M Slater determinants +|Ψk⟩ = +M +� +I=1 +CIk |ΦI⟩ , +(1) +where |ΦI⟩ represents different configurations built from a com- +mon set of molecular orbitals (MO) φp(x) and the CIk are the +variable CI coefficients for state k.89 Here, x = (r, σ) is the +combined spatial and spin electronic coordinate. The MOs +are constructed as linear combinations of n (nonorthogonal) +atomic orbitals (AO) χµ(x) as +φp(x) = +n +� +µ +χµ(x) cµ· +·p, +(2) +where we use the nonorthogonal tensor notation of Ref. 90 +and the cµ· +·p denote the variable MO coefficients. Normalisation +of the wave function, and orthogonalisation of the MOs, is +guaranteed by the constraints +M +� +I=1 +|CI|2 = 1 +and +n +� +µ=1 +(c∗)·µ +p· ⟨χµ|χν⟩ cν· +·q = δpq, +(3) +where ⟨χµ|χν⟩ denotes the AO overlap matrix elements. We +will only consider wave functions where CIk and cµ· +·p are real. +When every electronic configuration for a finite basis set +is included in an FCI expansion, the global minimum on the +parametrised electronic energy landscape corresponds to the +exact ground state.1 Excited states form saddle points of the +energy and the number of downhill directions increases with +each excitation.1,3,6,7 The FCI wave function is invariant to uni- +tary transformations of the MOs, but the number of configura- +tions scales exponentially with the system size. +The complete active space (CAS) framework builds a trun- +cated expansion using every configuration within a set of +“active orbitals” that describe the dominant static electron +correlation.53 The orbitals are partitioned into inactive and vir- +tual orbitals that are doubly occupied or empty in every config- +uration, respectively, and active orbitals with varying occupa- +tions. Simultaneously optimising the energy with respect to the + +3 +orbital and CI coefficients leads to the state-specific CASSCF +approach and gives true stationary points of the electronic +energy.53,54,74 If the CASSCF wave function targeting the kth +excited state is represented by the kth eigenstate of the cor- +responding CAS-CI expansion, then the Hylleraas-Undheim- +MacDonald theorem91,92 also provides a upper bound to the +excited-state energy.3 +B. +Differential geometry of the CASSCF energy +We exploit an exponential form of the CASSCF wave func- +tion that conserves the orthogonality constraints [Eq. (3)].70,72 +Starting from an initial CASSCF wave function |Ψ0⟩, an arbi- +trary step can be defined using unitary transformations as +|Ψ⟩ = e ˆRe ˆS |Ψ0⟩ , +(4) +where e ˆR and e ˆS account for orbital relaxation and transforma- +tions of the CI component, respectively. The ˆR operator is anti- +Hermitian and is defined using the second-quantised creation +and annihilation operators for the current MOs as70,93 +ˆR = +� +p>q +Rpq ˆE− +pq, +(5) +where the spin-adapted one-body anti-Hermitian replacement +operators are89 +ˆE− +pq = +� +σ∈{↑,↓} +ˆa† +qσˆapσ − ˆa† +pσˆaqσ. +(6) +The invariance of the energy with respect to inactive-inactive, +active-active, and virtual-virtual orbital transformations means +that Rpq can be further restricted to only excitations between +different sub-blocks. Similarly, e ˆS performs a unitary transfor- +mation between the CI component of |Ψ0⟩ and the remaining +orthogonal states |ΨK⟩ in the current CASCI space, with ˆS de- +fined as72 +ˆS = +� +K�0 +S K +� +|ΨK⟩ ⟨Ψ0| − |Ψ0⟩ ⟨ΨK| +� +. +(7) +Using the exponential parametrisation, the CASSCF energy +can be expressed as +E(R, S) = ⟨Ψ0|e− ˆS e− ˆR ˆHe ˆRe ˆS |Ψ0⟩ , +(8) +where R and S are vectors that gather the Rpq and S K coeffi- +cients in the orbital and CI transformations, respectively, and +ˆH is the electronic Hamiltonian. Stationary points of E, corre- +sponding to optimal CASSCF solutions, then occur when the +gradients with respect to orbital and CI transformations are si- +multaneously zero. Performing a Baker–Campbell–Hausdorff +expansion of the energy to second order gives72 +E ≈ ⟨Ψ0| ˆH|Ψ0⟩ + ⟨Ψ0|[ ˆH, ( ˆR + ˆS )]|Ψ0⟩ ++ 1 +2 ⟨Ψ|[[ ˆH, ( ˆR + ˆS )], ( ˆR + ˆS )]|Ψ0⟩ + . . . +(9) +Expressions for the first- and second-derivates of the energy +can then be identified as +∂E +∂Rpq +������R,S=0 += ⟨Ψ0|[ ˆH, ˆE− +pq]|Ψ0⟩ , +(10a) +∂E +∂S K +�����R,S=0 += 2 ⟨Ψ0| ˆH|ΨK⟩ , +(10b) +and +∂2E +∂Rpq∂Rrs +������R,S=0 += 1 +2(1 + Ppq,rs) ⟨Ψ0|[[ ˆH, ˆE− +pq], ˆE− +rs]|Ψ0⟩ , +(11a) +∂2E +∂Rpq∂S K +������R,S=0 += ⟨Ψ0|[ ˆH, ˆE− +pq]|ΨK⟩ , +(11b) +∂E +∂S L∂S K +�����R,S=0 += 2 ⟨ΨK| ˆH − E0|ΨL⟩ , +(11c) +where E0 is the energy at R, S = 0, Ppq,rs permutes the (pq) +and (rs) indices, and the Hermiticity of ˆH and [ ˆH, ˆE− +pq] have +been exploited. Explicit formulae for these expressions have +been summarised elsewhere [see Ref. 4] but are given in the +Supporting Information (Section S1) for completeness. +Note that the first and second derivatives can only be com- +puted when R = 0 and S = 0.93 Therefore, after taking a step +in the parameter space, the energy gradient and Hessian must +be computed using the new MOs and CI vectors corresponding +to the updated wave function. A similar shift in the reference +state after each step is also required for second-order HF opti- +misation algorithms.39,94 +C. +Characterising distinct solutions +The invariance to unitary transformations within each orbital +partition means that the same CASSCF wave function can +be identified with different CI or MO coefficients. We use +the overlap between two stationary solutions |xΨ⟩ and |wΨ⟩ to +define a positive semidefinite distance metric +d(x, w) = 1 − | ⟨xΨ|wΨ⟩ |. +(12) +The overlap for two arbitrary CI wave functions with Mx and +Mw configurations, respectively, is given by +⟨xΨ|wΨ⟩ = +Mx +� +I=1 +Mw +� +J=1 +xC∗ +I ⟨xΦI|wΦJ⟩ wCJ. +(13) +Since |xΨ⟩ and |wΨ⟩ have different sets of MOs, evaluating +the overlap matrix elements ⟨xΦI|wΦJ⟩ requires a nonorthogo- +nal framework. We compute these matrix elements using the +extended nonorthogonal Wick’s theory,95,96 which avoids the +computationally expensive generalized Slater–Condon rules.97 +To understand the MOs in a CASSCF solution, we canon- +icalise the inactive and virtual orbitals and construct natural +orbitals within the active space. The canonical inactive and +virtual orbitals, and their associated orbital energies, are identi- +fied by diagonalising the relevant sub-blocks of the Fock ma- +trix, defined as89 +Fpq = hpq + +� +rs +γrs +� +(pq|sr) − 1 +2(pr|sq) +� +. +(14) +Here, γpq denotes the one-body reduced density matrix ele- +ments in the MO basis, hrq are the one-electron Hamiltonian + +4 +matrix elements, and (pq|rs) are the two-electron repulsion in- +tegrals. The natural orbitals within the active space are the +eigenvectors of the one-body reduced density matrix and their +eigenvalues are the occupation numbers np.98 +D. +Optimization techniques +Since we are concerned with understanding the CASSCF +solution space, we require an algorithm capable of converg- +ing arbitrary stationary points on the energy landscape, includ- +ing minima and higher-index saddle points. Higher-energy +CASSCF stationary points are notoriously difficult to converge +due to the strong coupling between the orbital and CI degrees +of freedom,56,60,61,72,77 and the possibility of root flipping in +the configuration space.75,99 Therefore, we employ second- +order techniques that introduce the orbital-CI coupling through +the analytic Hessian matrix of second derivatives. These algo- +rithms are too computationally expensive to be practical for +larger systems, but they are sufficient for understanding the +CASSCF solutions in small molecules. +We search for multiple solutions using several initial guesses +generated using random orbital and CI rotations from the +ground state HF solution. The eigenvector-following technique +with analytic gradient and Hessian information was used to +target stationary points with a particular Hessian index.100,101 +While this method has been described in detail elsewhere [see +Ref. 102], we include a summary in the Supporting Informa- +tion (Section S2). Related mode-following methods have previ- +ously been applied to locate higher-energy electronic stationary +points in multiconfigurational2–4,103 and single-determinant39 +SCF calculations. The convergence behaviour was further im- +proved with a modified trust region approach based on the dog- +leg method.104 Trust region methods are a well-established ap- +proach for controlling the convergence of second-order meth- +ods in CASSCF calculations.77–80 Once a set of stationary +points have been identified, their evolution with changes in the +molecular structure can be determined by using the optimised +orbital and CI coefficients at one geometry to define an initial +guess at the next geometry. Since the Hessian index may not be +conserved along a reaction coordinate,2 these subsequent cal- +culations are performed using a trust region Newton–Raphson +algorithm, as described in the Supporting Information (Sec- +tion S3). +We have implemented this numerical optimisation in an ex- +tension to the PySCF software package.105 The convergence +threshold for the root-mean-squared value of the gradient am- +plitudes was universally set to 10−8 Eh. The canonical and nat- +ural orbitals for stationary points were subsequently computed +using PySCF and visualised using VMD.106 All other graphi- +cal figures were created using Mathematica 12.0.107 +III. +Results and Discussion +A. +Molecular H2 dissocation +We start by considering the H2 binding curve using the 6- +31G basis set.108 To identify all the CASSCF (2,2) solutions, a +comprehensive search was performed using up to 1000 random +starting points for target Hessian indices from 0 to 16. Solu- +tions were identified near the equilibrium geometry R = 1.0 a0 +and the dissociation limit R = 6.0 a0, and were then traced over +all bond lengths, as shown in Fig. 1. We believe that we have +found every stationary point on the landscape, although the na- +ture of non-convex optimisation means that this can never be +guaranteed. To the best of our knowledge, this study is the first +comprehensive enumeration of the CASSCF solutions for a +molecular system. +1. +Excitations near equilibrium +Near the equilibrium geometry, the ground state of H2 can +be accurately described using a single reference approximation. +We have identified 25 stationary points on the CASSCF (2,2) +energy landscape, corresponding to 19 singlet solutions and +6 triplet solutions (Table I). Each of the exact FCI states has +TABLE I: Energies of H2 at R = 1 a0 using the 6-31G basis +set for various formalism: FCI, SA-CASSCF (2,2), +SA-CASSCF (3,2), and SS-CASSCF (2,2). +State +FCI +SA(2,2) +SA(3,2) +SS(2,2) +⟨S 2⟩ +Index +0 +-1.09897 +-1.07170 +-1.08924 +-1.09225 +0 +0 +-1.08569 +0 +1 +-1.07871 +0 +2 +1 +-0.57616 +-0.57166 +-0.57406 +-0.57417 +2 +1 +2 +-0.46395 +-0.43494 +-0.44196 +-0.46368 +0 +2 +3 +-0.28180 +-0.27990 +-0.27990 +2 +2 +4 +-0.07450 +-0.06164 +-0.05946 +0 +3 +5 +0.32015 +0.33066 +0.32624 +0.31914 +0 +3 +0.31821 +0 +2 +0.31844 +0 +2 +0.32440 +0 +3 +6 +0.51519 +0.51654 +0.51638 +2 +3 +7 +0.57224 +0.61682 +0.61429 +0 +4 +8 +0.62520 +0.62401 +2 +3 +9 +0.86353 +0.86876 +0.86392 +0 +5 +0.85673 +0 +4 +0.86266 +0 +4 +10 +0.96373 +0.91147 +0 +4 +11 +1.30761 +1.30572 +2 +4 +12 +1.46479 +1.45704 +0 +5 +13 +1.61884 +1.61685 +2 +5 +14 +1.81277 +1.80747 +0 +6 +15 +2.71948 +2.71766 +0 +7 +2.70046 +0 +6 +2.69883 +0 +5 +a corresponding SS-CASSCF (2,2) counterpart, and the ener- +getic agreement between these solutions is consistent for all ex- +citations. We have also found several additional solutions that +appear to be less accurate approximations to the exact states, +which will be characterised in Sections III A 2 and III A 3. In +comparison, the SA-CASSCF (2,2) approach can only describe +the lowest triplet and the three lowest singlet states, while in- +creasing the number of active orbitals to a (3,2) active space +provides an approximation to the lowest nine excitations. +These results demonstrate two important features of state- +specific calculations. Firstly, they can describe more excited +states than state-averaged calculations by defining the active +space using only orbitals that are relevant for a particular ex- +citation. This property allows higher energy excitations to + +5 +H(2s2)− · · · H+ +H(2s1) · · · H(2s1) +H(1s12s1)− · · · H+ +H(1s1) · · · H(2s1) +H(1s2)− · · · H+ +H(1s1) · · · H(1s1) +FIG. 1: State-specific CASSCF (2,2) stationary points can be identified for every excited FCI state in H2. Additional solutions can +also be found that dissociate to an unphysical electronic state. +be predicted while avoiding large active spaces and the asso- +ciated increase in the configuration space. An upper bound +to the exact excited state energy is only provided by station- +ary points that correspond to the correct excitation within the +CASCI configuration space,3 although more accurate energies +are generally preferred even if they are not variational. Sec- +ondly, bespoke orbital optimisation for each state-specific so- +lutions can give more accurate total energies for the excited +states compared to the state-averaged approach. For exam- +ple, the mean absolute deviations (MAD) for the lowest four +states are 2.5 mEh and 17.8 mEh for the state-specific and state- +averaged CASSCF (2,2) approaches, respectively. +Using analytic second derivatives of the energy also allows +the nature of SS-CASSCF (2,2) stationary points to be charac- +terised according to their number of downhill directions. The +corresponding Hessian index for each solution is listed in Ta- +ble I. It is known that the exact n-th excited state should have +n downhill directions.1,2,4 We find that the SS-CASSCF (2,2) +excited states are all saddle points on the electronic energy +landscape and the Hessian index generally increases with the +energy, in common with the observations for other theoretical +approximations.22,25,37,39 However, except for the lowest three +exact states, the Hessian index does not provide a reliable in- +dicator of the corresponding exact excitation index. This mis- +match must always occur for higher-lying excited states as the +approximate CASSCF (2,2) wave function has fewer degrees +of freedom than the exact formulation. Consequently, if we +only consider stationary points of the correct Hessian index, +then we must forgo the advantages of capturing state-specific +excitations outside the state-averaged active space. +2. +Multiple ground state solutions +While Table I shows that a SS-CASSCF (2,2) approxima- +tion can be identified for each exact eigenstate, we also find +additional state-specific solutions. In particular, there are three +close-lying stationary points that can be considered as approxi- +mations to the ground state, with Hessian indices of 0, 1, and 2 +in order of ascending energy. This pattern of multiple solutions +is repeated for the (2σg)2 and (2σu)2 singlet configurations, +while the other closed-shell (1σu)2 configuration exhibits four +close-lying solutions. Choosing the most physical solution for +each eigenstate presents a challenge for state-specific CASSCF +approaches. Therefore, it is important that we understand their +mathematical origins and physical differences. +The natural orbitals in the active space provides a clear ex- +planation for the multiple H2 ground-state solutions. Figure 2A +compares the natural orbitals and occupation numbers for the +three lowest-energy singlet stationary points. Since the ground +state at the equilibrium geometry can be relatively well approx- +imated by a single closed-shell Slater determinant, the active +space for each of these solutions includes a (1σg)-like natural +orbital that is almost completely doubly occupied. This natural +orbital dominates the electronic wave function and the corre- +sponding energies are all relatively close approximations to the +exact ground state. However, the second active orbital, which +is almost completely unoccupied, is different for each solution, +corresponding to a (1σu), (2σg), or (2σu) orbital as the energy +increases, respectively. These higher-energy stationary points +have downhill orbital rotations that interconvert the multiple +ground state solutions and correspond to the negative eigenval- +ues of the Hessian. +Different choices for the nearly unoccupied active orbital +have only a small effect on the total H2 energy near equilibrium. + +SS-CASSCF (2,2) +Exact +1.0 +6 +0.5 +Energy +0.0 +-0.5 +-1.0 +2 +3 +4 +5 +6 +Bond Length / ao6 +(1σg) nocc = 1.9998 +(2σu) nocc = 0.0002 +E = −1.07871 Eh +A +(1σg) nocc = 1.9925 +(2σg) nocc = 0.0076 +E = −1.08569 Eh +(1σg) nocc = 1.9887 +(1σu) nocc = 0.0113 +E = −1.09225 Eh +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +1 +2 +3 +4 +5 +6 +-1.15 +-1.10 +-1.05 +-1.00 +-0.95 +-0.90 +-0.85 +-0.80 +B +FIG. 2: There are three SS-CASSCF(2,2) solutions that represent the exact ground state in H2. (A) Comparison of the natural +orbitals for each ground-state solution at R = 1.0 a0. (B) Only the lowest-energy solution dissociates correctly, while the +higher-energy solutions mirror the restricted Hartree–Fock binding curve. +However, the incorrect choice of the active space becomes +very significant as the bond is stretched towards dissociation. +Only the {1σg, 1σu} active space can correctly dissociate into +the H(1s) · · · H(1s) ground state of the dissociated fragments +(Fig. 2B). In contrast, the binding curves for the {1σg, 2σg} and +{1σg, 2σu} solutions mirrors the RHF energy as the correspond- +ing wave functions are close to a single Slater determinant at +all geometries, with (1σg) occupation numbers at dissociation +of 1.997 and 1.999, respectively. Notably, the stationary points +preserve the character of the active orbitals along the potential +energy surface, suggesting that SS-CASSCF solutions exhibit +some degree of diabatic character. +The same pattern of solutions is observed for the other +closed-shell solutions. However, the (1σu)2 configuration ex- +hibits an additional multiple solution where the nearly unoccu- +pied active orbital corresponds to a symmetry-broken 2s-like +orbital localised on either the left or right H atom. This sym- +metry breaking results in a two-fold degenerate pair of station- +ary points. +These results indicate that additional solutions can arise +from the free choice of virtual orbitals when the active space is +larger than required for the degree of static correlation. Mal- +rieu and co-works elegantly summarised this phenomenon by +stating that “the so-called valence CASSCF wave function does +not necessarily keep a valence character when the wave func- +tion concentrates on a closed-shell valence bond structure”.109 +Therefore, we expect that the number of ground state solu- +tions will increase combinatorially with the number of active +orbitals or the basis set size, and the number of unphysical so- +lutions can grow for larger active spaces even though the cor- +rect ground state solution will become more accurate. Table II +demonstrates this increase for H2 using the 6-311G basis set +with three basis functions for each hydrogen atom.110 Taking +the (3,2) active space as an example, there are two redundant +active orbitals beyond the 1σg that must be chosen from the +five remaining orbitals, giving a total of ten solutions through +the binomial coefficient +�5 +2 +� += 10. The relative energy order- +ing of these additional solutions will depend on the amount of +dynamic correlation captured by the redundant active orbitals, +which may not correspond with the same orbital required to +capture the static correlation in the dissociation limit. This +phenomenon has previously been described for MgO, where +oxygen-centred orbitals are preferred over the magnesium d +orbitals,87 and transition metal compounds where non-valence +d orbitals may be preferred over certain valence d orbitals.111 +TABLE II: Close-lying ground-state (n, 2) SS-CASSCF +energies (Eh) of H2 at R = 1 a0 using the 6-311G basis set for +various active space size n. +SS(1,2): HF +-1.08025 +SS(2,2) +-1.09429 -1.08866 -1.08074 -1.08033 -1.08026 +SS(3,2) +-1.10195 -1.09500 -1.09436 -1.09429 -1.08904 +-1.08886 -1.08867 -1.08082 -1.08075 -1.08034 +SS(4,2) +-1.10251 -1.10212 -1.10196 -1.09507 -1.09500 +-1.09437 -1.08923 -1.08905 -1.08886 -1.08083 +SS(5,2) +-1.10267 -1.10251 -1.10213 -1.09507 -1.08924 +SS(6,2): FCI -1.10267 +3. +Open-shell singlet and triplet excitations +The low-lying open-shell triplet and singlet (1σg)1(1σu)1 +configurations are represented by only one SS-CASSCF (2,2) + +7 +1 +2 +3 +4 +5 +6 +7 +8 +-0.2 +0.0 +0.2 +0.4 +0.6 +FIG. 3: Spontaneous symmetry breaking occurs when the +active space is not large enough to capture all the important +configurations in the physical wave function, as illustrated for +the 1s 2s states in the dissociation of H2 (6-31G). +solution across the full binding curve (Fig. 1). These single +solutions arise because all the active orbitals are required to +describe the two-configurational static correlation and there +is no flexibility for multiple solutions to exist. In addition, +SS-CASSCF (2,2) gives an accurate representation of the +open-shell (1σg/u)1(2σg/u)1 configurations. However, the accu- +racy of these solutions deteriorates in the dissociation limit, +where additional symmetry broken solutions can be identified +(Fig. 3). These additional solutions break spatial symmetry and +spontaneously appear at instability thresholds that are multi- +configurational analogues to the Coulson–Fischer points41 in +HF theory.42,112–115 Each stationary point is a pure singlet or +triplet state and has a two-fold degeneracy, reflecting the left- +right symmetry of the molecule. +The origin of this symmetry breaking is explained by con- +sidering the correlation processes involved in the excited dis- +sociation limit. These excited states dissociate to hydrogenic +(1s)1(2s)1 configurations, where the occupied 1s and 2s orbitals +can either be on the same or different atomic centres. Taking +the latter case as an example, the corresponding open-shell sin- +glet wave function at large nuclear separations has the form +|Ψ⟩ = 1 +2 +� +|1sL2sR⟩ + |1sR2sL⟩ +�� +|αβ⟩ − |βα⟩ +� +. +(15) +Correctly describing this wave function requires an active space +with four spatial orbitals {1sL, 1sR, 2sL, 2sR}, or equivalently +{1σg, 1σu, 2σg, 2σu}, and thus the SS-CASSCF (2,2) approx- +imation is insufficient for these correlation mechanisms. In- +stead, the symmetry breaking reduces the SS-CASSCF (2,2) +wave function to a subset of the dominant configurations, e.g. +| ˜Ψ⟩ = +1√ +2 +|1sL2sR⟩ +� +|αβ⟩ − |βα⟩ +� +. +(16) +The CASSCF configurations corresponding to each symmetry- +broken solution are assigned in Table III. This “pinning” of +State +Energy / Eh +⟨S 2⟩ +Configuration +A +0.543355 +0.00 +������� +|1sL2sL⟩ (|αβ⟩ − |βα⟩) +|1sR2sR⟩ (|αβ⟩ − |βα⟩) +B +0.293363 +2.00 +������� +|1sL2sL⟩ (|αβ⟩ + |βα⟩) +|1sR2sR⟩ (|αβ⟩ + |βα⟩) +C +-0.037221 +0.00 +������� +|1sL2sR⟩ (|αβ⟩ − |βα⟩) +|1sR2sL⟩ (|αβ⟩ − |βα⟩) +D +-0.037499 +2.00 +������� +|1sL2sR⟩ (|αβ⟩ + |βα⟩) +|1sR2sL⟩ (|αβ⟩ + |βα⟩) +TABLE III: The symmetry-broken CASSCF (2,2) solutions in +the dissociation of H2 are two-fold degenerate and represent +dominant configurations in the exact excitations. +the wave function onto a particular electronic configuration +is directly analogous to the symmetry breaking phenomena +observed in HF theory116,117 and demonstrates that the active +space is too small to fully account for the static correlation. +From the energy landscape perspective, the onset of +symmetry-broken CASSCF (2,2) states is associated with a +change in the Hessian index for the associated symmetry-pure +solutions. For example, the symmetry-broken state D (Ta- +ble III) emerges from the symmetry-pure (1σg)1(2σg)1 triplet +state at an instability threshold close to R = 2.28 a0. The Hes- +sian index of the symmetry-pure state changes from 2 to 3 at +this point, while the symmetry-broken solutions form index- +2 saddle points, leading to a higher-index analogue of a cusp +catastrophe.39,40,118 Practically, the emergence of a zero Hes- +sian eigenvalue at these instability thresholds may hinder the +numerical optimisation of second-order techniques onto these +higher-energy stationary points. It is also interesting to note +that, while the symmetry-broken solutions describe two degen- +erate FCI states at dissociation, they only connect to one of +the corresponding symmetry-pure solutions in the equilibrium +region. Consequently, one cannot rely on these additional so- +lutions to obtain an accurate and continuous representation of +every excited state across all geometries. +B. +Conical Intersection in methylene +We next consider the bending mode of methylene, which has +a diradical ground state with 3B1 symmetry and a low-lying +1 1A1 excited state. The bond length was fixed to the value +R(C−H) = 2.11 a0 identified by Bauschlicher and Taylor119,120 +and the 6-31G basis set was used.108 Methylene has a long his- +tory as a benchmark for electronic structure theory.121 One of +the primary questions is the description of the conical intersec- +tion between the low-lying 3B1 and 1 1A1 states. +1. +Local minima for the minimal (2,2) active space +A minimal two-configuration wave function is required to +qualitatively describe both the lowest-energy singlet S0 (1A1) +and diradical triplet T0 (3B1) states.119 Therefore, we begin +by analysing the SS-CASSCF (2,2) energy landscape. The S0 + +8 +80 +100 +120 +140 +-38.90 +-38.88 +-38.86 +-38.84 +-38.82 +nocc = 1.9398 +nocc = 0.0602 +E = −38.86871 Eh +nocc = 1.0000 +nocc = 1.0000 +E = −38.89125 Eh +nocc = 1.8749 +nocc = 0.1251 +E = −38.86793 Eh +nocc = 1.9789 +nocc = 0.0211 +E = −38.87070 Eh +C +H +H +A +B +C +D +E +FIG. 4: Low-lying SS-CASSCF (2,2) in the bending mode of methylene represent the 1 3B1 and 1 1A1 configurations. (A) Both +states remain local minima (solid purple) for a short region beyond the conical intersection before becoming an index-1 saddle point +(solid cyan). An additional spin-contaminated index-1 saddle point (dashed cyan) connects the two instability thresholds (black +dots). Two degenerate local minima exist everywhere along the bending curve (dashed purple) with an active space containing +C-H bonding σ and antibonding σ∗ orbitals. (B–E) The natural orbitals at a bond angle of 103.7◦ are illustrated for each solution. +and T0 are the ground state for small and large bond angles, +respectively, and provide an example of a conical intersection +separating the two regimes. At bond angles of 76◦, 102◦ and +130◦, a large number of stationary points can be identified +with a variety of Hessian indices. Therefore, we simplify our +analysis by focussing on a subset of low-energy solutions that +resemble the desired physical states (Fig. 4). +The energetic minimum of the S0 state occurs at a bond angle +of 103.7◦. While the S0 state is the first excited state at this +geometry, we find that the corresponding SS-CASSCF (2,2) +stationary point is a local minimum rather than an index-1 +saddle point. This incorrect Hessian index arises from a root +flip in the configuration space, where the singlet state is the +ground state for the corresponding active orbitals. When the +bond angle increases, this singlet state eventually becomes an +index-1 saddle point. Similarly, when the bond angle decreases +from 103.7◦, the T0 state remains a local minimum beyond the +conical intersection where it becomes the first excited state. +This process behaves like an unphysical hysteresis, where the +ground state remains a local minimum for a small region after +a conical intersection before becoming an index-1 saddle point +at an instability threshold. +An additional index-1 saddle point can be identified that +connects these two solutions and coalesces with each local +minimum at the two instability thresholds. This unphysical +index-1 stationary point is two-fold degenerate, has symmetry- +pure spatial orbitals, but is spin contaminated with an ⟨ ˆS 2⟩ +value that changes continuously from 0 to 2 as it connects +the singlet and triplet states. +Similar patterns of coalesc- +ing solutions have been observed in single determinant SCF +approximations,40,112,122,123 particularly in the generalised HF +representation of conical intersections between states with dif- +ferent ⟨ ˆS z⟩ values.124 In contrast to symmetry-broken SCF so- +lutions, the spin contamination observed here arises from the +mixture of singlet and triplet states in the configurational part +of the CASSCF wave function. +Since the S0 solution has only one significantly occupied +active orbital, we predict the existence of closely-related so- +lutions that have alternative redundant orbitals with nocc ≈ 0. +Indeed, there are a pair of degenerate local minima that lie +slightly lower in energy than the S0 solution. In contrast to the +H2 ground state, including the inactive space means that methy- +lene has multiple doubly occupied orbitals, and thus the active +orbital with nocc ≈ 2 may also change between different solu- +tions. The active orbitals for these symmetry-broken solutions +are localised bonding σ and anti-bonding σ∗ orbitals for one of +the two C–H bonds, and the degeneracy accounts for the two +possible ways to localise onto one bond. Notably, the symme- +try breaking here is associated with an active space that is too +large, in contrast to H2 where symmetry breaking arises from +an insufficient active space for the static correlation. These so- +lutions are local minima across all the bond angles considered. +While they provide an accurate energy for the S0 state near the +singlet equilibrium structure, this deteriorates for large angles +as the active space cannot describe the diradical open-shell 1Σ+ +g +state at the linear geometry. Their existence indicates that the +C–H σ/σ∗ configurations provide an important contribution +to static correlation and should ideally be included in the ac- +tive space, as suggested by Bauschlicher and Taylor.119,120 +2. +Full valence active space +Using the full-valence (6,6) active space, we find that the +symmetry-pure singlet state is now correctly represented by +an index-1 saddle point at a bond angle of 102° (Fig. 5). The +unique downhill direction corresponds to a rotation in the con- +figuration space only, as expected for the first excited state. De- + +9 +80 +100 +120 +140 +-38.95 +-38.94 +-38.93 +-38.92 +-38.91 +-38.90 +-38.89 +-38.88 +FIG. 5: Low-lying SS-CASSCF (6,6) for the bending mode of +methylene represent the 1 3B1 and 1 1A1 configurations. The +full valence (6,6) active space introduces more unphysical +solutions, but does not remove the spin-contaminated solution +that arises at the conical intersection. +spite the larger active space, a root flip still occurs as the states +approach the singlet–triplet conical intersection at 82.2°, with +the singlet state becoming a local minimum at 89.6° and the +triplet state becoming an index-1 saddle point at 77.0°. Like +the (2,2) active space, a degenerate pair of unphysical, spin- +contaminated index-1 saddle points connect the solutions that +cross at the conical intersection. This phenomenon occurs be- +cause the orbital optimisation can lower the energy of the tar- +get excited state below the corresponding ground state configu- +ration when the energy gap becomes small. Therefore, while +larger active spaces will reduce the range of molecular geome- +tries affected, these unphysical local minima will be common +for state-specific conical intersections. +While the larger active space alleviates root flipping, it also +causes more unphysical solutions associated with redundant ac- +tive orbitals. For example, the triplet ground state (dominated +by two configurations) is represented by one SS-CASSCF (2,2) +solution, but there are several higher-energy solutions in the +(6,6) active space. Analogously to the H2 ground state, these +additional solutions have a higher Hessian index, with two +index-1 and one index-2 saddle points represented in Fig. 5. +Again, the main difference from the true ground state are the +active orbitals with occupation numbers close to zero, as il- +lustrated for the global minimum and lowest-energy index-1 +saddle point in Fig. 6. Furthermore, we find an additional lo- +cal minimum and index-1 saddle point that represent the 1A1 +state. While all the triplet solutions give approximately the +same equilibrium bond angle, the unphysical stationary points +shift the conical intersection to coincide with the singlet equi- +librium geometry. This qualitative change in the energy sur- +face would create a near-barrierless decay from the singlet ex- +cited state to the triplet ground state, demonstrating the impor- +tance of verifying the physicality of state-specific solutions. +C. +Avoided crossing in Lithium Fluoride +1. +Physicality of multiple solutions +The LiF binding curve provides a typical example of an +avoided crossing. The ground state has ionic character at equi- +librium, but becomes a covalent state with almost no dipole mo- +ment in the dissociation limit. Multiple HF solutions are known +to behave “quasi-diabatically” and cross each other at the phys- +ical avoided crossing.47,125 On the other hand, Bauschlicher +and Langhoff demonstrated that this avoided crossing can lead +to discontinuities in the CASSCF ground- and excited-state +energy surfaces.126 Here, we start by considering the state- +specific singlet CASSCF solutions in the 6-31G basis set. +Using the minimal (2,2) active, we search for stationary +points with Hessian indices of 0 to 10 at R(Li−F) = 2.75 a0 +(near the equilibrium geometry), using 1000 random starting +points for each index. The active space for the SS-CASSCF +global minimum contains the valence bonding σ and anti- +bonding σ∗ orbitals with occupation numbers close to 2 and 0, +respectively (Fig. 7B). Because the exact wave function is dom- +inated by a single closed-shell configuration, there are many +additional solutions that are close to the ground-state energy +at the equilibrium geometry. For example, the second lowest +energy solution has an active space containing the out-of-plane +fluorine 2px/y and 3px/y orbitals with occupation numbers close +to 2 and 0, respectively (Fig. 7C). This active space accounts +for the radial correlation on the fluorine atom, providing a more +balanced description of F and F–.126 In contrast, the exact ex- +cited state is more multiconfigurational at short bond lengths +and is accurately represented by only one solution (Fig. 7D), +alongside a spurious symmetry-broken solution with diradical +character (Fig. 7E). These characteristics are reversed for bond +lengths longer than the avoided crossing, where the excited +state has closed-shell character with a large number of solu- +tions and the ground state is represented by only two solutions. +State-specific CASSCF solutions can behave both quasi- +diabatically and adiabatically in the vicinity of the avoided +crossing. As the bond length changes, the unphysical solutions +do not have the correct active orbitals to capture the strong +correlation at the avoided crossing. Therefore, the two lowest- +energy unphysical solutions intersect quasi-diabatically (dark +purple in Fig. 7A, corresponding to the solutions in Fig. 7C +and 7E). On the other hand, the physically meaningful solu- +tions behave adiabatically and correctly avoid each other (cyan +in Fig. 7A). In principle, a linear expansion of both the quasi- +diabatic and adiabatic states may provide a more accurate rep- +resentation of the avoided crossing by introducing some of +the dynamic correlation captured by the unphysical solutions. +This expansion would require a multiconfigurational variant of +nonorthogonal CI,125 where the Hamiltonian and overlap ma- +trix elements can be efficiently computed using the nonorthog- +onal framework developed in Refs. 95 and 96. +While a complete description of the avoided crossing re- +quires dynamic correlation,127 the advantage of state-specific +orbital relaxation is still clear in the dissociation limit. The +physical SS-CASSCF excited state tends towards the exact FCI +energy for the separated Li+ · · · F– configuration, while state- +averaged calculations (with an equal weighting for the two +states) overestimates the energy of this excited state (Fig. 7A). + +10 +nocc = 1.0000 +nocc = 0.0284 +nocc = 0.0224 +nocc = 1.9801 +nocc = 1.9677 +nocc = 1.0014 +E = −38.93241 Eh +A +nocc = 0.9978 +nocc = 0.0254 +nocc = 0.0029 +nocc = 1.9985 +nocc = 1.9745 +nocc = 1.0009 +E = −38.91416 Eh +B +FIG. 6: Comparison of the active orbitals for the two lowest energy triplet CASSCF solutions for CH2 (6-31G) using the full +valence (6,6) active space at a bond angle of 102◦. (A) The active orbitals for the local minimum represent the chemically +intuitive valence space. (B) For the unphysical index-1 saddle point, one of the antibonding C–H σ∗ orbitals with nocc ≈ 0 is +replaced by a carbon 3p orbital with nocc = 0.0029. The remaining σ and σ∗ orbitals localise onto the C–H bonds. +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● 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+●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +●●●●●●●●●●●●●●●●●●●●●●●●● +2 +4 +6 +8 +10 +12 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +nocc = 1.9814 +nocc = 0.0186 +E = −106.89419 Eh +nocc = 1.9908 +nocc = 0.0092 +E = −106.88883 Eh +nocc = 1.3806 +nocc = 0.6194 +E = −106.76134 Eh +nocc = 1.0000 +nocc = 1.0000 +E = −106.78556 Eh +A +B +C +D +E +FIG. 7: (A) The SS-CASSCF (2,2) appoach gives many solutions for the LiF binding curve (6-31G) when the ground or excited +state is dominated by a single configuration. Ground- and excited-state solutions with a suitable active space (B and D) behave +adiabatically at the avoided crossing (cyan lines). Additional solutions with unsuitable active orbitals can represent either the ionic +equilibrium configuration (C) or the covalent dissociation configuration (E), and behave quasi-diabatically at the avoided crossing +(purple lines). The active orbitals are plotted at R(Li−F) = 4 a0. Exact FCI and SA(2)-CASSCF (2,2) data are taken from Ref. 47. +In this SS-CASSCF solution, the σ and σ∗ orbitals (Fig. 7A) +both localise to give 2pz orbitals that accurately represent the +F– anion. Consequently, as expected, the state-specific for- +malism provides a more accurate representation of this charge +transfer excitation than a state-averaged approach. +2. +Elucidating the Bauschlicher–Langhoff discontinuity +The seminal CASSCF investigation of LiF, by Bauschlicher +and Langhoff, highlighted the presence of a discontinuity in +the ground-state dipole moment in the vicinty of the avoided +crossing.126 This discontinuity is a signature of a discontinuity +in the wave function, which manifests as a cusp in the corre- +sponding energy surface. This phenomenon, which we name +the “Bauschlicher–Langhoff discontinuity”, has long been used +as key evidence for the potential issues of state-specific cal- +culations in the vicinity of an avoided crossing. Malrieu and +co-workers attributed its origin to a near degeneracy between +the closed-shell ionic and the open-shell covalent configura- +tions, and described a lower-energy covalent state that emerges +from a potential symmetry-breaking point as the bond length +increases.109 The framework developed here, and the advance +in computing over the past 30 years, now allows this topologi- +cal characterisation to be rigorously tested. +To identify the relevant solutions, we searched for minima +and index-1 saddle points at a bond length of 8.50 a0 using +1000 random starting points, a (2, 2) active space, and the orig- + +::11 +4 +6 +8 +10 +12 +-106.95 +-106.90 +-106.85 +-106.80 +7.0 7.1 7.2 7.3 7.4 7.5 7.6 +-106.840 +-106.835 +-106.830 +FIG. 8: Topology of the low-energy SS-CASSCF (2, 2) +solutions near the Bausclicher–Langhoff discontinuity in +LiF,109,126 using the basis set defined in Ref. 126. A cusp in +the ground-state energy occurs when two local minima cross, +while the covalent structure coalesces with an index-1 saddle +point at a pair annihilation point (black dot). +inal basis set described in Ref. 126. At R(Li−F) = 8.5 a0, the +global minimum corresponds to the covalent structure identi- +fied in Ref. 109. In addition, two local minima and two index- +1 saddle points exist at higher energies, representing the ionic +configurations (Fig. 8). As the bond length is shortened, there +is a crossing between the lowest energy ionic and covalent min- +ima near R(Li−F) = 7.4 a0, which we believe corresponds to +the previously described discontinuity.109,126 +Topologically, two non-degenerate minima cannot coalesce +without the presence of an index-1 saddle point, and thus the +disappearance described by Malrieu and co-workers cannot +be the full picture.109 Instead, we find that the covalent struc- +ture crosses the two lowest-energy local minima and eventu- +ally coalesces with an index-1 saddle point representing the +ionic configurations. Following the downhill directions from +this index-1 saddle points reveals that it connects the covalent +local minimum with the lowest-energy ionic local minimum. +Furthermore, the downhill Hessian eigenvector has significant +orbital and CI components, which highlights the strong cou- +pling between the different degrees of freedom in the vicinity +of the avoided crossing. Both solutions disappear at this point +(black dot in Fig. 8), and thus there is no quasi-diabatic cova- +lent solution at shorter bond lengths. +In the mathematical framework of catastrophe theory,128 +this type of coalescence can be classified as a fold catastro- +phe, or a pair annihilation point. Singularities in this class have +previously been identified and characterised for multiple HF +solutions,112 where they most commonly occur in asymmetric +molecules, for example LiF,47,125 H–Z40 (for a partial nuclear +charge Z), and ethylene analogues.40 The discontinuous jump +in the energy at the pair annihilation point in LiF will create +issues for calculations that attempt to follow the covalent so- +lution across multiple bond lengths, making these solutions +unsuitable for techniques such as ab initio molecular dynam- +ics. Furthermore, since the lowest energy covalent and ionic +local minima cross rather than coalesce, the gradient of the +global minimum energy at the crossing point is discontinuous +and there is an unphysical cusp in the resulting energy surface. +The absence of this pair annihilation point using 6-31G com- +pared to Bauschlicher and Langhoff’s basis set demonstrates +how the topology of multiple CASSCF solutions can be af- +fected by the AO basis. We suspect that these differences arise +from the subtle changes in the underlying energy landscape +that affect the relative stability of different solutions. However, +these results demonstrate the danger of generalising conclu- +sions from one basis set to another, even for the same molecule. +IV. +Concluding Remarks +State-specific approximations promise to provide a more +balanced representation of electronic excitations by indepen- +dently optimising both the ground- and excited-state wave +functions. In this work, we have investigated the energy land- +scape for excited state-specific stationary points in the multi- +configurational CASSCF approach. We have shown how state- +specific approximations can accurately describe high-energy +and charge transfer excitations, beyond the reach of state- +averaged calculations with small active spaces. However, the +CASSCF energy landscape can have a large number of station- +ary points, which complicates the selection and interpretation +of physically relevant solutions. +Multiple stationary points in state-specific CASSCF calcu- +lations arise through two primary mechanisms. Firstly, many +solutions occur when the active space is too large for the static +correlation that must be described. In this case, the redundant +active orbitals with nocc ≈ 0 can be interchanged with virtual +orbitals without significantly changing the energy, creating a +series of stationary points with an increasing number of down- +hill Hessian eigendirections. Active orbitals with nocc ≈ 2 can +be interchanged with doubly occupied inactive orbitals in a +similar fashion. On the other hand, symmetry broken solutions +occur when the active space is too small to describe the static +correlation mechanisms, causing the CASSCF wave function +to become “pinned” onto a subset of the configurations in the +exact wave function. These results demonstrate the importance +of finding a “Goldilocks region”, where the active space is nei- +ther too large or too small, but just right. +Unphysical solutions can have important consequences for +the resulting potential energy surfaces. For example, while +choosing the wrong active space only introduces a small energy +error when the wave function is dominated by a single closed- +shell configuration, it can prevent the CASSCF wave function +from correctly capturing static correlation when the molecular +structure changes. The active space for stationary points does +not change significantly along a reaction coordinate, meaning + +12 +that the incorrect active orbitals remain for all geometries. For +ground-state calculations, one can rely on following downhill +directions away from saddle points to obtain a more suitable +local minimum, hopefully with the best active space. However, +it is hard to predict which Hessian index will give the most +physical stationary point for an excited state, and thus choosing +the most accurate excited-state stationary point is challenging +without prior chemical intuition. It has long been known that +the right choice of active orbitals is key to the success of +CASSCF, but the current results demonstrate the severity of +this challenge for state-specific excitations. +In addition, we have investigated the topology of SS- +CASSCF (2, 2) solutions near the singlet-triplet conical inter- +section in CH2 and the covalent-ionic avoided crossing in LiF. +We observe unphysical root flipping where the CH2 excited +state solution is a local minimum near the conical intersec- +tion, before becoming an index-1 saddle point further along +the reaction trajectory. This phenomenon occurs because the +state-specific orbital optimisation artificially stabilises the lo- +cal minima, and is still present in the full valence (6, 6) active +space. Furthermore, the change in Hessian index is associated +with an additional spin-contaminated index-1 saddle point that +connects the singlet and triplet stationary points. The pres- +ence of zero Hessian eigenvalues at these instability thresholds +may cause numerical issues for second-order optimisation al- +gorithms. On the other hand, for the LiF avoided crossing, we +have observed the coalescence of the local covalent minimum +with an index-1 saddle point representing the ionic state, which +both disappear entirely at shorter bond lengths. While this pair- +wise coalescence depends on the basis set, it would catastroph- +ically affect the applicability of SS-CASSCF for generating +smooth and continuous potential energy surfaces. +Moving forwards, SS-CASSCF calculations must overcome +the troublesome issues of multiple solutions. Practical solu- +tions may rely on the identification of suitable initial guesses +from more black-box techniques, or by focussing on optimisa- +tion algorithms that target desirable excited-state physical prop- +erties (e.g. dipole moments), such as the generalized variational +principles developed by Hanscam and Neuscamman.87 Alter- +natively, more bespoke excited-state wave function ansätze, +such as minimal configuration state functions69 or excited-state +mean-field theory,64,65,68 may remove unphysical solutions as- +sociated with redundant active orbitals and avoid the disappear- +ance of solutions at pairwise coalescence points. Surmounting +these issues will allow the benefits of state-specific calcula- +tions for computing excited states, with bespoke orbitals and +small active spaces, to be fully realised. +Supporting Information +Derivation of the gradient and second-derivatives for the +CASSCF energy, description of eigenvector-following and +Newton–Raphson optimisation algorithms used (PDF). +Acknowledgements +H.G.A.B was supported by New College, Oxford, through +the Astor Junior Research Fellowship. The authors thank David +Tew for support and computing resources. +References +1Burton, H. G. A. Energy Landscape of State-Specific Electronic Structure +Theory. J. Chem. Theory Comput. 2022, 18, 1512. +2Olsen, J.; Jørgensen, P.; Yeager, D. L. Multiconfigurational Hartree–Fock +studies of avoided curve crossing using the Newton–Raphson technique. J. +Chem. Phys. 1982, 76, 527. +3Golab, J. T.; Yeager, D. L.; Jørgensen, P. Proper characterization of MC +SCF stationary points. Chem. Phys. 1983, 78, 175. +4Olsen, J.; Yeager, D. L.; Jørgensen, P. Optimization and Characterization of +a Multiconfigurational Self-Consistent Field (MCSCF) State. In Adv. Chem. +Phys.; John Wiley and Sons, Ltd, 1983; pp 1–176. +5Golab, J. T.; Yeager, D. L.; Jørgensen, P. Multiple stationary point represen- +tations in MC SCF calculations. Chem. Phys. 1985, 93, 83. +6Bacalis, N. C.; Xiong, Z.; Zang, J.; Karaoulanis, D. Computing correct +truncated excited state wavefunctions. AIP Conf. Proc. 2016, 1790, 020007. +7Bacalis, N. C. If Truncated Wave Functions of Excited State Energy Saddle +Points Are Computed as Energy Minima, Where Is the Saddle Point? In +Theoretical Chemistry for Advanced Nanomaterials: Functional Analysis +by Computation and Experiment; Onishi, T., Ed.; Springer Singapore: +Singapore, 2020; p 465. +8Runge, E.; Gross, E. K. U. Density-Functional Theory for Time-Dependent +Systems. Phys. Rev. Lett. 1984, 52, 997. +9Dreuw, A.; Head-Gordon, M. Single-Reference ab Initio Methods for the +Calculation of Excited States of Large Molecules. Chem. Rev. 2005, 105, +4009. +10Burke, K.; Werschnik, J.; Gross, E. K. U. Time-dependent density func- +tional theory: Past, present, and future. J. Chem. Phys. 2005, 123, 062206. +11Hait, D.; Rettig, A.; Head-Gordon, M. Beyond the Coulson–Fischer point: +characterizing single excitation CI and TDDFT for excited states in single +bond dissociations. Phys. Chem. Chem. Phys. 2019, 21, 21761. +12Maitra, N. T.; Zhang, F.; Cave, R. J.; Burke, K. Double excitations within +time-dependent density functional theory linear response. J. Chem. Phys. +2004, 120, 5932. +13Schirmer, J. Beyond the random-phase approximation: A new approxima- +tion scheme for the polarization propagator. Phys. Rev. A 1982, 26, 2395. +14Dreuw, A.; Wormit, M. The algebraic diagrammatic construction scheme +for the polarization propagator for the calculation of excited states. WIREs +Comput. Mol. Sci. 2015, 5, 82. +15Stanton, J. F.; Bartlett, R. J. The equation of motion coupled-cluster method. +A systematic biorthogonal approach to molecular excitation energies, tran- +sition probabilities, and excited state properties. J. Chem. Phys. 1993, 98, +7029. +16Krylov, A. I. Equation-of-Motion Coupled-Cluster Methods for Open-Shell +and Electronically Excited Species: The Hitchhiker’s Guide to Fock Space. +Annu. Rev. Phys. Chem. 2008, 59, 433. +17Tozer, D. J. Relationship between Long-Range Charge-Transfer Excitation +Energy Error and Integer Discontinuity in Kohn–Sham Theory. J. Chem. +Phys. 2003, 119, 12697. +18Dreuw, A.; +Head-Gordon, M. Failure of Time-Dependent Den- +sity Functional Theory for Long-Range Charge-Transfer Excited +States: The Zincbateriochlorin–Bacteriochloring and Bacteriochlorophyll— +Spheroidene Complexes. J. Am. Chem. Soc. 2004, 126, 4007. +19McLachlan, A. D.; Ball, M. A. Time-Dependent Hartree–Fock Theory for +Molecules. Rev. Mod. Phys. 1964, 36, 844. +20Bartlett, R. J. Coupled-cluster theory and its equation-of-motion extensions. +WIREs Comput. Mol. Sci. 2012, 2, 126. +21Helmich-Paris, B. Benchmarks for Electronically Excited States with +CASSCF methods. J. Chem. Theory Comput. 2019, 15, 4170. +22Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-Consistent Field Calcu- +lations of Excited States Using the Maximum Overlap Method (MOM). J. +Phys. Chem. A 2008, 112, 13164. +23Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Communication: Hartree– +Fock description of excited states of H2. J. Chem. Phys. 2014, 141, 111104. +24Barca, G. M. J.; Gilbert, A. T. B.; Gill, P. M. W. Simple Models for Difficult +Electronic Excitations. J. Chem. Theory Comput. 2018, 14, 1501. +25Hait, D.; Head-Gordon, M. Orbital Optimized Density Functional Theory +for Electronic Excited States. J. Phys. Chem. Lett. 2021, 12, 4517. +26Hait, D.; Head-Gordon, M. Excited State Orbital Optimization via Mini- +mizing the Square of the Gradient: General Approach and Application to + +13 +Singly and Doubly Excited States via Density Functional Theory. J. Chem. +Theory Comput. 2020, 16, 1699. +27Carter-Fenk, K.; Herbert, J. M. State-Targeted Energy Projection: A Simple +and Robust Approach to Orbital Relaxation of Non-Aufbau Self-Consistent +Field Solutions. J. Chem. Theory Comput. 2020, 16, 5067. +28Levi, G.; Ivanov, A. V.; Jónsson, H. Variational Density Functional Calcula- +tions of Excited States via Direct Optimization. J. Chem. Theory Comput. +2020, 16, 6968. +29Levi, G.; Ivanov, A. V.; Jónsson, H. Variational calculations of excited +states via direct optimization of the orbitals in DFT. Faraday Discuss. 2020, +224, 448. +30Ivanov, A. V.; Gianluca Levi, E. O. J.; Jónsson, H. Method for Calculating +Excited Electronic States Using Density Functionals and Direct Orbital +Optimization with Real Space Grid or Plane-Wave Basis Set. J. Chem. +Theory Comput. 2021, 17, 5034. +31Shea, J. A. R.; Neuscamman, E. Size Consistent Excited States via Algo- +rithmic Transformations between Variational Principles. J. Chem. Theory +Comput. 2017, 13, 6078. +32Jankowski, K.; Kowalski, K.; Jankowski, P. Applicability of single- +reference coupled-cluster methods to excited states. A model study. Chem. +Phys. Lett. 1994, 222, 608. +33Jankowski, K.; Kowalski, K.; Jankowski, P. Multiple Solutions of the Single- +Reference Coupled-Cluster Equations. II. Alternative Reference States. Int. +J. Quantum Chem. 1994, 53, 501. +34Piecuch, P.; Kowalski, K. In Computational Chemistry: Reviews of Current +Trends; Leszczynski, J., Ed.; World Scientific, 2000; Vol. 5; Chapter 1. +In Search of the Relationship between Multiple Solutions Characterizing +Coupled-Cluster Theories, p 1. +35Mayhall, N. J.; Raghavachari, K. Multiple Solutions to the Single-Reference +CCSD Equations for NiH. J. Chem. Theory Comput. 2010, 6, 2714. +36Lee, J.; Head-Gordon, M. Distinguishing artifical and essential symmetry +breaking in a single determinant: approach and application to the C60, C36, +and C20 fullerenes. Phys. Chem. Chem. Phys. 2019, 21, 4763. +37Kossoski, F.; Marie, A.; Scemama, A.; Caffarel, M.; Loos, P.-F. Excited +States from State-Specific Orbital-Optimized Pair Coupled Cluster. J. Chem. +Theory Comput. 2021, 17, 4756. +38Marie, A.; Kossoski, F.; Loos, P.-F. Variational coupled cluster for ground +and excited states. J. Chem. Phys. 2021, 155, 104105. +39Burton, H. G. A.; Wales, D. J. Energy Landscapes for Electronic Structure. +J. Chem. Theory Comput. 2021, 17, 151. +40Burton, H. G. A.; Gross, M.; Thom, A. J. W. Holomorphic Hartree–Fock +Theory: The Nature of Two-Electron Problems. J. Chem. Theory Comput. +2018, 14, 607. +41Coulson, C. A.; Fischer, I. XXXIV. Notes on the molecular orbital treatment +of the hydrogen molecule. Philos. Mag. 1949, 40, 386. +42Fukutome, H. The Unrestricted Hartree–Fock Theory of Chemical Reac- +tions. III: Instability Conditions for Paramagnetic and Spin Density Wave +States and Application to Internal Rotation of Ethylene. Prog. Theor. Phys. +1973, 50, 1433. +43Fukutome, H. Theory of the Unrestricted Hartree–Fock Equation and Its +Solutions. III: Classification of Instabilities and Interconnection Relation +between the Eight Classes of UHF Solutions. Prog. Theor. Phys. 1974, 52, +1766. +44Fukutome, H. Theory of the Unrestricted Hartree–Fock Equation and Its +Solutions III: Classification and Characterization of UHF Solutions by +Their Behaviour for Spin Rotation and Time Reversal. Prog. Theor. Phys. +1974, 52, 115. +45Ye, H.-Z.; Welborn, M.; Ricke, N. D.; Van Voorhis, T. σ-SCF: A direct +energy-targeting method to mean-field excited states. J. Chem. Phys. 2017, +147, 214104. +46Thom, A. J. W.; Head-Gordon, M. Locating Multiple Self-Consistent Field +Solutions: An Approach Inspired by Metadynamics. Phys. Rev. Lett. 2008, +101, 193001. +47Burton, H. G. A.; Thom, A. J. W. Reaching Full Correlation through +Nonorthogonal Configuration Interaction: A Second-Order Perturbative +Approach. J. Chem. Theory Comput. 2020, 16, 5586. +48Jensen, K. T.; Benson, R. L.; Cardamone, S.; Thom, A. J. W. Modeling +Electron Transfers Using Quasidiabatic Hartree–Fock States. J. Chem. The- +ory Comput. 2018, 14, 4629. +49Vaucher, A. C.; Reiher, M. Steering Orbital Optimization out of Local +Minima and Saddle Points Toward Lower Energy. J. Chem. Theory Comput. +2017, 13, 1219. +50Dong, X.; Mahler, A. D.; Kempfer-Robertson, E. M.; Thompson, L. M. +Global Elucidation of Self-Consistent Field Solution Space Using Basin +Hopping. J. Chem. Theory Comput. 2020, 16, 5635. +51Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; Dover Publications +Inc., 1989. +52Das, G.; Wahl, A. C. Extended Hartree—Fock Wavefunctions: Optimized +Valence Configurations for H2 and Li2, Optimized Double Configurations +for F2. J. Chem. Phys. 1966, 44, 87. +53Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. M. A complete active space SCF +method (CASSCF) using a density matrix formulated super-CI. Chem. Phys. +1980, 48, 157. +54Roos, B. O. The Complete Active Space SCF method in a Fock-Matrix- +Based Super-CI Formulation. Int. J. Quantum Chem. 1980, 18, 175. +55Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Windmark, P.-O. +Multiconfigurational Quantum Chemistry; Wiley, 2016. +56Das, G. Multiconfiguration self-consistent field (MCSCF) theory for excited +states. J. Chem. Phys. 1973, 58, 5104. +57Krauss, M.; Neumann, D. B. The 5Σ+ +g states of N2. Mol. Phys. 1976, 32, +101. +58Bauschlicher Jr., C. W.; Yarkony, D. R. Electronic structure of CaO. I. J. +Chem. Phys. 1978, 68, 3990. +59Bauschlicher Jr., C. W.; Yarkony, D. R. MCSCF wave functions for excited +states of polar moleculars: Application to BeO. J. Chem. Phys. 1980, 72, +1138. +60Bauschlicher Jr., C. W.; Lengsfield III, B. H.; Yarkony, D. R. On the low +lying singlet states of BeO. J. Chem. Phys. 1980, 73, 5702. +61Bauschlicher Jr., C. W.; Silver, D. M.; Yarkony, D. R. An SCF and MCSCF +description of the low-lying states of MgO. J. Chem. Phys. 1980, 73, 2867. +62Guihery, N.; Malrieu, J.-P.; Maynau, D.; Handrick, K. Unexpected CASSCF +Bistability Phenomenon. Int. J. Quantum Chem. 1997, 61, 45. +63Angeli, C.; Calzado, C. J.; Cimiraglia, R.; Evangelista, S.; Mayna, D. +Multiple complete active space self-consistent field solutions. Mol. Phys. +2003, 101, 1937. +64Shea, J. A. R.; Neuscamman, E. A mean field platform for excited state +quantum chemistry. J. Chem. Phys. 2018, 149, 081101. +65Shea, J. A. R.; Gwin, E.; Neuscamman, E. A Generalized Variational +Principle with Applications to Excited State Mean Field Theory. J. Chem. +Theory Comput. 2020, 16, 1526. +66Zhao, L.; Neuscamman, E. Excited state mean-field theory without auto- +matic differentiation. J. Chem. Phys. 2020, 152, 204112. +67Zhao, L.; Neuscamman, E. Density Functional Extension to Excited-State +Mean-Field Theory. J. Chem. Theory Comput. 2020, 16, 164. +68Hardikar, T. S.; Neuscamman, E. A self-consistent field formulation of +excited state mean field theory. J. Chem. Phys. 2020, 153, 164108. +69Kossoski, F.; Loos, P.-F. “State-Specific Configuration Interaction for Ex- +cited States” 2022, +70Dalgaard, E.; Jørgensen, P. Optimization of orbitals for multiconfigurational +reference states. J. Chem. Phys. 1978, 69, 3833. +71Dalgaard, E. A quadratically convergent reference state optimization proce- +dure. Chem. Phys. Lett. 1979, 65, 559. +72Yeager, D. L.; Jørgensen, P. Convergency studies of second and approximate +second order multiconfigurational Hartree–Fock procedures. J. Chem. Phys. +1979, 71, 755. +73Lengsfield III, B. H. General second order MCSCF theory: A density matrix +directed algorithm. J. Chem. Phys. 1980, 73, 382. +74Seigbahn, P. E. M.; Almlöf, J.; Heiberg, A.; Roos, B. O. The complete +active space SCF (CASSCF) method in a Newton-Raphson formulation +with application to the HNO molecule. J. Chem. Phys. 1981, 74, 2384. +75Werner, H.-J.; Meyer, W. A quadratically convergent MCSCF method for +the simultaneous optimization of several states simultaneous optimization +of several states. J. Chem. Phys. 1981, 74, 5794. +76Werner, H.-J.; Knowles, P. J. A second order multiconfigurational SCF +procedure with optimum convergence. J. Chem. Phys. 1985, 82, 5053. +77Yeager, D. L.; Jørgensen, P. A numerical study of the convergency of second +and approximate second-order multiconfiguration Hartree–Fock procedures. +Mol. Phys. 1980, 39, 587. + +14 +78Yeager, D. L.; Albertsen, P.; Jørgensen, P. Mode damping in multiconfigu- +rational Hartree–Fock procedures. J. Chem. Phys. 1980, 73, 2811. +79Jørgensen, P.; Olsen, J.; Yeager, D. L. Generalizations of Newton–Raphson +and multiplicity independent Newton–Raphson approaches in multiconfigu- +rational Hartree–Fock theory. J. Chem. Phys. 1981, 75, 5802. +80Yeager, D. L.; Lynch, D.; Nichols, J.; Jørgensen, P.; Olsen, J. Newton– +Raphson Approaches and Generalizations in Multiconfigurational Self- +Consistent Field Calculations. J. Phys. Chem. 1982, 86, 2140. +81Sun, Q.; Yang, J.; Chan, G. K.-L. A General Second Order Complete Active +Space Self-Consistent-Field Solver for Large-Scale Systems. Chem. Phys. +Lett. 2017, 683, 291. +82Kreplin, D. A.; Knowles, P. J.; Werner, H.-J. Second-order MCSCF opti- +mization revisited. I. Improved algorithms for fast and robust second-order +CASSCF convergence. J. Chem. Phys. 2019, 150, 194106. +83Kreplin, D. A.; Knowles, P. J.; Werner, H.-J. MCSCF optimization revis- +ited. II. Combined first- and second-order orbital optimization for large +molecules. J. Chem. Phys. 2020, 152, 074102. +84Rizzo, A.; Yeager, D. L. Characteristics and some peculiarities of multi- +configurational self-consistent field stationary points of the Li– ground +state. J. Chem. Phys. 1990, 93, 8011. +85Zaitsevskii, A.; Malrieu, J.-P. The discontinuities of state-average MCSCF +potential surfaces. Chem. Phys. Lett. 1994, 228, 458. +86Tran, L. N.; Neuscamman, E. Improving Excited-State Potential Energy +Surfaces via Optimal Orbital Shapes. J. Phys. Chem. A 2020, 124, 8273. +87Hanscam, R.; Neuscamman, E. Applying Generalized Variational Princi- +ples to Excited-State-Specific Complete Active Space Self-consistent Field +Theory. J. Chem. Theory Comput. 2022, 18, 6608. +88Tran, L. N.; Shea, J. A. R.; Neuscamman, E. Tracking Excited States +in Wave Function Optimization Using Density Matrices and Variational +Principles. J. Chem. Theory Comput. 2019, 15, 4790. +89Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic-Structure The- +ory; John Wiley & Sons, 2000. +90Head-Gordon, M.; Maslen, P. E.; White, C. A. A tensor formulation of +many-electron theory iin a nonorthogonal single-particle basis. J. Chem. +Phys. 1998, 108, 616. +91Hylleraas, E. A.; Undheim, B. Numerische Berechnung der 2S -Terme von +Ortho- und Par-Helium. Z. Phys. 1930, 65, 759. +92MacDonald, J. K. L. Successive Approximations by the Rayleigh-Ritz +Variation Method. Phys. Rev. 1933, 43, 830. +93Douady, J.; Ellinger, Y.; Subra, R.; Levy, B. Exponential transformation of +molecular orbitals: A quadratically convergent SCF procedure. I. General +formulation and application to closed-shell ground states. J. Chem. Phys. +1980, 72, 1452. +94Van Voorhis, T.; Head-Gordon, M. A geometric approach to direct mini- +mization. Mol. Phys. 2002, 100, 1713. +95Burton, H. G. A. Generalized nonorthogonal matrix elements: Unifying +Wick’s theorem and the Slater–Condon rules. J. Chem. Phys. 2021, 154, +144109. +96Burton, H. G. A. Generalized nonorthogonal matrix elements. II: Extension +to arbitrary excitations. J. Chem. Phys. 2022, 157, 204109. +97Mayer, I. Simple Theorems, Proofs, and Derivations in Quantum Chemistry; +Springer, 2003. +98Löwdin, P.-O. Quantum Theory of Many-Particle Systems. I. Physical +Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and +Convergence Problems in the Method of Configurational Interaction. Phys. +Rev. 1955, 97, 1474. +99Docken, K. K.; Hinze, J. LiH Potential Curves and Wavefunctions for X +1Σ+, A 1Σ+, B 1Π, 3Σ+, and 3Π. J. Chem. Phys. 1972, 57, 4928. +100Cerjan, C. J.; Miller, W. H. On finding transition states. J. Chem. Phys. +1981, 75, 2800. +101Wales, D. J. Structural and Topological Consequences of Anisotropic Inter- +actions in Clusters. Faraday Discuss. 1990, 86, 3505. +102Wales, D. J. Energy Landscapes: Applications to Clusters, Biomolecules +and Glasses; Cambridge University Press: Cambridge, 2004. +103Hoffmann, M. R.; Sherrill, C. D.; Leininger, M. L.; Schaefer III, H. F. Opti- +mization of MCSCF excited states using directions of negative curvature. +Chem. Phys. Lett. 2002, 355, 183. +104Nocedal, J.; Wright, S. Numerical Optimization; Springer-Verlag, 2006. +105Sun, Q. et al. Recent developments in the PySCF program package. J. Chem. +Phys. 2020, 153, 024109. +106Humphrey, W.; Dalke, A.; Schulten, K. VMD – Visual Molecular Dynamics. +J. Mol. Graph. 1996, 14, 33. +107Wolfram Research, Inc., Mathematica, Version 12.0.0. https://www. +wolfram.com/mathematica, Champaign, IL, 2021. +108Ditchfield, R.; Hehre, W. J.; Pople, J. A. Self-Consistent Molecular-Orbital +Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital +Studies of Organic Molecules. J. Chem. Phys. 1971, 54, 724. +109Sanchez de Meras, A.; Lepetit, M.-B.; Malrieu, J.-P. Discontinuity of +valence CASSCF wave functions around weakly avoided crossing between +valence configurations. Chem. Phys. Lett. 1990, 172, 163. +110Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. Self-consistent Molec- +ular Orbital Methods. XX. A Basis Set for Correlated Wave Functions. J. +Chem. Phys. 1980, 72, 650. +111Andersson, K.; Roos, B. O. Excitation energies in the nickel atom studied +with the complete active space SCF method and second-order perturbation +theory. Chem. Phys. Lett. 1992, 191, 507. +112Fukutome, H. Theory of the Unrestricted Hartree–Fock Equation and Its +Solutions. IV: Behavior of UHF Solutions in the Vicinity of Interconnecting +Instability Threshold. Prog. Theor. Phys. 1975, 53, 1320. +113Mestechkin, M. M. Restricted Hartree–Fock Method Instability. Int. J. +Quantum Chem. 1978, 13, 469. +114Mestechkin, M. Instability Threshold and Peculiar Solutions of Hartree– +Fock Equations. Int. J. Quantum Chem. 1979, 15, 601. +115Mestechkin, M. Potential Energy Surface near the Hartree–Fock Instability +Threshold. J. Mol. Struct.: THEOCHEM 1988, 181, 231. +116Trail, J. R.; Towler, M. D.; Needs, R. J. Unrestricted Hartree–Fock theory +of Wigner crystals. Phys. Rev. B 2003, 68, 045107. +117Burton, H. G. A. Hartree–Fock critical nuclear charge in two-electron atoms. +J. Chem. Phys. 2021, 154, 111103. +118Gilmore, R. Catastrophe Theory for Scientists and Engineers, 1st ed.; Dover +Publications Inc., 1993. +119Bauschlicher Jr., C. W.; Taylor, P. R. Benchmark full configuration- +interaction calculations on H2O, F, and F– . J. Chem. Phys. 1986, 85, 2779. +120Bauschlicher Jr., C. W.; Taylor, P. R. A full CI treatment of the 1A1–3B1 +separation in methylene. J. Chem. Phys. 1986, 85, 6510. +121Schaefer, H. F. Methylene: A Paradigm for Computational Quantum Chem- +istry. Science 1986, 231, 1100. +122Zarotiadis, R. A.; Burton, H. G. A.; Thom, A. J. W. Towards a Holomorphic +Density Functional Theory. J. Chem. Theory Comput. 2020, 16, 7400. +123Huynh, B. C.; Thom, A. J. W. Symmetry in Multiple Self-Consistent-Field +Solutions of Transition-Metal Complexes. J. Chem. Theory Comput. 2020, +16, 904. +124Jiménez-Hoyos, C. A.; Henderson, T. M.; Scuseria, G. E. Generalized +Hartree–Fock Description of Molecular Dissociation. J. Chem. Theory +Comput. 2011, 7, 2667. +125Thom, A. J. W.; Head-Gordon, M. Hartree–Fock solutions as a quasidiabatic +basis for nonorthogonal configuration interaction. J. Chem. Phys. 2009, +131, 124113. +126Bauschlicher, C. W.; Langhoff, S. R. Full configuration-interaction study of +the ionic-neutral curve crossing in LiF. J. Chem. Phys. 1988, 89, 4246. +127Malrieu, J.-P.; Heully, J.-L.; Zaitsevskii, A. Multiconfigurational second- +order perturbative methods: Overview and comparison of basic properties. +Theor. Chim. Acta 1995, 90, 167. +128Thom, R. Structural Stability and Morphogenesis, 1st ed.; Westview Press, +1994. + diff --git a/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/load_file.txt b/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc11b4cb9876b61badcc0e48775d19b6a4179344 --- /dev/null +++ b/qdFKT4oBgHgl3EQfHy2j/content/tmp_files/load_file.txt @@ -0,0 +1,2069 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf,len=2068 +page_content='Excited states, symmetry breaking, and unphysical solutions in state-specific CASSCF theory Antoine Marie1, 2 and Hugh G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Burton1, a) 1)Physical and Theoretical Chemical Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2)Current address: Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, Toulouse, France (Dated: 30 January 2023) State-specific electronic structure theory provides a route towards balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiconfigurational wave function approximations can describe both closed- and open- shell excited states and avoid the issues associated with state- averaged approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We investigate the existence of higher- energy solutions in complete active space self-consistent field (CASSCF) theory and characterise their topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large, or symmetry breaking when the active space is too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Furthermore, we investigate the conical intersection in CH2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Introduction Electronic excited states are fundamentally higher-energy so- lutions to the time-independent Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' “State- specific” representations can be identified using higher-energy stationary points of the electronic energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1 The ex- act excited states in full configuration interaction (FCI) corre- spond to energy saddle points and the number of downhill Hes- sian eigenvalues increases with each energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1–7 Higher- energy stationary points also exist in non-linear wave function approximations, but the development of practical state-specific methods has been hindered by the challenges of non-ground- state optimisation, the non-linearity of the electronic energy landscape, and the presence of unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Instead, the workhorse of modern excited-state electronic struture theory is linear-response time-dependent density func- tional theory (LR-TDDFT), which predicts excitation energies from the response of the ground-state electron density to a weak external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='8–10 Despite its computational efficiency, LR-TDDFT inherits the failures of approximate Kohn-Sham (KS) exchange-correlation functionals, creating large errors for bond dissociation or open-shell electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='11 Further- more, the ubiquitous adiabatic approximation excludes double excitations and their associated avoided crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10,12 Alter- native single-reference methods, such as algebraic diagram- matic construction13,14 (ADC) and equation-of-motion coupled cluster15,16 (EOM-CC) can provide more accurate excitation energies at a greater computational cost, but depend strongly a)Electronic mail: hgaburton@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='com on the quality of the reference determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The strong influ- ence of the ground-state orbitals can also create an unbalanced description of charge transfer and Rydberg excitations,17,18 where significant electronic relaxation can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='8–10,19–21 These challenges have encouraged researchers to revisit ex- cited state-specific approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For higher-energy SCF calculations (∆SCF), this progress has been catalysed by the development of new optimisation algorithms that avoid variational collapse to the ground state, including the maxi- mum overlap method,22–24 square-gradient optimisation,25,26 state-targeted energy projection,27 quasi-Newton direct or- bital optimisation,28–30 and generalised variational principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='31 Recent calculations have shown that higher-energy Hartree– Fock (HF) and KS-DFT solutions can accurately describe charge transfer and double excitations at a low computa- tional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='22,26 Beyond SCF approximations, higher-energy variational or projective coupled-cluster (∆CC) solutions can provide more accurate double and double-core exci- tations by incorporating dynamic electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='32–38 While ∆SCF and ∆CC are successful for double and charge transfer excitations, these single-reference methods can- not describe open-shell excited states and statically corre- lated ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The onset of this failure usually be- comes apparent through spin contamination,39,40 spontaneous symmetry breaking,11,23,39,41–45 and additional unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='32–35,37–39 Furthermore, the solutions of interest can disappear as the molecular structure changes, creating discon- tinuous excited-state energy surfaces or gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='38,39,46–50 Multiconfigurational SCF (MCSCF) methods,51 particu- larly the complete-active space self-consistent field (CASSCF) formulation,52–54 are the state-of-the-art for describing stati- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='11731v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='chem-ph] 27 Jan 2023 State-Specific CASSCF2 cally correlated electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='55 The CASSCF wave func- tion is a linear expansion of all the configurations that can be constructed from a set of partially occupied “active orbitals”, and the energy is optimised with respect to the configuration interaction (CI) and orbital coefficients simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='53 It has long been known that higher-energy MCSCF solutions can represent electronic excited states,56–61 and that multiple symmetry-broken CASSCF solutions can occur for an inad- equate active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='62,63 More recently, MCSCF expansions truncated to single excitations have shown promise for singly excited charge transfer states,64–68 while state-specific config- uration interaction with higher degrees of truncation can han- dle challenging multireference problems, singly-, and doubly- excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='69 However, the strong coupling between the or- bital and CI degrees of freedom makes the optimisation chal- lenging, and second-order optimisation algorithms are gener- ally required to reach convergence in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='70–83 Extensive research in the 1980s focused on characterising higher-energy MCSCF solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' It was originally suggested that an nth excited state approximation should be the nth state in the configuration expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='73 However, this requirement is often not achieved, resulting in “root flipping”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2,56,75 Further- more, several stationary points satisfying this condition can often be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='3,5,84 The enormous complexity of the mul- ticonfigurational solution space led Golab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' to conclude that “selecting an MCSCF stationary point is a very severe problem.”3 Instead, the state-averaged (SA) approach is gener- ally used, where a weighted average energy of the n lowest CI states constructed from one set of orbitals is optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='75 While this approach has become the method of choice for excited- state CASSCF, it has several disadvantages: discontinuities can occur on the SA-CASSCF potential energy surface if two states require orbitals with significantly different character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='85 the number of states is limited by the size of the active space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' large active spaces are required to target high-lying states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' and the Helmann-Feynamn theorem cannot be applied to compute nuclear gradients because individual SA-CASSCF solutions are not stationary points of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Recently, the limitations of SA-CASSCF and the develop- ment of non-ground-state SCF optimisation algorithms has in- spired several new investigations into state-specific CASSCF excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In particular, Neuscamman and co-workers have developed generalized variational principles86,87 and the WΓ approach inspired by MOM-SCF,88 demonstrating that the is- sues of root flipping and variational collapse to the ground state can be successfully avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Despite these advances, we still do not have a complete understanding of the multiple station- ary points on the SS-CASSCF energy landscape and several practical questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For example, how many stationary points are there and how does this change with the active space or basis set size?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Where do unphysical solutions arise, what are their characteristics, and when does symmetry breaking oc- cur?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' And finally, do state-specific excitations behave diabati- cally or adiabatically as the molecular structure evolves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Our aim in this work is to answer these questions and estab- lish a theoretical foundation for practical excited state-specific calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Using second-order optimisation techniques, we investigate the existence and properties of multiple CASSCF solutions in typical molecular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Our numerical optimi- sation exploits analytic gradients and second derivatives of the CASSCF energy, and the relevant differential geometry is sum- marised below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Using these techniques, we comprehensively enumerate the multiple CASSCF solutions in H2 (6-31G) and characterise the resulting unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We find that state-specific calculations can accurately describe high-lying excitations with fewer active orbitals than state-averaged for- malisms, and reveal that multiple solutions can arise from ac- tive spaces that are too large or too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We then investigate the conical intersection in CH2 (6-31G) and the avoided cross- ing of LiF (6-31G), demonstrating the importance and diffi- culty of selecting the correct physical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Exploring the multiconfigurational energy landscape A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Defining the CASSCF wave function A multiconfigurational wave function is defined as the linear combination of M Slater determinants |Ψk⟩ = M � I=1 CIk |ΦI⟩ , (1) where |ΦI⟩ represents different configurations built from a com- mon set of molecular orbitals (MO) φp(x) and the CIk are the variable CI coefficients for state k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='89 Here, x = (r, σ) is the combined spatial and spin electronic coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The MOs are constructed as linear combinations of n (nonorthogonal) atomic orbitals (AO) χµ(x) as φp(x) = n � µ χµ(x) cµ· p, (2) where we use the nonorthogonal tensor notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 90 and the cµ· p denote the variable MO coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Normalisation of the wave function, and orthogonalisation of the MOs, is guaranteed by the constraints M � I=1 |CI|2 = 1 and n � µ=1 (c∗)·µ p· ⟨χµ|χν⟩ cν· q = δpq, (3) where ⟨χµ|χν⟩ denotes the AO overlap matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We will only consider wave functions where CIk and cµ· p are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' When every electronic configuration for a finite basis set is included in an FCI expansion, the global minimum on the parametrised electronic energy landscape corresponds to the exact ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1 Excited states form saddle points of the energy and the number of downhill directions increases with each excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1,3,6,7 The FCI wave function is invariant to uni- tary transformations of the MOs, but the number of configura- tions scales exponentially with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The complete active space (CAS) framework builds a trun- cated expansion using every configuration within a set of “active orbitals” that describe the dominant static electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='53 The orbitals are partitioned into inactive and vir- tual orbitals that are doubly occupied or empty in every config- uration, respectively, and active orbitals with varying occupa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Simultaneously optimising the energy with respect to the 3 orbital and CI coefficients leads to the state-specific CASSCF approach and gives true stationary points of the electronic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='53,54,74 If the CASSCF wave function targeting the kth excited state is represented by the kth eigenstate of the cor- responding CAS-CI expansion, then the Hylleraas-Undheim- MacDonald theorem91,92 also provides a upper bound to the excited-state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Differential geometry of the CASSCF energy We exploit an exponential form of the CASSCF wave func- tion that conserves the orthogonality constraints [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='70,72 Starting from an initial CASSCF wave function |Ψ0⟩, an arbi- trary step can be defined using unitary transformations as |Ψ⟩ = e ˆRe ˆS |Ψ0⟩ , (4) where e ˆR and e ˆS account for orbital relaxation and transforma- tions of the CI component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The ˆR operator is anti- Hermitian and is defined using the second-quantised creation and annihilation operators for the current MOs as70,93 ˆR = � p>q Rpq ˆE− pq, (5) where the spin-adapted one-body anti-Hermitian replacement operators are89 ˆE− pq = � σ∈{↑,↓} ˆa† qσˆapσ − ˆa† pσˆaqσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (6) The invariance of the energy with respect to inactive-inactive, active-active, and virtual-virtual orbital transformations means that Rpq can be further restricted to only excitations between different sub-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Similarly, e ˆS performs a unitary transfor- mation between the CI component of |Ψ0⟩ and the remaining orthogonal states |ΨK⟩ in the current CASCI space, with ˆS de- fined as72 ˆS = � K�0 S K � |ΨK⟩ ⟨Ψ0| − |Ψ0⟩ ⟨ΨK| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (7) Using the exponential parametrisation, the CASSCF energy can be expressed as E(R, S) = ⟨Ψ0|e− ˆS e− ˆR ˆHe ˆRe ˆS |Ψ0⟩ , (8) where R and S are vectors that gather the Rpq and S K coeffi- cients in the orbital and CI transformations, respectively, and ˆH is the electronic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Stationary points of E, corre- sponding to optimal CASSCF solutions, then occur when the gradients with respect to orbital and CI transformations are si- multaneously zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Performing a Baker–Campbell–Hausdorff expansion of the energy to second order gives72 E ≈ ⟨Ψ0| ˆH|Ψ0⟩ + ⟨Ψ0|[ ˆH, ( ˆR + ˆS )]|Ψ0⟩ + 1 2 ⟨Ψ|[[ ˆH, ( ˆR + ˆS )], ( ˆR + ˆS )]|Ψ0⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (9) Expressions for the first- and second-derivates of the energy can then be identified as ∂E ∂Rpq ������R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='S=0 = ⟨Ψ0|[ ˆH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ˆE− pq]|Ψ0⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (10a) ∂E ∂S K �����R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='S=0 = 2 ⟨Ψ0| ˆH|ΨK⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (10b) and ∂2E ∂Rpq∂Rrs ������R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='S=0 = 1 2(1 + Ppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='rs) ⟨Ψ0|[[ ˆH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ˆE− pq],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ˆE− rs]|Ψ0⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (11a) ∂2E ∂Rpq∂S K ������R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='S=0 = ⟨Ψ0|[ ˆH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ˆE− pq]|ΨK⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (11b) ∂E ∂S L∂S K �����R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='S=0 = 2 ⟨ΨK| ˆH − E0|ΨL⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (11c) where E0 is the energy at R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' S = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ppq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='rs permutes the (pq) and (rs) indices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' and the Hermiticity of ˆH and [ ˆH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ˆE− pq] have been exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Explicit formulae for these expressions have been summarised elsewhere [see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 4] but are given in the Supporting Information (Section S1) for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Note that the first and second derivatives can only be com- puted when R = 0 and S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='93 Therefore, after taking a step in the parameter space, the energy gradient and Hessian must be computed using the new MOs and CI vectors corresponding to the updated wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A similar shift in the reference state after each step is also required for second-order HF opti- misation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='39,94 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Characterising distinct solutions The invariance to unitary transformations within each orbital partition means that the same CASSCF wave function can be identified with different CI or MO coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We use the overlap between two stationary solutions |xΨ⟩ and |wΨ⟩ to define a positive semidefinite distance metric d(x, w) = 1 − | ⟨xΨ|wΨ⟩ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (12) The overlap for two arbitrary CI wave functions with Mx and Mw configurations, respectively, is given by ⟨xΨ|wΨ⟩ = Mx � I=1 Mw � J=1 xC∗ I ⟨xΦI|wΦJ⟩ wCJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (13) Since |xΨ⟩ and |wΨ⟩ have different sets of MOs, evaluating the overlap matrix elements ⟨xΦI|wΦJ⟩ requires a nonorthogo- nal framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We compute these matrix elements using the extended nonorthogonal Wick’s theory,95,96 which avoids the computationally expensive generalized Slater–Condon rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='97 To understand the MOs in a CASSCF solution, we canon- icalise the inactive and virtual orbitals and construct natural orbitals within the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The canonical inactive and virtual orbitals, and their associated orbital energies, are identi- fied by diagonalising the relevant sub-blocks of the Fock ma- trix, defined as89 Fpq = hpq + � rs γrs � (pq|sr) − 1 2(pr|sq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (14) Here, γpq denotes the one-body reduced density matrix ele- ments in the MO basis, hrq are the one-electron Hamiltonian 4 matrix elements, and (pq|rs) are the two-electron repulsion in- tegrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The natural orbitals within the active space are the eigenvectors of the one-body reduced density matrix and their eigenvalues are the occupation numbers np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='98 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Optimization techniques Since we are concerned with understanding the CASSCF solution space, we require an algorithm capable of converg- ing arbitrary stationary points on the energy landscape, includ- ing minima and higher-index saddle points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Higher-energy CASSCF stationary points are notoriously difficult to converge due to the strong coupling between the orbital and CI degrees of freedom,56,60,61,72,77 and the possibility of root flipping in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='75,99 Therefore, we employ second- order techniques that introduce the orbital-CI coupling through the analytic Hessian matrix of second derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These algo- rithms are too computationally expensive to be practical for larger systems, but they are sufficient for understanding the CASSCF solutions in small molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We search for multiple solutions using several initial guesses generated using random orbital and CI rotations from the ground state HF solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The eigenvector-following technique with analytic gradient and Hessian information was used to target stationary points with a particular Hessian index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='100,101 While this method has been described in detail elsewhere [see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 102], we include a summary in the Supporting Informa- tion (Section S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Related mode-following methods have previ- ously been applied to locate higher-energy electronic stationary points in multiconfigurational2–4,103 and single-determinant39 SCF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The convergence behaviour was further im- proved with a modified trust region approach based on the dog- leg method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='104 Trust region methods are a well-established ap- proach for controlling the convergence of second-order meth- ods in CASSCF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='77–80 Once a set of stationary points have been identified, their evolution with changes in the molecular structure can be determined by using the optimised orbital and CI coefficients at one geometry to define an initial guess at the next geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Since the Hessian index may not be conserved along a reaction coordinate,2 these subsequent cal- culations are performed using a trust region Newton–Raphson algorithm, as described in the Supporting Information (Sec- tion S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We have implemented this numerical optimisation in an ex- tension to the PySCF software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='105 The convergence threshold for the root-mean-squared value of the gradient am- plitudes was universally set to 10−8 Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The canonical and nat- ural orbitals for stationary points were subsequently computed using PySCF and visualised using VMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='106 All other graphi- cal figures were created using Mathematica 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='107 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Results and Discussion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Molecular H2 dissocation We start by considering the H2 binding curve using the 6- 31G basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='108 To identify all the CASSCF (2,2) solutions, a comprehensive search was performed using up to 1000 random starting points for target Hessian indices from 0 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Solu- tions were identified near the equilibrium geometry R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 a0 and the dissociation limit R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 a0, and were then traced over all bond lengths, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We believe that we have found every stationary point on the landscape, although the na- ture of non-convex optimisation means that this can never be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' To the best of our knowledge, this study is the first comprehensive enumeration of the CASSCF solutions for a molecular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Excitations near equilibrium Near the equilibrium geometry, the ground state of H2 can be accurately described using a single reference approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We have identified 25 stationary points on the CASSCF (2,2) energy landscape, corresponding to 19 singlet solutions and 6 triplet solutions (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Each of the exact FCI states has TABLE I: Energies of H2 at R = 1 a0 using the 6-31G basis set for various formalism: FCI, SA-CASSCF (2,2), SA-CASSCF (3,2), and SS-CASSCF (2,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' State FCI SA(2,2) SA(3,2) SS(2,2) ⟨S 2⟩ Index 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09897 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='07170 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08924 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='71948 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='71766 0 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='70046 0 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='69883 0 5 a corresponding SS-CASSCF (2,2) counterpart, and the ener- getic agreement between these solutions is consistent for all ex- citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We have also found several additional solutions that appear to be less accurate approximations to the exact states, which will be characterised in Sections III A 2 and III A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In comparison, the SA-CASSCF (2,2) approach can only describe the lowest triplet and the three lowest singlet states, while in- creasing the number of active orbitals to a (3,2) active space provides an approximation to the lowest nine excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These results demonstrate two important features of state- specific calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Firstly, they can describe more excited states than state-averaged calculations by defining the active space using only orbitals that are relevant for a particular ex- citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This property allows higher energy excitations to 5 H(2s2)− · · · H+ H(2s1) · · · H(2s1) H(1s12s1)− · · · H+ H(1s1) · · · H(2s1) H(1s2)− · · · H+ H(1s1) · · · H(1s1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1: State-specific CASSCF (2,2) stationary points can be identified for every excited FCI state in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Additional solutions can also be found that dissociate to an unphysical electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' be predicted while avoiding large active spaces and the asso- ciated increase in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' An upper bound to the exact excited state energy is only provided by station- ary points that correspond to the correct excitation within the CASCI configuration space,3 although more accurate energies are generally preferred even if they are not variational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Sec- ondly, bespoke orbital optimisation for each state-specific so- lutions can give more accurate total energies for the excited states compared to the state-averaged approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For exam- ple, the mean absolute deviations (MAD) for the lowest four states are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 mEh and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='8 mEh for the state-specific and state- averaged CASSCF (2,2) approaches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Using analytic second derivatives of the energy also allows the nature of SS-CASSCF (2,2) stationary points to be charac- terised according to their number of downhill directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The corresponding Hessian index for each solution is listed in Ta- ble I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' It is known that the exact n-th excited state should have n downhill directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1,2,4 We find that the SS-CASSCF (2,2) excited states are all saddle points on the electronic energy landscape and the Hessian index generally increases with the energy, in common with the observations for other theoretical approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='22,25,37,39 However, except for the lowest three exact states, the Hessian index does not provide a reliable in- dicator of the corresponding exact excitation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This mis- match must always occur for higher-lying excited states as the approximate CASSCF (2,2) wave function has fewer degrees of freedom than the exact formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Consequently, if we only consider stationary points of the correct Hessian index, then we must forgo the advantages of capturing state-specific excitations outside the state-averaged active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple ground state solutions While Table I shows that a SS-CASSCF (2,2) approxima- tion can be identified for each exact eigenstate, we also find additional state-specific solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In particular, there are three close-lying stationary points that can be considered as approxi- mations to the ground state, with Hessian indices of 0, 1, and 2 in order of ascending energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This pattern of multiple solutions is repeated for the (2σg)2 and (2σu)2 singlet configurations, while the other closed-shell (1σu)2 configuration exhibits four close-lying solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Choosing the most physical solution for each eigenstate presents a challenge for state-specific CASSCF approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Therefore, it is important that we understand their mathematical origins and physical differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The natural orbitals in the active space provides a clear ex- planation for the multiple H2 ground-state solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Figure 2A compares the natural orbitals and occupation numbers for the three lowest-energy singlet stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Since the ground state at the equilibrium geometry can be relatively well approx- imated by a single closed-shell Slater determinant, the active space for each of these solutions includes a (1σg)-like natural orbital that is almost completely doubly occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This natural orbital dominates the electronic wave function and the corre- sponding energies are all relatively close approximations to the exact ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, the second active orbital, which is almost completely unoccupied, is different for each solution, corresponding to a (1σu), (2σg), or (2σu) orbital as the energy increases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These higher-energy stationary points have downhill orbital rotations that interconvert the multiple ground state solutions and correspond to the negative eigenval- ues of the Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Different choices for the nearly unoccupied active orbital have only a small effect on the total H2 energy near equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' SS-CASSCF (2,2) Exact 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 2 3 4 5 6 Bond Length / ao6 (1σg) nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9998 (2σu) nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0002 E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='07871 Eh A (1σg) nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9925 (2σg) nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0076 E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08569 Eh (1σg) nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9887 (1σu) nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0113 E = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09225 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='Eh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='80 B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2: There are three SS-CASSCF(2,2) solutions that represent the exact ground state in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (A) Comparison of the natural orbitals for each ground-state solution at R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (B) Only the lowest-energy solution dissociates correctly, while the higher-energy solutions mirror the restricted Hartree–Fock binding curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, the incorrect choice of the active space becomes very significant as the bond is stretched towards dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Only the {1σg, 1σu} active space can correctly dissociate into the H(1s) · · · H(1s) ground state of the dissociated fragments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In contrast, the binding curves for the {1σg, 2σg} and {1σg, 2σu} solutions mirrors the RHF energy as the correspond- ing wave functions are close to a single Slater determinant at all geometries, with (1σg) occupation numbers at dissociation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='997 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Notably, the stationary points preserve the character of the active orbitals along the potential energy surface, suggesting that SS-CASSCF solutions exhibit some degree of diabatic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The same pattern of solutions is observed for the other closed-shell solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, the (1σu)2 configuration ex- hibits an additional multiple solution where the nearly unoccu- pied active orbital corresponds to a symmetry-broken 2s-like orbital localised on either the left or right H atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This sym- metry breaking results in a two-fold degenerate pair of station- ary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These results indicate that additional solutions can arise from the free choice of virtual orbitals when the active space is larger than required for the degree of static correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mal- rieu and co-works elegantly summarised this phenomenon by stating that “the so-called valence CASSCF wave function does not necessarily keep a valence character when the wave func- tion concentrates on a closed-shell valence bond structure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='109 Therefore, we expect that the number of ground state solu- tions will increase combinatorially with the number of active orbitals or the basis set size, and the number of unphysical so- lutions can grow for larger active spaces even though the cor- rect ground state solution will become more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Table II demonstrates this increase for H2 using the 6-311G basis set with three basis functions for each hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='110 Taking the (3,2) active space as an example, there are two redundant active orbitals beyond the 1σg that must be chosen from the five remaining orbitals, giving a total of ten solutions through the binomial coefficient �5 2 � = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The relative energy order- ing of these additional solutions will depend on the amount of dynamic correlation captured by the redundant active orbitals, which may not correspond with the same orbital required to capture the static correlation in the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This phenomenon has previously been described for MgO, where oxygen-centred orbitals are preferred over the magnesium d orbitals,87 and transition metal compounds where non-valence d orbitals may be preferred over certain valence d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='111 TABLE II: Close-lying ground-state (n, 2) SS-CASSCF energies (Eh) of H2 at R = 1 a0 using the 6-311G basis set for various active space size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' SS(1,2): HF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08025 SS(2,2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09429 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08866 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08074 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08033 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08026 SS(3,2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10195 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09500 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09436 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09429 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08904 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08886 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08867 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08082 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08075 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08034 SS(4,2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10251 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10212 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10196 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09507 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09437 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08923 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08905 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08886 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08083 SS(5,2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10267 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10251 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10213 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='09507 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='08924 SS(6,2): FCI -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10267 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Open-shell singlet and triplet excitations The low-lying open-shell triplet and singlet (1σg)1(1σu)1 configurations are represented by only one SS-CASSCF (2,2) 7 1 2 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 3: Spontaneous symmetry breaking occurs when the active space is not large enough to capture all the important configurations in the physical wave function, as illustrated for the 1s 2s states in the dissociation of H2 (6-31G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' solution across the full binding curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These single solutions arise because all the active orbitals are required to describe the two-configurational static correlation and there is no flexibility for multiple solutions to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In addition, SS-CASSCF (2,2) gives an accurate representation of the open-shell (1σg/u)1(2σg/u)1 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, the accu- racy of these solutions deteriorates in the dissociation limit, where additional symmetry broken solutions can be identified (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These additional solutions break spatial symmetry and spontaneously appear at instability thresholds that are multi- configurational analogues to the Coulson–Fischer points41 in HF theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='42,112–115 Each stationary point is a pure singlet or triplet state and has a two-fold degeneracy, reflecting the left- right symmetry of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The origin of this symmetry breaking is explained by con- sidering the correlation processes involved in the excited dis- sociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These excited states dissociate to hydrogenic (1s)1(2s)1 configurations, where the occupied 1s and 2s orbitals can either be on the same or different atomic centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Taking the latter case as an example, the corresponding open-shell sin- glet wave function at large nuclear separations has the form |Ψ⟩ = 1 2 � |1sL2sR⟩ + |1sR2sL⟩ �� |αβ⟩ − |βα⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (15) Correctly describing this wave function requires an active space with four spatial orbitals {1sL, 1sR, 2sL, 2sR}, or equivalently {1σg, 1σu, 2σg, 2σu}, and thus the SS-CASSCF (2,2) approx- imation is insufficient for these correlation mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In- stead, the symmetry breaking reduces the SS-CASSCF (2,2) wave function to a subset of the dominant configurations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' | ˜Ψ⟩ = 1√ 2 |1sL2sR⟩ � |αβ⟩ − |βα⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (16) The CASSCF configurations corresponding to each symmetry- broken solution are assigned in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This “pinning” of State Energy / Eh ⟨S 2⟩ Configuration A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='543355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 ������� |1sL2sL⟩ (|αβ⟩ − |βα⟩) |1sR2sR⟩ (|αβ⟩ − |βα⟩) B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='293363 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 ������� |1sL2sL⟩ (|αβ⟩ + |βα⟩) |1sR2sR⟩ (|αβ⟩ + |βα⟩) C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='037221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 ������� |1sL2sR⟩ (|αβ⟩ − |βα⟩) |1sR2sL⟩ (|αβ⟩ − |βα⟩) D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='037499 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 ������� |1sL2sR⟩ (|αβ⟩ + |βα⟩) |1sR2sL⟩ (|αβ⟩ + |βα⟩) TABLE III: The symmetry-broken CASSCF (2,2) solutions in the dissociation of H2 are two-fold degenerate and represent dominant configurations in the exact excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' the wave function onto a particular electronic configuration is directly analogous to the symmetry breaking phenomena observed in HF theory116,117 and demonstrates that the active space is too small to fully account for the static correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' From the energy landscape perspective, the onset of symmetry-broken CASSCF (2,2) states is associated with a change in the Hessian index for the associated symmetry-pure solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For example, the symmetry-broken state D (Ta- ble III) emerges from the symmetry-pure (1σg)1(2σg)1 triplet state at an instability threshold close to R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='28 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The Hes- sian index of the symmetry-pure state changes from 2 to 3 at this point, while the symmetry-broken solutions form index- 2 saddle points, leading to a higher-index analogue of a cusp catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='39,40,118 Practically, the emergence of a zero Hes- sian eigenvalue at these instability thresholds may hinder the numerical optimisation of second-order techniques onto these higher-energy stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' It is also interesting to note that, while the symmetry-broken solutions describe two degen- erate FCI states at dissociation, they only connect to one of the corresponding symmetry-pure solutions in the equilibrium region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Consequently, one cannot rely on these additional so- lutions to obtain an accurate and continuous representation of every excited state across all geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Conical Intersection in methylene We next consider the bending mode of methylene, which has a diradical ground state with 3B1 symmetry and a low-lying 1 1A1 excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The bond length was fixed to the value R(C−H) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='11 a0 identified by Bauschlicher and Taylor119,120 and the 6-31G basis set was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='108 Methylene has a long his- tory as a benchmark for electronic structure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='121 One of the primary questions is the description of the conical intersec- tion between the low-lying 3B1 and 1 1A1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Local minima for the minimal (2,2) active space A minimal two-configuration wave function is required to qualitatively describe both the lowest-energy singlet S0 (1A1) and diradical triplet T0 (3B1) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='119 Therefore, we begin by analysing the SS-CASSCF (2,2) energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The S0 8 80 100 120 140 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='90 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='88 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='86 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='84 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='82 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9398 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0602 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='86871 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0000 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0000 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='89125 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='8749 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1251 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='86793 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9789 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0211 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='87070 Eh C H H A B C D E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 4: Low-lying SS-CASSCF (2,2) in the bending mode of methylene represent the 1 3B1 and 1 1A1 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (A) Both states remain local minima (solid purple) for a short region beyond the conical intersection before becoming an index-1 saddle point (solid cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' An additional spin-contaminated index-1 saddle point (dashed cyan) connects the two instability thresholds (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Two degenerate local minima exist everywhere along the bending curve (dashed purple) with an active space containing C-H bonding σ and antibonding σ∗ orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (B–E) The natural orbitals at a bond angle of 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='7◦ are illustrated for each solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' and T0 are the ground state for small and large bond angles, respectively, and provide an example of a conical intersection separating the two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' At bond angles of 76◦, 102◦ and 130◦, a large number of stationary points can be identified with a variety of Hessian indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Therefore, we simplify our analysis by focussing on a subset of low-energy solutions that resemble the desired physical states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The energetic minimum of the S0 state occurs at a bond angle of 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While the S0 state is the first excited state at this geometry, we find that the corresponding SS-CASSCF (2,2) stationary point is a local minimum rather than an index-1 saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This incorrect Hessian index arises from a root flip in the configuration space, where the singlet state is the ground state for the corresponding active orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' When the bond angle increases, this singlet state eventually becomes an index-1 saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Similarly, when the bond angle decreases from 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='7◦, the T0 state remains a local minimum beyond the conical intersection where it becomes the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This process behaves like an unphysical hysteresis, where the ground state remains a local minimum for a small region after a conical intersection before becoming an index-1 saddle point at an instability threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' An additional index-1 saddle point can be identified that connects these two solutions and coalesces with each local minimum at the two instability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This unphysical index-1 stationary point is two-fold degenerate, has symmetry- pure spatial orbitals, but is spin contaminated with an ⟨ ˆS 2⟩ value that changes continuously from 0 to 2 as it connects the singlet and triplet states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Similar patterns of coalesc- ing solutions have been observed in single determinant SCF approximations,40,112,122,123 particularly in the generalised HF representation of conical intersections between states with dif- ferent ⟨ ˆS z⟩ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='124 In contrast to symmetry-broken SCF so- lutions, the spin contamination observed here arises from the mixture of singlet and triplet states in the configurational part of the CASSCF wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Since the S0 solution has only one significantly occupied active orbital, we predict the existence of closely-related so- lutions that have alternative redundant orbitals with nocc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Indeed, there are a pair of degenerate local minima that lie slightly lower in energy than the S0 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In contrast to the H2 ground state, including the inactive space means that methy- lene has multiple doubly occupied orbitals, and thus the active orbital with nocc ≈ 2 may also change between different solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The active orbitals for these symmetry-broken solutions are localised bonding σ and anti-bonding σ∗ orbitals for one of the two C–H bonds, and the degeneracy accounts for the two possible ways to localise onto one bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Notably, the symme- try breaking here is associated with an active space that is too large, in contrast to H2 where symmetry breaking arises from an insufficient active space for the static correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These so- lutions are local minima across all the bond angles considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While they provide an accurate energy for the S0 state near the singlet equilibrium structure, this deteriorates for large angles as the active space cannot describe the diradical open-shell 1Σ+ g state at the linear geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Their existence indicates that the C–H σ/σ∗ configurations provide an important contribution to static correlation and should ideally be included in the ac- tive space, as suggested by Bauschlicher and Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='119,120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Full valence active space Using the full-valence (6,6) active space, we find that the symmetry-pure singlet state is now correctly represented by an index-1 saddle point at a bond angle of 102° (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The unique downhill direction corresponds to a rotation in the con- figuration space only, as expected for the first excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' De- 9 80 100 120 140 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='95 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='94 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='93 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='92 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='91 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='90 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='89 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='88 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 5: Low-lying SS-CASSCF (6,6) for the bending mode of methylene represent the 1 3B1 and 1 1A1 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The full valence (6,6) active space introduces more unphysical solutions, but does not remove the spin-contaminated solution that arises at the conical intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' spite the larger active space, a root flip still occurs as the states approach the singlet–triplet conical intersection at 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2°, with the singlet state becoming a local minimum at 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='6° and the triplet state becoming an index-1 saddle point at 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Like the (2,2) active space, a degenerate pair of unphysical, spin- contaminated index-1 saddle points connect the solutions that cross at the conical intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This phenomenon occurs be- cause the orbital optimisation can lower the energy of the tar- get excited state below the corresponding ground state configu- ration when the energy gap becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Therefore, while larger active spaces will reduce the range of molecular geome- tries affected, these unphysical local minima will be common for state-specific conical intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While the larger active space alleviates root flipping, it also causes more unphysical solutions associated with redundant ac- tive orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For example, the triplet ground state (dominated by two configurations) is represented by one SS-CASSCF (2,2) solution, but there are several higher-energy solutions in the (6,6) active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Analogously to the H2 ground state, these additional solutions have a higher Hessian index, with two index-1 and one index-2 saddle points represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Again, the main difference from the true ground state are the active orbitals with occupation numbers close to zero, as il- lustrated for the global minimum and lowest-energy index-1 saddle point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Furthermore, we find an additional lo- cal minimum and index-1 saddle point that represent the 1A1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While all the triplet solutions give approximately the same equilibrium bond angle, the unphysical stationary points shift the conical intersection to coincide with the singlet equi- librium geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This qualitative change in the energy sur- face would create a near-barrierless decay from the singlet ex- cited state to the triplet ground state, demonstrating the impor- tance of verifying the physicality of state-specific solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Avoided crossing in Lithium Fluoride 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Physicality of multiple solutions The LiF binding curve provides a typical example of an avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The ground state has ionic character at equi- librium, but becomes a covalent state with almost no dipole mo- ment in the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple HF solutions are known to behave “quasi-diabatically” and cross each other at the phys- ical avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='47,125 On the other hand, Bauschlicher and Langhoff demonstrated that this avoided crossing can lead to discontinuities in the CASSCF ground- and excited-state energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='126 Here, we start by considering the state- specific singlet CASSCF solutions in the 6-31G basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Using the minimal (2,2) active, we search for stationary points with Hessian indices of 0 to 10 at R(Li−F) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='75 a0 (near the equilibrium geometry), using 1000 random starting points for each index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The active space for the SS-CASSCF global minimum contains the valence bonding σ and anti- bonding σ∗ orbitals with occupation numbers close to 2 and 0, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Because the exact wave function is dom- inated by a single closed-shell configuration, there are many additional solutions that are close to the ground-state energy at the equilibrium geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For example, the second lowest energy solution has an active space containing the out-of-plane fluorine 2px/y and 3px/y orbitals with occupation numbers close to 2 and 0, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This active space accounts for the radial correlation on the fluorine atom, providing a more balanced description of F and F–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='126 In contrast, the exact ex- cited state is more multiconfigurational at short bond lengths and is accurately represented by only one solution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7D), alongside a spurious symmetry-broken solution with diradical character (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These characteristics are reversed for bond lengths longer than the avoided crossing, where the excited state has closed-shell character with a large number of solu- tions and the ground state is represented by only two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' State-specific CASSCF solutions can behave both quasi- diabatically and adiabatically in the vicinity of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' As the bond length changes, the unphysical solutions do not have the correct active orbitals to capture the strong correlation at the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Therefore, the two lowest- energy unphysical solutions intersect quasi-diabatically (dark purple in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7A, corresponding to the solutions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7C and 7E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' On the other hand, the physically meaningful solu- tions behave adiabatically and correctly avoid each other (cyan in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In principle, a linear expansion of both the quasi- diabatic and adiabatic states may provide a more accurate rep- resentation of the avoided crossing by introducing some of the dynamic correlation captured by the unphysical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This expansion would require a multiconfigurational variant of nonorthogonal CI,125 where the Hamiltonian and overlap ma- trix elements can be efficiently computed using the nonorthog- onal framework developed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 95 and 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While a complete description of the avoided crossing re- quires dynamic correlation,127 the advantage of state-specific orbital relaxation is still clear in the dissociation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The physical SS-CASSCF excited state tends towards the exact FCI energy for the separated Li+ · · · F– configuration, while state- averaged calculations (with an equal weighting for the two states) overestimates the energy of this excited state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 10 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0000 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0284 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0224 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9801 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9677 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0014 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='93241 Eh A nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9978 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0254 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0029 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9985 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9745 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0009 E = −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='91416 Eh B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 6: Comparison of the active orbitals for the two lowest energy triplet CASSCF solutions for CH2 (6-31G) using the full valence (6,6) active space at a bond angle of 102◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (A) The active orbitals for the local minimum represent the chemically intuitive valence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' (B) For the unphysical index-1 saddle point, one of the antibonding C–H σ∗ orbitals with nocc ≈ 0 is replaced by a carbon 3p orbital with nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The remaining σ and σ∗ orbitals localise onto the C–H bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='15 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9814 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0186 E = −106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='89419 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='9908 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0092 E = −106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='88883 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='3806 nocc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='6194 E = −106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='76134 Eh nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0000 nocc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0000 E = −106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='78556 Eh A B C D E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7: (A) The SS-CASSCF (2,2) appoach gives many solutions for the LiF binding curve (6-31G) when the ground or excited state is dominated by a single configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ground- and excited-state solutions with a suitable active space (B and D) behave adiabatically at the avoided crossing (cyan lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Additional solutions with unsuitable active orbitals can represent either the ionic equilibrium configuration (C) or the covalent dissociation configuration (E), and behave quasi-diabatically at the avoided crossing (purple lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The active orbitals are plotted at R(Li−F) = 4 a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Exact FCI and SA(2)-CASSCF (2,2) data are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In this SS-CASSCF solution, the σ and σ∗ orbitals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7A) both localise to give 2pz orbitals that accurately represent the F– anion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Consequently, as expected, the state-specific for- malism provides a more accurate representation of this charge transfer excitation than a state-averaged approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Elucidating the Bauschlicher–Langhoff discontinuity The seminal CASSCF investigation of LiF, by Bauschlicher and Langhoff, highlighted the presence of a discontinuity in the ground-state dipole moment in the vicinty of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='126 This discontinuity is a signature of a discontinuity in the wave function, which manifests as a cusp in the corre- sponding energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This phenomenon, which we name the “Bauschlicher–Langhoff discontinuity”, has long been used as key evidence for the potential issues of state-specific cal- culations in the vicinity of an avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Malrieu and co-workers attributed its origin to a near degeneracy between the closed-shell ionic and the open-shell covalent configura- tions, and described a lower-energy covalent state that emerges from a potential symmetry-breaking point as the bond length increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='109 The framework developed here, and the advance in computing over the past 30 years, now allows this topologi- cal characterisation to be rigorously tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' To identify the relevant solutions, we searched for minima and index-1 saddle points at a bond length of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='50 a0 using 1000 random starting points, a (2, 2) active space, and the orig- ::11 4 6 8 10 12 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='95 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='90 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='85 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='6 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='840 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='835 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='830 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 8: Topology of the low-energy SS-CASSCF (2, 2) solutions near the Bausclicher–Langhoff discontinuity in LiF,109,126 using the basis set defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A cusp in the ground-state energy occurs when two local minima cross, while the covalent structure coalesces with an index-1 saddle point at a pair annihilation point (black dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' inal basis set described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' At R(Li−F) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='5 a0, the global minimum corresponds to the covalent structure identi- fied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In addition, two local minima and two index- 1 saddle points exist at higher energies, representing the ionic configurations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' As the bond length is shortened, there is a crossing between the lowest energy ionic and covalent min- ima near R(Li−F) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='4 a0, which we believe corresponds to the previously described discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='109,126 Topologically, two non-degenerate minima cannot coalesce without the presence of an index-1 saddle point, and thus the disappearance described by Malrieu and co-workers cannot be the full picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='109 Instead, we find that the covalent struc- ture crosses the two lowest-energy local minima and eventu- ally coalesces with an index-1 saddle point representing the ionic configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Following the downhill directions from this index-1 saddle points reveals that it connects the covalent local minimum with the lowest-energy ionic local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Furthermore, the downhill Hessian eigenvector has significant orbital and CI components, which highlights the strong cou- pling between the different degrees of freedom in the vicinity of the avoided crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Both solutions disappear at this point (black dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 8), and thus there is no quasi-diabatic cova- lent solution at shorter bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In the mathematical framework of catastrophe theory,128 this type of coalescence can be classified as a fold catastro- phe, or a pair annihilation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Singularities in this class have previously been identified and characterised for multiple HF solutions,112 where they most commonly occur in asymmetric molecules, for example LiF,47,125 H–Z40 (for a partial nuclear charge Z), and ethylene analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='40 The discontinuous jump in the energy at the pair annihilation point in LiF will create issues for calculations that attempt to follow the covalent so- lution across multiple bond lengths, making these solutions unsuitable for techniques such as ab initio molecular dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Furthermore, since the lowest energy covalent and ionic local minima cross rather than coalesce, the gradient of the global minimum energy at the crossing point is discontinuous and there is an unphysical cusp in the resulting energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The absence of this pair annihilation point using 6-31G com- pared to Bauschlicher and Langhoff’s basis set demonstrates how the topology of multiple CASSCF solutions can be af- fected by the AO basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We suspect that these differences arise from the subtle changes in the underlying energy landscape that affect the relative stability of different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, these results demonstrate the danger of generalising conclu- sions from one basis set to another, even for the same molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Concluding Remarks State-specific approximations promise to provide a more balanced representation of electronic excitations by indepen- dently optimising both the ground- and excited-state wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In this work, we have investigated the energy land- scape for excited state-specific stationary points in the multi- configurational CASSCF approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We have shown how state- specific approximations can accurately describe high-energy and charge transfer excitations, beyond the reach of state- averaged calculations with small active spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, the CASSCF energy landscape can have a large number of station- ary points, which complicates the selection and interpretation of physically relevant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple stationary points in state-specific CASSCF calcu- lations arise through two primary mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Firstly, many solutions occur when the active space is too large for the static correlation that must be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In this case, the redundant active orbitals with nocc ≈ 0 can be interchanged with virtual orbitals without significantly changing the energy, creating a series of stationary points with an increasing number of down- hill Hessian eigendirections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Active orbitals with nocc ≈ 2 can be interchanged with doubly occupied inactive orbitals in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' On the other hand, symmetry broken solutions occur when the active space is too small to describe the static correlation mechanisms, causing the CASSCF wave function to become “pinned” onto a subset of the configurations in the exact wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' These results demonstrate the importance of finding a “Goldilocks region”, where the active space is nei- ther too large or too small, but just right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Unphysical solutions can have important consequences for the resulting potential energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For example, while choosing the wrong active space only introduces a small energy error when the wave function is dominated by a single closed- shell configuration, it can prevent the CASSCF wave function from correctly capturing static correlation when the molecular structure changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The active space for stationary points does not change significantly along a reaction coordinate, meaning 12 that the incorrect active orbitals remain for all geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' For ground-state calculations, one can rely on following downhill directions away from saddle points to obtain a more suitable local minimum, hopefully with the best active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' However, it is hard to predict which Hessian index will give the most physical stationary point for an excited state, and thus choosing the most accurate excited-state stationary point is challenging without prior chemical intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' It has long been known that the right choice of active orbitals is key to the success of CASSCF, but the current results demonstrate the severity of this challenge for state-specific excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In addition, we have investigated the topology of SS- CASSCF (2, 2) solutions near the singlet-triplet conical inter- section in CH2 and the covalent-ionic avoided crossing in LiF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' We observe unphysical root flipping where the CH2 excited state solution is a local minimum near the conical intersec- tion, before becoming an index-1 saddle point further along the reaction trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' This phenomenon occurs because the state-specific orbital optimisation artificially stabilises the lo- cal minima, and is still present in the full valence (6, 6) active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Furthermore, the change in Hessian index is associated with an additional spin-contaminated index-1 saddle point that connects the singlet and triplet stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The pres- ence of zero Hessian eigenvalues at these instability thresholds may cause numerical issues for second-order optimisation al- gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' On the other hand, for the LiF avoided crossing, we have observed the coalescence of the local covalent minimum with an index-1 saddle point representing the ionic state, which both disappear entirely at shorter bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' While this pair- wise coalescence depends on the basis set, it would catastroph- ically affect the applicability of SS-CASSCF for generating smooth and continuous potential energy surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Moving forwards, SS-CASSCF calculations must overcome the troublesome issues of multiple solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Practical solu- tions may rely on the identification of suitable initial guesses from more black-box techniques, or by focussing on optimisa- tion algorithms that target desirable excited-state physical prop- erties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' dipole moments), such as the generalized variational principles developed by Hanscam and Neuscamman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='87 Alter- natively, more bespoke excited-state wave function ansätze, such as minimal configuration state functions69 or excited-state mean-field theory,64,65,68 may remove unphysical solutions as- sociated with redundant active orbitals and avoid the disappear- ance of solutions at pairwise coalescence points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Surmounting these issues will allow the benefits of state-specific calcula- tions for computing excited states, with bespoke orbitals and small active spaces, to be fully realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Supporting Information Derivation of the gradient and second-derivatives for the CASSCF energy, description of eigenvector-following and Newton–Raphson optimisation algorithms used (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Acknowledgements H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='B was supported by New College, Oxford, through the Astor Junior Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The authors thank David Tew for support and computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' References 1Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Energy Landscape of State-Specific Electronic Structure Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2022, 18, 1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2Olsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiconfigurational Hartree–Fock studies of avoided curve crossing using the Newton–Raphson technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1982, 76, 527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 3Golab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Proper characterization of MC SCF stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1983, 78, 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 4Olsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Optimization and Characterization of a Multiconfigurational Self-Consistent Field (MCSCF) State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' John Wiley and Sons, Ltd, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' pp 1–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 5Golab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple stationary point represen- tations in MC SCF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1985, 93, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 6Bacalis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Zang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Karaoulanis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Computing correct truncated excited state wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2016, 1790, 020007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 7Bacalis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' If Truncated Wave Functions of Excited State Energy Saddle Points Are Computed as Energy Minima, Where Is the Saddle Point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In Theoretical Chemistry for Advanced Nanomaterials: Functional Analysis by Computation and Experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Onishi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Springer Singapore: Singapore, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' p 465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 8Runge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gross, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Density-Functional Theory for Time-Dependent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1984, 52, 997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 9Dreuw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2005, 105, 4009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 10Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Werschnik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gross, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Time-dependent density func- tional theory: Past, present, and future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2005, 123, 062206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 11Hait, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rettig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Beyond the Coulson–Fischer point: characterizing single excitation CI and TDDFT for excited states in single bond dissociations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2019, 21, 21761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 12Maitra, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Cave, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Double excitations within time-dependent density functional theory linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2004, 120, 5932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 13Schirmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Beyond the random-phase approximation: A new approxima- tion scheme for the polarization propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A 1982, 26, 2395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 14Dreuw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Wormit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The algebraic diagrammatic construction scheme for the polarization propagator for the calculation of excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' WIREs Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2015, 5, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 15Stanton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The equation of motion coupled-cluster method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A systematic biorthogonal approach to molecular excitation energies, tran- sition probabilities, and excited state properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1993, 98, 7029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 16Krylov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker’s Guide to Fock Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2008, 59, 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 17Tozer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Relationship between Long-Range Charge-Transfer Excitation Energy Error and Integer Discontinuity in Kohn–Sham Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2003, 119, 12697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 18Dreuw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Failure of Time-Dependent Den- sity Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbateriochlorin–Bacteriochloring and Bacteriochlorophyll— Spheroidene Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2004, 126, 4007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 19McLachlan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ball, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Time-Dependent Hartree–Fock Theory for Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1964, 36, 844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 20Bartlett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Coupled-cluster theory and its equation-of-motion extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' WIREs Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2012, 2, 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 21Helmich-Paris, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Benchmarks for Electronically Excited States with CASSCF methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2019, 15, 4170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 22Gilbert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Besley, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Self-Consistent Field Calcu- lations of Excited States Using the Maximum Overlap Method (MOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A 2008, 112, 13164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 23Barca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gilbert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Communication: Hartree– Fock description of excited states of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2014, 141, 111104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 24Barca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gilbert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Simple Models for Difficult Electronic Excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2018, 14, 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 25Hait, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Orbital Optimized Density Functional Theory for Electronic Excited States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 12, 4517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 26Hait, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Excited State Orbital Optimization via Mini- mizing the Square of the Gradient: General Approach and Application to 13 Singly and Doubly Excited States via Density Functional Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 1699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 27Carter-Fenk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Herbert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' State-Targeted Energy Projection: A Simple and Robust Approach to Orbital Relaxation of Non-Aufbau Self-Consistent Field Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 5067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 28Levi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ivanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jónsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Variational Density Functional Calcula- tions of Excited States via Direct Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 6968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 29Levi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ivanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jónsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Variational calculations of excited states via direct optimization of the orbitals in DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Faraday Discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 224, 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 30Ivanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gianluca Levi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jónsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane-Wave Basis Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 17, 5034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 31Shea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Size Consistent Excited States via Algo- rithmic Transformations between Variational Principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2017, 13, 6078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 32Jankowski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jankowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Applicability of single- reference coupled-cluster methods to excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A model study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1994, 222, 608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 33Jankowski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jankowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple Solutions of the Single- Reference Coupled-Cluster Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Alternative Reference States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1994, 53, 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 34Piecuch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Kowalski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In Computational Chemistry: Reviews of Current Trends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Leszczynski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' World Scientific, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' In Search of the Relationship between Multiple Solutions Characterizing Coupled-Cluster Theories, p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 35Mayhall, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Raghavachari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple Solutions to the Single-Reference CCSD Equations for NiH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2010, 6, 2714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 36Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Distinguishing artifical and essential symmetry breaking in a single determinant: approach and application to the C60, C36, and C20 fullerenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2019, 21, 4763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 37Kossoski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Marie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Scemama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Caffarel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Loos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 17, 4756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 38Marie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Kossoski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Loos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Variational coupled cluster for ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 155, 104105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 39Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Wales, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Energy Landscapes for Electronic Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 17, 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 40Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gross, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Holomorphic Hartree–Fock Theory: The Nature of Two-Electron Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2018, 14, 607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 41Coulson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Fischer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' XXXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Notes on the molecular orbital treatment of the hydrogen molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1949, 40, 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 42Fukutome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The Unrestricted Hartree–Fock Theory of Chemical Reac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' III: Instability Conditions for Paramagnetic and Spin Density Wave States and Application to Internal Rotation of Ethylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1973, 50, 1433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 43Fukutome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory of the Unrestricted Hartree–Fock Equation and Its Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' III: Classification of Instabilities and Interconnection Relation between the Eight Classes of UHF Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1974, 52, 1766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 44Fukutome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory of the Unrestricted Hartree–Fock Equation and Its Solutions III: Classification and Characterization of UHF Solutions by Their Behaviour for Spin Rotation and Time Reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1974, 52, 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 45Ye, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Welborn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ricke, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Van Voorhis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' σ-SCF: A direct energy-targeting method to mean-field excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2017, 147, 214104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 46Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Locating Multiple Self-Consistent Field Solutions: An Approach Inspired by Metadynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2008, 101, 193001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 47Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 5586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 48Jensen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Benson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Cardamone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Modeling Electron Transfers Using Quasidiabatic Hartree–Fock States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The- ory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2018, 14, 4629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 49Vaucher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Reiher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Steering Orbital Optimization out of Local Minima and Saddle Points Toward Lower Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2017, 13, 1219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 50Dong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mahler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Kempfer-Robertson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thompson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Global Elucidation of Self-Consistent Field Solution Space Using Basin Hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 5635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 51Szabo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ostlund, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Modern Quantum Chemistry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Dover Publications Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 52Das, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Wahl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Extended Hartree—Fock Wavefunctions: Optimized Valence Configurations for H2 and Li2, Optimized Double Configurations for F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1966, 44, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 53Roos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Taylor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Sigbahn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A complete active space SCF method (CASSCF) using a density matrix formulated super-CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 48, 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 54Roos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The Complete Active Space SCF method in a Fock-Matrix- Based Super-CI Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 18, 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 55Roos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lindh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Malmqvist, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Veryazov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Windmark, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiconfigurational Quantum Chemistry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Wiley, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 56Das, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiconfiguration self-consistent field (MCSCF) theory for excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1973, 58, 5104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 57Krauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neumann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The 5Σ+ g states of N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1976, 32, 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 58Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yarkony, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Electronic structure of CaO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1978, 68, 3990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 59Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yarkony, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' MCSCF wave functions for excited states of polar moleculars: Application to BeO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 72, 1138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 60Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lengsfield III, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yarkony, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' On the low lying singlet states of BeO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 73, 5702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 61Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Silver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yarkony, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' An SCF and MCSCF description of the low-lying states of MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 73, 2867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 62Guihery, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Malrieu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Maynau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Handrick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Unexpected CASSCF Bistability Phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1997, 61, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 63Angeli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Calzado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Cimiraglia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Evangelista, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mayna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiple complete active space self-consistent field solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2003, 101, 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 64Shea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A mean field platform for excited state quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2018, 149, 081101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 65Shea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Gwin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A Generalized Variational Principle with Applications to Excited State Mean Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 1526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 66Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Excited state mean-field theory without auto- matic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 152, 204112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 67Zhao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Density Functional Extension to Excited-State Mean-Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 68Hardikar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A self-consistent field formulation of excited state mean field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 153, 164108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 69Kossoski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Loos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' “State-Specific Configuration Interaction for Ex- cited States” 2022, 70Dalgaard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Optimization of orbitals for multiconfigurational reference states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1978, 69, 3833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 71Dalgaard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A quadratically convergent reference state optimization proce- dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1979, 65, 559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 72Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Convergency studies of second and approximate second order multiconfigurational Hartree–Fock procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1979, 71, 755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 73Lengsfield III, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' General second order MCSCF theory: A density matrix directed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 73, 382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 74Seigbahn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Almlöf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Heiberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Roos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The complete active space SCF (CASSCF) method in a Newton-Raphson formulation with application to the HNO molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1981, 74, 2384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 75Werner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Meyer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A quadratically convergent MCSCF method for the simultaneous optimization of several states simultaneous optimization of several states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1981, 74, 5794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 76Werner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Knowles, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A second order multiconfigurational SCF procedure with optimum convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1985, 82, 5053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 77Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A numerical study of the convergency of second and approximate second-order multiconfiguration Hartree–Fock procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 39, 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 14 78Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Albertsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mode damping in multiconfigu- rational Hartree–Fock procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 73, 2811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 79Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Olsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Generalizations of Newton–Raphson and multiplicity independent Newton–Raphson approaches in multiconfigu- rational Hartree–Fock theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1981, 75, 5802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 80Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lynch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Olsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Newton– Raphson Approaches and Generalizations in Multiconfigurational Self- Consistent Field Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1982, 86, 2140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 81Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A General Second Order Complete Active Space Self-Consistent-Field Solver for Large-Scale Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2017, 683, 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 82Kreplin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Knowles, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Werner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Second-order MCSCF opti- mization revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Improved algorithms for fast and robust second-order CASSCF convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2019, 150, 194106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 83Kreplin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Knowles, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Werner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' MCSCF optimization revis- ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Combined first- and second-order orbital optimization for large molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 152, 074102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 84Rizzo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Yeager, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Characteristics and some peculiarities of multi- configurational self-consistent field stationary points of the Li– ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1990, 93, 8011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 85Zaitsevskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Malrieu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' The discontinuities of state-average MCSCF potential surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1994, 228, 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 86Tran, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Improving Excited-State Potential Energy Surfaces via Optimal Orbital Shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A 2020, 124, 8273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 87Hanscam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Applying Generalized Variational Princi- ples to Excited-State-Specific Complete Active Space Self-consistent Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2022, 18, 6608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 88Tran, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Shea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Neuscamman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Tracking Excited States in Wave Function Optimization Using Density Matrices and Variational Principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2019, 15, 4790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 89Helgaker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Jørgensen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Olsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Molecular Electronic-Structure The- ory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' John Wiley & Sons, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 90Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Maslen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' White, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A tensor formulation of many-electron theory iin a nonorthogonal single-particle basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1998, 108, 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 91Hylleraas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Undheim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Numerische Berechnung der 2S -Terme von Ortho- und Par-Helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1930, 65, 759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 92MacDonald, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Successive Approximations by the Rayleigh-Ritz Variation Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1933, 43, 830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 93Douady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Ellinger, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Subra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Levy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Exponential transformation of molecular orbitals: A quadratically convergent SCF procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' General formulation and application to closed-shell ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 72, 1452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 94Van Voorhis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A geometric approach to direct mini- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2002, 100, 1713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 95Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Generalized nonorthogonal matrix elements: Unifying Wick’s theorem and the Slater–Condon rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 154, 144109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 96Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Generalized nonorthogonal matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' II: Extension to arbitrary excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2022, 157, 204109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 97Mayer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Simple Theorems, Proofs, and Derivations in Quantum Chemistry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Springer, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 98Löwdin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Theory of Many-Particle Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1955, 97, 1474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 99Docken, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Hinze, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' LiH Potential Curves and Wavefunctions for X 1Σ+, A 1Σ+, B 1Π, 3Σ+, and 3Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1972, 57, 4928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 100Cerjan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Miller, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' On finding transition states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1981, 75, 2800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 101Wales, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Structural and Topological Consequences of Anisotropic Inter- actions in Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Faraday Discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1990, 86, 3505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 102Wales, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Energy Landscapes: Applications to Clusters, Biomolecules and Glasses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Cambridge University Press: Cambridge, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 103Hoffmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Sherrill, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Leininger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Schaefer III, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Opti- mization of MCSCF excited states using directions of negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2002, 355, 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 104Nocedal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Wright, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Numerical Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Springer-Verlag, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 105Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Recent developments in the PySCF program package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 153, 024109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 106Humphrey, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Dalke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Schulten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' VMD – Visual Molecular Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1996, 14, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 107Wolfram Research, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', Mathematica, Version 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='com/mathematica, Champaign, IL, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 108Ditchfield, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Hehre, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Self-Consistent Molecular-Orbital Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1971, 54, 724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 109Sanchez de Meras, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lepetit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Malrieu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Discontinuity of valence CASSCF wave functions around weakly avoided crossing between valence configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1990, 172, 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 110Krishnan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Binkley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Seeger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Pople, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Self-consistent Molec- ular Orbital Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A Basis Set for Correlated Wave Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1980, 72, 650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 111Andersson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Roos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Excitation energies in the nickel atom studied with the complete active space SCF method and second-order perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1992, 191, 507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 112Fukutome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory of the Unrestricted Hartree–Fock Equation and Its Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' IV: Behavior of UHF Solutions in the Vicinity of Interconnecting Instability Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1975, 53, 1320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 113Mestechkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Restricted Hartree–Fock Method Instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1978, 13, 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 114Mestechkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Instability Threshold and Peculiar Solutions of Hartree– Fock Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1979, 15, 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 115Mestechkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Potential Energy Surface near the Hartree–Fock Instability Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' : THEOCHEM 1988, 181, 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 116Trail, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Towler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Needs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Unrestricted Hartree–Fock theory of Wigner crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' B 2003, 68, 045107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 117Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Hartree–Fock critical nuclear charge in two-electron atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2021, 154, 111103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 118Gilmore, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Catastrophe Theory for Scientists and Engineers, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Dover Publications Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 119Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Taylor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Benchmark full configuration- interaction calculations on H2O, F, and F– .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1986, 85, 2779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 120Bauschlicher Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Taylor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A full CI treatment of the 1A1–3B1 separation in methylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1986, 85, 6510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 121Schaefer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Methylene: A Paradigm for Computational Quantum Chem- istry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Science 1986, 231, 1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 122Zarotiadis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Burton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Towards a Holomorphic Density Functional Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 7400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 123Huynh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Symmetry in Multiple Self-Consistent-Field Solutions of Transition-Metal Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2020, 16, 904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 124Jiménez-Hoyos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Henderson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Scuseria, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Generalized Hartree–Fock Description of Molecular Dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2011, 7, 2667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 125Thom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Hartree–Fock solutions as a quasidiabatic basis for nonorthogonal configuration interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 2009, 131, 124113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 126Bauschlicher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Langhoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Full configuration-interaction study of the ionic-neutral curve crossing in LiF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 1988, 89, 4246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 127Malrieu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Heully, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Zaitsevskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Multiconfigurational second- order perturbative methods: Overview and comparison of basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Chim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Acta 1995, 90, 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' 128Thom, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Structural Stability and Morphogenesis, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} +page_content=' Westview Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFKT4oBgHgl3EQfHy2j/content/2301.11731v1.pdf'} diff --git a/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/2301.03482v1.pdf.txt b/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/2301.03482v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b296c10c259b03204365d1be4e8ea6fe251cb7a7 --- /dev/null +++ b/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/2301.03482v1.pdf.txt @@ -0,0 +1,5724 @@ +A general maximal projection approach to uniformity +testing on the hypersphere +Jaroslav Borodavka and Bruno Ebner +January 10, 2023 +Abstract +We propose a novel approach to uniformity testing on the d-dimensional unit hypersphere Sd−1 +based on maximal projections. This approach gives a unifying view on the classical uniformity +tests of Rayleigh and Bingham, and it links to measures of multivariate skewness and kurtosis. We +derive the limiting distribution under the null hypothesis using limit theorems for Banach space +valued stochastic processes and we present strategies to simulate the limiting processes by applying +results on the theory of spherical harmonics. We examine the behavior under contiguous and fixed +alternatives and show the consistency of the testing procedure for some classes of alternatives. For +the first time in uniformity testing on the sphere, we derive local Bahadur efficiency statements. +We evaluate the theoretical findings and empirical powers of the procedures in a broad competitive +Monte Carlo simulation study and, finally, apply the new tests to a data set on midpoints of large +craters on the moon. +1 +Introduction +Testing uniformity on the circle, the sphere and the hypersphere Sd−1 = {x ∈ Rd : ∥x∥ = 1}, d ∈ N, +d ≥ 2, of Rd, endowed with the Euclidean norm ∥x∥ = +√ +x⊤x, are classical and still up-to-date research +fields in directional statistics. Here and in the following, ⊤ stands for the transpose of a matrix or a +vector. We numerate just a small subset of fields, where data on the surface of the unit hypersphere +Sd−1 is applied: meteorology, geology, paleomagnetism, political sciences, text mining and wildfire +MSC 2010 subject classifications. Primary 62G10 Secondary 62H15 +Key words and phrases uniformity tests, maximal projections, directional data, stochastic processes in Banach spaces, +contiguous alternatives, Bahadur efficiency, Monte Carlo simulations +1 +arXiv:2301.03482v1 [math.ST] 9 Jan 2023 + +orientation, for examples of such datasets, see [30] and the contributions therein. The first step to +serious statistical inference on Sd−1 is to check whether or not a sample of unit vectors stems from +the uniform law, since this distribution characterizes the absence of structure in directional data. To +be specific, we model the observed data by independent identically distributed (iid.) column random +vectors U, U1, . . . , Un taking values in Sd−1. The testing problem of interest is whether or not the +hypothesis +H0 : PU = U +� +Sd−1� +holds, against general alternatives. Here, PU stands for the distribution of U and U(·) denotes the +uniform distribution. This problem has been extensively studied in the literature: Lord Rayleigh pre- +sented the first test of uniformity in [38] based on the norm of the arithmetic mean. Rayleigh’s test +was followed by circular tests based on the classical goodness-of-fit measures of Kolmogorov-Smirnov +type in [27] and of Cram´er-von Mises type in [43]. Later, Bingham developed a test of uniformity in +[9] based on the sample scatter matrix and Gin´e, see [21], introduced the so-called Sobolev-tests. We +refer to [25, 33] for more details on these tests and to [24] for some new developments. More recently, +[10] proposed a Kolmogorov-Smirnov type test based on random projections, [16] suggest a procedure +using powers of volumes of nearest-neighbor spheres, and [19] consider the Cram´er-von Mises coun- +terpart to [10]. For details on this approach as well as more recent developments in uniformity testing +of axial data see [29], chapter 6. The authors of the review article [20] give an overview of uniformity +tests on the hypersphere. Comparative Monte Carlo simulation studies are found in [14] for d = 3 and +for higher dimensions in [17]. +A well-known characterizing property of U +� +Sd−1� +is invariance with respect to rotations about the +origin. Any test (say) Tn of uniformity should therefore inherit this structure and as such be invariant +under rotations, i.e. +Tn(AU1, . . . , AUn) = Tn(U1, . . . , Un) +holds for all A ∈ SO(d), +(1.1) +where SO(d) is the d-dimensional rotation group, i.e., for d × d-matrices A ∈ SO(d) we have AA⊤ = +A⊤A = Id and det(A) = 1. We denote the identity matrix by Id, and det(·) is the notation for the +determinant of a matrix. In the following, we call the property (1.1) rotational invariance of the test +statistic Tn. +We propose a novel class of statistics Tn, β based on powers of maximal projections. In this spirit +assume U ∼ U +� +Sd−1� +and by [7], we have using the rotational invariance of the uniform distribution +2 + +and symmetry arguments for every b ∈ Sd−1 and β ∈ N +ψd(β) = E(b⊤U)β = +����������� +Γ ((β + 1)/2) Γ (d/2) +√πΓ ((β + d)/2) +, +if β is even, +0, +if β is odd, +(1.2) +where Γ(·) denotes the Gamma function. Hence ψd(β) is independent of the choice of b, a property +that likewise follows by the rotation invariance of the uniform law on the sphere. Next, we define the +family of statistics +Tn, β = Tn, β(U1, . . . , Un) = n max +b∈Sd−1 +��������� +1 +n +n +� +j=1 +(b⊤U j)β − ψd(β) +��������� +2 +, +β ∈ N. +(1.3) +It is obvious that Tn, β is rotational invariant for every β due to the rotational invariance of the maximum +functional. +Interestingly, Tn, β has close connections to well-known classical tests such as the Rayleigh test, +the Bingham test and to measures of multivariate skewness and kurtosis by Malkovich and Afifi. First, +notice that with the sample mean of the observations Un = 1 +n +�n +j=1 U j we have +Tn,1 = n max +b∈Sd−1 +��������� +1 +n +n +� +j=1 +b⊤U j +��������� +2 += n∥Un∥2 max +b∈Sd−1 +������b⊤ Un +∥Un∥ +������ +2 += n∥Un∥2, +since the scalar product in the maximum is the cosine of the angle between the two unit vectors, which +takes its maximum for b = +Un +∥Un∥. Hence we have an equivalent test as the classical Rayleigh test, see +[38], given by Rn = 2nd∥Un∥2. +Second, with the sample scatter matrix S = 1 +n +�n +j=1 U jU⊤ +j we have +Tn,2 = n max +b∈Sd−1 +��������� +1 +n +n +� +j=1 +(b⊤U j)2 − 1 +d +��������� +2 += n max +b∈Sd−1 +� +b⊤S b − 1 +d +�2 += n max +b∈Sd−1 +� +b⊤ +� +S − 1 +d Id +� +b +�2 +, +Notice that Tn,2 is the squared spectral norm of S − E(UU⊤) for U ∼ U +� +Sd−1� +, hence it compares the +scatter matrix to the covariance matrix of U, which is in the same spirit as the Bingham test, see [9]. +Note that by the Courant–Fischer–Weyl min-max principle from linear algebra, we have +Tn,2 = n(max(|λmin|, |λmax|))2, +where λmin and λmax are the minimal and maximal eigenvalues of the symmetric matrix S − 1 +d Id. +Third, we have +Tn,3 = n max +b∈Sd−1 +��������� +1 +n +n +� +j=1 +(b⊤U j)3 +��������� +2 +and +Tn,4 = n max +b∈Sd−1 +��������� +1 +n +n +� +j=1 +(b⊤U j)4 − +3 +d(d + 2) +��������� +2 +, +(1.4) +3 + +which can be interpreted as analogs to the multivariate sample skewness and sample kurtosis by +Malkovich and Afifi, for a definition see [31]. For Tn, β, β > 2, no explicit closed form and easy to +calculate formula is known. The authors of [31] suggest using the Newton-Raphson method to obtain +a good approximation of the maximal value in (1.3). Since for such a numerical routine, the choice +of some good start values is not straightforward, we suggest to use a random approach, see Section 6, +which is related to the idea of random projections as suggested in [10]. +The rest of the paper is organized as follows: We present asymptotic theory under the null hy- +pothesis in Section 2. In Section 3 we derive the behaviour of Tn,β for contiguous alternatives. We +show consistency of the tests against some classes of fixed alternatives in Section 4. Afterwards, we +establish local approximate and exact asymptotic relative efficiency statements in the Bahadur sense +in Section 5. We examine the theoretical findings by a Monte Carlo simulation study in Section 6 and +provide a real data application to midpoints of large craters on the moon in Section 7. Conclusions as +well as an outlook are provided in Section 8. We finish the article by three Appendices A, B and C +that contain facts on d-dimensional Legendre polynomials and spherical harmonics, as well as some +technical Lemmas and proofs. +2 +Asymptotic null distribution of Tn, β +Let C(Sd−1, R) be the Banach space of continuous functions f : Sd−1 → R, equipped with the norm +∥f∥∞ = supb∈Sd−1 |f(b)|. We introduce the stochastic process +Zn,β(b) = √n +��������� +1 +n +n +� +j=1 +(b⊤U j)β − ψd(β) +��������� , +b ∈ Sd−1. +For the covariance structure in the following theorem, we write +ηβ(b, c) = E(b⊤U) β(c⊤U) β = +β +� +j=0 +(cj, d(β))2 +νd(j) +P d +j (b⊤c), +b, c ∈ Sd−1. +(2.1) +Here, P d +j (·) is the d-dimensional Legendre polynomial of order j, for a definition see (A.4), νd(j) is +the dimension of the space of d-dimensional spherical harmonics of order j, see (A.1), and cj,d(β) are +constants only depending on j, d and β, compare with (A.5) and Proposition A.8. An explicit way of +calculation can be found in Appendix B. +Theorem 2.1. Let U1, . . . , Un be iid. with U1 ∼ U +� +Sd−1� +. For fixed β ∈ N there exists a centred +4 + +Gaussian process Zβ(b), b ∈ Sd−1 with continuous sample paths and covariance kernel +ρβ(b, c) = ηβ(b, c) − ψ2 +d(β), +b, c ∈ Sd−1. +(2.2) +Regarding Zβ(·) as a random element of C(Sd−1, R), we have +Zn,β(·) +D +−→ Zβ(·). +Remark 2.2. For the special cases of Section 1 and some higher powers β, we have +ρ1(b, c) += +1 +d(b⊤c), +ρ2(b, c) += +1 +d(d + 2) +� +2(b⊤c)2 + 1 +� +− 1 +d2 , +ρ3(b, c) += +1 +d(d + 2)(d + 4) +� +6(b⊤c)3 + 9(b⊤c) +� +, +ρ4(b, c) += +� +24(b⊤c)4 + 72(b⊤c)2 + 9 +� +3 +� +j=0 +(d + 2 j)−1 − +9 +d2(d + 2)2 , +ρ5(b, c) += +� +120(b⊤c)5 + 600(b⊤c)3 + 225(b⊤c) +� +4 +� +j=0 +(d + 2 j)−1, +ρ6(b, c) += +� +720(b⊤c)6 + 5400(b⊤c)4 + 4050(b⊤c)2 + 225 +� +5 +� +j=0 +(d + 2 j)−1 − +225 +d2(d + 2)2(d + 4)2 , +and thus explicit formulas for the covariance kernel in Theorem 2.1. +Note that the covariance kernel ρβ(b, c) solely depends on the scalar product b⊤c and hence can be +written as a function (say) ρβ(b, c) = Q(b⊤c), where Q ∈ C([−1, 1], R) is a polynomial of degree β. +Kernels of this particular structure are called zonal kernels, for an application of Gaussian processes +with zonal covariance kernel in machine learning see [15]. The fact that |ρβ(b, c)| ≤ 1 for all β ∈ +N follows by the inequalities of Cauchy–Schwarz and Popoviciu, since the projections are bounded +random variables. +Define the integral operators Kβ for β ∈ N given by +Kβ f(x) = +1 +|Sd−1| +� +Sd−1 ρβ(ω, x)f(ω) dσ(ω), +x ∈ Sd−1, f ∈ L2(Sd−1, dσ), +(2.3) +where integration is with respect to the unique spherical Lebesgue measure σ on Sd−1. Since ρβ is +continuous on a compact set of Rd, the operator Kβ is compact from L2(Sd−1, dσ) to L2(Sd−1, dσ). Due +to the zonal covariance structure we can even show that Kβ is a finite-rank operator, i.e., an operator +whose range is finite-dimensional. The latter, and other properties, are presented and proved in the next +5 + +proposition. In the following, we denote by Hk(Sd−1) the space of d-dimensional spherical harmonic +functions of order k ∈ N0, for a definition see [22]. +Proposition 2.3. Let β ∈ N and Kβ be defined as in (2.3). +i) For any spherical harmonic φ ∈ Hk(Sd−1) of order k ∈ N0, we have Kβφ = λkφ, where +λk = +��������������� +�ck, d(β) +νd(k) +�2 +, +for 0 < k ≤ β, +0 +, +for k = 0 or k > β, +(2.4) +with constants ck, d(β) ∈ R, depending only on k, d and β, compare with Proposition A.8, and +νd(k) = dim(Hk(Sd−1)). +ii) Kβ is a finite-rank operator. +iii) The spectrum of Kβ consists of 0 and the eigenvalues in (2.4). +iv) Kβ is positive, i.e., we have +� +Kβ f, f +� +L2 ≥ 0 for all f ∈ L2(Sd−1, dσ). +In the spirit of [8], we thus have alternative representations of the limiting Gaussian process for our +special cases. +Proposition 2.4. Let νd(k) be the dimension of the space of d-dimensional spherical harmonics of order +k ∈ N, see (A.1). +i) If β is odd, the limiting Gaussian process Zβ(b), b ∈ Sd−1, can be represented in the form +Zβ(b) = +� +|Sd−1| +β +� +k=1 +k odd +� +λk +νd(k) +� +j=1 +φk, j(b)Nk,j, +b ∈ Sd−1. +Here, Nk,j, k = 1, 3, . . . and j = 1, 2, . . . , νd(k), is an array of independent unit normal vari- +ables, λk is the eigenvalue in (2.4), and ϕk,j are j = 1, 2, . . . , νd(k) linearly independent surface +harmonics of degree k being orthonormal with respect to σ/|Sd−1|, compare with the proof of +Proposition 2.3, ii). +ii) If β is even, the limiting Gaussian process Zβ(b), b ∈ Sd−1, can be represented in the form +Zβ(b) = +� +|Sd−1| +β +� +k=1 +k even +� +λk +νd(k) +� +j=1 +φk,j(b)Nk,j, +b ∈ Sd−1. +6 + +Here, Nk,j, k = 0, 2, 4, . . . and j = 1, 2, . . . , νd(k), is an array of independent unit normal vari- +ables, λk is the eigenvalue in (2.4), and ϕk,j are j = 1, 2, . . . , νd(k) linearly independent spherical +harmonics of degree k being orthonormal with respect to σ/|Sd−1|, compare with the proof of +Proposition 2.3, ii). +Proposition 2.4 shows an easy way to simulate Gaussian random processes on the sphere with a +polynomial covariance kernel. What is essentially needed are three ingredients: the positive eigenval- +ues (which can be calculated explicitly), an array of independent unit normal variables and an imple- +mentation of spherical harmonics, see Section 6 for more details. For a generation method of a suitable +basis of spherical harmonics, see [3], Section 2.11, or [13], Theorem 1.1.9. The package HFT.m in +Mathematica, see [23], provides a direct way to calculate an orthonormal basis of spherical harmon- +ics in any dimension d and any order k based on Theorem 5.25 in [4]. Note that explicit versions of +orthonormal systems up to order 4 in any dimensions can be found in [32], Tables 1 and 2. +Remark 2.5. +• Case β = 1: We have νd(1) = d and u �→ uk ∈ H1(Sd−1) for all k = 1, . . . , d, where +u = (u1, . . . , ud) ∈ Sd−1. These functions form an orthogonal system of H1(Sd−1), see [22], +Lemma 3.2.3. Normalization w.r.t. σ yields the orthonormal basis functions +u �→ +� +d +|Sd−1|uk ∈ H1(Sd−1), +k = 1, . . . , d. +We have a single positive eigenvalue λ1 = 1/d2. With Proposition 2.4 it follows +Z1(u) = +1√ +d +d +� +j=1 +ujN j, +u ∈ Sd−1, +N1, . . . , Nd +uiv∼ N(0, 1). +Moreover, putting N = (N1, . . . , Nd) ∼ Nd(0, Id) the Cauchy–Schwarz Inequality yields +Z2 +1(u) = 1 +d +d +� +j,k=1 +ujukNjNk = 1 +d(u⊤N)2 ≤ 1 +d∥N∥2. +Hence we obtain +d max +u∈Sd−1 Z2 +1(u) = ∥N∥2 ∼ χ2 +d, +thus recovering the limit result of the Rayleigh-Test. +• Case β = 2: By (A.1) it is νd(2) = (d + 2)(d − 1)/2. Straightforward calculations yield the single +positive eigenvalue +λ2 = +� +2 +d(d + 2) +�2 +. +7 + +Let φ2,1, . . . , φ2,νd(2) be an orthonormal basis of H2(Sd−1), see [32], Table 1, for an explicit +representation, and set φ2(u) = (φ2,1(u), . . . , φ2,νd(2)(u)), u ∈ Sd−1. With Proposition A.4 and A.6 +it follows that +∥φ2(u)∥2 = +νd(2) +� +i=1 +φ2,i(u)φ2,i(u) = νd(2) +|Sd−1|P d +2 (u⊤u) = νd(2) +|Sd−1|, +u ∈ Sd−1. +Therefore, putting N = (N1, . . . , Nνd(2)) ∼ Nνd(2) +�0, Iνd(2) +�, gives us again using the Cauchy– +Schwarz Inequality +Z2 +2(u) = +4|Sd−1| +d2(d + 2)2 +��������� +d +� +j=1 +φ2,j(u)Nj +��������� +2 +≤ +4νd(2) +d2(d + 2)2 ∥N∥2 = 2(d − 1) +d2(d + 2)∥N∥2. +Since ∥N∥2 ∼ χ2 +νd(2), a comparison of 95% quantiles of 2(d − 1)∥N∥2/(d2(d + 2)) with Table 2 +shows that this upper bound is only a good approximation for d = 2. +In the following we give a list of the non-null eigenvalues λk, k = 0, . . . , β, in (2.4) corresponding +to higher values of β. +• Case β = 3: λ1 = (3/(d(d + 2)))2 and λ3 = (6/(d(d + 2)(d + 4)))2. +• Case β = 4: λ2 = (12/(d(d + 2)(d + 4)))2 and λ4 = (24/(d(d + 2)(d + 4)(d + 6)))2. +• Case β = 5: λ1 = (15/(d(d + 2)(d + 4)))2, λ3 = (60/(d(d + 2)(d + 4)(d + 6)))2, and λ5 = +(120/(d(d + 2)(d + 4)(d + 6)(d + 8)))2. +• Case β = 6: λ2 = (90/(d(d + 2)(d + 4)(d + 6)))2, λ4 = (360/(d(d + 2)(d + 4)(d + 6)(d + 8)))2, +as well as λ6 = (720/(d(d + 2)(d + 4)(d + 6)(d + 8)(d + 10)))2. +Since Sd−1 is compact, a direct application of the continuous mapping theorem and Theorem 2.1 +prove the following Corollary to Theorem 2.1. +Corollary 2.6. Let U1, . . . , Un be iid. with U1 ∼ U +� +Sd−1� +. Then we have +Tn, β +D +−→ max +b∈Sd−1Z2 +β(b), +where Zβ(·) is the limiting Gaussian process of Theorem 2.1. +The resulting limit random variable in Corollary 2.6 is not of pure theoretic interest, since the +distribution and hence the asymptotic critical value can be approximated, see Section 6. +8 + +3 +Contiguous alternatives +In this section, we consider a triangular array Un1, . . . , Unn of rowwise identically independent dis- +tributed random vectors on Sd−1 having the density function fn(x) = µ(x) +� +1 + h(x)/ √n +� +, x ∈ Sd−1, +where µ(·) denotes the density of the uniform distribution with respect to the spherical Lebesgue mea- +sure σ, and h is a bounded measurable function satisfying +� +Sd−1 h(x)µ(x) dσ(x) = 0. We consider n +large enough to assure the non-negativity of fn. +First, define +P(n) = +n +� +j=1 +(µ σ) +and +A(n) = +n +� +j=1 +(fn σ) +on the measurable space (Xn, Bn) = +n +� +j=1 +(Sd−1, M), where M denotes the class of subsets of Sd−1 that +are measurable with respect to σ. Further, denote the likelihood ratio with Ln = dA(n) +dP(n) and write +S β : Sd−1 × Sd−1 → R, +(a, b) �→ S β(a, b) = (a⊤b) β − ψd(β), +where ψd(·) is defined in (1.2). +Theorem 3.1. Under the standing assumptions we have for the triangular array Un1, . . . , Unn +Zn,β(·) +D +−→ Zβ(·) + S ∗ +β(·) +under A(n) +in C(Sd−1, R), where Zβ(·) is a centred Gaussian process in C(Sd−1, R) having covariance kernel ρβ +from Theorem 2.1. The shift function S ∗ +β(·) is given by +S ∗ +β(b) = +1 +|Sd−1| +� +Sd−1 S β(u, b)h(u) dσ(u), b ∈ Sd−1. +(3.1) +As a direct consequence of Theorem 3.1 and the continuous mapping theorem we have the follow- +ing Corollary. +Corollary 3.2. Under the conditions of Theorem 3.1, we have +Tn, β +D +−→ max +b∈Sd−1 +� +Zβ(b) + S ∗ +β(b) +�2 +under A(n). +Example 3.3. As an example we consider the alternatives where +h(x) = hm,θ(x) = P d +m(θ⊤x), +x ∈ Sd−1, m ≥ 1, +9 + +is the Legendre polynomial of degree m and θ ∈ Sd−1 is fixed. Note that x �→ P d +m(θ⊤x) is a spherical +harmonic function of degree m such that the orthogonality property of spherical harmonics, see [22], +Section 3.2, implies +� +Sd−1 h(x)dσ(x) = +� +Sd−1 P d +m(θ⊤x)P d +0 (θ⊤x) dσ(x) = 0, +since P d +0 (θ⊤·) = 1 is the spherical harmonic of degree 0. If X follows the law given by the density fn +we have for any orthogonal d × d-matrix A with Aθ = θ that the distribution of AX is the same as the +distribution of X, hence these types of alternatives are rotationally symmetric about θ. An application +of the Funk/Hecke-Theorem A.3 shows for b ∈ Sd−1 +S ∗ +β(b) += +1 +|Sd−1| +� +Sd−1 S β(u, b)h(u) dσ(u) += +1 +|Sd−1| +� +Sd−1 +� +(b⊤u)β − ψd(β) +� +P d +m(θ⊤u) dσ(u) += +λd(β, m)P d +m(θ⊤b), +where +λd(β, m) = |Sd−2| +|Sd−1| +� 1 +−1 +P d +m(t) +� +tβ − ψd(β) +� +(1 − t2) +d−3 +2 dt. +It follows with (A.5) and Proposition A.5 +λd(β, m) = |Sd−2| +|Sd−1| +��������� +β +� +j=0 +cj, d(β)⟨P d +m, P d +j ⟩ − ψd(β)⟨P d +m, P d +0 ⟩ +��������� = cm, d(β) +νd(m) , +so that +S ∗ +β(b) = cm, d(β) +νd(m) P d +m(θ⊤b), +b ∈ Sd−1. +(3.2) +Note that λd(β, m) = 0, if β + m is odd or m > β, because then the coefficients cm, d(β) in (A.5) equal +zero. Hence, in these cases we have the same asymptotic behaviour under contiguous alternatives as +under the null hypothesis. We can conclude that the tests Tn, β are not able to detect the alternatives for +such a combination of β and m. For the shift function we have S ∗ +β(θ) = cm, d(β)/νd(m), so that there is a +non-negative shift in the limiting distribution under contiguous alternatives as long as we can show that +the coefficients cm, d(β) are non-negative. We conjecture that this is indeed the case, as all examples +after Proposition fulfill this property. That in turn means that, under the assumption of this conjecture, +there is a positive shift if β + m is even and m ≤ β. Thus, Tn, β is a family of testing procedures which is +able to detect the contiguous alternatives. As indicated in [11], Section 2, the famous von Mises–Fisher +distribution (see [33], Section 9.3, for a definition) with mean direction θ and concentration parameter +κ ≥ 0 falls into a comparable class of contiguous alternatives. We expect to see matchable power +performances of Tn, β in the simulation study, see Section 6. +10 + +4 +Consistency +In this short section we consider spherical random vectors U, U1, . . . , Un with a distribution having a +continuous density f w.r.t. the spherical Lebesgue measure σ. We adopt the reasoning in [8] to argue +that the considered tests are consistent against a large class of alternatives. If for β ∈ N there is a unit +vector (say) b0 ∈ Sd−1 such that +ζ(b0) = +� +E(b⊤ +0 U)β − ψd(β) +�2 > 0, +the strong law of large numbers shows +lim +n→∞ ζn(b0) = lim +n→∞ +��������� +1 +n +n +� +j=1 +(b⊤ +0 U j)β − ψd(β) +��������� +2 += ζ(b0) +a.s. +and since Tn, β/n ≥ ζn(b0) we have +lim +n→∞ Tn, β = ∞ +a.s. +This reasoning shows that the tests Tn, β are consistent against each such alternative. Nonetheless, as +we have already seen in the last section, Tn, β is not consistent against any arbitrary alternative class. +For certain combinations of β and m, the order of the Legendre polynomial, Tn, β exhibits the same +asymptotic behaviour under the alternatives as under the null hypothesis. Another indication for the +inconsistency of Tn, β can be seen in the case β = 1, which essentially concerns the Rayleigh test. The +authors of [18] have shown that, in the rather general context of rotationally symmetric alternatives +with a location and concentration parameter and a defining angular function, the Rayleigh test is blind +against certain local alternatives. These local alternatives show polynomial decrease of the concen- +tration parameter towards zero (hence yielding the null hypothesis) and the odd-order derivatives of +their angular function vanish at zero. An example of such an alternative is the well-known Watson +distribution. +5 +Bahadur efficiencies +In this section we present some interesting insights into the Bahadur asymptotic relative efficiencies +(ARE) of the statistics (Tn, β)β∈N. For an elaborate and comprehensive introduction to the concept of +Bahadur efficiency we refer the reader to [5] and [35]. +We consider alternative classes whose defining density f(· | κ) w.r.t. σ is parameterized through a +non-negative number κ ≥ 0, where the uniform distribution on Sd−1 is only obtained for the limit case +11 + +κ → 0+ in L1(Sd−1, dσ), i.e. +lim +κ→0+ ||f(· | κ) − f(· | 0)||L1 = lim +κ→0+ +� +Sd−1 |f(x | κ) − f(x | 0)| dσ(x) = 0. +(5.1) +Hence, the testing problem can be reformulated as +H0 : κ = 0 against H1 : κ > 0. +(5.2) +In order to properly apply the Bahadur theory, we consider the family of equivalent test statistics +�Tn, β = �Tn, β, β ∈ N. In the following we mainly focus our attention to the local approximate and the +local exact Bahadur ARE. For two statistics T (1) +n +and T (2) +n +these are defined as follows. +The local approximate Bahadur ARE is given by +Λa +T (1),T (2) = lim +κ→0+ +c a +T (1)(κ) +c a +T (2)(κ), +where c a +T denotes the approximate Bahadur slope of a statistic Tn, see [35], page 10. +The local exact Bahadur ARE is defined by +Λex +T (1),T (2) = lim +κ→0+ +cT (1)(κ) +cT (2)(κ), +where cT denotes the exact Bahadur slope of a statistic Tn, see [35], Section 1.2. In many cases the +approximate and exact Bahadur slopes coincide in the proximity of the null hypothesis, i.e. in the limit +case. This can also be observed for (�Tn, β)β∈N in the next proposition. +Proposition 5.1. Let β ∈ N. The approximate Bahadur slope of �Tn, β is given by +c a +�Tβ(κ) = +max +b∈Sd−1γ2 +κ(b) +max +b∈Sd−1ρβ(b, b) = +max +b∈Sd−1γ2 +κ(b) +�β +j=1 λjνd(j) +, +κ > 0, +with the eigenvalues λ j from Proposition 2.3, νd(j) as in (A.1) and +γκ(b) = Eκ(b⊤U) β − ψd(β), +b ∈ Sd−1, +where U ∼ f(· | κ), κ > 0. Furthermore, the exact Bahadur slope of �Tn, β is for sufficiently small κ > 0 +given by +c�Tβ(κ) = +max +b∈Sd−1γ2 +κ(b) +�β +j=1 λjνd(j) ++ o +� +max +b∈Sd−1γ2 +κ(b) +� +. +12 + +The Bahadur slopes of �Tn, β apparently coincide locally, so that we do not distinguish between +them anymore. In the following, we consider some explicit alternative classes and determine the local +Bahadur ARE of �Tn, β w.r.t. Mn = −2 log(Λn), where Λn is the likelihood-ratio test. It is well-known +that the exact and approximate Bahadur slope of Mn are the same and are given by +c a +M(κ) = cM(κ) = 2KL(κ, 0), +κ > 0, +where +KL(κ, κ0) = Eκ +� +log +� f(U | κ) +f(U | κ0 +�� +, +κ, κ0 ≥ 0, +is the Kullback–Leibler information number for U ∼ f(· | κ). The proof for the following quite technical +calculations can be found in Appendix B. +Example 5.2. A random vector U with values in Sd−1 has a von Mises–Fisher distribution vMF(θ, κ) +with mean direction θ ∈ Sd−1 and concentration parameter κ ≥ 0 if the density w.r.t. σ is given by +f(x | κ) = +(κ/2)d/2−1 +2πd/2I d +2 −1(κ) exp(κx⊤θ), +x ∈ Sd−1, +(5.3) +where I d +2 −1 is the modified Bessel function of the first kind and order d/2 − 1, see (B.2). In case of the +von Mises–Fisher alternative class vMF(θ, κ) with a fixed mean direction θ ∈ Sd−1 and κ > 0 we have +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λ1νd(1), +whereby the local Bahadur ARE is +Λex +�Tβ,M = Λa +�Tβ,M = lim +κ→0+ +c a +�Tβ(κ) +c a +M(κ) = +1 +�β +j=1 λjνd(j) +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += +λ1νd(1) +�β +j=1 λjνd(j) +. +The special case of β = 1 yields the local asymptotic optimality of �T1 in the Bahadur sense (see [35], +page 9, for this concept) +Λex +�T1,M = Λa +�T1,M = 1. +This is not surprising since the Rayleigh test is exactly the likelihood-ratio test in the von Mises–Fisher +model. On the contrary, if β is even, then +Λex +�Tβ,M = Λa +�Tβ,M = 0, +since the eigenvalue λ1 equals zero in this case. +13 + +β +d +2 +3 +5 +10 +β +d +2 +3 +5 +10 +vMF +1 +1.00 +1.00 +1.00 +1.00 +LP1 +1 +1.00 +1.00 +1.00 +1.00 +3 +0.90 +0.84 +0.77 +0.70 +3 +0.90 +0.84 +0.77 +0.70 +5 +0.79 +0.67 +0.54 +0.41 +5 +0.79 +0.67 +0.54 +0.41 +W +2 +1.00 +1.00 +1.00 +1.00 +LP2 +2 +1.00 +1.00 +1.00 +1.00 +4 +0.94 +0.92 +0.89 +0.84 +4 +0.94 +0.92 +0.89 +0.84 +6 +0.86 +0.80 +0.72 +0.61 +6 +0.86 +0.80 +0.72 +0.61 +LP3 +3 +0.10 +0.16 +0.23 +0.30 +LP4 +4 +0.06 +0.08 +0.11 +0.16 +5 +0.20 +0.31 +0.43 +0.54 +6 +0.14 +0.19 +0.27 +0.37 +LP5 +5 +0.01 +0.02 +0.03 +0.06 +LP6 +6 +0.004 +0.01 +0.01 +0.02 +Table 1: Non-trivial local Bahadur ARE of �Tn, β w.r.t. Mn for the alternative classes of the ex- +amples 5.2 to 5.4 with dimension d ∈ {2, 3, 5, 10} and order m ∈ {1, . . . , 6} of the LP alternative +class. +Example 5.3. A random vector U with values in Sd−1 has a Watson distribution W(θ, κ) with mean +direction θ ∈ Sd−1 and concentration parameter κ ∈ R if the density w.r.t. σ is given by +f(x | κ) = +Γ(d/2) +2πd/2M(1/2, d/2, κ) exp(κ(x⊤θ)2), +x ∈ Sd−1, +(5.4) +where M(·, ·, ·) is the Kummer function, see B.11. In the following, we only consider the case κ ≥ 0. +In case of the Watson alternative class W(θ, κ) with a fixed mean direction θ ∈ Sd−1 and κ > 0 we have +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λ2νd(2). +Thus the local Bahadur ARE equals +Λex +�Tβ,M = Λa +�Tβ,M = +λ2νd(2) +�β +j=1 λjνd(j) +. +This time the special case of β = 2 yields the local asymptotic optimality of �T2 in the Bahadur sense +Λex +�T2,M = Λa +�T2,M = 1, +whereas if β is odd, then +Λex +�Tβ,M = Λa +�Tβ,M = 0. +14 + +Example 5.4. In concordance with Example 3.3 we shall define the alternative class LPm(θ, κ) of order +m ∈ N with direction θ ∈ Sd−1 and κ ∈ [0, 1]. LP stands for Legendre polynomial in this context. Let +this class be given by the density +f(x | κ) = +1 +|Sd−1| +� +1 + κP d +m(θ⊤x) +� +, +x ∈ Sd−1. +(5.5) +For fixed order m and direction θ we have +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λmνd(m), +so that the local Bahadur ARE equals +Λex +�Tβ,M = Λa +�Tβ,M = +λmνd(m) +�β +j=1 λjνd(j) +. +We obtain non-trivial local Bahadur AREs only for combinations of β and m, where m ≤ β and β + m +is even. In particular, the special case β = m = 1 or β = m = 2 gives the local asymptotic optimality of +�T1 or �T2, respectively, in the Bahadur sense. +6 +Simulations +We present a competitive Monte-Carlo simulation study, that was implemented and performed in the +statistical computing environment R, see [36]. The maximum on the hypersphere in (1.3) cannot be +calculated analytically, and therefore one has to approximate it with a computationally fast method. We +suggest to use a uniform random cover of the hypersphere: Simulate a large number m of uniformly +distributed points on Sd−1, B1, . . . , Bm (say), evaluate the so chosen centered and squared projections +�1 +n +�n +j=1(B⊤ +k U j) β − ψd(β) +�2, k = 1, . . . , m, and approximate the maximum value in (1.3) by the discrete +maximum over all k. Critical values for Tn, β under H0 have been simulated with 20000 replications +and a random cover of m = 5000 points for d = 2, 3 and with 20000 replications and m = 20000 points +for d = 5, 10, see Table 2. +The critical values in the rows in Table 2 denoted by ”∞” and ”∞∗” represent approximations +of the limit random element maxb∈Sd−1 Z2 +β(b) in Corollary 2.6 via two methods. The first method, +which corresponds to the rows with ”∞”, simulates the same random cover of the sphere as above, +and it considers a large number (say) ℓ of random variables Z j = max(X2 +j), j = 1, . . . , ℓ, with iid. +X j ∼ Nm(0, Σβ), where Nm is the m-variate normal distribution and Σβ = +� +ρβ(Bk1, Bk2) +� +k1,k2∈{1,...,m} is +a singular m × m-covariance matrix for m ≥ d and ρβ is the covariance kernel in (2.2) for which we +15 + +have already summarized explicit formulas in Remark 2.2. Here x2 is shorthand for the vector of +squared components of x. Next, we calculate the empirical 95% quantile of Z1, . . . , Zℓ, where each +approximation was simulated with ℓ = 100000 and m = 1000 for d = 2, 3 as well as ℓ = 10000 and +m = 5000 for d = 5, 10. The second method utilizes the alternative representation of the Gaussian +process from Proposition 2.4. For this purpose, we need orthonormal bases of the spaces Hk(Sd−1) +for k = 0, . . . , β and the corresponding eigenvalues λk. We have already presented an explicit list +of these eigenvalues for β = 1, . . . , 6 at the end of section 2. In each replication step of the Monte- +Carlo simulation we generate an array Nk,j, j ∈ {1, . . . , νd(k)} of independent unit normal random +variables, cover Sd−1, once again, with m uniformly distributed points B1, . . . , Bm and calculate Yi = +� +|Sd−1| �β +k=0 +√λk +�νd(k) +j=1 φk,j(b)Nk,j, i = 1, . . . , m. Repeating this step for the number of set replications +ℓ yields an approximation of the limit distribution of Tn, β in the same fashion as before. However, +so far there is no library with a stable implementation of orthonormal spherical harmonics in higher +dimensions and orders, which is why we restricted the simulation with this method to the case of d = 2. +We used the package HFT.m in Mathematica in order to implement an orthonormal basis in R. Each +approximation was performed with ℓ = 20000 and m = 2500. +Table 2 shows empirical and approximated 0.95 quantiles of Tn, β under the null hypothesis. It +is interesting to compare the approximated critical values with the 0.95 quantiles of χ2 +d/d for Tn,1 +(respectively the Rayleigh test), which are 2.996 for d = 2, 2.605 for d = 3, 2.214 for d = 5, and +1.830 for d = 10. Evidently, the approximation with the random covering and the limiting process +is close to the theoretical asymptotic critical values for the dimensions d = 2, 3, 5, but it gets less +accurate for dimensions greater than 5. This behaviour can be explained by the curse of dimensionality, +indicating that more points on the unit sphere in the random covering should be considered to increase +the accuracy of the approximation. A similar behaviour can be observed for β = 2 and 2(d −1)/(d2(d + +2)) χ2 +νd(2) with νd(2) = (d − 1)(d + 2)/2, where, of course, the latter random variable is only an upper +bound for the limit distribution of Tn,2 as we have seen in Remark 2.5. Nevertheless, the numerical +results support the theoretical findings of Section 2. +We consider testing for uniformity on the unit circle S1, on the unit sphere S2 and on the hy- +persphere S5, and we divide the presentation of the simulation study into two parts, since different +competing tests are considered in these cases. +Generating uniformly distributed random numbers on Sd−1 can be done efficiently, since for a +random vector N ∼ Nd(0, Id), where Nd stands for the d-variate normal distribution on Rd, we have +N +||N|| ∼ U +� +Sd−1� +. +16 + +n +β +1 +2 +3 +4 +5 +6 +d = 2 +20 +2.906 +0.746 +2.037 +0.917 +1.761 +0.941 +50 +2.968 +0.730 +2.051 +0.914 +1.734 +0.934 +100 +3.004 +0.752 +2.031 +0.906 +1.730 +0.936 +500 +3.033 +0.735 +2.081 +0.923 +1.724 +0.939 +∞ +2.986 +0.750 +2.050 +0.924 +1.729 +0.944 +∞∗ +2.944 +0.753 +2.047 +0.923 +1.735 +0.945 +d = 3 +20 +2.582 +0.864 +1.306 +0.794 +0.994 +0.738 +50 +2.578 +0.875 +1.319 +0.751 +0.932 +0.666 +100 +2.562 +0.881 +1.293 +0.734 +0.922 +0.632 +500 +2.585 +0.869 +1.298 +0.733 +0.901 +0.614 +∞ +2.605 +0.866 +1.283 +0.724 +0.895 +0.606 +d = 5 +20 +2.183 +0.736 +0.729 +0.519 +0.444 +0.399 +50 +2.157 +0.695 +0.674 +0.451 +0.378 +0.318 +100 +2.203 +0.685 +0.654 +0.407 +0.352 +0.280 +500 +2.173 +0.663 +0.632 +0.363 +0.324 +0.232 +∞ +2.161 +0.638 +0.620 +0.340 +0.306 +0.206 +d = 10 +20 +1.567 +0.419 +0.252 +0.165 +0.107 +0.090 +50 +1.580 +0.368 +0.209 +0.124 +0.074 +0.060 +100 +1.586 +0.342 +0.190 +0.105 +0.059 +0.046 +500 +1.605 +0.314 +0.173 +0.080 +0.045 +0.030 +∞ +1.485 +0.277 +0.153 +0.062 +0.036 +0.019 +Table 2: Empirical and approximated 0.95 quantiles of Tn, β under H0 for dimensions d ∈ +{2, 3, 5, 10}, sample sizes n ∈ {20, 50, 100, 500} and β ∈ {1, . . . , 6}. Here, ∞ denotes the ap- +proximation of the limit distribution of Tn, β via covariance kernel and ∞∗ the approximation via +spherical harmonics. +17 + +This property is merely a consequence of the rotational invariance of N and the fact that the uniform +distribution is the only rotationally invariant distribution on Sd−1. We consider the following alterna- +tives to the uniform distribution: +• von Mises–Fisher distribution: +This alternative class was already introduced in Example 5.2. The density is given by +Sd−1 ∋ x �→ +(κ/2)d/2−1 +2πd/2I d +2 −1(κ) exp(κx⊤θ), +where I d +2 −1 is the modified Bessel function of the first kind and order d/2 − 1. This class is +denoted with vMF(θ, κ). +• Mix of von Mises–Fisher distributions with two centers: +Let U be uniformly distributed on (0, 1), p ∈ (0, 1) and Yi ∼ vMF(θi, κi) with corresponding lo- +cation and concentration parameters for i = 1, 2. Let U, Y1 and Y2 be stochastically independent. +Then we generate a random sample X according to +X = Y11{U 0 is the regularized, incomplete Beta +function and sign(·) the usual sign function on R. +• Cram´er-von Mises type test, [19]: +Lastly, we present another test that is based on a projection approach and comes from Garc´ıa- +Portugu´es et al., see [19]. It is, to a certain extent, the Cram´er-von Mises counterpart to the test +by Cuesta-Albertos et al. and, thus, considers the expected value +CvMn = n EH +�� 1 +−1 +|Fn,H(y) − Fd−1(y)|2 dFd−1(y) +� +. +Here, H +∼ +U +� +Sd−1� +, Fn,H is the empirical distribution function based on U1⊤H, . . . , +Un⊤H, and Fd−1 is the distribution function in (6.1). This test statistic can be written as a +U-statistic for a practical implementation in R according to +CvMn = 2 +n +� +1≤i 0), +ν3(k) = 2k + 1, +(A.2) +see [22], Section 3, for details on spherical harmonics. Two important theoretical results in the theory of +spherical harmonics are the density of finite linear combinations of spherical harmonics in L2(Sd−1, dσ) +and the orthogonality property of spherical harmonics of different order, which can be phrased as +follows. +Theorem A.1 ([42], Chapter 4, Corollary 2.3). The set of finite linear combinations of elements of +�∞ +k=0 Hk(Sd−1) is dense in L2(Sd−1, dσ). +Proposition A.2 ([42], Chapter 4, Corollary 2.4). For Φ ∈ Hk(Sd−1), Ψ ∈ Hl(Sd−1) with k � l, we +have +⟨Φ, Ψ⟩L2 = +� +Sd−1 Φ(ω)Ψ(ω) dσ(ω) = 0. +With these statements, it is possible to represent a function f ∈ L2(Sd−1, dσ) uniquely as a series of +spherical harmonics. For this purpose, we consider an orthonormal basis {φk,1, . . . , φk,νd(k)} of Hk(Sd−1) +for each k ∈ N0. Then �∞ +k=0{φk,1, . . . , φk,νd(k)} is an orthonormal basis of L2(Sd−1, dσ), and we can write +f +L2 += +∞ +� +k=0 +Ψk +(A.3) +with Ψk = �νd(k) +j=1 ⟨f, φk,j⟩L2φk,j ∈ Hk(Sd−1). +The following Theorem is called the Funk–Hecke- +Theorem and is used frequently. +Theorem A.3 (Funk–Hecke-Theorem, [22], Theorem 3.4.1). Let k ∈ N0 and u ∈ Sd−1. If Λ is a +bounded, integrable function on [−1, 1] and φ ∈ Hk(Sd−1), then the function Λ(u⊤·): Sd−1 → R; x �→ +Λ(u⊤x) is integrable and +� +Sd−1 Λ(u⊤x) φ(x) dσ(x) = λkφ(u) +with +λk = |Sd−2| +� 1 +−1 +P d +k (t)Λ(t)(1 − t2)(d−3)/2 dt, +where P d +k is the d-dimensional Legendre polynomial of order k. +The existence and uniqueness of higher dimensional Legendre polynomials are stated in the fol- +lowing Theorem. +34 + +Theorem A.4 ([22], Theorem 3.3.3). For each k ∈ N0, there is exactly one polynomial P d +k on [−1, 1] +with the property: If {φ1, . . . , φνd(k)} is an orthonormal basis of Hk(Sd−1), then +νd(k) +� +i=1 +φi(u)φi(v) = νd(k) +|Sd−1|P d +k (u⊤v), +u, v ∈ Sd−1. +The degree of P d +k is k, and the function P d +k (u⊤·): Sd−1 → R; v �→ P d +k (u⊤v) is for fixed u ∈ Sd−1 +a d-dimensional spherical harmonic of order k. Furthermore, P d +k is an even function, whenever k is +even, and an odd function, whenever k is odd. +The polynomial P d +k is called the d-dimensional Legendre polynomial of order k. Legendre polyno- +mials of different orders fulfill certain orthogonality properties. To state these properties, we introduce +the weighted scalar product +⟨ f, g⟩ = +� 1 +−1 +f(t)g(t)(1 − t2)(d−3)/2 dt +(A.4) +for bounded and integrable functions f, g on [−1, 1]. The following proposition shows that Legendre +polynomials are orthogonal w.r.t. this scalar product. +Proposition A.5 ([22], Proposition 3.3.6). Let k, l ∈ N0. If P d +k and P d +l are d-dimensional Legendre +polynomials of order k and l, respectively, then +� +P d +k , P d +l +� += δkl +|Sd−1| +νd(k) |Sd−2| = δkl +√π Γ((d − 1)/2) +νd(k) Γ(d/2) +. +Remark A.6 ([22], Lemma 3.3.5). Let k ∈ N0. For the d-dimensional Legendre polynomial of order k, +we have |P d +k (t)| ≤ 1 for all t ∈ [−1, 1] and P d +k (1) = 1. +The next result gives two explicit formulas for the calculation of Legendre polynomials of arbitrary +dimension and order. +Proposition A.7 ([34], Section 1.2, S.16 and Lemma 1.6.1). For k ∈ N0, we have +P d +k (t) = k! Γ +�d − 1 +2 +� ⌊ k +2 ⌋ +� +l=0 +� +−1 +4 +�l +(1 − t2)lt k−2l +l! (k − 2l)! Γ(l + d−1 +2 ) += +⌊ k +2⌋ +� +l=0 +a2l,ktk−2l, +t ∈ [−1, 1], +with +a0,0 = 1, +a2l,k = +� +−1 +4 +�l Γ(d − 1) +Γ(d/2) +2k−1k! +(k + d − 3)! +Γ(k − l + (d − 2)/2) +l!(k − 2l)! +, +l ∈ {0, . . . , ⌊k/2⌋}, k > 0, +where ⌊·⌋ is the lower Gauss bracket. +35 + +With these formulas, it is straightforward to obtain the first seven Legendre polynomials, which are +given by +P d +0 (t) = 1, +P d +1 (t) = t, +P d +2 (t) = +1 +d − 1[dt2 − 1], +P d +3 (t) = +1 +d − 1[(d + 2)t3 − 3t], +P d +4 (t) = +1 +(d − 1)(d + 1)[(d + 2)(d + 4)t4 − 6(d + 2)t2 + 3], +P d +5 (t) = +1 +(d − 1)(d + 1)[(d + 4)(d + 6)t5 − 10(d + 4)t3 + 15t], +P d +6 (t) = +1 +(d − 1)(d + 1)(d + 3)[(d + 4)(d + 6)(d + 8)t6 − 15(d + 4)(d + 6)t4 + 45(d + 4)t2 − 15]. +In a reverse conclusion, we can write any mth power, m ∈ N0, of a number t ∈ [−1, 1] as a linear +combination of Legendre polynomials +tm = +m +� +j=0 +cj, d(m)P d +j (t), +t ∈ [−1, 1], +(A.5) +with coefficients c j, d(m), which only depend on j, d and m. An elaborate calculation discloses the +explicit form of these coefficients. +Proposition A.8. Let k, l ∈ N0 with k ≥ 2l and [l] = 1, . . . , l. Then, with the coefficients a2l,k from +Proposition A.7, we have +ck−2l, d(k) = +����������������������� +1 +a0,k +, +for l = 0, +l� +r=1 +� +(l1,...,lr)∈[l]r +l1+···+lr=l +(−1)r a2l1,ka2l2,k−2l1 · · · a2lr,k−2(l1+···+lr−1) +a0,ka0,k−2l1 · · · a0,k−2l +, +for l > 0, +where l0 = 0. These are the only nonzero coefficients in (A.5) for given k ∈ N0. By the proof of +Proposition 2.3 i), we have in particular, c0, d(β) = ψd(β) for all β ∈ N. +Proof. The claim is obvious for k = 0, hence let k ∈ N. If l = 0, the second representation in +Proposition A.7 yields +tk = +1 +a0,k +P d +k (t) − a2,k +a0,k +tk−2 − · · · − a2g(k),k +a0,k +, +(A.6) +36 + +where g(k) = ⌊k/2⌋. It follows that ck, d(k) = 1/a0,k, since monomials of lower degree do not contain +Legendre polynomials of order k by (A.5). Let the claim hold true for all j ∈ {0, 1, ..., l − 1} for some +l ∈ N with k ≥ 2l. Next, we will prove the claim for this l and conclude the statement by means of the +principle of strong induction. The dots · · · are occasionally used for the sake of readability. +We repeatedly insert equation (A.6) for lower powers into (A.6). The induction hypothesis gives +tk = ck, d(k)P d +k (t) − a2,k +a0,k +� +1 +a0,k−2 +P d +k−2(t) − a2,k−2 +a0,k−2 +tk−4 − · · · − a2(l−1),k−2 +a0,k−2 +tk−2l − · · · − a2g(k−2),k−2 +a0,k−2 +� +− a4,k +a0,k +tk−4 − · · · − a2l,k +a0,k +tk−2l − · · · − a2g(k),k +a0,k += ck, d(k)P d +k (t) + ck−2, d(k)P d +k−2(t) − +�a4,k +a0,k ++ ck−2, d(k)a2,k−2 +� +tk−4 +− · · · − +�a2l,k +a0,k ++ ck−2, d(k)a2(l−1),k−2 +� +tk−2l − · · · , +since +ck−2, d(k) = +1 +� +r=1 +� +(l1,...,lr)∈[1]r +l1+···+lr=1 +(−1)r a2l1,ka2l2,k−2l1 · · · a2lr,k−2(l1+···+lr−1) +a0,ka0,k−2l1 · · · a0,k−2l += − +a2,k +a0,ka0,k−2 +. +Furthermore, we have +ck−4, d(k) = +2 +� +r=1 +� +(l1,...,lr)∈[2]r +l1+···+lr=2 +(−1)r a2l1,ka2l2,k−2l1 · · · a2lr,k−2(l1+···+lr−1) +a0,ka0,k−2l1 · · · a0,k−2l += − +a4,k +a0,ka0,k−4 ++ +a2,ka2,k−2 +a0,ka0,k−2a0,k−4 += − +�a4,k +a0,k ++ ck−2,d(k)a2,k−2 +� +1 +a0,k−4 +, +so that +tk = ck, d(k)P d +k (t) + ck−2, d(k)P d +k−2(t) + ck−4, d(k)P d +k−4(t) +− · · · − +�a2l,k +a0,k ++ ck−2, d(k)a2(l−1),k−2 + ck−4, d(k)a2(l−2),k−4 +� +tk−2l − · · · +If we continue this procedure iteratively, then we arrive at +tk = ck, d(k)P d +k (t) + · · · − +�������� +a2l,k +a0,ka0,k−2l ++ +l−1 +� +i=1 +ck−2i, d(k)a2(l−i),k−2i +a0,k−2l +�������� P d +k−2l(t) − · · · . +For the following, we abbreviate the quotient from the Proposition as +ql(l1, . . . , lr) = a2l1,ka2l2,k−2l1 · · · a2lr,k−2(l1+···+lr−1) +a0,ka0,k−2l1 · · · a0,k−2l +. +37 + +Once again, with the induction hypothesis it follows for each i ∈ N with i ≤ l − 1 +ck−2i, d(k)a2(l−i),k−2i +a0,k−2l += +i� +s=1 +� +(i1,...,is)∈[i]s +i1+···+is=i +(−1)sqi(i1, . . . , is)a2(l−i),k−2i +a0,k−2l += +i� +s=1 +� +(i1,...,is)∈[i]s×{l−i} +i1+···+is+1=l +(−1)sql(i1, . . . , is+1) +and, thus, finally +ck−2l, d(k) = − +�������� +a2l,k +a0,ka0,k−2l ++ +l−1 +� +i=1 +ck−2i,d(k)a2(l−i),k−2i +a0,k−2l +�������� += − +�������������� +a2l,k +a0,ka0,k−2l ++ +l−1 +� +i=1 +i� +s=1 +� +(i1,...,is)∈[i]s×{l−i} +i1+···+is+1=l +(−1)sql(i1, . . . , is+1) +�������������� += +l� +r=1 +� +(l1,...,lr)∈[l]r +l1+···+lr=l +(−1)rql(l1, . . . , lr), +since the first summand satisfies +− +a2l,k +a0,ka0,k−2l += +� +(l1)∈[l]1 +l1=l +(−1)1ql(l1). +In the second summand we sum over all tuples (i1, . . . , is+1) ∈ [i]s × {l − i} satisfying i1 + · · · + is+1 = l +for s = 1, . . . , i; i = 1, . . . , l − 1. This coincides exactly with the sum over all tuples (l1, . . . , lr) ∈ [l]r +satisfying l1 + · · · + lr = l for r = 2, . . . , l. +□ +For the first couple of powers, we have +t0 = P d +0 (t), +t = P d +1 (t), +t2 = d − 1 +d +P d +2 (t) + 1 +d, +t3 = d − 1 +d + 2P d +3 (t) + +3 +d + 2P d +1 (t), +t4 = (d − 1)(d + 1) +(d + 2)(d + 4)P d +4 (t) + 6(d − 1) +d(d + 4)P d +2 (t) + +3 +d(d + 2), +38 + +t5 = (d − 1)(d + 1) +(d + 4)(d + 6)P d +5 (t) + +10(d − 1) +(d + 2)(d + 6)P d +3 (t) + +15 +(d + 2)(d + 4)P d +1 (t), +t6 = (d − 1)(d + 1)(d + 3) +(d + 4)(d + 6)(d + 8)P d +6 (t) + +15(d − 1)(d + 1) +(d + 2)(d + 4)(d + 8)P d +4 (t) ++ +45(d − 1) +d(d + 4)(d + 6)P d +2 (t) + +15 +d(d + 2)(d + 4). +B +Technical Lemmas and Proofs +Lemma B.1. If U ∼ U +� +Sd−1� +, then +ηβ(b, c) = E(b⊤U) β(c⊤U) β = +β +� +j=0 +(cj, d(β))2 +νd(j) +P d +j (b⊤c), +b, c ∈ Sd−1, β ∈ N. +Proof. It follows by the Funk–Hecke-Theorem, Theorem A.3, applied to the spherical harmonic P d +j (c⊤·): Sd−1 → +R, c ∈ Sd−1, and the representation of a monomial via (A.5) +ηβ(b, c) = E(b⊤U) β(c⊤U) β = +1 +|Sd−1| +� +Sd−1(b⊤ω) β(c⊤ω) β dσ(ω) += +1 +|Sd−1| +� +Sd−1(b⊤ω) β +β +� +j=0 +c j, d(β) P d +j (ω⊤c) dσ(ω) += |Sd−2| +|Sd−1| +β +� +j=0 +cj, d(β) P d +j (b⊤c) +� 1 +−1 +t βP d +j (t)(1 − t2)(d−3)/2 dt += |Sd−2| +|Sd−1| +β +� +j=0 +β +� +l=0 +c j, d(β) cl, d(β) P d +j (b⊤c) +� 1 +−1 +P d +l (t)P d +j (t)(1 − t2)(d−3)/2 dt. +Due to the orthogonality properties of the d-dimensional Legendre polynomials, see Proposition A.5, +we can conclude +ηβ(b, c) = |Sd−2| +|Sd−1| +β +� +j=0 +β +� +l=0 +cj, d(β) cl, d(β) P d +j (b⊤c) +� +P d +l , P d +j +� += +β +� +j=0 +β +� +l=0 +cj, d(β) cl, d(β) P d +j (b⊤c) δjl +νd(j) = +β +� +j=0 +(cj, d(β))2 +νd(j) +P d +j (b⊤c). +□ +39 + +Lemma B.2. Under (P(n))n∈N we have for n → ∞ +log Ln +D +−→ N +� +−τ2 +2 , τ2 +� +, +where τ2 = +� +Sd−1 h2(x)σ(x) dx < ∞. +Proof. Using a Taylor expansion of the logarithm around x0 = 1 we have +log Ln(Un1, . . . , Unn) += +n +� +j=1 +log +� +1 + h(Un j) +√n +� += +n +� +j=1 +����� +h(Un j) +√n +− h2(Un j) +2n ++ Rn, j +����� , +with +Rn,j = 1 +3! +2 +� +1 + ηn,j +�3 +h3(Un,j) +n +3 +2 +, +(B.1) +where |ηn,j| ≤ |h(Un,j)| +√n +. Since h is bounded, we have +n +� +j=1 +Rn, j = oP(n)(1), and +E +��������� +1 +n +n +� +j=1 +h2 � +Un,j +� +��������� −→ τ2 as well as V +��������� +1 +n +n +� +j=1 +h2 � +Un,j +� +��������� −→ 0 +for n → ∞. The claim is a consequence of the Lindeberg–Feller CLT. +□ +The next Proposition deals with the technical calculations of Examples 5.2, 5.3 and 5.4. Before we +provide the proof, we present some special functions and their properties, compare with [33], Appendix +1. +Modified Bessel function of the first kind and order p ≥ 0: +Ip(κ) = +(κ/2)p +Γ(p + 1/2)Γ(1/2) +� 1 +−1 +eκt(1 − t2)p−1/2 dt, +κ > 0. +(B.2) +The function Ip satisfies +Ip(κ) = +∞ +� +r=0 +1 +Γ(p + r + 1)Γ(r + 1) +�κ +2 +�2r+p +, +p ≥ 0, κ > 0. +(B.3) +For κ > 0 and p ≥ 1 we have +κI′ +p(κ) = pIp(κ) + κIp+1(κ), +I′ +0(κ) = I1(κ). +(B.4) +40 + +For κ > 0 let +Ad(κ) = +Id/2(κ) +Id/2−1(κ). +(B.5) +For sufficiently small κ > 0 we have +Ad(κ) = κ +d − +κ3 +d2(d + 2) + O(κ5). +(B.6) +For κ > 0 we have +A′ +d(κ) = 1 − A2 +d(κ) − d − 1 +κ +Ad(κ). +(B.7) +For κ > 0 let +ad(κ) = 2πd/2 �κ +2 +�1−d/2 +Id/2−1(κ). +(B.8) +With the series expansion in (B.3) we obtain +ad(κ) = 2πd/2 �κ +2 +�1−d/2 � +1 +Γ(d/2)Γ(1) +�κ +2 +�d/2−1 ++ O(κd/2+1) +� +κ→0+ +→ |Sd−1|+. +(B.9) +An application of (B.4) yields for κ > 0 +∂κ[log ad(κ)] = +a′ +d(κ) +ad(κ) = Ad(κ). +(B.10) +Kummer function for non-negative κ: +M(a, b, κ) = +1 +B(a, b − a) +� 1 +−1 +eκt2t2a−1(1 − t2)b−a−1 dt, +b > a > 0, κ ≥ 0. +(B.11) +We have +M(a, b, κ) = +∞ +� +r=0 +Γ(a + r)Γ(b) +Γ(a)Γ(b + r) +κr +r!, +b > a > 0, κ ≥ 0. +(B.12) +For κ ≥ 0 and b > a > 0 we have +M′(a, b, κ) = a +b M(a + 1, b + 1, κ). +(B.13) +With the series expansion in (B.12) we obtain for b > a > 0 +M(a, b, κ) = 1 + O(κ) +κ→0+ +→ 1+. +(B.14) +For κ ≥ 0 let +Dd(κ) = M(3/2, d/2 + 1, κ) +dM(1/2, d/2, κ) . +(B.15) +41 + +For sufficiently small κ ≥ 0 we have +Dd(κ) = 1 +d + 2(d − 1) +d2(d + 2)κ + O(κ2). +(B.16) +For κ > 0 we have +D′ +d(κ) = +3 +d+2 M(5/2, d/2 + 2, κ)M(1/2, d/2, κ) − 1 +d M(3/2, d/2 + 1, κ) +dM2(1/2, d/2, κ) +. +(B.17) +With (B.14) we obtain +lim +κ→0+ D′ +d(κ) = 2(d − 1) +d2(d + 2) +(B.18) +For κ ≥ 0 let +dd(κ) = 2πd/2 +Γ(d/2) M(1/2, d/2, κ). +(B.19) +Due to the series expansion in (B.12) we have for κ > 0 +1 − M(1/2, d/2, κ) = − +∞ +� +r=1 +Γ(1/2 + r)Γ(d/2) +Γ(1/2)Γ(d/2 + r) +κr +r! < 0. +(B.20) +An application of (B.13) immediately gives for κ > 0 +∂κ[log dd(κ)] = +d′ +d(κ) +dd(κ) = Dd(κ). +(B.21) +We come to the announced Proposition. The notation corresponds as far as possible to the one of +Chapter 5. +Proposition B.3. Let κ > 0, and let f(· | κ) be a continuous density w.r.t the surface measure σ, which +is parameterized via κ. The limit case κ → 0+ is assumed to yield the uniform distribution on Sd−1. +Further, let U be a random vector, which is distributed according to f(· | κ). Then +i) For the von Mises–Fisher alternative vMF(θ, κ), with θ ∈ Sd−1 fixed: +(a) +KL(κ, 0) = Ad(κ)κ − log ad(κ) + log |Sd−1|, +(b) +γκ(b) = |Sd−2| +ad(κ) +∞ +� +l=0 +κl +l! +β +� +j=0 +cj, d(β)∆j(l)Pd +j(θ⊤b) − ψd(β), +b ∈ Sd−1, +∆j(l) = +� 1 +−1 +Pd +j(t)tl(1 − t2)(d−3)/2 dt, +j = 1, . . . , β, l ∈ N0. +42 + +(c) +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λ1νd(1). +ii) For the Watson alternative W(θ, κ), with θ ∈ Sd−1 fixed: +(a) +KL(κ, 0) = Dd(κ)κ − log dd(κ) + log |Sd−1|, +(b) +γκ(b) = |Sd−2| +dd(κ) +∞ +� +l=0 +κl +l! +β +� +j=0 +cj, d(β)∆j(2l)Pd +j(θ⊤b) − ψd(β), +b ∈ Sd−1, +(c) +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λ2νd(2). +iii) For the Legendre polynomial alternative LPm(θ, κ), with m ∈ N and θ ∈ Sd−1 fixed and κ ∈ [0, 1]: +(a) +KL(κ, 0) = +κ2 +νd(m) +� +1 − +1 +2(1 + ξκ)2 +� +− +κ3 +2(1 + ξκ)2 +|Sd−2| +|Sd−1| +� 1 +−1 +(Pd +m(t))3(1 − t2)(d−3)/2 dt, +with an intermediate point ξκ satisfying |ξκ| ≤ κ. +(b) +γκ(b) = κ2 λm +νd(m)Pd +m(θ⊤b), +b ∈ Sd−1, +(c) +lim +κ→0+ +maxb∈Sd−1 γ2 +κ(b) +2KL(κ, 0) += λmνm(2). +Proof. +i) By definition of the von Mises–Fisher distribution, see (5.3), we have for fixed θ ∈ Sd−1 +f(x | κ) = +1 +ad(κ) exp(κθ⊤x), +x ∈ Sd−1. +We compute for κ > 0 +KL(κ, 0) = Eκ +� +log +� f(U | κ) +f(U | 0 +�� += +� +Sd−1 log +� +|Sd−1|f(x | κ) +� +f(x | κ) dσ(x) += +κ +ad(κ) +� +Sd−1 θ⊤x exp(κ θ⊤x) dσ(x) − log ad(κ) + log |Sd−1| += κ|Sd−2| +ad(κ) +� +Sd−1 teκt(1 − t2)(d−3)/2 dt − log ad(κ) + log |Sd−1|, +43 + +where the last equality follows from the Funk–Hecke-Theorem for Pd +0 ≡ 1. Moreover, we obtain +by virtue of Γ((d + 1)/2) = Γ((d − 1)/2)(d − 1)/2 and an integration by parts +Id/2(κ) = +(κ/2)d/2 +Γ((d + 1)/2)Γ(1/2) +� 1 +−1 +eκt(1 − t2)(d−1)/2 dt += +(κ/2)d/2−1 +Γ((d − 1)/2)Γ(1/2) +� 1 +−1 +teκt(1 − t2)(d−3)/2 dt, +so that statement a) follows from +KL(κ, 0) = |Sd−2| +ad(κ) +Γ((d − 1)/2)Γ(1/2) +(κ/2)d/2−1 +Id/2(κ)κ − log ad(κ) + log |Sd−1| += +2πd/2Γ((d − 1)/2)−1 +2πd/2(κ/2)1−d/2Id/2−1(κ) +Γ((d − 1)/2) +(κ/2)d/2−1 Id/2(κ)κ − log ad(κ) + log |Sd−1| += +Id/2(κ) +Id/2−1(κ)κ − log ad(κ) + log |Sd−1| += Ad(κ)κ − log ad(κ) + log |Sd−1|. +For statement b) we observe for κ > 0 and b ∈ Sd−1 +Eκ(b⊤U) β = +1 +ad(κ) +� +Sd−1(b⊤x) β exp(κ θ⊤x) dσ(x) += +1 +ad(κ) +β +� +j=0 +cj, d(β) +� +Sd−1 Pd +j(b⊤x) exp(κ θ⊤x) dσ(x), +where the last equality is due to (A.5). The function Sd−1 ∋ x �→ Pd +j(b⊤x) is a d-dimensional +spherical harmonic of order j, so that, once again, the Funk–Hecke-Theorem yields +Eκ(b⊤U) β = |Sd−2| +ad(κ) +β +� +j=0 +cj, d(β)Pd +j(θ⊤b) +� 1 +−1 +Pd +j(t)eκt(1 − t2)(d−3)/2 dt += |Sd−2| +ad(κ) +β +� +j=0 +cj, d(β)Pd +j(θ⊤b) +∞ +� +l=0 +κl +l! ∆j(l) += |Sd−2| +ad(κ) +∞ +� +l=0 +κl +l! +β +� +j=0 +cj, d(β)∆j(l)Pd +j(θ⊤b). +Since γκ(b) = Eκ(b⊤U) β − ψd(β), formula b) follows. In particular, Proposition A.5 and the +44 + +examples after Proposition A.8 for j = 1, . . . , β yield +∆j(0) = +� +P d +j , P d +0 +� += δj 0 +|Sd−1| +νd(0) |Sd−2| = δj 0 +|Sd−1| +|Sd−2|, +(B.22) +∆j(1) = +� +P d +j , P d +1 +� += δj 1 +|Sd−1| +νd(1) |Sd−2|, +(B.23) +∆j(2) = 1 +d +� +P d +j , P d +0 +� ++ d − 1 +d +� +P d +j , P d +2 +� += δ j 0 +1 +d +|Sd−1| +|Sd−2| + δj 2 +d − 1 +d +|Sd−1| +νd(2) |Sd−2|. +(B.24) +With the fact that c0, d(β) = ψd(β), see proof of Proposition 2.3 i), a brief calculation gives +γκ(b) = +�|Sd−1| +ad(κ) − 1 +� +ψd(β) + κ|Sd−1| +ad(κ) +c1, d(β) +νd(1) Pd +1(θ⊤b) ++ |Sd−2| +ad(κ) +∞ +� +l=2 +κl +l! +β +� +j=0 +c j, d(β)∆j(l)Pd +j(θ⊤b). +(B.25) +In order to prove the last statement in i) we assume that κ ∈ (0, R) for some R > 0. This is no +severe restrictions, since we are solely interested in the asymptotics near 0. As the initial step +we want to determine +lim +κ→0+ +γκ(b) +√2KL(κ, 0) +, +b ∈ Sd−1. +For this purpose, we distinguish the cases l = 0, l = 1 and l ≥ 2 as in equation (B.25). +Consider l = 0 and define αd(κ) = +� +|Sd−1| +ad(κ) − 1 +� +. With (B.9) we realize that limκ→0+ αd(κ) = 0. +With an application of de L’Hospital’s rule, we calculate +lim +κ→0+ +2KL(κ, 0) +α2 +d(κ) += lim +κ→0+ +2 +� +Ad(κ)κ − log ad(κ) + log |Sd−1| +� +α2 +d(κ) += lim +κ→0+ +2[A′ +d(κ)κ + Ad(κ) − Ad(κ)] +2α′ +d(κ)αd(κ) += +1 +|Sd−1| lim +κ→0+ +A′ +d(κ)κ +− +a′ +d(κ) +a2 +d(κ)αd(κ) += +1 +|Sd−1| lim +κ→0+ +A′ +d(κ)κ +Ad(κ) +a2 +d(κ) +�ad(κ) − |Sd−1|� += +1 +|Sd−1| lim +κ→0+ +a2 +d(κ) +����������� +κ +Ad(κ) − Ad(κ)κ − d + 1 +����������� +ad(κ) − |Sd−1| += ∞. +Here, the second and the second to last equality follow from (B.10), and the last equation follows +from (B.7). Then, the enumerator converges to |Sd−1|2 due to limκ→0+ Ad(κ)κ = 0 and with (B.6) +due to +lim +κ→0+ +Ad(κ) +κ += lim +κ→0+ +�1 +d − +κ2 +d2(d + 2) + O(κ4) +� += 1 +d. +45 + +On the other hand, the denominator converges to 0+ due to (B.9). Hence, it follows for l = 0 +lim +κ→0+ +����������� +|Sd−1| +ad(κ) − 1 +����������� ψd(β) +√2KL(κ, 0) += − lim +κ→0+ +ψd(β) +� +� +� +�2KL(κ, 0) +α2 +d(κ) += 0. +Consider l = 1. We have +lim +κ→0+ +κ +√2KL(κ, 0) += lim +κ→0+ +1 +� +� +2 +� +Ad(κ)κ − log ad(κ) + log |Sd−1| +� +κ2 += +√ +d. +De L’Hospital’s rule and (B.7) yield +lim +κ→0+ +2 +� +Ad(κ)κ − log ad(κ) + log |Sd−1| +� +κ2 += lim +κ→0+ A′ +d(κ) += lim +κ→0+ +� +1 − A2 +d(κ) − d − 1 +κ +Ad(κ) +� += 1 +d. +Together with νd(1) = d it follows that +lim +κ→0+ +κ +√2KL(κ, 0) +|Sd−1| +ad(κ) +c1, d(β) +νd(1) Pd +1(θ⊤b) = c1, d(β) +√νd(1) +Pd +1(θ⊤b), +b ∈ Sd−1. +Consider l ≥ 2. Then we immediately have +lim +κ→0+ +κl +√2KL(κ, 0) += lim +κ→0+ +κ +√2KL(κ, 0) +κl−1 = 0, +because the first factor is bounded. For the following, we define the function series +S N : (0, R) × Sd−1 → R; +(κ, b) �→ +N +� +l=2 +1 +l! +κl +√2KL(κ, 0) +β +� +j=0 +cj, d(β)∆j(l)Pd +j(θ⊤b) +for N ∈ N, N ≥ 2. For fixed κ ∈ (0, R), the series is continuous on Sd−1. Furthermore, it +converges uniformly on (0, R) as N → ∞ and for fixed b ∈ Sd−1, since for κ ∈ (0, R) we have +κl +√2KL(κ, 0) += +κ +√2KL(κ, 0) +κl−1 ≤ max{C, R}l +with a constant C > 0 bounding κ/ √2KL(κ, 0) from above. Due to | Pd +j | ≤ 1, see Proposition +A.6, and |∆ j(l)| ≤ |Sd−1|/|Sd−2| for j = 1, . . . , β, l ∈ N0, it then follows +�������� +∞ +� +l=2 +1 +l! +κl +√2KL(κ, 0) +β +� +j=0 +cj, d(β)∆j(l)Pd +j(θ⊤b) +�������� +≤ β max +j=1,...,β |c j, d(β)||Sd−1| +|Sd−2| +∞ +� +l=2 +max{C, R}l +l! +, +46 + +and the last series converges. The uniform convergence yields for b ∈ Sd−1 +lim +κ→0+ +γκ(b) +√2KL(κ, 0) += lim +κ→0+ +���������������������������� +����������� +|Sd−1| +ad(κ) − 1 +����������� ψd(β) +√2KL(κ, 0) ++ +κ +√2KL(κ, 0) +|Sd−1| +ad(κ) +c1, d(β) +νd(1) Pd +1(θ⊤b) +���������������������������� ++ lim +κ→0+ +|Sd−2| +ad(κ) +∞ +� +l=2 +1 +l! +κl +√2KL(κ, 0) +β +� +j=0 +c j, d(β)∆j(l)Pd +j(θ⊤b) += 0 + c1, d(β) +√νd(1) +Pd +1(θ⊤b) + 0 = c1, d(β) +√νd(1) +Pd +1(θ⊤b). +We may see here, especially, that γκ(·)/ √2KL(κ, 0) converges pointwise to c1, d(β)/ √νd(1)Pd +1(θ⊤·) +for κ → 0+. As a matter of fact, the convergence is even uniform on Sd−1, because | Pd +j | ≤ 1. +Thus we can finally conclude +lim +κ→0+ +max +b∈Sd−1γ2 +κ(b) +2KL(κ, 0) = lim +κ→0+ max +b∈Sd−1 +� +γκ(b) +√2KL(κ, 0) +�2 += max +b∈Sd−1 +� +lim +κ→0+ +γκ(b) +√2KL(κ, 0) +�2 += (c1, d(β))2 +νd(1) +max +b∈Sd−1| Pd +1(θ⊤b) |2 = λ1νd(1). +Here, the last equality is due to (2.4) and the fact that max +b∈Sd−1| Pd +1(θ⊤b) |2 = 1, see Remark A.6. +ii) In the case of a Watson alternative, we are not going to perform such a detailed proof as before +since the relevant quantities are quite similar to those in i) and, otherwise, many arguments would +be repeated. Recall that the density of a Watson distribution with fixed parameter θ ∈ Sd−1 is +given by +f(x | κ) = +1 +dd(κ) exp(κθ⊤x), +x ∈ Sd−1. +Let κ > 0. First of all, the Kullback–Leibler information number satisfies +KL(κ, 0) = κ|Sd−2| +dd(κ) +� +Sd−1 t2 exp(κt2)(1 − t2)(d−3)/2 dt − log dd(κ) + log |Sd−1|. +By the definition of the Kummer function +M(3/2, d/2 + 1, κ) = +1 +B(3/2, (d − 1)/2) +� 1 +−1 +t2eκt2(1 − t2)(d−3)/2 dt += +d Γ(d/2) +√π Γ((d − 1)/2) +� 1 +−1 +t2eκt2(1 − t2)(d−3)/2 dt, +47 + +so that after a short calculation it follows that +KL(κ, 0) = κ|Sd−2| +dd(κ) +√π Γ((d − 1)/2) +d Γ(d/2) +M(3/2, d/2 + 1, κ) − log dd(κ) + log |Sd−1| += Dd(κ)κ − log dd(κ) + log |Sd−1|. +For γκ the exact same arguments as in i) yield the formula +γκ(b) = |Sd−2| +dd(κ) +∞ +� +l=0 +κl +l! +β +� +j=0 +cj, d(β)∆j(2l)Pd +j(θ⊤b) − ψd(β), +b ∈ Sd−1. +Therefore, +γκ(b) = +�|Sd−1| +dd(κ) − 1 +� +ψd(β) + κ|Sd−2| +dd(κ) +β +� +j=0 +cj, d(β)∆j(2)Pd +j(θ⊤b) ++ |Sd−2| +dd(κ) +∞ +� +l=2 +κl +l! +β +� +j=0 +cj, d(β)∆j(2l)Pd +j(θ⊤b) += +�|Sd−1| +dd(κ) − 1 +� +ψd(β) + κ|Sd−1| +dd(κ) +�ψd(β) +d ++ c2, d(β) +νd(2) +d − 1 +d +Pd +2(θ⊤b) +� ++ |Sd−2| +dd(κ) +∞ +� +l=2 +κl +l! +β +� +j=0 +cj, d(β)∆j(2l)Pd +j(θ⊤b). +The last equality is due to (B.22). Again we distinguish the cases l = 0, l = 1 and l ≥ 2 and +assume that κ > 0 is sufficiently small. Consider l = 0 and define δd(κ) = +� +|Sd−1| +dd(κ) − 1 +� +. Then +δd(κ) = +� +1 +M(1/2,d/2,κ) − 1 +� +< 0 by (B.20). Thus we calculate +lim +κ→0+ +2KL(κ, 0) +�����������1 − +1 +M(1/2, d/2, κ) +����������� +2 = lim +κ→0+ +2 +� +Dd(κ)κ − log dd(κ) + log |Sd−1| +� +�����������1 − +1 +M(1/2, d/2, κ) +����������� +2 += lim +κ→0+ +D′ +d(κ)κ +M′(1/2, d/2, κ) +M2(1/2, d/2, κ) +M(1/2, d/2, κ) − 1 +M(1/2, d/2, κ) += lim +κ→0+ M2(1/2, d/2, κ) +D′ +d(κ)κ +Dd(κ)(M(1/2, d/2, κ) − 1) += lim +κ→0+ +D′ +d(κ) +Dd(κ) lim +κ→0+ +κ +M(1/2, d/2, κ) − 1 +48 + += lim +κ→0+ +D′ +d(κ) +Dd(κ) lim +κ→0+ +1 +M′(1/2, d/2, κ). +Here, the second and the last equality are due to de L’Hospital’s rule, and the third and fourth +equality follow from (B.21) and (B.14), respectively. By (B.13), (B.14), (B.16) and (B.18), it +follows that +lim +κ→0+ +����������� +|Sd−1| +dd(κ) − 1 +����������� ψd(β) +√2KL(κ, 0) += − lim +κ→0+ +ψd(β) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +2KL(κ, 0) +�����������1 − +1 +M(1/2, d/2, κ) +����������� +2 += −ψd(β) +√ +2 +� +d + 2 +d − 1. +If l = 1, then +lim +κ→0+ +κ +√2KL(κ, 0) += lim +κ→0+ +1 +� +� +2 +� +Dd(κ)κ − log dd(κ) + log |Sd−1| +� +κ2 += +d√ +2 +� +d + 2 +d − 1, +since +lim +κ→0+ +2 +� +Dd(κ)κ − log ad(κ) + log |Sd−1| +� +κ2 += lim +κ→0+ D′ +d(κ) = 2(d − 1) +d2(d + 2), +where the last equation is due to (B.18). The expressions for l ≥ 2 vanish for κ → 0+ by the +same argument as in i). +These results yield for b ∈ Sd−1 +lim +κ→0+ +γκ(b) +√2KL(κ, 0) += lim +κ→0+ +����������� +|Sd−1| +dd(κ) − 1 +����������� ψd(β) +√2KL(κ, 0) ++ lim +κ→0+ +κ +√2KL(κ, 0) +|Sd−1| +dd(κ) +�ψd(β) +d ++ c2, d(β) +νd(2) +d − 1 +d +Pd +2(θ⊤b) +� ++ lim +κ→0+ +|Sd−2| +dd(κ) +∞ +� +l=2 +1 +l! +κl +√2KL(κ, 0) +β +� +j=0 +cj, d(β)∆j(2l)Pd +j(θ⊤b) += −ψd(β) +√ +2 +� +d + 2 +d − 1 + d√ +2 +� +d + 2 +d − 1 +�ψd(β) +d ++ c2, d(β) +νd(2) +d − 1 +d +Pd +2(θ⊤b) +� ++ 0 +49 + += +� +(d + 2)(d − 1) +2 +c2, d(β) +νd(2) Pd +2(θ⊤b) = c2, d(β) +√νd(2) +Pd +2(θ⊤b). +Using (2.4) once again, it follows that +lim +κ→0+ +max +b∈Sd−1γ2 +κ(b) +2KL(κ, 0) = lim +κ→0+ max +b∈Sd−1 +� +γκ(b) +√2KL(κ, 0) +�2 += max +b∈Sd−1 +� +lim +κ→0+ +γκ(b) +√2KL(κ, 0) +�2 += (c2, d(β))2 +νd(2) +max +b∈Sd−1| Pd +2(θ⊤b) |2 = λ2νd(2). +iii) Recall the density +f(x | κ) = +1 +|Sd−1| +� +1 + κPd +m(θ⊤x) +� +, +x ∈ Sd−1, +of the Legendre polynomial alternative with fixed m ∈ N and θ ∈ Sd−1. We calculate for κ ∈ [0, 1] +KL(κ, 0) = +1 +|Sd−1| +� +Sd−1 log +� +1 + κPd +m(θ⊤x) +� � +1 + κPd +m(θ⊤x) +� +dσ(x) += |Sd−2| +|Sd−1| +� 1 +−1 +log +� +1 + κPd +m(t) +� � +1 + κPd +m(t) +� +(1 − t2)(d−3)/2 dt. +A Taylor expansion of order 1 of t �→ log(1 + t) around 0 yields +KL(κ, 0) = |Sd−2| +|Sd−1| +� 1 +−1 +����������κPd +m(t) − +� +κPd +m(t) +�2 +2(1 + ξκ)2 +���������� +� +1 + κPd +m(t) +� +(1 − t2)(d−3)/2 dt += |Sd−2| +|Sd−1| +� +κ +� +P d +m, P d +0 +� ++ κ2 � +P d +m, P d +m +� +− +κ2 +2(1 + ξκ)2 +� +P d +m, P d +m +� +− +κ3 +2(1 + ξκ)2 +�� +P d +m +�2 , P d +m +�� += +κ2 +νd(m) +� +1 − +1 +2(1 + ξκ)2 +� +− +κ3 +2(1 + ξκ)2 +|Sd−2| +|Sd−1| +� 1 +−1 +� +P d +m(t) +�3 (1 − t2)(d−3)/2 dt. +Here, ξκ is an intermediate point, which satisfies |ξκ| ≤ κ| Pd +m(t) | ≤ κ for t ∈ [−1, 1]. Therefore, +we have +KL(κ, 0) = +κ2 +νd(m) +� +1 − +1 +2(1 + ξκ)2 +� ++ O(κ3), +κ → 0+. +For γκ we compute for b ∈ Sd−1 +Eκ(b⊤U) β = +1 +|Sd−1| +� +Sd−1(b⊤x) β � +1 + κPd +m(θ⊤x) +� +dσ(x) += ψd(β) + κPd +m(θ⊤b)|Sd−2| +|Sd−1| +� 1 +−1 +P d +m(t)t β(1 − t2)(d−3)/2 dt +50 + += ψd(β) + κPd +m(θ⊤b)|Sd−2| +|Sd−1| +β +� +j=0 +cj, d(β) +� +P d +j , P d +m +� += ψd(β) + κcm, d(β) +νd(m) Pd +m(θ⊤b), +where the second and third equality are due to (1.2) and (A.5), respectively. Hence, +max +b∈Sd−1γ2 +κ(b) = κ2λm max +b∈Sd−1| Pd +m(θ⊤b) |2 = κ2λm. +In view of these results and the fact, that limκ→0+ ξκ = 0, we conclude +lim +κ→0+ +max +b∈Sd−1γ2 +κ(b) +2KL(κ, 0) = lim +κ→0+ +κ2λm +2κ2 +νd(m) +�����������1 − +1 +2(1 + ξκ)2 +����������� + O(κ3) += lim +κ→0+ +λm +2 +νd(m) +�����������1 − +1 +2(1 + ξκ)2 +����������� + O(κ) += +λm +2 +νd(m) +�����������1 − +1 +2 +����������� += λmνd(m). +□ +C +Proofs of main results +Proof of Theorem 2.1. +Proof. The proof uses the methods presented in the proof of Theorem 2.1 in [8], although the co- +variance structure is a generalization. Putting W(b) = (b⊤U1)β − ψd(β), b ∈ Sd−1, we have (using +xβ − yβ = (x − y) �β−1 +j=0 x jyβ−1− j, x, y ∈ R, β ∈ N) with the Cauchy–Schwarz inequality +|W(b) − W(c)| ≤ β∥b − c∥, +b, c ∈ Sd−1, +and a direct application of the CLT in Banach spaces, see [2], Corollary 7.17, yields the claim. The +Corollary is applicable, since the metric space +� +Sd−1, ∥ · ∥ +� +clearly satisfies the stated entropy condition. +To finish the proof it suffices to calculate the covariance structure E(W(b)W(c)) by taking advantage of +Lemma B.1. +□ +51 + +Proof of Proposition 2.3. +Proof. +i) Let β ∈ N and k ∈ N0. Since ρβ(b, c) = Q(b⊤c) for all b, c ∈ Sd−1, it follows by the +Funk–Hecke-Theorem A.3 for x ∈ Sd−1 and φ ∈ Hk(Sd−1) +Kβφ(x) = +1 +|Sd−1| +� +Sd−1 ρβ(ω, x)φ(ω) dσ(ω) = +1 +|Sd−1| +� +Sd−1 Q(ω⊤x)φ(ω) dσ(ω) = λkφ(x) +with +λk = |Sd−2| +|Sd−1| +� 1 +−1 +P d +k (t)Q(t)(1 − t2)(d−3)/2dt, +where the expression for λk is derived from the Funk–Hecke-Theorem. By Proposition A.5 we +have the following orthogonality property of the Legendre polynomials w.r.t the scalar product +in (A.4) +� +Pd +k, Pd +j +� += δk j +|Sd−1| +νd(k) |Sd−2|, +∀k, j ∈ N0. +In view of (2.1) and (2.2) we have +Q(t) = +β +� +j=0 +(cj, d(β))2 +νd(j) +P d +j (t) − ψ2 +d(β), +t ∈ [−1, 1]. +We calculate for k ≤ β with Pd +0 ≡ 1 +� 1 +−1 +P d +k (t)Q(t)(1 − t2)(d−3)/2 = +β +� +j=0 +(c j, d(β))2 +νd(j) +� +Pd +k, Pd +j +� +− +� 1 +−1 +P d +k (t)ψ2 +d(β)(1 − t2)(d−3)/2dt += |Sd−1| +|Sd−2| +β +� +j=0 +�cj, d(β) +νd(j) +�2 +δk j − ψ2 +d(β) +� +Pd +k, Pd +0 +� += |Sd−1| +|Sd−2| +������� +�ck, d(β) +νd(k) +�2 +− δk0ψ2 +d(β) +������� . +Thereby we obtain +λk = |Sd−2| +|Sd−1| +� 1 +−1 +P d +k (t)Q(t)(1 − t2)(d−3)/2dt = +�ck, d(β) +νd(k) +�2 +− δk0ψ2 +d(β), +k ≤ β. +Evidently is λk = 0 for k > β by the orthogonality of the Legendre polynomials. Consider the +case k = 0. With νd(0) = 1 we already know that +λk = (c0, d(β))2 − ψ2 +d(β) = (c0, d(β) − ψd(β))(c0, d(β) + ψd(β)). +52 + +If β is odd, then ψd(β) = c0, d(β) = 0 due to (2.1) and Proposition A.8. Thus let β be even. +Then the Funk–Hecke-Theorem for 1 ∈ H0(Sd−1), equation (A.5), and the orthogonality of the +Legendre polynomials yield +ψd(β) = E(b⊤U) β = +1 +|Sd−1| +� +Sd−1(b⊤ω) β dσ(ω) = |Sd−2| +|Sd−1| +� 1 +−1 +t β(1 − t2)(d−3)/2 dt += |Sd−2| +|Sd−1| +β +� +j=0 +cj, d(β) +� 1 +−1 +P d +j (t)(1 − t2)(d−3)/2 dt = |Sd−2| +|Sd−1| +β +� +j=0 +cj, d(β) +� +Pd +j, Pd +0 +� += |Sd−2| +|Sd−1| +β +� +j=0 +cj, d(β) δj 0|Sd−1| +νd(0) |Sd−2| = c0, d(β). +ii) Let {φk,1, . . . , φk,νd(k)} be an orthonormal basis of Hk(Sd−1). Then �∞ +k=0{φk,1, . . . , φk,νd(k)} is an +orthonormal basis of L2(Sd−1, dσ) and we get the unique series expansion for f ∈ L2(Sd−1, dσ) +f +L2 += +∞ +� +k=0 +Ψk, +(C.1) +with Ψk = �νd(k) +j=1 +� +f, φk,j +� +L2 φk,j ∈ Hk(Sd−1), compare with (A.3). Set ak,j = +� +f, φk,j +� +L2 for all +j ∈ {1, . . . , νd(k)}, k ∈ N0. It follows for x ∈ Sd−1 +Kβ f(x) = +1 +|Sd−1| +� +Sd−1 ρβ(ω, x)f(ω) dσ(ω) = +1 +|Sd−1| +� +ρβ(·, x), f +� +L2 += +1 +|Sd−1| +∞ +� +k=0 +νd(k) +� +j=1 +ak,j +� +ρβ(·, x), φk,j +� +L2 +i)= +∞ +� +k=0 +νd(k) +� +j=1 +ak,jλkφk,j(x)1{k ≤ β} += +β +� +k=1 +λk +νd(k) +� +j=1 +ak,jφk,j(x). +Hence, Kβ is a finite-rank operator, and it has the representation +Kβ f(x) = +β +� +k=1 +λk +νd(k) +� +j=1 +ak,jφk,j(x), +x ∈ Sd−1. +(C.2) +iii) Let λ ∈ C, λ � {0} ∪ {λk | k = 1, . . . , β}, and consider the equation λf − Kβ f = 0 f¨ur f ∈ +L2(Sd−1, dσ) as in (C.1). Observe for i ∈ {1, . . . , νd(m)}, m ∈ {1, . . . , β} +λ � f, φm,i +� +L2 = +� +Kβ f, φm,i +� +L2 = +β +� +k=1 +λk +νd(k) +� +j=1 +ak,j +� +φk,j, φm,i +� +L2 += +β +� +k=1 +λk +νd(k) +� +j=1 +ak,jδmkδi j = +β +� +k=1 +λkδmkak,i = λmam,i = λm +�f, φm,i +� +L2 . +53 + +Since λ � λm, it immediately follows that � f, φm,i +� +L2 = 0. Hence, �f, φm,i +� +L2 = 0 for all i ∈ +{1, . . . , νd(m)}, m ∈ {1, . . . , β} and therewith Kβ f = 0. We conclude λ f = 0 and, since λ � 0, it +follows that f = 0. This, in turn, means that λ is a resolvent point. +iv) With f ∈ L2(Sd−1, dσ) like in (C.1) and by (C.2) we conclude +⟨Kβ f, f⟩L2 = +β +� +k=1 +λk +νd(k) +� +j=1 +ak,j⟨φk,j, f⟩L2 = +β +� +k=1 +λk +νd(k) +� +j=1 +a2 +k,j ≥ 0, +since all eigenvalues are non-negative. +□ +Proof of Proposition 2.4. +Proof. We prove the claim only for the case of β odd as the other case can be done analogously. +Since a centred Gaussian process is defined by the covariance kernel, we merely have to show that the +covariance kernels of Zβ(·) and the process on the right-hand side, denoted by Y(·), coincide. Clearly, +after some calculations we first see that Y(·) is a centred Gaussian process. Furthermore, +EY(b)Y(c) = |Sd−1| +β +� +k,m=1 +k,m odd +� +λkλm +νd(k) +� +j=1 +νd(m) +� +i=1 +φk,j(b) φm,i(c) ENk,jNm,i += |Sd−1| +β +� +k,m=1 +k,m odd +� +λkλm +νd(k) +� +j=1 +νd(m) +� +i=1 +φk,j(b) φm,i(c) δkmδi j += |Sd−1| +β +� +k=1 +k odd +λk +νd(k) +� +j=1 +φk,j(b) φk,j(c) = ρβ(b, c), +where the last equality follows from Mercer’s theorem, see [41], Theorem 2.10, which is applicable +due to our previously obtained results. Note that the eigenvalues in (2.4) equal zero for even indices, +if β is odd, for the occurring constants ck, d(β) are zero in these cases. Compare also with Proposition +A.8. +□ +Proof of Theorem 3.1. +Proof. Lemma B.2 yields by Le Cams first Lemma, see [29] Proposition 5.2.1, that P(n) and A(n) are +mutually contiguous. Straightforward calculations show for b ∈ Sd−1 under P(n) +i) lim +n→∞ Cov +� +Zn,β(b), log Ln +� += S ∗ +β(b), +54 + +ii) for l ∈ N, a1, . . . , al ∈ Sd−1 the distribution of (Zn,β(a1), . . . , Zn,β(al), log Ln)⊤ converges to +Nl+1 +��������� +� +0, . . . , 0, −τ2 +2 +�⊤ +, +��������� +Σa1,...,al +z +z⊤ +τ2 +��������� +��������� , +where Σa1,...,al = +� +ρβ +� +aj1, aj2 +�� +1≤j1,j2≤l is a l×l-matrix, z = +� +S ∗ +β(a1), . . . , S ∗ +β(al) +�⊤ is a l-dimensional +vector, and τ is defined in Lemma B.2. +Le Cam’s third lemma shows that, under A(n) the finite-dimensional distributions of Zn,β converge +weakly to the corresponding distributions of the shifted Gaussian process Zβ + S ∗ +β. Since Zn,β is tight +under P(n) and A(n) is contiguous to P(n) we have tightness of Zn,β and Zn,β +D +−→ Zβ + S ∗ +β under A(n). +□ +Proof of Theorem 5.1. +Proof. For the first statement, we use [6], Theorem 7.2, and verify the two conditions therein for the +case of the approximate Bahadur slope, compare with [5], Section 6. First of all, due to the strong law +of large numbers we have +1 +n +n +� +j=1 +(b⊤U j) β − ψd(β) +Pκ-f.s. +→ γκ(b), +b ∈ Sd−1. +Hence we immediately obtain +�Tn, β +√n +Pκ→ max +b∈Sd−1γ2 +κ(b), +κ > 0. +Under H0, Corollary 2.6 yields +�Tn, β = +� +Tn, β +D +−→ max +b∈Sd−1| Zβ(b) |. +With F(t) = P( maxb∈Sd−1 | Zβ(b) | < t), t ∈ R, an application of [28], Corollary 3.2, gives +lim +t→∞ +log(1 − F(t)) +t2 += lim +t→∞ +log P +� +maxb∈Sd−1 | Zβ(b) | ≥ t +� +t2 += lim +t→∞ +log P +� +∥ Zβ(·) ∥∞ ≥ t +� +t2 += − +1 +2 max +b∈Sd−1ρβ(b, b). +Thus, the approximate Bahadur slope is +c a +�Tβ(κ) = +max +b∈Sd−1γ2 +κ(b) +max +b∈Sd−1ρβ(b, b), +κ > 0. +55 + +For the second statement, we utilize again Theorem 7.2 in [6] and prove the second condition for +sufficiently small κ > 0 with the help of [37], Lemma 2.1 and Lemma 2.2. Let us consider, once again, +the C(Sd−1, R)-valued random elements W j(b) = (b⊤U j) β − ψd(β), b ∈ Sd−1, which are centered under +H0 and due to their compact support fulfill the integrability condition of [37], Lemma 2.1. Hence, with +Fn(t) = P0 +��Tn, β < t +� +, t ∈ R, we obtain for sufficiently small ϵ > 0 +lim +n→∞ +log(1 − Fn( √nϵ)) +n += lim +n→∞ +log P0 +��Tn, β ≥ √nϵ +� +n += lim +n→∞ +log P0 +� ���� 1 +n +�n +j=1 W j(·) +����∞ ≥ ϵ +� +n += − +ϵ2 +2 sup +b∈Sd−1EW2 +1(b) + o(ϵ2). +According to the proof of Theorem 2.1, W1(·) defines the covariance kernel ρβ of the Gaussian process +Zβ(·), so that indeed +lim +n→∞ +log(1 − Fn( √nϵ)) +n += − +ϵ2 +2 max +b∈Sd−1ρβ(b, b) + o(ϵ2) +for sufficiently small ϵ > 0. Due to the assumed condition (5.1), it follows that +|γκ(b)| = +����� +� +Sd−1(b⊤x) β f(x | κ) dσ(x) − ψd(β) +����� = +����� +� +Sd−1(b⊤x) β (f(x | κ) − f(x | 0)) dσ(x) +����� +≤ ||f(· | κ) − f(· | 0)||L1 +κ→0+ +−→ 0. +Lebesgue’s dominated convergence theorem shows that γκ ∈ C(Sd−1; R) for each κ > 0, and the +last calculation justifies the uniform convergence of γκ against 0 on Sd−1 for κ → 0+. This implies +limκ→0+ max +b∈Sd−1γ2 +κ(b) = 0. Thus, the exact Bahadur slope is +c�Tβ(κ) = +max +b∈Sd−1γ2 +κ(b) +max +b∈Sd−1ρβ(b, b) + o +� +max +b∈Sd−1γ2 +κ(b) +� +for sufficiently small κ > 0. Finally, notice that for each b ∈ Sd−1, we have +ρβ(b, b) = ηβ(b, b) − ψ2 +d(β) = +β +� +j=0 +(cj, d(β))2 +νd(j) +P d +j (b⊤b) − ψ2 +d(β) = +β +� +j=0 +(c j, d(β))2 +νd( j) +− ψ2 +d(β) += (c0, d(β))2 +νd(0) ++ +β +� +j=1 +λ jνd(j) − ψ2 +d(β) = +β +� +j=1 +λ jνd(j). +Here, the second and third equality follow from (2.1) and Remark A.6, respectively. +□ +56 + +J. Borodavka, +Steinbuch Centre for Computing, +Karlsruhe Institute of Technology (KIT), +Zirkel 2, D-76131 Karlsruhe. +E-mail: Jaroslav.Borodavka@kit.edu +B. Ebner, +Institute of Stochastics, +Karlsruhe Institute of Technology (KIT), +Englerstr. 2, D-76128 Karlsruhe. +E-mail: Bruno.Ebner@kit.edu +57 + diff --git a/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/load_file.txt b/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f7550aa1ba87977b659c4e210d7053b290983c2 --- /dev/null +++ b/rNE1T4oBgHgl3EQf2wV_/content/tmp_files/load_file.txt @@ -0,0 +1,1821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf,len=1820 +page_content='A general maximal projection approach to uniformity testing on the hypersphere Jaroslav Borodavka and Bruno Ebner January 10, 2023 Abstract We propose a novel approach to uniformity testing on the d-dimensional unit hypersphere Sd−1 based on maximal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This approach gives a unifying view on the classical uniformity tests of Rayleigh and Bingham, and it links to measures of multivariate skewness and kurtosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We derive the limiting distribution under the null hypothesis using limit theorems for Banach space valued stochastic processes and we present strategies to simulate the limiting processes by applying results on the theory of spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We examine the behavior under contiguous and fixed alternatives and show the consistency of the testing procedure for some classes of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For the first time in uniformity testing on the sphere, we derive local Bahadur efficiency statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We evaluate the theoretical findings and empirical powers of the procedures in a broad competitive Monte Carlo simulation study and, finally, apply the new tests to a data set on midpoints of large craters on the moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 1 Introduction Testing uniformity on the circle, the sphere and the hypersphere Sd−1 = {x ∈ Rd : ∥x∥ = 1}, d ∈ N, d ≥ 2, of Rd, endowed with the Euclidean norm ∥x∥ = √ x⊤x, are classical and still up-to-date research fields in directional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Here and in the following, ⊤ stands for the transpose of a matrix or a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We numerate just a small subset of fields, where data on the surface of the unit hypersphere Sd−1 is applied: meteorology, geology, paleomagnetism, political sciences, text mining and wildfire MSC 2010 subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Primary 62G10 Secondary 62H15 Key words and phrases uniformity tests, maximal projections, directional data, stochastic processes in Banach spaces, contiguous alternatives, Bahadur efficiency, Monte Carlo simulations 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='03482v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='ST] 9 Jan 2023 orientation, for examples of such datasets, see [30] and the contributions therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The first step to serious statistical inference on Sd−1 is to check whether or not a sample of unit vectors stems from the uniform law, since this distribution characterizes the absence of structure in directional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' To be specific, we model the observed data by independent identically distributed (iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=') column random vectors U, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un taking values in Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The testing problem of interest is whether or not the hypothesis H0 : PU = U � Sd−1� holds, against general alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Here, PU stands for the distribution of U and U(·) denotes the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This problem has been extensively studied in the literature: Lord Rayleigh pre- sented the first test of uniformity in [38] based on the norm of the arithmetic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Rayleigh’s test was followed by circular tests based on the classical goodness-of-fit measures of Kolmogorov-Smirnov type in [27] and of Cram´er-von Mises type in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Later, Bingham developed a test of uniformity in [9] based on the sample scatter matrix and Gin´e, see [21], introduced the so-called Sobolev-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We refer to [25, 33] for more details on these tests and to [24] for some new developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' More recently, [10] proposed a Kolmogorov-Smirnov type test based on random projections, [16] suggest a procedure using powers of volumes of nearest-neighbor spheres, and [19] consider the Cram´er-von Mises coun- terpart to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For details on this approach as well as more recent developments in uniformity testing of axial data see [29], chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The authors of the review article [20] give an overview of uniformity tests on the hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Comparative Monte Carlo simulation studies are found in [14] for d = 3 and for higher dimensions in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' A well-known characterizing property of U � Sd−1� is invariance with respect to rotations about the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Any test (say) Tn of uniformity should therefore inherit this structure and as such be invariant under rotations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Tn(AU1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , AUn) = Tn(U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un) holds for all A ∈ SO(d), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) where SO(d) is the d-dimensional rotation group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=', for d × d-matrices A ∈ SO(d) we have AA⊤ = A⊤A = Id and det(A) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We denote the identity matrix by Id, and det(·) is the notation for the determinant of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following, we call the property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) rotational invariance of the test statistic Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We propose a novel class of statistics Tn, β based on powers of maximal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In this spirit assume U ∼ U � Sd−1� and by [7], we have using the rotational invariance of the uniform distribution 2 and symmetry arguments for every b ∈ Sd−1 and β ∈ N ψd(β) = E(b⊤U)β = ����������� Γ ((β + 1)/2) Γ (d/2) √πΓ ((β + d)/2) , if β is even, 0, if β is odd, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2) where Γ(·) denotes the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Hence ψd(β) is independent of the choice of b, a property that likewise follows by the rotation invariance of the uniform law on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Next, we define the family of statistics Tn, β = Tn, β(U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un) = n max b∈Sd−1 ��������� 1 n n � j=1 (b⊤U j)β − ψd(β) ��������� 2 , β ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3) It is obvious that Tn, β is rotational invariant for every β due to the rotational invariance of the maximum functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Interestingly, Tn, β has close connections to well-known classical tests such as the Rayleigh test, the Bingham test and to measures of multivariate skewness and kurtosis by Malkovich and Afifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' First, notice that with the sample mean of the observations Un = 1 n �n j=1 U j we have Tn,1 = n max b∈Sd−1 ��������� 1 n n � j=1 b⊤U j ��������� 2 = n∥Un∥2 max b∈Sd−1 ������b⊤ Un ∥Un∥ ������ 2 = n∥Un∥2, since the scalar product in the maximum is the cosine of the angle between the two unit vectors, which takes its maximum for b = Un ∥Un∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Hence we have an equivalent test as the classical Rayleigh test, see [38], given by Rn = 2nd∥Un∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Second, with the sample scatter matrix S = 1 n �n j=1 U jU⊤ j we have Tn,2 = n max b∈Sd−1 ��������� 1 n n � j=1 (b⊤U j)2 − 1 d ��������� 2 = n max b∈Sd−1 � b⊤S b − 1 d �2 = n max b∈Sd−1 � b⊤ � S − 1 d Id � b �2 , Notice that Tn,2 is the squared spectral norm of S − E(UU⊤) for U ∼ U � Sd−1� , hence it compares the scatter matrix to the covariance matrix of U, which is in the same spirit as the Bingham test, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Note that by the Courant–Fischer–Weyl min-max principle from linear algebra, we have Tn,2 = n(max(|λmin|, |λmax|))2, where λmin and λmax are the minimal and maximal eigenvalues of the symmetric matrix S − 1 d Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Third, we have Tn,3 = n max b∈Sd−1 ��������� 1 n n � j=1 (b⊤U j)3 ��������� 2 and Tn,4 = n max b∈Sd−1 ��������� 1 n n � j=1 (b⊤U j)4 − 3 d(d + 2) ��������� 2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4) 3 which can be interpreted as analogs to the multivariate sample skewness and sample kurtosis by Malkovich and Afifi, for a definition see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For Tn, β, β > 2, no explicit closed form and easy to calculate formula is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The authors of [31] suggest using the Newton-Raphson method to obtain a good approximation of the maximal value in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Since for such a numerical routine, the choice of some good start values is not straightforward, we suggest to use a random approach, see Section 6, which is related to the idea of random projections as suggested in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The rest of the paper is organized as follows: We present asymptotic theory under the null hy- pothesis in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In Section 3 we derive the behaviour of Tn,β for contiguous alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We show consistency of the tests against some classes of fixed alternatives in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Afterwards, we establish local approximate and exact asymptotic relative efficiency statements in the Bahadur sense in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We examine the theoretical findings by a Monte Carlo simulation study in Section 6 and provide a real data application to midpoints of large craters on the moon in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Conclusions as well as an outlook are provided in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We finish the article by three Appendices A, B and C that contain facts on d-dimensional Legendre polynomials and spherical harmonics, as well as some technical Lemmas and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 2 Asymptotic null distribution of Tn, β Let C(Sd−1, R) be the Banach space of continuous functions f : Sd−1 → R, equipped with the norm ∥f∥∞ = supb∈Sd−1 |f(b)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We introduce the stochastic process Zn,β(b) = √n ��������� 1 n n � j=1 (b⊤U j)β − ψd(β) ��������� , b ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For the covariance structure in the following theorem, we write ηβ(b, c) = E(b⊤U) β(c⊤U) β = β � j=0 (cj, d(β))2 νd(j) P d j (b⊤c), b, c ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) Here, P d j (·) is the d-dimensional Legendre polynomial of order j, for a definition see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4), νd(j) is the dimension of the space of d-dimensional spherical harmonics of order j, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1), and cj,d(β) are constants only depending on j, d and β, compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5) and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' An explicit way of calculation can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un be iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' with U1 ∼ U � Sd−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For fixed β ∈ N there exists a centred 4 Gaussian process Zβ(b), b ∈ Sd−1 with continuous sample paths and covariance kernel ρβ(b, c) = ηβ(b, c) − ψ2 d(β), b, c ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2) Regarding Zβ(·) as a random element of C(Sd−1, R), we have Zn,β(·) D −→ Zβ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For the special cases of Section 1 and some higher powers β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' we have ρ1(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = 1 d(b⊤c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ρ2(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = 1 d(d + 2) � 2(b⊤c)2 + 1 � − 1 d2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ρ3(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = 1 d(d + 2)(d + 4) � 6(b⊤c)3 + 9(b⊤c) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ρ4(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = � 24(b⊤c)4 + 72(b⊤c)2 + 9 � 3 � j=0 (d + 2 j)−1 − 9 d2(d + 2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ρ5(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = � 120(b⊤c)5 + 600(b⊤c)3 + 225(b⊤c) � 4 � j=0 (d + 2 j)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ρ6(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' c) = � 720(b⊤c)6 + 5400(b⊤c)4 + 4050(b⊤c)2 + 225 � 5 � j=0 (d + 2 j)−1 − 225 d2(d + 2)2(d + 4)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' and thus explicit formulas for the covariance kernel in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Note that the covariance kernel ρβ(b, c) solely depends on the scalar product b⊤c and hence can be written as a function (say) ρβ(b, c) = Q(b⊤c), where Q ∈ C([−1, 1], R) is a polynomial of degree β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Kernels of this particular structure are called zonal kernels, for an application of Gaussian processes with zonal covariance kernel in machine learning see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The fact that |ρβ(b, c)| ≤ 1 for all β ∈ N follows by the inequalities of Cauchy–Schwarz and Popoviciu, since the projections are bounded random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Define the integral operators Kβ for β ∈ N given by Kβ f(x) = 1 |Sd−1| � Sd−1 ρβ(ω, x)f(ω) dσ(ω), x ∈ Sd−1, f ∈ L2(Sd−1, dσ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3) where integration is with respect to the unique spherical Lebesgue measure σ on Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Since ρβ is continuous on a compact set of Rd, the operator Kβ is compact from L2(Sd−1, dσ) to L2(Sd−1, dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Due to the zonal covariance structure we can even show that Kβ is a finite-rank operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=', an operator whose range is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The latter, and other properties, are presented and proved in the next 5 proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following, we denote by Hk(Sd−1) the space of d-dimensional spherical harmonic functions of order k ∈ N0, for a definition see [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let β ∈ N and Kβ be defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' i) For any spherical harmonic φ ∈ Hk(Sd−1) of order k ∈ N0, we have Kβφ = λkφ, where λk = ��������������� �ck, d(β) νd(k) �2 , for 0 < k ≤ β, 0 , for k = 0 or k > β, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4) with constants ck, d(β) ∈ R, depending only on k, d and β, compare with Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='8, and νd(k) = dim(Hk(Sd−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ii) Kβ is a finite-rank operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' iii) The spectrum of Kβ consists of 0 and the eigenvalues in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' iv) Kβ is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=', we have � Kβ f, f � L2 ≥ 0 for all f ∈ L2(Sd−1, dσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the spirit of [8], we thus have alternative representations of the limiting Gaussian process for our special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let νd(k) be the dimension of the space of d-dimensional spherical harmonics of order k ∈ N, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' i) If β is odd, the limiting Gaussian process Zβ(b), b ∈ Sd−1, can be represented in the form Zβ(b) = � |Sd−1| β � k=1 k odd � λk νd(k) � j=1 φk, j(b)Nk,j, b ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Here, Nk,j, k = 1, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , νd(k), is an array of independent unit normal vari- ables, λk is the eigenvalue in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4), and ϕk,j are j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , νd(k) linearly independent surface harmonics of degree k being orthonormal with respect to σ/|Sd−1|, compare with the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3, ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' ii) If β is even, the limiting Gaussian process Zβ(b), b ∈ Sd−1, can be represented in the form Zβ(b) = � |Sd−1| β � k=1 k even � λk νd(k) � j=1 φk,j(b)Nk,j, b ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 6 Here, Nk,j, k = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , νd(k), is an array of independent unit normal vari- ables, λk is the eigenvalue in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4), and ϕk,j are j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , νd(k) linearly independent spherical harmonics of degree k being orthonormal with respect to σ/|Sd−1|, compare with the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3, ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4 shows an easy way to simulate Gaussian random processes on the sphere with a polynomial covariance kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' What is essentially needed are three ingredients: the positive eigenval- ues (which can be calculated explicitly), an array of independent unit normal variables and an imple- mentation of spherical harmonics, see Section 6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For a generation method of a suitable basis of spherical harmonics, see [3], Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='11, or [13], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The package HFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='m in Mathematica, see [23], provides a direct way to calculate an orthonormal basis of spherical harmon- ics in any dimension d and any order k based on Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='25 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Note that explicit versions of orthonormal systems up to order 4 in any dimensions can be found in [32], Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 1: We have νd(1) = d and u �→ uk ∈ H1(Sd−1) for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , d, where u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , ud) ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' These functions form an orthogonal system of H1(Sd−1), see [22], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Normalization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' σ yields the orthonormal basis functions u �→ � d |Sd−1|uk ∈ H1(Sd−1), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We have a single positive eigenvalue λ1 = 1/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' With Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4 it follows Z1(u) = 1√ d d � j=1 ujN j, u ∈ Sd−1, N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Nd uiv∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Moreover, putting N = (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Nd) ∼ Nd(0, Id) the Cauchy–Schwarz Inequality yields Z2 1(u) = 1 d d � j,k=1 ujukNjNk = 1 d(u⊤N)2 ≤ 1 d∥N∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Hence we obtain d max u∈Sd−1 Z2 1(u) = ∥N∥2 ∼ χ2 d, thus recovering the limit result of the Rayleigh-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 2: By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) it is νd(2) = (d + 2)(d − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Straightforward calculations yield the single positive eigenvalue λ2 = � 2 d(d + 2) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 7 Let φ2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , φ2,νd(2) be an orthonormal basis of H2(Sd−1), see [32], Table 1, for an explicit representation, and set φ2(u) = (φ2,1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , φ2,νd(2)(u)), u ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' With Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='6 it follows that ∥φ2(u)∥2 = νd(2) � i=1 φ2,i(u)φ2,i(u) = νd(2) |Sd−1|P d 2 (u⊤u) = νd(2) |Sd−1|, u ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Therefore, putting N = (N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Nνd(2)) ∼ Nνd(2) �0, Iνd(2) �, gives us again using the Cauchy– Schwarz Inequality Z2 2(u) = 4|Sd−1| d2(d + 2)2 ��������� d � j=1 φ2,j(u)Nj ��������� 2 ≤ 4νd(2) d2(d + 2)2 ∥N∥2 = 2(d − 1) d2(d + 2)∥N∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Since ∥N∥2 ∼ χ2 νd(2), a comparison of 95% quantiles of 2(d − 1)∥N∥2/(d2(d + 2)) with Table 2 shows that this upper bound is only a good approximation for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following we give a list of the non-null eigenvalues λk, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , β, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4) corresponding to higher values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 3: λ1 = (3/(d(d + 2)))2 and λ3 = (6/(d(d + 2)(d + 4)))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 4: λ2 = (12/(d(d + 2)(d + 4)))2 and λ4 = (24/(d(d + 2)(d + 4)(d + 6)))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 5: λ1 = (15/(d(d + 2)(d + 4)))2, λ3 = (60/(d(d + 2)(d + 4)(d + 6)))2, and λ5 = (120/(d(d + 2)(d + 4)(d + 6)(d + 8)))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Case β = 6: λ2 = (90/(d(d + 2)(d + 4)(d + 6)))2, λ4 = (360/(d(d + 2)(d + 4)(d + 6)(d + 8)))2, as well as λ6 = (720/(d(d + 2)(d + 4)(d + 6)(d + 8)(d + 10)))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Since Sd−1 is compact, a direct application of the continuous mapping theorem and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1 prove the following Corollary to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un be iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' with U1 ∼ U � Sd−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Then we have Tn, β D −→ max b∈Sd−1Z2 β(b), where Zβ(·) is the limiting Gaussian process of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The resulting limit random variable in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='6 is not of pure theoretic interest, since the distribution and hence the asymptotic critical value can be approximated, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 8 3 Contiguous alternatives In this section, we consider a triangular array Un1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Unn of rowwise identically independent dis- tributed random vectors on Sd−1 having the density function fn(x) = µ(x) � 1 + h(x)/ √n � , x ∈ Sd−1, where µ(·) denotes the density of the uniform distribution with respect to the spherical Lebesgue mea- sure σ, and h is a bounded measurable function satisfying � Sd−1 h(x)µ(x) dσ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We consider n large enough to assure the non-negativity of fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' First, define P(n) = n � j=1 (µ σ) and A(n) = n � j=1 (fn σ) on the measurable space (Xn, Bn) = n � j=1 (Sd−1, M), where M denotes the class of subsets of Sd−1 that are measurable with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Further, denote the likelihood ratio with Ln = dA(n) dP(n) and write S β : Sd−1 × Sd−1 → R, (a, b) �→ S β(a, b) = (a⊤b) β − ψd(β), where ψd(·) is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Under the standing assumptions we have for the triangular array Un1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Unn Zn,β(·) D −→ Zβ(·) + S ∗ β(·) under A(n) in C(Sd−1, R), where Zβ(·) is a centred Gaussian process in C(Sd−1, R) having covariance kernel ρβ from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The shift function S ∗ β(·) is given by S ∗ β(b) = 1 |Sd−1| � Sd−1 S β(u, b)h(u) dσ(u), b ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) As a direct consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1 and the continuous mapping theorem we have the follow- ing Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1, we have Tn, β D −→ max b∈Sd−1 � Zβ(b) + S ∗ β(b) �2 under A(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' As an example we consider the alternatives where h(x) = hm,θ(x) = P d m(θ⊤x), x ∈ Sd−1, m ≥ 1, 9 is the Legendre polynomial of degree m and θ ∈ Sd−1 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Note that x �→ P d m(θ⊤x) is a spherical harmonic function of degree m such that the orthogonality property of spherical harmonics, see [22], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2, implies � Sd−1 h(x)dσ(x) = � Sd−1 P d m(θ⊤x)P d 0 (θ⊤x) dσ(x) = 0, since P d 0 (θ⊤·) = 1 is the spherical harmonic of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' If X follows the law given by the density fn we have for any orthogonal d × d-matrix A with Aθ = θ that the distribution of AX is the same as the distribution of X, hence these types of alternatives are rotationally symmetric about θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' An application of the Funk/Hecke-Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3 shows for b ∈ Sd−1 S ∗ β(b) = 1 |Sd−1| � Sd−1 S β(u, b)h(u) dσ(u) = 1 |Sd−1| � Sd−1 � (b⊤u)β − ψd(β) � P d m(θ⊤u) dσ(u) = λd(β, m)P d m(θ⊤b), where λd(β, m) = |Sd−2| |Sd−1| � 1 −1 P d m(t) � tβ − ψd(β) � (1 − t2) d−3 2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' It follows with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5) and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5 λd(β, m) = |Sd−2| |Sd−1| ��������� β � j=0 cj, d(β)⟨P d m, P d j ⟩ − ψd(β)⟨P d m, P d 0 ⟩ ��������� = cm, d(β) νd(m) , so that S ∗ β(b) = cm, d(β) νd(m) P d m(θ⊤b), b ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2) Note that λd(β, m) = 0, if β + m is odd or m > β, because then the coefficients cm, d(β) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5) equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Hence, in these cases we have the same asymptotic behaviour under contiguous alternatives as under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We can conclude that the tests Tn, β are not able to detect the alternatives for such a combination of β and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For the shift function we have S ∗ β(θ) = cm, d(β)/νd(m), so that there is a non-negative shift in the limiting distribution under contiguous alternatives as long as we can show that the coefficients cm, d(β) are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We conjecture that this is indeed the case, as all examples after Proposition fulfill this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' That in turn means that, under the assumption of this conjecture, there is a positive shift if β + m is even and m ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Thus, Tn, β is a family of testing procedures which is able to detect the contiguous alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' As indicated in [11], Section 2, the famous von Mises–Fisher distribution (see [33], Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3, for a definition) with mean direction θ and concentration parameter κ ≥ 0 falls into a comparable class of contiguous alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We expect to see matchable power performances of Tn, β in the simulation study, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 10 4 Consistency In this short section we consider spherical random vectors U, U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Un with a distribution having a continuous density f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' the spherical Lebesgue measure σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We adopt the reasoning in [8] to argue that the considered tests are consistent against a large class of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' If for β ∈ N there is a unit vector (say) b0 ∈ Sd−1 such that ζ(b0) = � E(b⊤ 0 U)β − ψd(β) �2 > 0, the strong law of large numbers shows lim n→∞ ζn(b0) = lim n→∞ ��������� 1 n n � j=1 (b⊤ 0 U j)β − ψd(β) ��������� 2 = ζ(b0) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' and since Tn, β/n ≥ ζn(b0) we have lim n→∞ Tn, β = ∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This reasoning shows that the tests Tn, β are consistent against each such alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Nonetheless, as we have already seen in the last section, Tn, β is not consistent against any arbitrary alternative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For certain combinations of β and m, the order of the Legendre polynomial, Tn, β exhibits the same asymptotic behaviour under the alternatives as under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Another indication for the inconsistency of Tn, β can be seen in the case β = 1, which essentially concerns the Rayleigh test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The authors of [18] have shown that, in the rather general context of rotationally symmetric alternatives with a location and concentration parameter and a defining angular function, the Rayleigh test is blind against certain local alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' These local alternatives show polynomial decrease of the concen- tration parameter towards zero (hence yielding the null hypothesis) and the odd-order derivatives of their angular function vanish at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' An example of such an alternative is the well-known Watson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 5 Bahadur efficiencies In this section we present some interesting insights into the Bahadur asymptotic relative efficiencies (ARE) of the statistics (Tn, β)β∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For an elaborate and comprehensive introduction to the concept of Bahadur efficiency we refer the reader to [5] and [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We consider alternative classes whose defining density f(· | κ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' σ is parameterized through a non-negative number κ ≥ 0, where the uniform distribution on Sd−1 is only obtained for the limit case 11 κ → 0+ in L1(Sd−1, dσ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' lim κ→0+ ||f(· | κ) − f(· | 0)||L1 = lim κ→0+ � Sd−1 |f(x | κ) − f(x | 0)| dσ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) Hence, the testing problem can be reformulated as H0 : κ = 0 against H1 : κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2) In order to properly apply the Bahadur theory, we consider the family of equivalent test statistics �Tn, β = �Tn, β, β ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following we mainly focus our attention to the local approximate and the local exact Bahadur ARE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For two statistics T (1) n and T (2) n these are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The local approximate Bahadur ARE is given by Λa T (1),T (2) = lim κ→0+ c a T (1)(κ) c a T (2)(κ), where c a T denotes the approximate Bahadur slope of a statistic Tn, see [35], page 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The local exact Bahadur ARE is defined by Λex T (1),T (2) = lim κ→0+ cT (1)(κ) cT (2)(κ), where cT denotes the exact Bahadur slope of a statistic Tn, see [35], Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In many cases the approximate and exact Bahadur slopes coincide in the proximity of the null hypothesis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' in the limit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This can also be observed for (�Tn, β)β∈N in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let β ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The approximate Bahadur slope of �Tn, β is given by c a �Tβ(κ) = max b∈Sd−1γ2 κ(b) max b∈Sd−1ρβ(b, b) = max b∈Sd−1γ2 κ(b) �β j=1 λjνd(j) , κ > 0, with the eigenvalues λ j from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3, νd(j) as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='1) and γκ(b) = Eκ(b⊤U) β − ψd(β), b ∈ Sd−1, where U ∼ f(· | κ), κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Furthermore, the exact Bahadur slope of �Tn, β is for sufficiently small κ > 0 given by c�Tβ(κ) = max b∈Sd−1γ2 κ(b) �β j=1 λjνd(j) + o � max b∈Sd−1γ2 κ(b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 12 The Bahadur slopes of �Tn, β apparently coincide locally, so that we do not distinguish between them anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following, we consider some explicit alternative classes and determine the local Bahadur ARE of �Tn, β w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Mn = −2 log(Λn), where Λn is the likelihood-ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' It is well-known that the exact and approximate Bahadur slope of Mn are the same and are given by c a M(κ) = cM(κ) = 2KL(κ, 0), κ > 0, where KL(κ, κ0) = Eκ � log � f(U | κ) f(U | κ0 �� , κ, κ0 ≥ 0, is the Kullback–Leibler information number for U ∼ f(· | κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The proof for the following quite technical calculations can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' A random vector U with values in Sd−1 has a von Mises–Fisher distribution vMF(θ, κ) with mean direction θ ∈ Sd−1 and concentration parameter κ ≥ 0 if the density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' σ is given by f(x | κ) = (κ/2)d/2−1 2πd/2I d 2 −1(κ) exp(κx⊤θ), x ∈ Sd−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3) where I d 2 −1 is the modified Bessel function of the first kind and order d/2 − 1, see (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In case of the von Mises–Fisher alternative class vMF(θ, κ) with a fixed mean direction θ ∈ Sd−1 and κ > 0 we have lim κ→0+ maxb∈Sd−1 γ2 κ(b) 2KL(κ, 0) = λ1νd(1), whereby the local Bahadur ARE is Λex �Tβ,M = Λa �Tβ,M = lim κ→0+ c a �Tβ(κ) c a M(κ) = 1 �β j=1 λjνd(j) lim κ→0+ maxb∈Sd−1 γ2 κ(b) 2KL(κ, 0) = λ1νd(1) �β j=1 λjνd(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The special case of β = 1 yields the local asymptotic optimality of �T1 in the Bahadur sense (see [35], page 9, for this concept) Λex �T1,M = Λa �T1,M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This is not surprising since the Rayleigh test is exactly the likelihood-ratio test in the von Mises–Fisher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' On the contrary, if β is even, then Λex �Tβ,M = Λa �Tβ,M = 0, since the eigenvalue λ1 equals zero in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 13 β d 2 3 5 10 β d 2 3 5 10 vMF 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='00 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='54 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='37 LP5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='06 LP6 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='02 Table 1: Non-trivial local Bahadur ARE of �Tn, β w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Mn for the alternative classes of the ex- amples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4 with dimension d ∈ {2, 3, 5, 10} and order m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , 6} of the LP alternative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' A random vector U with values in Sd−1 has a Watson distribution W(θ, κ) with mean direction θ ∈ Sd−1 and concentration parameter κ ∈ R if the density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' σ is given by f(x | κ) = Γ(d/2) 2πd/2M(1/2, d/2, κ) exp(κ(x⊤θ)2), x ∈ Sd−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4) where M(·, ·, ·) is the Kummer function, see B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In the following, we only consider the case κ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In case of the Watson alternative class W(θ, κ) with a fixed mean direction θ ∈ Sd−1 and κ > 0 we have lim κ→0+ maxb∈Sd−1 γ2 κ(b) 2KL(κ, 0) = λ2νd(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Thus the local Bahadur ARE equals Λex �Tβ,M = Λa �Tβ,M = λ2νd(2) �β j=1 λjνd(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This time the special case of β = 2 yields the local asymptotic optimality of �T2 in the Bahadur sense Λex �T2,M = Λa �T2,M = 1, whereas if β is odd, then Λex �Tβ,M = Λa �Tβ,M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 14 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In concordance with Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3 we shall define the alternative class LPm(θ, κ) of order m ∈ N with direction θ ∈ Sd−1 and κ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' LP stands for Legendre polynomial in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let this class be given by the density f(x | κ) = 1 |Sd−1| � 1 + κP d m(θ⊤x) � , x ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5) For fixed order m and direction θ we have lim κ→0+ maxb∈Sd−1 γ2 κ(b) 2KL(κ, 0) = λmνd(m), so that the local Bahadur ARE equals Λex �Tβ,M = Λa �Tβ,M = λmνd(m) �β j=1 λjνd(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We obtain non-trivial local Bahadur AREs only for combinations of β and m, where m ≤ β and β + m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In particular, the special case β = m = 1 or β = m = 2 gives the local asymptotic optimality of �T1 or �T2, respectively, in the Bahadur sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 6 Simulations We present a competitive Monte-Carlo simulation study, that was implemented and performed in the statistical computing environment R, see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The maximum on the hypersphere in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3) cannot be calculated analytically, and therefore one has to approximate it with a computationally fast method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We suggest to use a uniform random cover of the hypersphere: Simulate a large number m of uniformly distributed points on Sd−1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Bm (say), evaluate the so chosen centered and squared projections �1 n �n j=1(B⊤ k U j) β − ψd(β) �2, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , m, and approximate the maximum value in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='3) by the discrete maximum over all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Critical values for Tn, β under H0 have been simulated with 20000 replications and a random cover of m = 5000 points for d = 2, 3 and with 20000 replications and m = 20000 points for d = 5, 10, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The critical values in the rows in Table 2 denoted by ”∞” and ”∞∗” represent approximations of the limit random element maxb∈Sd−1 Z2 β(b) in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='6 via two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The first method, which corresponds to the rows with ”∞”, simulates the same random cover of the sphere as above, and it considers a large number (say) ℓ of random variables Z j = max(X2 j), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , ℓ, with iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' X j ∼ Nm(0, Σβ), where Nm is the m-variate normal distribution and Σβ = � ρβ(Bk1, Bk2) � k1,k2∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=',m} is a singular m × m-covariance matrix for m ≥ d and ρβ is the covariance kernel in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2) for which we 15 have already summarized explicit formulas in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Here x2 is shorthand for the vector of squared components of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Next, we calculate the empirical 95% quantile of Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Zℓ, where each approximation was simulated with ℓ = 100000 and m = 1000 for d = 2, 3 as well as ℓ = 10000 and m = 5000 for d = 5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The second method utilizes the alternative representation of the Gaussian process from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' For this purpose, we need orthonormal bases of the spaces Hk(Sd−1) for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , β and the corresponding eigenvalues λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We have already presented an explicit list of these eigenvalues for β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , 6 at the end of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' In each replication step of the Monte- Carlo simulation we generate an array Nk,j, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , νd(k)} of independent unit normal random variables, cover Sd−1, once again, with m uniformly distributed points B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , Bm and calculate Yi = � |Sd−1| �β k=0 √λk �νd(k) j=1 φk,j(b)Nk,j, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Repeating this step for the number of set replications ℓ yields an approximation of the limit distribution of Tn, β in the same fashion as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' However, so far there is no library with a stable implementation of orthonormal spherical harmonics in higher dimensions and orders, which is why we restricted the simulation with this method to the case of d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We used the package HFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='m in Mathematica in order to implement an orthonormal basis in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Each approximation was performed with ℓ = 20000 and m = 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Table 2 shows empirical and approximated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='95 quantiles of Tn, β under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' It is interesting to compare the approximated critical values with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='95 quantiles of χ2 d/d for Tn,1 (respectively the Rayleigh test), which are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='996 for d = 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='605 for d = 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='214 for d = 5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='830 for d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Evidently, the approximation with the random covering and the limiting process is close to the theoretical asymptotic critical values for the dimensions d = 2, 3, 5, but it gets less accurate for dimensions greater than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This behaviour can be explained by the curse of dimensionality, indicating that more points on the unit sphere in the random covering should be considered to increase the accuracy of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' A similar behaviour can be observed for β = 2 and 2(d −1)/(d2(d + 2)) χ2 νd(2) with νd(2) = (d − 1)(d + 2)/2, where, of course, the latter random variable is only an upper bound for the limit distribution of Tn,2 as we have seen in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Nevertheless, the numerical results support the theoretical findings of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We consider testing for uniformity on the unit circle S1, on the unit sphere S2 and on the hy- persphere S5, and we divide the presentation of the simulation study into two parts, since different competing tests are considered in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Generating uniformly distributed random numbers on Sd−1 can be done efficiently, since for a random vector N ∼ Nd(0, Id), where Nd stands for the d-variate normal distribution on Rd, we have N ||N|| ∼ U � Sd−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 16 n β 1 2 3 4 5 6 d = 2 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='746 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='917 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='941 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='968 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='944 ∞∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='753 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='923 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='735 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='606 d = 5 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='519 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='030 ∞ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='019 Table 2: Empirical and approximated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='95 quantiles of Tn, β under H0 for dimensions d ∈ {2, 3, 5, 10}, sample sizes n ∈ {20, 50, 100, 500} and β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' , 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Here, ∞ denotes the ap- proximation of the limit distribution of Tn, β via covariance kernel and ∞∗ the approximation via spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' 17 This property is merely a consequence of the rotational invariance of N and the fact that the uniform distribution is the only rotationally invariant distribution on Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' We consider the following alterna- tives to the uniform distribution: von Mises–Fisher distribution: This alternative class was already introduced in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' The density is given by Sd−1 ∋ x �→ (κ/2)d/2−1 2πd/2I d 2 −1(κ) exp(κx⊤θ), where I d 2 −1 is the modified Bessel function of the first kind and order d/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' This class is denoted with vMF(θ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Mix of von Mises–Fisher distributions with two centers: Let U be uniformly distributed on (0, 1), p ∈ (0, 1) and Yi ∼ vMF(θi, κi) with corresponding lo- cation and concentration parameters for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Let U, Y1 and Y2 be stochastically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQf2wV_/content/2301.03482v1.pdf'} +page_content=' Then we generate a random sample X according to X = Y11{U 0, +µS,N +t += 1 +N +N +� +i=1 +1{Ei +t=S}δXi +t, +µI,N +t += 1 +N +N +� +i=1 +1{Ei +t=I}δXi +t, +µR,N +t += 1 +N +N +� +i=1 +1{Ei +t=R}δXi +t. +Since the law of large numbers and the central limit theorem of the initial sequence +(µI,N +0 +, µS,N +0 +)N≥1 has already been studied in [7], under the assumption (H0) that the law of X1 +0, +is absolutly continuous with respect to the Lebesgue measure, in this paper, we will first write +the equation of evolution of (µS,N +t +, µI,N +t +, µR,N +t +), when the size of the population N is fixed. We +shall next study the law of large numbers and the central limit theorem of those sequences. +The law of large number result will be a convergence result in the space of measure valued pro- +cesses. The convergence proof will start with tightness in the appropriate space, identification +of the limit of any vaguely converging subsequence with the unique deterministic solution of a + +2 +PRELIMINARIES +4 +system of PDEs, from which the weak convergence, then in probability of the whole sequence +will follow. +The central limit theorem is technically more involved. +The first difficulty comes from the +fact that our domain is not compact. The approximating sequence lives in the space of signed +measures valued processes and one of the main problems to overcome is to find a suitable space +in which this sequence, as well as its limit, take their values. +We prove that the approximating sequence converges in the Skorokhod space [D(R+, H−s,σ(Rd))]3, +(σ > d/2, 1 + ⌈d +2⌉ < s < 2 + ⌈d +2⌉), to a continuous process characterized as the unique solution +of a linear Gaussian processes valued stochastic partial differential equation (abbreviated below +SPDE). The weighted Sobolev spaces H−s,σ(Rd) we consider here were introduced by Métivier +[28], for the integer values of s. Mélérad [24] uses that space for the study of the central limit +theorem of a sequence of empirical (random) measures of interacting particle systems. Clé- +mençon and all [11] also use that space to study a central limit theorem for a specific stochastic +epidemic model accounting for the effect of contact-tracing on the spread of an infectious dis- +ease. The work of Löfstrom [22] and [23] allows us to extend that space to the non interger +values of s, by using real interpolation techniques. +The paper is organized as follows. In section 2 we recall some results that will be useful in the +sequel. In section 3, we first establish the evolution equations of the measure-valued processes +µS,N, µI,N and µI,N then we show that the sequence {(µS,N +t +, µI,N +t +, µR,N +t +), t ≥ 0} converges in +probability as N → ∞ towards (µS, µI, µR), the unique solution of a system of parabolic +PDEs. In section 5 we study the convergence of the sequence of fluctuations processes (UN = +√ +N(µS,N − µ), V N = +√ +N(µI,N − µI), W N = +√ +N(µR,N − µR)) as N → ∞. +2 +Preliminaries +Notation: For any metric space E, +• MF(E) denotes the space of finite measures on E. +• For any integer k ≥ 0, Ck(E) (resp. +Ck +c (E)) denotes the space of continuous and k +times continuously differentiable real valued functions defined on E, (resp. with compact +support). For k = 0, we write C(E) (resp .Cc(E)) instead of C0(E). (resp. C0 +c (E)). +• For any integer k ≥ 0, Ck +b (E) denotes the space of real valued functions of class Ck on E +with bounded derivatives up to order k (order 0 included ). +• C0(E) denotes the space of continuous functions on E vanishing at infinity. +• For µ ∈ MF(E) and ϕ ∈ C(E), we denote the integral +� +E ϕ(x)µ(dx) by (µ, ϕ). +• In the following, the letter C will denote a (constant) positive real number which can +change from line to line. +• We equip MF(E) with the topology of weak convergence. +• Let E be a complete separable metric space, C(R+, E) (resp. +D(R+, E) is a space of +continuous (resp. +càdlàg) functions from R+ to E, equipped with the locally uniform +(resp. Skorokhod) topology. We refer the reader to section 12 of [6] for a presentation of +the Skorokhod topology and its associated metric. + +2 +PRELIMINARIES +5 +2.1 +Weighted Spaces of functions +For every nonnegative integer m and σ ∈ R+, we consider the space of all real valued functions +ϕ defined on Rd, with partial derivative up to oder m such that: +∥ϕ∥m,σ= +� � +|γ|≤m +� +Rd +|Dγϕ(x)|2 +1 + |x|2σ dx +� +< +∞, +where |.| denotes the euclidian norm on Rd, and for γ = (γ1, γ2.....γd) ∈ Nd, |γ|= +d� +i=1 +γi and +Dγϕ = (∂|γ|ϕ)/(∂xγ1 +1 ∂xγ2 +2 .....∂xγd +d ). +Let W m,σ +0 +(Rd) be the closure of the set of functions of class C∞ with compact support for this +norm. W m,σ +0 +(Rd) is a Hilbert space for the norm ∥.∥m,σ. For a nonnegative real number s we +extend the above space as follows. +Let J s be the potential operator defined on Rd by (J sϕ) = F −1[(1 + |.|2)s/2 �ϕ] (where the +Fourier transform �ϕ of ϕ is well defined, F −1 denotes the inverse of the Fourier transform). +Hs,σ(Rd) denotes the space of functions ϕ which satisfy the following. +∥ϕ∥s,σ= ∥(J sϕ)∥0,σ< ∞. +It is shown in [22] that Hm,σ(Rd) = W m,σ +0 +(Rd), for any nonnegative integer m. +We denote by H−s,σ(Rd) the dual space of Hs,σ(Rd). +Let Cm,σ(Rd) be the space of functions ϕ with continuous partial derivatives up to oder m +and such that +lim +|x|−→∞|Dγϕ(x)|2/1 + |x|2σ= 0 for all |γ|≤ m. +This space is normed with +∥ϕ∥Cm,σ= +� +|γ|≤m +sup +x∈Rd +|Dγϕ(x)| +1 + |x|σ . +Let C−m,σ(Rd) denotes the dual of Cm,σ(Rd). +We have the following continuous embeddings (see [1] and [28]). +Cm+j,σ ֒→ Cm,σ+r +m ≥ 0, +j ≥ 0, +σ > 0, +r ≥ 0. +(2.1) +Hs,σ(Rd) ֒→ Cℓ,σ(Rd), +s > d/2 + ℓ, +ℓ ≥ 0, +σ > 0. +(2.2) +Cm,σ(Rd) ֒→ W m,σ+η +0 +(Rd), +η > d/2, +m ≥ 0, +σ > 0. +(2.3) +Lemma 2.1. (A special case of theorem 2.1 in [19]) +Let (H, ∥.∥H) be a separable Hilbert space, M be an H−valued locally square integrable càdlàg +martingale and T(t) a contraction semigroup operator of L(H). Then there exists a finite +constant C depending only on the Hilbert norm ∥.∥H such that for all T > 0. +E +� +sup +0≤t≤T +��� +� t +0 +T(t − r)dMr +��� +2 +H +� +≤ Ce4σT E +� +∥MT∥2 +H +� +, +where σ is a real number such that ∥T(t)∥L≤ eσt. +Definition 2.2. (White noise) +White noise on Rd is a random distribution W defined on a probability space (Ω, F, P) which +is such that the mapping ϕ �→ (W, ϕ) is linear and continuous from L2(Rd) into L2(Ω) and +{(W, ϕ), ϕ ∈ L2(Rd)} is a centered Gaussian generalized process satisfying: +E((W, ϕ)(W, φ)) = (ϕ, φ)L2, for any ϕ, φ ∈ L2(Rd). +Where (., .)L2 denotes a scalar product on L2(Rd). +Space-time white noise is a white noise on R+ × Rd. + +3 +LAW OF LARGE NUMBERS +6 +3 +Law of Large Numbers +The aim of this section is to study the convergence of (µS,N, µI,N, µR,N), as N → ∞ under +Assumption (H1) below. +To this end we are going to: +• Write the system of evolution equations of (µS,N, µI,N, µR,N). +• Study the tightness of (µS,N, µI,N, µR,N)N≥1 in Skorokhod’s space [D(R+, (MF(Rd), v))]3, +where (MF(Rd), v) is the space of finite measure on Rd, equipped with the vague topology. +• Find the system of evolution equations satisfies by the limit in law (µS, µI, µR) of a +convergent subsequence of (µS,NµI,N, µR,N)N≥1 +• Show that the system of PDEs verified by (µS, µI, µR) admits a unique solution in +Λ = {(µ1, µ2, µ3)/0 ≤ (µi, 1) ≤ 1, i ∈ {1, 2, 3}}. . +The following is assumed to hold throughout this section. +Assumption (H1) +• The law of X1 +0 is absolutly continuous with respect to the Lebesgue measure and its +density is bounded. +• K ∈ Cc(Rd × Rd). +• For any A ∈ {S, I, R}, and x ∈ Rd, the matrix (θθt)(A, x) is invertible. +. +3.1 +System of evolution equations of (µS,N, µI,N, µR,N) +In this subsection we shall establish the following result. +Proposition 3.1. For any ϕ ∈ C2 +c (Rd), {(µS,N +t +, ϕ), (µI,N +t +, ϕ), (µR,N +t +, ϕ)} satisfies, +(µS,N +t +, ϕ) = (µS,N +0 +, ϕ) + +� t +0 +(µS,N +r +, QSϕ)dr − +� t +0 +� +µS,N +r +, ϕ(µI,N +r +, K) +� +dr + MN,ϕ +t +, +(3.1) +(µI,N +t +, ϕ) = (µI,N +0 +, ϕ) + +� t +0 +(µI,N +r +, QIϕ)dr + +� t +0 +� +µS,N +r +, ϕ(µI,N +r +, K) +� +dr − α +� t +0 +(µI,N +r +, ϕ)dr + LN,ϕ +t +, +(3.2) +(µR,N +t +, ϕ) = +� t +0 +(µR,N +r +, QRϕ)dr + α +� t +0 +(µI,N +r +, ϕ)dr + Y N,ϕ +t +. +(3.3) +Where +� +µS,N +r +, ϕ(µI,N +r +, K) +� += +� +Rd ϕ(x) +� +Rd K(x, y)µI,N +r +(dy)µS,N +r +(dx); +QAϕ(x) = m(A, x). ▽ ϕ(x) + 1 +2 +� +1≤ℓ,u≤d +(θ θt)ℓ,u(S, x) ∂2ϕ +∂xℓxu +(x), + +3 +LAW OF LARGE NUMBERS +7 +and with {Mi}1≤i≤N, {Qi}1≤i≤N two collections of standard (i.e. +with mean the Lebesgue +measure) Poisson random measures (abbreviated below PRM) on R2 ++, which are such that +B1, M1, Q1, . . . , BN, MN, QN are mutually independent, and denoting by M +i and Q +i the com- +pensated PRMs associated to Mi and Qi, we have +MN,ϕ +t += − 1 +N +N +� +i=1 +� t +0 +� ∞ +0 +1{Ei +r−=S}ϕ(Xi +r)1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du) ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=S} ▽ ϕ(Xi +r)θ(S, Xi +r)dBi +r. +LN,ϕ +t += 1 +N +N +� +i=1 +� t +0 +� ∞ +0 +1{Ei +r−=S}ϕ(Xi +r)1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du)) ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=I} ▽ ϕ(Xi +r)θ(I, Xi +r)dBi +r − 1 +N +N +� +i=1 +� t +0 +� α +0 +1{Eir=I}ϕ(Xi +r)Q +i(dr, du). +Y N,ϕ +t += 1 +N +N +� +i=1 +� t +0 +1{Eir=R} ▽ ϕ(Xi +r)θ(R, Xi +r)dBi +r + 1 +N +N +� +i=1 +� t +0 +� α +0 +1{Ei +r−=I}ϕ(Xi +r)Q +i(dr, du). +Proof. Let us first recall that for any t ≥ 0, +Xi +t = Xi +0 + +� t +0 +m(Ei +r, Xi +r)dr + +� t +0 +θ(Ei +r, Xi +t)dBi +r. +Let ϕ ∈ C2 +c (Rd), according to Itô’s formula we have, +ϕ(Xi +t) = ϕ(Xi) + +� t +0 +▽ϕ(Xi +r).m(Ei +r, Xi +r)dr + 1 +2 +� t +0 +� +1≤ℓ,u≤d +(θ θt)ℓ,u(Ei +r, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ +� t +0 +▽ϕ(Xi +r)θ(Ei +r, Xi +r)dBi +r. +(3.4) +On the other hand +1{Ei +t=S} = 1{Ei +0=S} − +� t +0 +� ∞ +0 +1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}1{Ei +r−=S}Mi(du, dr). +(3.5) +Hence using (3.4) and (3.5), we have +1{Ei +t=S}ϕ(Xi +t) = 1{Ei +0=S}ϕ(Xi) + +� t +0 +1{Eir=S} ▽ ϕ(Xi +r).m(S, Xi +r)dr ++ 1 +2 +� t +0 +1{Eir=S} +� +1≤ℓ,u≤d +(θ θt)ℓ,u(S, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ +� t +0 +1{Eir=S} ▽ ϕ(Xi +r)θ(S, Xi +r)dBi +r +− +� t +0 +� ∞ +0 +1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}1{Ei +r−=S}ϕ(Xi +r)Mi(du, dr). +Taking the sum over i and multiplying by +1 +N , we obtain +1 +N +N +� +i=1 +1{Ei +t=S}ϕ(Xi +t) = 1 +N +N +� +i=1 +1{Ei +0=S}ϕ(Xi) ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=S} ▽ ϕ(Xi +r).m(S, Xi +r)dr + +3 +LAW OF LARGE NUMBERS +8 ++ 1 +2N +N +� +i=1 +� t +0 +1{Eir=S} +� +1≤ℓ,u≤2 +(θ θt)ℓ,u(S, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=S} ▽ ϕ(Xi +r)θ(S, Xi +r)dBi +r +− 1 +N +N +� +i=1 +� t +0 +� ∞ +0 +1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}1{Ei +r−=S}ϕ(Xi +r)M +i(du, dr) +− 1 +N +N +� +i=1 +� t +0 +1 +N +N +� +j=1 +K(Xi +r, Xj +r)1{Ej +r=I}1{Eir=S}ϕ(Xi +r)dr, +from which (3.1) follows. Similarly, with again ϕ ∈ C2 +c (Rd), {1{Ei +t=I}ϕ(Xi +t), t ≥ 0} is a jump +process satisfying, +1{Ei +t=I}ϕ(Xi +t) = 1{Ei +0=I}ϕ(Xi) + +� t +0 +1{Eir=I} ▽ ϕ(Xi +r).m(I, Xi +r)dr ++ 1 +2 +� t +0 +1{Eir=I} +� +1≤ℓ,u≤2 +(θ θt)ℓ,u(I, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ +� t +0 +1{Eir=I} ▽ ϕ(Xi +r)θ(I, Xi +r)dBi +r ++ +� t +0 +� ∞ +0 +1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}1{Ei +r−=S}ϕ(Xi +r)Mi(du, dr) +− +� t +0 +� α +0 +1{Ei +r−=I}ϕ(Xi +r)Qi(du, dr). +Summing over i and multiplying by +1 +N , we obtain +1 +N +N +� +i=1 +1{Ei +t=I}ϕ(Xi +t) = 1 +N +N +� +i=1 +1{Ei +0=I}ϕ(Xi) ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=I} ▽ ϕ(Xi +r).m(I, Xi +r)dr ++ 1 +2N +N +� +i=1 +� t +0 +1{Eir=I} +� +1≤ℓ,u≤d +(θ θt)ℓ,u(I, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=I} ▽ ϕ(Xi +r)θ(I, Xi +r)dBi +r +− 1 +N +N +� +i=1 +� t +0 +� ∞ +0 +1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}1{Ei +r−=S}ϕ(Xi +r)M +i(du, dr) +− 1 +N +N +� +i=1 +� t +0 +1 +N +N +� +j=1 +K(Xi +r, Xj +r)1{Ej +r=I}1{Eir=S}ϕ(Xi +r)dr +− 1 +N +N +� +i=1 +� t +0 +� α +0 +1{Ei +r−=I}ϕ(Xi +r)Q +i(du, dr) − α +N +N +� +i=1 +� t +0 +1{Eir=I}ϕ(Xi +r)dr, +from which (3.2) follows. Similarly, with once again ϕ ∈ C2 +c (Rd), {1{Ei +t=R}ϕ(Xi +t), t ≥ 0} is a +jump processes satisfying, + +3 +LAW OF LARGE NUMBERS +9 +1{Ei +t=R}ϕ(Xi +t) = +� t +0 +1{Eir=R} ▽ ϕ(Xi +r).m(R, Xi +r)dr ++ +� t +0 +1{Eir=R} +� +1≤ℓ,u≤d +(θ θt)ℓ,u(R, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ +� t +0 +1{Eir=R} ▽ ϕ(Xi +r)θ(R, Xi +r)dBi +r + +� t +0 +� α +0 +1{Ei +r−=I}ϕ(Xi +r)Qi(du, dr). +Summing over i and multiplying by +1 +N we obtain, +1 +N +N +� +i=1 +1{Ei +t=R}ϕ(Xi +t) = 1 +N +N +� +i=1 +� t +0 +1{Eir=R} ▽ ϕ(Ei +r, Xi +r).m(R, Xi +r)dr ++ 1 +2N +N +� +i=1 +� t +0 +1{Eir=R} +� +1≤ℓ,u≤d +(θ θt)ℓ,u(R, Xi +r) ∂2ϕ +∂xℓxu +(Xi +r)dr ++ 1 +N +N +� +i=1 +� t +0 +1{Eir=R} ▽ ϕ(Xi +r)θ(R, Xi +r)dBi +r ++ 1 +N +N +� +i=1 +� t +0 +� α +0 +1{Ei +r−=I}ϕ(Xi +r)Q +i(du, dr) + α +N +N +� +i=1 +� t +0 +1{Eir=I}ϕ(Xi +r)dr, +from which (3.3) follows. +3.2 +Tightness and Convergence of (µS,N, µI,N, µR,N) in +[D(R+, MF (Rd))]3 +Recall that we equip MF(Rd) with the topology of weak convergence and the Skorokhod space +of cádlág function from R+ into MF(Rd) with the Skorokhod topology (we refer to page 63 of +[18] for an explicit definition). +Remark 3.2. For A ∈ {S, I, R}, we have (µA,N +t +, 1Rd) = 1 +N +�N +i=1 1{Ei +t=A} ≤ 1, +thus for any ϕ ∈ Cc(Rd), A ∈ {S, I, R}, |(µA,N +t +, ϕ)|≤ ∥ϕ∥∞. +We can now establish the wished tightness. +Proposition 3.3. The sequences (µS,N)N≥1; (µI,N)N≥1 and (µR,N)N≥1 are tight in +D(R+, (MF(Rd), v)), where MF(Rd) is equipped with the topology of vague convergence. +Proof. Let us prove that (µS,N)N≥1 is tight in D(R+, (MF(Rd), v)), where MF(Rd) is equipped +with the vague topology. We refer to Theorem 2.2 of Roelly [32]. Let Ξ be a dense subset of +C0(Rd), a sufficient condition for (µS,N)N≥1 to be tight in D(R+, (MF(Rd), v)) is that: +for any +ϕ ∈ Ξ, +{(µS,N +t +, ϕ), +t ≥ 0, +N ≥ 1} is tight in D(R+, R). +We choose Ξ = C∞ +c (Rd) (= the space of infinitely differentiable functions with compact support). +Let ϕ ∈ C∞ +c (Rd), we have +(µS,N +t +, ϕ) = (µS,N +0 +, ϕ) + +� t +0 +(µS,N +r +, QSϕ)dr − +� t +0 +� +µS,N +r +, ϕ(µI,N +r +, K) +� +dr + MN,ϕ +t +, +We notice that (µS,N, ϕ) is a semi-martingale since MN,ϕ is a square integrable martingale. +Indeed, MN,ϕ is a local martingale as the sum of local martingales and + +3 +LAW OF LARGE NUMBERS +10 +< MN,ϕ >t = 1 +N +� t +0 +� +µS,N +r +, ϕ2(µI,N +r +, K) +� +dr ++ 1 +N +� t +0 +� +µS,N +r +, +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr, +≤ 1 +N +� t +0 +���� +� +X +ϕ2(x) +� +X×X +K(x, y)µI,N +r +(dy)µS,N +r +(dx) +���� dr ++ t +N +� +1≤ℓ,u≤d +��� ∂ϕ +∂xℓ +��� +2 +∞ +��θℓ,u(S, .) +��2 +∞ + 2t +N +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +��� ∂ϕ +∂xℓ +��� +∞ +��� ∂ϕ +∂xu +��� +∞∥θℓ,e(S, .)∥∞∥θu,e(S, .)∥∞, +≤ t∥ϕ∥2 +∞∥K∥∞ +N ++ t +N C. +thus E(< MN,ϕ >t) < ∞, ∀t ≥ 0. +On other hand +(µS,N +t +, ϕ) = (µS,N +0 +, ϕ) + +� t +0 +ωN,ϕ +r +dr + MN,ϕ +t +with < MN,ϕ >t= +� t +0 +̟N,ϕ +r +dr, +and +ωN,ϕ +t += +� +µS,N +t +, m(A, .). ▽ ϕ(.) + 1 +2 +� +1≤ℓ,u≤d +(θ θt)ℓ,u(S, .) ∂2ϕ +∂xℓxu +(.) +� +− +� +µS,N +t +, ϕ(µI,N +t +, K) +� +, +̟N,ϕ +t += 1 +N +� +µS,N +t +, ϕ2(µI,N +t +, K) +� ++ 1 +N +� +µS,N +t +, +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +. +Furthermore ωN,ϕ and ̟N,ϕ are progressively measurable since the are adapted and right con- +tinuous, so according to proposition 37 of [30] a sufficient condition for {(µS,N +t +, ϕ), t ≥ 0, N ≥ 1} +to be tight in D(R+, R) is that both: +• (µS,N +0 +, ϕ)N≥1 is tight in R, +• ∀T ≥ 0, sup +0≤t≤T +(| ωN,ϕ +t +| +̟N,ϕ +t +) is tight in R. +These follow readily from the facts that: +− |(µS,N +0 +, ϕ)|≤ ∥ϕ∥∞. +− | ωN,ϕ +t +|≤ +� +1≤ℓ≤d +∥mℓ∥∞∥ ∂ϕ +∂xℓ(x1...., xd)∥∞+ 1 +2 +� +1≤ℓ,u≤d +∥(θ θt)ℓ,u(S, .)∥∞∥ ∂2ϕ +∂xℓxu∥∞+∥ϕ∥∞∥K∥∞. +⩽ C +− ̟N,ϕ +t +≤ ∥ϕ∥2 +∞∥K∥∞ +N ++ C ≤ C. +The same arguments yield the tightness of {µI,N +t +, t ≥ 0, N ≥ 1} and {µR,N +t +, t ≥ 0, N ≥ 1} in +D(R+, (MF(Rd), v))). +The following Proposition follows from the fact that the jump of µA,N are order of 1/N( see +the proof of Proposition 3.3 in [7] for the explicit proof). +Proposition 3.4. The limit points (µS), (µI) and (µR) of the sequences (µS,N)N≥1, (µI,N)N≥1 +and (µR,N)N≥1 are elements of C(R+, MF(Rd)). + +3 +LAW OF LARGE NUMBERS +11 +Theorem 3.5. The sequence (µS,N, µI,N, µR,N)N≥1 converges in probability, in +� +D(R+, MF(Rd)) +�3 +towards (µS, µI, µR) ∈ +� +C(R+, MF(Rd)) +�3 which is the unique solution of the following system +of equations. For any ϕ ∈ C2 +c (Rd), +(µS +t , ϕ) = (µS,N +0 +, ϕ) + +� t +0 +(µS +r , QSϕ)dr − +� t +0 +� +µS +r , ϕ(µI +r, K) +� +dr, +(3.6) +(µI +t, ϕ) = (µI +0, ϕ) + +� t +0 +(µI +r, QIϕ)dr + +� t +0 +� +µS +r , ϕ(µI +r, K) +� +dr − α +� t +0 +(µI +r, ϕ)dr, +(3.7) +(µR +t , ϕ) = +� t +0 +(µR +r , QRϕ)dr + α +� t +0 +(µI +r, ϕ)dr. +(3.8) +3.2.1 +Proof of Theorem 3.5 +By Proposition 3.3, the sequence (µS,N, µI,N, µR,N)N≥1 is tight in +� +D(R+, (MF(Rd), v)) +�3, thus +according to Prokhorov’s Theorem there exists a subsequence of (µS,N, µI,N, µR,N)N≥ still de- +noted (µS,N, µI,N, µR,N)N≥1 which converges in law in +� +D(R+, (MF(Rd), v)) +�3 towards (µS, µI, µR), +where MF(Rd) is equipped with the vague topology. +Hence to complete the proof of Theorm 3.5 it remains to: +• Find the system of PDEs satisfes by {(µS +t , µI +t, µR +t ), t ≥ 0} +• Show that the system verifies by (µS +t , µI +t, µR +t ) admits a unique solution on +Λ = {(µ1, µ2, µ3)/0 ≤ (µi, 1Rd) ≤ 1, i ∈ {1, 2, 3}}. +• Conclude. +It is so easy to obtain the following Lemma, therefore we omit the proof. +Lemma 3.6. If we let Σ = {(µ1, µ2) ∈ MF(Rd), (µi, 1Rd) ≤ 1, ∀i ∈ {1, 2}}, for any ϕ, ψ ∈ +Cc(Rd), the following map is continuous. +Gϕ,ψ : (Σ, v) × (Σ, v) → (MF(Rd × Rd), v) +(µ, ν) �−→ (µ ⊗ ν, ϕψ) +where (µ ⊗ ν, ϕψ) = +� +Rd×Rd ϕ(x)ψ(y)ν(dy)µ(dx). +The following Proposition establishes the system of equations satisfied by (µS, µI, µR). +Proposition 3.7. The processes (µS, µI, µR) satisfies the system formed by the equations (3.6), +(3.7) and (3.8). +Proof. We prove this Proposition by taking the limit in the equations (3.1), (3.2) and (3.3). +Let us establish (3.6). +1- It has been shwon in [7] that the sequence {(µS,N +0 +, µI,N +0 +), N ⩾ 1} converges a.s. towards the +pair of deterministic measures (µS +0, µI +0). +2- Since the map x ∈ Rd �→ QSϕ(x) = m(A, x). ▽ ϕ(x) + 1 +2 +� +1≤ℓ,u≤d +(θ θt)ℓ,u(S, x) ∂2ϕ +∂xℓxu +(x) is con- +tinuous with compact support, +� t +0 +� +µS,N +r +, QSϕ +� +dr converges in law towards +� t +0 +� +µS +r , QSϕ +� +dr. +2- Form Proposition 3.3, the sequence µS,N ⊗µI,N is also tight, thus from Prokorov’s theorem it + +3 +LAW OF LARGE NUMBERS +12 +is possible to extract a sub-sequence still denotes (µS,N ⊗ µI,N)N≥1 such that (µS,N ⊗ µI,N)N≥1 +and the above subsequences (µS,N, µI,N, µR,N)N≥1 converges towards χS,I and (µS, µI, µR) re- +spectively. Furthermore the fact that for any t ≥ 0, χS,I +t += µS +t ⊗ µI +t follows from Lemma 3.6. +Consequently, we have +� t +0 +� +µS,N +r +, ϕ(µI,N +r +, K) +� +dr = +� t +0 +� +µS,N +r +⊗ µI,N +r +, ϕK +� +dr +L−→ +� t +0 +� +µS +r ⊗ µI +r, ϕK +� +dr. +3- Let us prove that the sequences MN,ϕ +t +; LN,ϕ +t +and Y N +t +converge to 0 in Probability. +− Convergence of MN,ϕ +t +. We have seen above that +E(| MN,ϕ +t +|2)= E(< MN,ϕ >t) +≤ t +N ∥ϕ∥2 +∞∥k∥∞+tC +N +N→∞ +−−−→ 0, +consequently MN,ϕ +t +converges to 0 in L2, so also in probability. By similar arguments, we obtain +the convergences in probability to 0 of the sequences LN,ϕ +t +and Y N,ϕ +t +. +Hence (3.6) follows from 1-, 2-, 3-. Similar arguments yield (3.7) and (3.8). +Let us now prove the following proposition which will be useful to show that the system of +equations (3.6), (3.7) and (3.8) admits a unique solution. +Lemma 3.8. For A ∈ {S, I, R}, ΥA(t) denotes the Markovian semi-group of the diffussion +process with diffusion matrix θ(A, .) and drift coefficient m(A, .). For any ϕ ∈ Cc(Rd), we have +(µS +t , ϕ) = (µS +0 , ΥS(t)ϕ) − +� t +0 +� +µS +r , ΥS(t − r)ϕ(µI +r, K) +� +dr, +(3.9) +(µI +t, ϕ) = (µI +0, ΥI(t)ϕ) + +� t +0 +� +µS +r , ΥI(t − r)ϕ(µI +r, K) +� +dr − α +� t +0 +(µI +r, ΥI(t − r)ϕ)dr, +(3.10) +(µR +t , ϕ) = α +� t +0 +(µI +r, ΥR(t − r)ϕ)dr. +(3.11) +Proof. We may classically derive from (3.6) a similar formula where the test function ϕ(x) is +replaced by ψr(x) = ψ(r, x) which is of class C1,2 on [0, t] × Rd: +(µS +t , ψt(.)) = (µS +0, ψ0(.)) + +� t +0 +� +µS +r , ∂ +∂rψr(.) +� +dr + +� t +0 +(µS +r , m(S, .). ▽ ϕ)dr ++ 1 +2 +� t +0 +(µS +r , Tr[(θ θt)(S, .)D2ϕ])dr − +� t +0 +� +µS +r , ψr(.)(µI +r, K) +� +dr. +(3.12) +Let us now consider a continuous fonctions ϕ on Rd, with compact support and fix a time +t ∈ R+. We define for (r, x) ∈ [0, t] × Rd, ψr(x) = ΥA(t − r)ϕ(x). +Then ψ is the solution of the following equation +dψr(x) +dr ++ QSϕ(x) = 0 +on [0, t] × Rd. +Equation (3.12) applied to this function ψ yields (3.9). We obtain (3.10) and (3.11) by similar +arguments. +Proposition 3.9. The system formed by the equations (3.9), (3.10) and (3.11), admits a unique +solution on the set Λ = {(µ1, µ2, µ3) ∈ [MF(Rd)]3/(µi, 1) ≤ 1 +∀i ∈ {1, 2, 3}}. +Proof. Let us recall that the distance in total variation on MF(Rd) is defined by +∥µ − ν∥V T=sup{|(µ − ν, ϕ)|, ϕ continuous with compact support and ∥ϕ∥∞≤ 1}. +Now let (µ1 +t, µ2 +t, µ3 +t) and (ν1 +t , ν2 +t , ν3 +t ) be two solutions of the system of equations (3.9), (3.10) and + +3 +LAW OF LARGE NUMBERS +13 +(3.11) with the same initial condition and ϕ ∈ Cc(Rd). Since ΥS(t) is a contraction semi-group +on Cc(Rd), we have +��� (µ1 +r, ΥS(t − r)ϕ(µ2 +r, K)) − (ν1 +r, ΥS(t − r)ϕ(ν2 +r, K)) +��� += +��� +� +µ1 +r − ν1 +r, ΥS(t − r)ϕ(µ2 +r, K) +� +− +� +ν1 +r, ΥS(t − r)ϕ(ν2 +r − µ2 +r, K) +� ���, +≤ +��� +� +Rd ΥS(t − r)ϕ(x)(µ2 +r, K(x, .))(µ1 +r − ν1 +r)(dx) +��� ++ +��� +� +Rd ΥS(t − r)ϕ(x) +� +Rd K(x, y)(µ2 +r − ν2 +r)(dy)ν1 +r(dx) +���, +≤ ∥ΥS(t − r)ϕ(µ2 +r, K)∥∞∥µ1 +r − ν1 +r∥V T+∥ΥS(t − r)ϕ∥∞∥K∥∞∥µ2 +r − ν2 +r∥V T, +≤ ∥ϕ∥∞∥K∥∞ +� +∥µ1 +r − ν1 +r∥V T+∥µ2 +r − ν2 +r∥V T +� +. +Thus using the equations (3.9), (3.10) and (3.11) respectively we obtain +sup +∥ϕ∥∞≤1 +|(µ1 +t − ν1 +t , ϕ)| ≤ ∥K∥∞ +� t +0 +� +∥µ1 +r − ν1 +r∥V T+∥µ2 +r − ν2 +r∥V T +� +dr, +(3.13) +sup +∥ϕ∥∞≤1 +|(µ2 +t − ν2 +t , ϕ)| ≤ ∥K∥∞(1 + α) +� t +0 +� +∥µ1 +r − ν1 +r∥V T+∥µ2 +r − ν2 +r∥V T +� +dr, +(3.14) +sup +∥ϕ∥∞≤1 +|(µ3 +t − ν3 +t , ϕ)| ≤ α +� t +0 +∥µ2 +r − ν2 +r∥V T, +(3.15) +where the suppremum is taken over continuous functions with compact support. Consequently +summing the equations (3.13), (3.14) and (3.15) the result follows from Gronwall’s Lemma. +We can now complete the proof of Theorem 3.5. +Since (µS,N, µI,N, µR,N)N≥1 is tight in [D(R+, (MF(Rd), v))]3, and all converging subsequences +of the sequence (µS,N, µI,N, µR,N)N≥1 converge in law in [D(R+, (MF(Rd), v))]3 to the same limit +(µS, µI, µR), where MF(Rd) is equipped with the vague topology, the sequence (µS,N, µI,N, µR,N)N≥1 +converge in law in [D(R+, (MF(Rd), v))]3 towards (µS, µI, µR). To extend this result to the weak +topology, we use a criterion (Proposition 3) proved in [25]. Since from Proposition 3.4 the lim- +iting process (µS, µI, µR) is continuous, it suffices to prove that the sequence +� +(µS,N, 1), (µI,N, 1), (µR,N, 1) +� +N≥1 converges in law to +� +(µS, 1), (µI, 1), (µR, 1) +� +in [D(R+, R)]3, +which follows from and Proposition 3.9 and the fact that: +• Proposition 3.1 remains true for the functions ϕ ∈ C2 +b (Rd). +• Following the Proof of Proposition 3.3, we see that the sequence +� +(µS,N, 1), (µI,N, 1), (µR,N, 1) +� +N≥1 is tight in [D(R+, R)]3. +• From Prokorov’s Theorem we deduce the existence of a subsequence which converge in +law towards +� +(µS, 1), (µI, 1), (µR, 1) +� +. +• Proposition 3.8 remains true when the test function ϕ is a constant. +Finally since the sequence (µS,N, µI,N, µR,N)N≥1 weakly converge in (D(R+, MF(Rd)))3 to- +wards (µS, µI, µR) and (µS, µI, µR) is deterministic, we have convergence in probability. + +3 +LAW OF LARGE NUMBERS +14 +3.3 +Existence of densities +The third assumptions in (H1), allow us to obtain the following result (see Theorem 22 in [26]). +Remark 3.10. Under the assumption (H1), For A ∈ {S, I, R}, there exists a measurable +function ΥA(t)(x, y), defined on Rd × Rd, which is a density in y ∈ Rd and such that for each +continuous function ϕ defined on Rd, one has +ΥA(t)ϕ(x) = +� +Rd ΥA(x, y)ϕ(y)dy. +Proposition 3.11. There exists (f S, f I, f R) ∈ L∞ +loc(R+, (L1(Rd)3) which satisfies: +∂tf S +t (x) = Q∗ +Sf S +t (x) − f S +t (x) +� +Rd K(x, y)f I +t (y)dy, +∂tf I +t (x) = Q∗ +If I +t (x) + f S +t (x) +� +Rd K(x, y)f I +t (y)dy − αf I +t (x), +∂tf R +t (x) = Q∗ +Rf R +t (x) + αf I +t (x), +(3.16) +where Q∗ +A is the adjoint operator. +Proof. Let us recall that the initial measures µS +0, µI +0 are absolutly continuous with respect to +the Lebesgue measure (see Theorem 3.1 in [7]). +We construct by inductions a sequences of functions (f n, gn, hn), satisfying in a weak sense the +following system. +∂tf n+1 +t +(x) = Q∗ +Sf n+1 +t +(x) − f n+1 +t +(x) +� +Rd K(x, y)gn +t (y)dy, +∂tgn+1 +t +(x) = Q∗ +Ign+1 +t +(x) + f n+1 +t +(x) +� +Rd K(x, y)gn +t (y)dy − αgn+1 +t +(x), +∂thn+1 +t +(x) = Q∗ +Rhn+1 +t +(x) + αgn+1 +t +(x), +f n+1 +0 +(x) = f S +0 (x), gn+1 +0 +(x) = f I +0 (x), hn+1 +0 +(x) = 0. +(3.17) +Thanks to the nonnegativity of f S +0 and applying the Feyman-Kac formula, we show that f n +is nonnegtive. The non-negativity of gn, follows by recurrence and by using the comparaison +principle, the Feyman Kac formula and the fact that f I +0 is non-negative. Finaly hn is non- +negative by using the comparaison principle, the Feyman Kac formula. +From system (3.17), it is easy to see that for any ϕ ∈ Cc(Rd), +(f n+1 +t +, ϕ) = +� +Rd f S +0 (x) +� +Rd ΥS(t)(x, y)ϕ(y)dydx +− +� t +0 +� +Rd f n+1 +r +(x) +� +Rd K(x, u)gn +r (u)du +� +Rd ΥS(t − r)(x, y)ϕ(y)dydxdr, +(gn+1 +t +, ϕ) = +� +Rd f I +0 (x) +� +Rd ΥI(t)(x, y)ϕ(y)dydx ++ +� t +0 +� +Rd f n+1 +r +(x) +� +Rd K(x, u)gn +r (u)du +� +Rd ΥI(t − r)(x, y)ϕ(y)dydxdr +− α +� t +0 +� +Rd gn+1 +r +(x) +� +Rd ΥI(t − r)(x, y)ϕ(y)dydxdr, +(hn+1 +t +, ϕ) = α +� t +0 +� +Rd gn+1 +r +(x) +� +Rd ΥR(t − r)(x, y)ϕ(y)dydxdr. + +3 +LAW OF LARGE NUMBERS +15 +Fubini’s theorem yields, +f n+1 +t +(y) = +� +Rd f S +0 (x)ΥS(t)(x, y)dx − +� t +0 +� +Rd f n+1 +r +(x) +� +Rd K(x, u)gn +r (u)duΥS(t − r)(x, y)dxdr, +(3.18) +gn+1 +t +(y) = +� +Rd f I +0 (x)ΥI(t)(x, y)dx + +� t +0 +� +Rd f n+1 +r +(x) +� +Rd K(x, u)gn +r (u)duΥI(t − r)(x, y)dxdr +− α +� t +0 +� +Rd gn+1 +r +(x)ΥI(t − r)(x, y)dxdr, +(3.19) +hn+1 +t +(y) = α +� t +0 +� +Rd gn+1 +r +(x)ΥR(t − r)(x, y)dxdr. +(3.20) +Fisrt of all, since ∀n ∈ N, f n +t ≥ 0 and gn +t ≥ 0, f n+1 +t +(y) ≤ +� +Rd f S +0 (x)ΥS(t)(x, y)dx. +Thus integrating over y ∈ Rd, using Fubini’s Theorem and the fact that +� +Rd ΥS(t)(x, y)dy = 1, +∀t ≥ 0, we see that +sup +n +sup +0≤t≤T +∥f n +t ∥L1 ≤ ∥f S +0 ∥L1≤ 1. +(3.21) +The last inequality follows from the fact that if H(x) is the density of the law of X1 +0, +f S +0 (x) = {(1−p)1A(x)+1Ac(x)}H(x) (see Theorem 3 1 in [7]), where A and p have been defined +in the introduction. +Moreover, since ∀n ∈ N, gn +t ≥ 0, +gn+1 +t +(y) ≤ +� +Rd f I +0 (x)ΥI(t)(x, y)dx + +� t +0 +� +Rd f n+1 +r +(x) +� +Rd K(x, u)gn +r (u)duΥI(t − r)(x, y)dxdr. +Thus integrating over y ∈ Rd, using (3.21), Fubini’s Theorem and Gronwall’s Lemma, we easily +deduce that +sup +n +sup +0≤t≤T +∥gn +t ∥L1 ≤ ∥f I +0 ∥L1exp(T∥K∥∞) ≤ exp(T∥K∥∞), +(3.22) +where the last inequality follows from the fact that if H(x) is the density of the law of X1 +0, +f I +0 (x) = p1A(x)H(x). +On the other hand, with the same argument as above, we deduce from (3.20) and (3.22) that +sup +n +sup +0≤t≤T +∥hn +t ∥L1≤ αTexp(T∥K∥∞). +(3.23) +Let us now show the convergence of the sequence (f n, gn, hn) in L∞ +loc(R+, [L1(Rd)]3). +A straightforward computation using (3.18), and similar arguments as above yields +f n+1 +t +(y) − f n +t (y) = − +� t +0 +� +Rd(f n+1 +r +(x) − f n +r (x)) +� +Rd K(x, u)gn +r (u)duΥS(t − r)(x, y)dxdr ++ +� t +0 +� +Rd f n +r (x) +� +Rd K(x, u)(gn−1 +r +(u) − gn +r (u))duΥS(t − r)(x, y)dxdr +∥f n+1 +t +− f n +t ∥L1≤ ∥K∥∞{sup +n +sup +0≤t≤T +∥f n +t ∥L1+sup +n +sup +0≤t≤T +∥gn +t ∥L1} + +4 +CENTRAL LIMIT THEOREM +16 +× +� t +0 +{∥f n+1 +r +− f n +r ∥L1+∥gn +r − gn−1 +r +∥L1}dr +≤ C(T)∥K∥∞ +� t +0 +{∥f n+1 +r +− f n +r ∥L1+∥gn +r − gn−1 +r +∥L1}dr. +(3.24) +Similarly, we have +∥gn+1 +t +− gn +t ∥L1≤ C(T)(1 + α)∥K∥∞ +� t +0 +{∥gn +r − gn−1 +r +∥L1+∥gn+1 +r +− gn +r ∥L1+∥f n+1 +r +− f n +r ∥L1}dr, +(3.25) +∥hn+1 +t +− hn +t ∥L1≤ α +� t +0 +∥gn+1 +r +− gn +r ∥L1dr. +(3.26) +Summing (3.24), (3.25) and (3.26) and using Gronwall’s Lemma, we have +sup +0≤r≤t +� +∥f n+1 +r +− f n +r ∥L1+∥gn+1 +r +− gn +r ∥L1+∥hn+1 +r +− hn +r ∥L1 +� +≤ C(t) +� t +0 +sup +0≤u≤r +� +∥f n +u −f n−1 +u +∥L1+∥gn +u −gn−1 +u +∥L1+∥hn +r −hn−1 +u +∥L1 +� +dr, +Picard’s Lemma then yields +� +n +sup +0≤t≤T +� +∥f n+1 +t +− f n +t ∥L1+∥gn+1 +t +− gn +t ∥L1+∥hn+1 +t +− hn +t ∥L1 +� +< ∞, for any T > 0. +Therefore the sequence (f n, gn, hn)n converge in L∞ +loc(R+, (L1)3) towards (f S, f I, f R) which sat- +isfies by (3.21), (3.22) and (3.23) respectively +sup +0≤t≤T +∥f S +t ∥L1≤ 1, +sup +0≤t≤T +∥f I +t ∥L1≤ exp(T∥K∥∞) and sup +0≤t≤T +∥f R +t ∥L1≤ Tαexp(T∥K∥∞∥). +Moreover it is easy to see that (f S, f I, f R) satisfies in the weak sense the system (3.16). +Since (µS +t , µI +t, µR +t ) satisfy (3.6), (3.7) and (3.8), from Proposition 3.11 and 3.9, we deduce +the following result. +Corollary 3.12. For any t ≥ 0, the measure µS +t , µI +t and µR +t are aboslutely continuous with +respect to the Lebesgue measure. Their density (f S, f I, f R) ∈ L∞ +loc(R+, (L1(Rd)3). +Remark 3.13. From the system of equations in Theorem 3.5 above, one can also prove that +the measure µS +t , µI +t and µR +t are aboslutely continuous with respect to the Lebesgue measure, +using this time assumption (H1) and the Feyman-kac formula and the fact that the law of the +markovian process having Q∗ +S or Q∗ +I or Q∗ +R as infinitesimal generator is absolutely continuous +with respect to the Lebesgue measure. +4 +Central Limit Theorem +In this section, we will study the convergence of the sequence of fluctuations processes +(UN = +√ +N(µS,N − µS), V N = +√ +N(µI,N − µI), W N = +√ +N(µR,N − µR)), +as N → ∞, under the assuption (H2) below. Note that the trajectories of these processes +belong to (D(R+, E(Rd)))3, where E(Rd) is the space of signed measures on Rd. However, since +the limit processes may be less regular than their approximations we will first: + +4 +CENTRAL LIMIT THEOREM +17 +• Formulated the equations verified by (UN, V N, W N). +• Fix the space in which the convergence results will be established. +Then we will study the convergence of the above sequence. +Letting D = ⌈d/2⌉ (where ⌈.⌉ is the upper integer part), the following is supposed to hold +throughout this section. +Assumption (H2): +• For any A ∈ {S, I, R}, for any ℓ, u ∈ {1, 2...d}, both functions θℓ,u(A, .) and mℓ(A, .) +belong to C3+D +b +(Rd). +• K ∈ C2+D +c +(Rd × Rd). +4.1 +System of evolution equations of the Processes (U N, V N, W N) +Let ϕ ∈ C2 +b (Rd), we have +(µS,N +t +, ϕ) = (µS,N +0 +, ϕ) + +� t +0 +(µS,N +r +, QSϕ)dr − +� t +0 +� +µS,N +r +, ϕ(µI,N +r +, K) +� +dr + MN,ϕ +t +, +(µS +t , ϕ) = (µS +0 , ϕ) + +� t +0 +(µS +r , QSϕ)dr − +� t +0 +� +µS +r , ϕ(µI +r, K) +� +dr. +Thus +(UN +t , ϕ) = (UN +0 , ϕ) + +� t +0 +(UN +r , QSϕ)dr − +� t +0 +(UN +r , ϕ(µI,N +r +, K))dr − +� t +0 +(µS +r , ϕ(V N +r , K))dr + +√ +NMN,ϕ +t +. +Hence letting � +MN,ϕ +t += +√ +NMN,ϕ +t +, we have +(UN +t , ϕ) = (UN +0 , ϕ) + +� t +0 +(UN +r , QSϕ)dr − +� t +0 +(UN +r , GI,N +r +ϕ)dr − +� t +0 +(V N +r , GS +r ϕ)dr + � +MN,ϕ +t +, +(4.1) +and also +(V N +t , ϕ) = (V N +0 , ϕ) + +� t +0 +(V N +r , QIϕ)dr + +� t +0 +(UN +r , GI,N +r +ϕ)dr + +� t +0 +(V N +r , GS +r ϕ)dr − α +� t +0 +(V N +r , ϕ)dr ++ �LN,ϕ +t +, +(4.2) +and +(W N +t , ϕ) = +� t +0 +(W N +r , QRϕ)dr + α +� t +0 +(V N +r , ϕ)dr + �Y N,ϕ +t +. +(4.3) +Where ∀x, y ∈ Rd, +GI,N +r +ϕ(x) = ϕ(x)(µI,N +r +, K(x, .)) = ϕ(x) +� +Rd K(x, y)µI,N +r +(dy), +GS +r ϕ(y) = (µS +r , ϕK(., y)) = +� +Rd ϕ(x)K(x, y)µS +r (dx). + +4 +CENTRAL LIMIT THEOREM +18 +4.2 +The Space of convergence of the sequences (U N, V N, W N) +Throughout this Subsection σ is an arbitrary positive real number and the following is assumed +to hold: +Assumption (H3): E(|X1 +0|2σ) < ∞. +The next lemma follows easilly from the definition of Xi +t (see (1.1)), the fact that the functions +m and θ are bounded and the inequality of Burkholder-Davis-Gundy. +Lemma 4.1. Under the assumption (H3), for any T > 0, there exists C(T) > 0 such that, +sup +1≤i≤N +E( sup +0≤t≤T +|Xi +t|2σ) < C(T). +Corollary 4.2. Under the assumption (H3), for any T > 0, there exists C(T) > 0, such that +sup +N≥1 +E( sup +0≤t≤T +(µS,N +t +, |.|2σ)) < C(T), and sup +0≤t≤T +(µS +t , |.|2σ) < C(T). +Lemma 4.3. For every fixed y ∈ Rd, ℓ ∈ {1, 2......d}, the mapping δy, Pℓ +y : Hs,σ → R defined by +δyϕ = ϕ(y) and Pℓ +yϕ = +∂ +∂yℓϕ(y) are continuous for s > d/2 and s > 1 + d/2 respectively and +∥δy∥−s,σ ≤ C(1 + |y|σ) +if s>d/2, +(4.4) +∥Pℓ +y∥−s,σ ≤ C(1 + |y|σ) +if s>d/2+1. +(4.5) +Proof. Continuity follows easily from Sobolev injections. On the other hand for every function +ϕ ∈ Hs,σ(Rd) one deduce from (2.2) (see subsection 2.1) that: +|ϕ(y)|≤ (1 + |y|σ)∥ϕ∥C0,σ≤ C(1 + |y|σ)∥ϕ∥s,σ. +The inequality (4.5) is proved in a similar way. +Corollary 4.4. If (ϕp)p≥1 is a complete orthonormal basis in Hs,σ(Rd), we have +∥δy∥2 +−s,σ= � +p≥1 +(ϕp(y))2 ≤ C(1 + |y|2σ), +if s > d/2 +∥Pℓ +y∥2 +−s,σ= � +p≥1 +( ∂ +∂yℓϕp(y))2 ≤ C(1 + |y|2σ), +if s > d/2 + 1. +Proposition 4.5. Under the assumption (H3), every limit point M1 of the seguence (� +MN)N≥1 +satisfies, +∀T ≥ 0, +sup +0≤t≤T +E(∥M1 +t∥2 +−s,σ) < ∞ +if s > 1 + d/2. +Proof. We have +� +MN,ϕ +t += − 1 +√ +N +N +� +i=1 +� t +0 +� ∞ +0 +1{Ei +r−=S}ϕ(Xi +r)1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du) ++ +1 +√ +N +N +� +i=1 +� t +0 +1{Eir=S} ▽ ϕ(Xi +r)θ(S, Xi +r)dBi +r, +< � +MN,ϕ >t = +� t +0 +� +µS,N +r +, ϕ2(µI,N +r +, K) +� +dr ++ +� t +0 +� +µS,N +r +, +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +1≤e≤d +1≤u≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr, + +4 +CENTRAL LIMIT THEOREM +19 +and it follows from Theorem 3.5, that +< � +MN,ϕ >t +P−→ +� t +0 +� +µS +r , ϕ2(µI +r, K) +� +dr ++ +� t +0 +� +µS +r , +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr. +Furthemore +� t +0 +� +µS +r , ϕ2(µI +r, K) +� +dr ++ +� t +0 +� +µS +r , +� +1≤ℓ≤d +1≤u≤d +� ∂ϕ +∂xℓ +�2θ2 +l,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr, +being the quadratic variation of a Gaussian martingale (we refer to Proposition 4.17 below for +the Gaussian property ) of the form (M1, ϕ), our aim is to find the smallest value of s for which +E(∥M1 +t∥2 +−s,σ) < ∞. With again (ϕp)p≥1 an orthonormal basis of Hs,σ(Rd), we have +E(∥M1 +t∥2 +−s,σ) = E(� +p≥1 +|(M1 +t, ϕp)|2) =� +p≥1 +E(< (M1, ϕp) >t). +From Corollary 4.2 and Corollary 4.4 and Asumption (H3), we have +� +p≥1 +< M1, ϕp >t= +� +p≥1 +� � t +0 +� +µS +r , ϕ2 +p(µI +r, K) +� +dr ++ +� t +0 +� +µS +r , +� +1≤ℓ≤d +�∂ϕp +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕp +∂xℓ +∂ϕp +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr +� +, += +� t +0 +� +Rd +�� +Rd K(x, y)µI +r(dy) +� � +p≥1 +ϕ2 +p(x)µS +r (dx)dr ++ +� t +0 +� +Rd +� � +1≤ℓ≤d +� +1≤u≤d +θ2 +ℓ,u(S, x) +� +p≥1 +�∂ϕp +∂xℓ +�2+2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +θℓ,e(S, x)θu,e(S, x) +� +p≥1 +∂ϕp +∂xℓ +(x)∂ϕp +∂xu +(x) +� +µS +r (dx)dr, +≤ ∥K∥∞ +� t +0 +� +Rd +� +p≥1 +ϕ2 +p(x)µS +r (dx)dr + +� t +0 +� +Rd +� +1≤ℓ≤d +1≤u≤d +∥θ2 +ℓ,u∥∞ +� +p≥1 +�∂ϕp +∂xℓ +�2(x)µS +r (dx)dr ++ 4 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∥θℓ,e∥∞∥θu,e∥∞ +� t +0 +� +Rd +� +p≥1 +��∂ϕp +∂xℓ +(x) +�2 ++ +�∂ϕp +∂xu +(x) +�2� +µS +r (dx)dr, +≤ C +� t +0 +� +Rd(1 + |x|2σ)µS +r (dx)dr + C(d) +� t +0 +� +Rd(1 + |x|2σ)µS +r (dx)dr, +≤ C(d, T). +By Doob’s inequality and by calculations similar to those done above we obtain the following +result. +Corollary 4.6. Under the asumption (H4), for any T > 0, s > 1 + D, ∃ C(T) > 0, such that: +sup +N≥1 +E( sup +0≤t≤T +∥� +MN +t ∥2 +−s,σ) ≤ C(T), sup +N≥1 +E( sup +0≤t≤T +∥�LN +t ∥2 +−s,σ) ≤ C(T), sup +N≥1 +E( sup +0≤t≤T +∥�Y N +t ∥2 +−s,σ) ≤ C(T). + +4 +CENTRAL LIMIT THEOREM +20 +In the rest of this section we arbitrarily choose σ > d/2 and 1 + D < s < 2 + D. +Furthermore, in all the sequel, the assumption (H3) is supposed to hold for that value of σ. +Thus we will prove that the sequences (UN, V N, W N)N≥1 converges in [D(R+, H−s,σ)]3, where +we have equipped D(R+, H−s,σ) with the Skorokhod topology (we refer to [6] for the explicit +definition of this topology). +Remark 4.7. Note that the assumption (H3) and the fact that σ > d/2, yield: +sup +x ∥K(x, .)∥2+D,σ< ∞ and sup +y ∥K(., y)∥2+D,σ< ∞. +4.3 +Convergence of (U N, V N, W N)N≥1 +We first derive an estimate of the norm of the fluctuation processes UN; V N and W N, which is +not uniform in N. +Lemma 4.8. For any N ≥ 1, T > 0, there exists a constant C(T) > 0, such that +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) ≤ C(T)N, +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) ≤ C(T)N and E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) ≤ C(T)N. +Proof. Let us prove the result for UN. We first recall that C1,σ ֒→ C0,2σ. Moreover since +s > 1 + D, Hs,σ ֒→ C1,σ. Now we have +|(UN +t , ϕ)| = +√ +N +��� 1 +N +N +� +i=1 +1{Ei +t=S}ϕ(Xi +t) − (µS +t , ϕ) +���, +≤ +√ +N +� 1 +N +N +� +i=1 +1{Ei +t=S}(1 + |Xi +t|2σ) |ϕ(Xi +t)| +1 + |Xi +t|2σ + +� +µS +t , (1 + |.|2σ) |ϕ(.)| +1 + |.|2σ +�� +, +≤ +√ +N∥ϕ∥C0,2σ +�� +µS,N +t +, 1 + |.|2σ� ++ +� +µS +t , 1 + |.|2σ�� +, +≤ +√ +N∥ϕ∥s,σ +�� +µS,N +t +, 1 + |.|2σ� ++ +� +µS +t , 1 + |.|2σ�� +. +This inequality combined with Corollary 4.2 and the fact that ∥UN +t ∥−s,σ= +sup +ϕ̸=0,ϕ∈Hs,σ +|(UN +t ,ϕ)| +∥ϕ∥s,σ +yields +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) ≤ 4C(T)N. +By the same arguments we obtain the same results for V N and W N. +We now give the estimates for the fluctuations at time 0. It is uniform in N. +Proposition 4.9. For any s > d/2, there exists C such that +sup +N≥1 +E(∥UN +0 ∥2 +−s,σ) < C and sup +N≥1 +E(∥V N +0 ∥2 +−s,σ) < C. +Proof. We only prove that sup +N≥1 +E(∥V N +0 ∥2 +−s,σ) < C. The other estimate follows by similar argu- +ments. Since 1A(Xj)ξjδXj are i.i.d with law µI +0, from Corollary 4.4 and from assumption (H3), + +4 +CENTRAL LIMIT THEOREM +21 +if s>d/2, we have +E(∥V N +0 ∥2 +−s,σ) = E +� � +p≥1 +(V N +0 , ϕp)2� +, += N +� +p≥1 +E +�� +(µI,N +0 +, ϕp) − (µI +0, ϕp) +�2� +, += 1 +N +� +i,n1,n2 +E + + +� N +� +j=1 +[1A(Xj)ξjϕp(Xj) − (µI +0, ϕp)] +�2 + , += 1 +N +� +p≥1 +N +� +j=1 +E +�� +1A(Xj)ξjϕp(Xj) − (µI +0, ϕp) +�2� +≤ 1 +N +� +p≥1 +N +� +j=1 +E +� +[1A(Xj)ξjϕp(Xj)]2� +, +≤ p +� +A +� +p≥1 +ϕ2 +p(x)dPX1 +0(x), +≤ pC +� +A +(1 + |x|2σ)dPX1 +0(x) ≤ C. +Remark 4.10. Using Proposition 4.9 and following the Proof of Theorem 3.3 in [7], we prove +easily that for any s > d/2, the sequence (UN +0 , V N +0 )N converges in law in H−s,σ(Rd) towards +(U0, V0), where for any ϕ, ψ ∈ Hs,σ, the expression of the Gaussian vector ((U0, ϕ), (V0, ψ)) is +given by +(U0, ϕ) = W1[ϕ√g{(1 − p)1A + 1Ac} − (1 − p)W1(√g) +� +A +ϕ(x)g(x)dx − W1(√g) +� +Ac ϕ(x)g(x)dx ++ W2(1Aϕ +� +(p − p2)g), +(4.6) +(V0, ψ) = pW1(1Aψ√g) − pW1(√g) +� +A +ψ(x)g(x)dx − W2(1Aψ +� +(p − p2)g), +(4.7) +(Z0, φ) = W1(φ√g) − W1(√g) +�� +Rd φ(x)g(x)dx +� +, +(4.8) +where g is the density of the law of X1 +0 and W1 and W2 are mutually independent two dimen- +sional white noises. +The proof of the next Lemma can be found in [7] (see the proof of Lemma 5.15 in [7], using +this time Corollary 5.2 below). +Lemma 4.11. For each N ≥ 1, the processes UN, V N and W N belong to D(R+, H−s,σ). +Let us give the main result of this section. +Theorem 4.12. Under (H2) and (H3), the sequence of processes (UN, V N, W N)N≥1 con- +verges in law in (D(R+, H−s,σ))3 towards the process (U, V, W) whose trajectories belong to + +4 +CENTRAL LIMIT THEOREM +22 +(C(R+, H−s,σ))3 and which satisfies for any t ≥ 0, +Ut = U0 + +� t +0 +Q∗ +SUrdr − +� t +0 +(GI +r)∗Urdr − +� t +0 +(GS +r )∗Vrdr + M1 +t, +Vt = V0 + +� t +0 +Q∗ +IVrdr + +� t +0 +(GI +r)∗Urdr + +� t +0 +((GS +r )∗ − αId)Vrdr + M2 +t, +Wt = +� t +0 +Q∗ +RWrdr + α +� t +0 +Vrdr + M3 +t, +where for any r > 0, GI +r and GS +r are defined as in subsection 4.1 (replacing µS,N +r +and µI,N +r +by µS +r +and µI +r respectively ), and ∀ϕ, ψ, φ ∈ Hs,σ, (M1, ϕ), (M2, ψ), (M3, φ)) is a centered Gaussian +martingale satisfying: +< (M1, ϕ) >t = +� t +0 +� +µS +r , ϕ2(µI +r, K) +� +dr ++ +� t +0 +� +µS +r , +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr, +< (M2, ψ) >t = +� t +0 +� +µS +r , ψ2(µI +r, K) +� +dr ++ +� t +0 +� +µI +r, +� +1≤ℓ≤d +� ∂ψ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(I, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ψ +∂xℓ +∂ψ +∂xu +θℓ,e(I, .)θu,e(I, .) +� +dr, +< (M3, φ) >t = α +� t +0 +(µR +r , φ2)dr ++ +� t +0 +� +µI +r, +� +1≤ℓ≤d +� ∂φ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(I, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂φ +∂xℓ +∂φ +∂xu +θℓ,e(I, .)θu,e(I, .) +� +dr, +< (M1, ϕ), (M2, ψ) >t = − +� t +0 +� +µS +r , ϕψ(µI +r, K) +� +dr, +< (M2, ψ), (M3, φ) >t = α +� t +0 +(µI +r, ψφ)dr, and < (M1, ϕ), (M3, φ) >t= 0. +Before we prove this Theorem we first state a condition of Aldous type for the tightness of +a sequence of H−s,σ-valued c`adl`ag processes, exploiting the fact that H−s,σ is a Hilbert space +(see Definition 2.2.1 of [27]). +Proposition 4.13. Let (ϑn)n be a sequence of H−s,σ-valued c`adl`ag processes, their laws ( �P n) +form a tight sequence in D(R+, H−s,σ) if: +(T1) +For each t in a dense subset T of R+, the sequence (ϑn +t )n is tight in H−s,σ. +(T2) +For each T > 0, ∀ε1, ε2 > 0, there exist δ > 0, n0 ≥ 1 such that for any collection of +stopping times τ n ≤ T, +sup +n≥n0 +̺≤δ +P(∥ϑn +(τ n+̺) − ϑn +τ n∥H> ε1) ≤ ε2. + +4 +CENTRAL LIMIT THEOREM +23 +The proof of Theorem 4.12 is the content of subsection 4.3.3 below, however let us first +prove a few preliminary results. +4.3.1 +Preliminary results +Proposition 4.14. The sequences (� +MN)N≥1, (�LN)N≥1 and (�Y N)N≥1 are tight in D(R+, H−s,σ). +Proof. − Tightness of (� +MN)N≥1. +Let us prove that (� +MN)N≥1 satisfies the conditions (T1) and (T2) of Proposition 4.13. +− To show (T1) it is enough to prove that: +∀t ≥ 0, ∀ε > 0 there exists a compact subset K of H−s,σ such that P(� +MN +t +/∈ K) < ε. +This follows readily from the fact that for each 1 + D < s′ = s+1+D +2 +< s, σ′ > σ > d/2, there +exists C(T) such that +E( sup +0≤t≤T +∥� +MN +t ∥2 +−s′,σ′) ≤ C(T) (see Corollary 4.6). +Indeed, since for any 1 + D < s′ = s+1+D +2 +< s, the embedding H−s′,σ′(Rd) ֒→ H−s,σ(Rd) is +compact (see Corollary 5.2, in the Appendix below), B−s′,σ′(R) = {µ ∈ H−s′,σ′; ∥µ∥H−s′,σ′≤ R}, +which is a closed and bounded subset of H−s′,σ′ is a compact subset of H−s,σ. Thus +P(� +MN +t +/∈ B−s′,σ′(R)) = P(∥� +MN +t ∥−s′,σ′> R) +≤ 1 +R2E(∥� +MN +t ∥2 +−s′,σ′) +≤ C(T) +R2 , +for any N ≥ 1. By choosing R arbitrarily large, we make the right hand side as small as we +wish, which yields the result. +− Proof of (T2). Note first that < � +MN,ϕ >t= +� t +0 +ΓN +r (ϕ)dr, where +ΓN +r (ϕ) = +� +µS,N +r +, ϕ2(µI,N +r +, K) +� ++ +� +µS,N +r +, +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +. +According to Theorem 2.3.2 in [27] it is enough to prove that +∀T > 0 +∀ε1, ε2 > 0 +∃δ > 0, N0 ≥ 1 such as for any stopping times τ N ≤ T, +sup +N≥N0 +sup +̺≤δ +P(|< � +MN >(τ N+̺) − < � +MN >τ N |> ε1) < ε2 +(4.9) +Where < � +M > is the increasing, continuous processes such that, ∥� +Mt∥2 +H−s,σ< � +M >t is a +martingale. +Let T > 0, ε1, ε2 > 0, ℓ > 1, we find δ > 0 such that τ N + δ ≤ ℓT and such that 4.9 holds. + +4 +CENTRAL LIMIT THEOREM +24 +We have +|< � +MN >(τ N+̺) − < � +MN >τ N | = | +� +p≥1 +{< � +MN,ϕp >(τ N+̺) − < � +MN,ϕp >τ N}| += +��� +� +p≥1 +� (τ N+̺) +τ N +ΓN +r (ϕp)dr +��� += +��� +� +p≥1 +� ̺ +0 +ΓN +(τ N +r)(ϕp)dr +��� +≤ C(∥K∥∞, d) +� ̺ +0 +� +Rd{1 + |x|2σ}µS,N +τ N+r(dx)dr +≤ ̺C(∥K∥∞, d)sup +N≥1 +sup +0≤t≤ℓT +� +µS,N +t +, 1 + |.|2σ� +. +(4.10) +Hence it follows from the Markov inequality and from Lemma 4.2 and (4.10) that +P(|< � +MN >(τ N+̺) − < � +MN >τ N |> ε1) ≤ E(|< � +MN >(τ N+̺) − < � +MN >τ N |) +ε1 +≤ Cδ +ε1 +. +(T2) follows. +We conclude from (T1) and (T2) that (� +MN)N≥1 is tight in D(R+, H−s,σ). The same arguments +yield the tightness of (�LN)N≥1 and (�Y N)N≥1 in D(R+, H−s,σ). +Lemma 4.15. (We refer to the proof of Proposition 6.17 in [7], for the proof) +Every limit point (M1, M2, M3) of the sequence (� +MN, �LN, �Y N)N≥1 is such that for any ϕ, ψ, φ ∈ +Hs,σ, ((M1, ϕ), (M2, ψ), (M3, φ) is a martingale. +The main argument for obtaining the following result is that the jumps of UN, V N and W N +respectively are of the order of +1 +√ +N . +Proposition 4.16. (We refer to the proof of Proposition 6.16 in [7], for the detail proof) +Every limit point (M1, M2, M3) of the sequence (� +MN, �LN, �Y N)N≥1 belongs to of (C(R+, H−s,σ))3. +Proposition 4.17. The sequence (� +MN, �LN, �Y N)N≥1 converges in law in (D(R+, H−s,σ))3 to- +wards the process (M1, M2, M3) ∈ (C(R+, H−s,σ))3 where ∀ϕ, ψ, φ ∈ Hs,σ, + +4 +CENTRAL LIMIT THEOREM +25 +((M1, ϕ), (M2, ψ), (M2, φ)) is a centered Gaussian martingale having the same law as +(M1 +t, ϕ) = − +� t +0 +� +Rd +� +fS(r, x) +� +Rd fI(r, y)K(x, y)dyϕ(x)W1(dr, dx) ++ +d +� +ℓ=1 +� t +0 +� +Rd +� +fS(r, x) +� � +1≤u≤d +∂ϕ +∂xu +(x)θu,l(S, x) +� +Wℓ+1(dr, dx), +(4.11) +(M2 +t, ψ) = +� t +0 +� +Rd +� +fS(r, x) +� +Rd fI(r, y)K(x, y)dyψ(x)W1(dr, dx) ++ +d +� +ℓ=1 +� t +0 +� +Rd +� +fI(r, x) +� � +1≤u≤d +∂ψ +∂xu +(x)θu,l(I, x) +� +Wℓ+1+d(dr, dx) +− +� t +0 +� +Rd ψ(x) +� +αfI(r, x)W3d+2(dr, dx), +(4.12) +(M3 +t, φ) = + +d +� +ℓ=1 +� t +0 +� +Rd +� +fR(r, x) +� � +1≤u≤d +∂φ +∂xu +(x)θu,l(R, x) +� +Wℓ+1+2d(dr, dx) ++ +� t +0 +� +Rd φ(x) +� +αfI(r, x)W3d+2(dr, dx), +(4.13) +where W1, W2,........,W3d+1, W3d+2 are independent spatio-temporal standard white noises. +Proof. From Proposition 4.14, (� +MN, �LN, �Y N)N≥1 is tight in (D(R+, H−s,σ))3, hence according +to Prokhorov’s Theorem there exists a subsequence still denoted (� +MN, �LN, �Y N)N≥1 which con- +verges in law in (D(R+, H−s,σ))3 towards (M1, M2, M3). By Lemma 4.15 and Proposition 4.16, +∀ϕ, ψ, φ ∈ Hs,σ, ((M1, ϕ), (M2, ψ), (M3, φ) is a continuous martingale, thus we end the proof +of Proposition 4.17 by showing that the centered, continuous martingale +((M1, ϕ), (M2, ψ), (M3, φ)) is Gaussian and satisfies (4.11), (4.12) and (4.13). +For any ϕ, ψ, ψ ∈ C2 +b , we have +� +Mt +N,ϕ = − 1 +√ +N +N +� +i=1 +� t +0 +� ∞ +0 +1{Ei +r−=S}ϕ(Xi +r)1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du) ++ +1 +√ +N +N +� +i=1 +� t +0 +1{Eir=S} ▽ ϕ(Xi +r)θ(S, Xi +r)dBi +r +=−M1,N,ϕ +t ++ M2,N,ϕ +t +, +�Lt +N,ψ = +1 +√ +N +N +� +i=1 +� t +0 +� ∞ +0 +1{Ei +r−=S}ψ(Xi +r)1{u≤ 1 +N +�N +j=1 K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du)) + ++ +1 +√ +N +N +� +i=1 +� t +0 +1{Eir=I} ▽ ψ(Xi +r)θ(I, Xi +r)dBi +r − 1 +√ +N +N +� +i=1 +� t +0 +� α +0 +1{Ei +r−=I}ψ(Xi +r)Q +i(dr, du) +=M1,N,ψ +t ++ M3,N,ψ +t +− M4,N,ψ +t +, +�Yt +N,φ = +1 +√ +N +N +� +i=1 +� t +0 +1{Eir=R} ▽ φ(Xi +r)θ(R, Xi +r)dBi +r + +1 +√ +N +N +� +i=1 +� t +0 +� α +0 +1{Ei +r−=I}φ(Xi +r)Q +i(dr, du) += M5,N,φ +t ++ M4,N,φ +t +. +Consider for ϕ, ψ, φ ∈ C2 +c , the following sequence of martingales +� +Mt +N,ϕ + �Lt +N,ψ + �Yt +N,φ = −M1,N,ϕ +t ++ M2,N,ϕ +t ++ M1,N,ψ +t ++ M3,N,ψ +t +− M4,N,ψ +t ++ M4,N,φ +t ++ M5,N,φ +t + +4 +CENTRAL LIMIT THEOREM +26 +The martingales M1,N,ϕ +t +, M2,N,ϕ +t +, M3,N,ψ +t +, M4,N,ψ +t +, M5,N,φ +t +being two by two orthogonal, +< � +MN,ϕ+�LN,ψ >t=< M1,N,ϕ >t + < M2,N,ϕ >t + < M1,N,ψ >t + < M3,N,ψ >t + < M4,N,ψ >t ++ < M4,N,φ >t + < M5,N,φ >t −2 < M1,N,ϕ, M1,N,ψ >t −2 < M4,N,ψ, M4,N,φ >t . +In addition we have the following convergences in probability +< M1,N,ϕ >t +P−→ +� t +0 +� +µS +r , ϕ2(µI +r, K) +� +dr, +< M2,N,ϕ >t +P−→ +� t +0 +� +µS +r , +� +1≤ℓ≤d +� ∂ϕ +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕ +∂xℓ +∂ϕ +∂xu +θℓ,e(S, .)θu,e(S, .) +� +dr. +On the other hand: +− � +MN,ϕ + �LN,ψ + �Y N,φ +L−→ (M1, ϕ) + (M2, ψ) + (M3, φ) along a subsequence since +(� +MN,ϕ, �LN,ψ, �LN,φ) +L−→ ((M1, ϕ), (M2, ψ), (M3, φ)) +− (M1, ϕ)+(M2, ψ)+(M3, φ) is a continuous martingale since (M1, ϕ), (M2, ψ), and (M3, φ) +have this property. +Thus (M1, ϕ) + (M2, ψ) + (M3, φ) is a time changed Brownian motion. +The quadratic variation +< (M1, ϕ) + (M2, ψ) + (M3, φ) >t += +� t +0 +� � +µS +r , ϕ2(µI +r, K) +� ++ +� +µS +r , ψ2(µI +r, K) +� +− 2 +� +µS +r , ϕψ(µI +r, K) +� � +dr ++ +� +A∈{S,I,R} +� t +0 +� +µA +r , +� +1≤ℓ≤d +1≤u≤d +�∂ϕA +∂xℓ +�2θ2 +ℓ,u(A, .) + 2 +� +1≤ℓ≤d−1 +ℓ+1≤u≤d +1≤e≤d +∂ϕA +∂xℓ +∂ϕA +∂xu +θℓ,e(A, .)θu,e(A, .) +� +dr ++ α +� t +0 +� +(µI +r, ψ2) + (µI +r, φ2) − 2(µI +r, ψφ) +� +dr, +(where we have let ϕS = ϕ, ϕI = ψ and ϕR = φ) of (M1, ϕ) + (M2, ψ) + (M3, φ) being +deterministic then we conclude that +(M1, ϕ) + (M2, ψ) + (M3, φ) is a Gaussian martingale having the same law as +Nt = +� t +0 +� +Rd +� +fS(r, x) +� +Rd fI(r, y)K(x, y)dy(ψ(x) − ϕ(x))W1(dr, dx) ++ +d +� +1=1 +� t +0 +� +Rd +� +fS(r, x) +� � +1≤u≤d +∂ϕ +∂xu +(x)θu,l(S, x) +� +Wl+1(dr, dx) ++ +d +� +1=1 +� t +0 +� +Rd +� +fI(r, x) +� � +1≤u≤d +∂ψ +∂xu +(x)θu,l(I, x) +� +Wl+d+1(dr, dx) ++ +d +� +1=1 +� t +0 +� +Rd +� +fR(r, x) +� � +1≤u≤d +∂φ +∂xu +(x)θu,l(R, x) +� +W2d+1(dr, dx) ++ +� t +0 +� +Rd +� +αfI(r, x)(φ(x) − ψ(x))W3d+2(dr, dx), +where W1, W2, .............. W3d+1, W3d+2 are independent spatio-temporal white noises. +So taking (ψ ≡ 0, φ ≡ 0), (ϕ ≡ 0, φ ≡ 0) and (ϕ ≡ 0, ψ ≡ 0) respectively, in the above equation +we see that (M1, ϕ), (M2, ψ) and (M3, φ) satisfy (4.11), (4.12) and (4.13). + +4 +CENTRAL LIMIT THEOREM +27 +Proposition 4.18. There exists a constant C > 0, such that for any ϕ ∈ Hs,σ(Rd), +∥GI,N +r +ϕ∥s,σ ≤ C∥ϕ∥s,σsup +y∈Rd∥K(., y)∥2+D,σ, +(4.14) +∥GS +r ϕ∥s,σ ≤ C∥ϕ∥s,σsup +y∈Rd∥K(., y)∥2+D,σ, +(4.15) +∥GI +rϕ∥s,σ ≤ C∥ϕ∥s,σsup +y∈Rd∥K(., y)∥2+D,σ. +(4.16) +Proof. We first recall that ∀x, y ∈ Rd, +GI,N +r +ϕ(x) = ϕ(x)(µI,N +r +, K(x, .)) = ϕ(x) +� +Rd K(x, y)µI,N +r +(dy), +GS +r ϕ(y) = (µS +r , ϕK(., y)) = +� +Rd ϕ(x)K(x, y)µS +r (dx). +Proof of (4.14). Since Hs,σ is a Banach algebra (see Remark 5.4 in the Appendix below), we +have +∥GI,N +r +ϕ∥s,σ≤ C∥ϕ∥s,σ∥(µI,N +r +, K)∥s,σ≤ C∥ϕ∥s,σ∥(µI,N +r +, K)∥2+D,σ. +(4.17) +Furthermore +∥(µI,N +r +, K)∥2 +2+D,σ = +� +|γ|≤2+D +� +Rd +|Dγ(µI,N +r +, K(x, .))|2 +1 + |x|2σ +dx, += +� +|γ|≤2+D +� +Rd +��� +� +Rd DγK(x, y)µI,N +r +(dy) +��� +2 +1 + |x|2σ +dx, +≤ +� +Rd +� +|γ|≤2+D +� +Rd +|DγK(x, y)|2 +1 + |x|2σ +dxµI,N +r +(dy), +≤ sup +y∈Rd∥K(., y)∥2 +2+D,σ. +(4.18) +Thus (4.14) follows from (4.17) and (4.18). +Proof of 4.15. Once again since H2+D,σ ֒→ Hs,σ, we have +∥GS +r ϕ∥s,σ= ∥(µS +r , ϕK)∥s,σ≤ C∥(µS +r , ϕK)∥2+D,σ. +(4.19) +Furthermore from Corollary 4.2, we have +∥(µS +r , ϕK)∥2 +D,σ = +� +|γ|≤2+D +� +Rd +|Dγ(µS +r , ϕK(., y))|2 +1 + |y|2σ +dx, += +� +|γ|≤2+D +� +Rd +��� +� +Rd ϕ(x)DγK(x, y)µS +r (dx) +��� +2 +1 + |y|2σ +dx, +≤ +� +Rd ϕ2(x)µS +r (dx) +� +Rd +� +|γ|≤2+D +� +Rd +|DγK(x, y)|2 +1 + |y|2σ +dyµS +r (dx), +≤ ∥ϕ∥2 +C0,σ sup +x∈Rd∥K(x, .)∥2 +2+D,σ. +(4.20) +So (4.15) follows from (4.19) and (4.20) and the embedding Hs,σ ֒→ C0,σ. +The proof of (4.16) is similar to that of (4.14). + +4 +CENTRAL LIMIT THEOREM +28 +Corollary 4.19. There exists a constant C > 0, such that for any U ∈ H−s,σ, we have +∥(GI,N +r +)∗U∥−s,σ ≤ C sup +y∈Rd∥K(., y)∥2+D,σ∥U∥−s,σ, +(4.21) +∥(GS +r )∗U∥−s,σ ≤ C sup +x∈Rd∥K(x, .)∥2+D,σ∥U∥−s,σ, +(4.22) +∥(GI +r)∗U∥−s,σ ≤ C sup +y∈Rd∥K(., y)∥2+D,σ∥U∥−s,σ. +(4.23) +4.3.2 +The evolution semi group +Let us define the evolution semigroup as a semi group of bounded linear operators in a Banach +space. Let us assume, for the moment in a general context, that for any A ∈ {S, I, R}, the +coeficients m(A, .) and θ(A, .) are in Cj+1 +b +, where j is a positive integer. +Let (Bt)t≥0 be a +standard Brownian motion on Rd. For any A ∈ {S, I, R}, one defined (see for example Kunita +[20] ) the flow of diffeomorphisms (of class Cj) as the unique solution of the Itô stochastic +differential equation started from x ∈ Rd at time u : +XA,x +u,t = x + +� t +u +m(A, XA,x +u,t )dr + +� t +u +θ(A, XA,x +u,t )dBr. +(4.24) +Moereover for any measurable and bounded function ϕ, A ∈ {S, I, R}, we define +ΥA(t − u)ϕ(x) = E(ϕ(XA,x +u,t )) and in the folowing Υ∗ +A(t) denotes the adjoint operator. +When u = 0, XA,x +u,t is denotes XA,x +t +. +We note that under the Assumptions (H2), for any 0 ≤ u < t the map x ∈ Rd �−→ XA,x +u,t is of +class C2+D, and the following results hold true. +− (Thank to Corrolary 4.6.7 in [20]) For any 0 ≤ |γ|≤ 2 + D, for any p ≥ 1, there exits a +constant C independent of t, such that +sup +x +E(|DγXA,x +t +|2p) ≤ C. +(4.25) +− (See Lemma 4.5.3 in [20]) For any real p, there exists a positive constant Cp, such that +E[(1 + |XA,x +t +|2)p] ≤ Cp(1 + |x|2)p, +∀x ∈ Rd. +(4.26) +Now we have the following result. +Lemma 4.20. Under the Asumption (H2), for any A ∈ {S, I, R}, t > 0, m ∈ {0, 1, ....2 + D}, +ϕ ∈ W m,σ +0 +(Rd), there exists a positive constant C, such that: +∥ΥA(t)ϕ∥m,σ≤ CeCt∥ϕ∥m,σ +Proof. We have +∥ΥA(t)ϕ∥2 +m,σ= +� +|γ|≤m +� +Rd +|DγΥA(t)ϕ(x)|2 +1 + |x|2σ +dx, +furthermore for |γ|= 0, and by using (4.26) and Lemma 5.3 in the Appendix below, we have +� +Rd +|ΥA(t)ϕ|2 +1 + |x|2σ dx = +� +Rd +|E(ϕ(XA,x +t +))|2 +1 + |x|2σ +dx ≤ +� +Rd +1 +1 + |x|2σ E[(1 + |XA,x +t +|σ)2]E +� +|ϕ(XA,x +t +)|2 +(1 + |XA,x +t +|σ)2 +� +dx, +≤ C +� +Rd +(1 + |x|2)σ +1 + |x|2σ +� +Rd ΥA(t)(x, y) |ϕ(y)|2 +1 + |y|2σ dydx, +≤ C(σ)∥ϕ∥L2,σ sup +y∈Rd +� � +Rd ΥS(t)(x, y)dx +� +, +≤ C(σ)∥ϕ∥m,σeCt. + +4 +CENTRAL LIMIT THEOREM +29 +For γ = (1, 0, 0...0) and by using (4.25) and Lemma 5.3, we have +� +Rd +|DγΥA(t)ϕ|2 +1 + |x|2σ +dx = +� +Rd +|E(Dγϕ(XA,x +t +))|2 +1 + |x|2σ +dx, +≤ +� +Rd +1 +1 + |x|2σ E[{DγXA,x +t +(1 + |XA,x +t +|σ)}2]E +� |∇ϕ(XA,x +t +)|2 +(1 + |XA,x +t +|σ)2 +� +dx, +≤ C +� +Rd +(1 + |x|2)σ +1 + |x|2σ E[(DγXA,x +t +)4]1/2E[(1 + |XA,x +t +|σ)4]1/2 +× +� +Rd ΥA(t)(x, y)|∇ϕ(y)|2 +1 + |y|2σ dydx, +≤ C(σ, d)∥ϕ∥1,σ sup +y∈Rd +� � +Rd ΥA(t)(x, y)dx +� +, +≤ C(σ, d)∥ϕ∥m,σeCt. +Similar argument allow us to have +� +Rd +|DγΥS(t)ϕ|2 +1 + |x|2σ +dx ≤ C(σ, d)∥ϕ∥m,σeCt, for all values of +|γ|≤ m. So the proof is complete. +Corollary 4.21. For any positive noninteger 1 + D < s < 2 + D, ϕ ∈ Hs,σ, there exists a +positive contant C, such that ∥ΥS(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ. +Proof. We prove this result by using the definition by interpolation of the space Hs,σ. +For any noninterger s > 0, there exists ρ ∈]0, 1[ such that s = (1 − ρ)(1 + D) + ρ(2 + D) and +(W 1+D,σ +0 +, W 2+D,σ +0 +)ρ,2 = Hs,σ, for the definition of the space (., .)ρ,q we refer to [22] or [37]. +So using Lemma 3.1.1 in [23], we have an equivalent norm on Hs,σ, which is given by: +∥ϕ∥s,σ= +�� ∞ +0 +{t−ρK(t, 1 + D, 2 + D)}2dt +t +�1/2 +, +where K(t, 1 + D, 2 + D) = +inf +ϕ=ϕ1+ϕ2{∥ϕ1∥1+D,σ+t∥ϕ2∥2+D,σ}. +So it is easy to see that the result follows from Lemma 4.20 and the above definition. +Let us prove the following results which will be useful to prove the tightness of the sequence +(UN, V N, W N)N in D(R+, H−s,σ). +Lemma 4.22. The sequence of processes (UN, V N, W N) satisfies ∀ 0 ≤ u < t, +UN +t = Υ∗ +S(t − u)UN +u − +� t +u +Υ∗ +S(t − r)(GI,N +r +)∗UN +r dr − +� t +u +Υ∗ +S(t − r)(GS +r )∗V N +r dr ++ +� t +u +Υ∗ +S(t − r)d� +MN +r , +(4.27) +V N +t += Υ∗ +I(t − u)V N +u + +� t +u +Υ∗ +I(t − r)(GI,N +r +)∗UN +r dr + +� t +u +Υ∗ +I(t − r)[(GI,N +r +)∗ − α]V N +r dr ++ +� t +u +Υ∗ +I(t − r)d�LN +r , +(4.28) +W N +t += Υ∗ +R(t − u)W N +u + α +� t +u +Υ∗ +R(t − r)V N +r dr + +� t +u +Υ∗ +R(t − r)d�Y N +r . +(4.29) + +4 +CENTRAL LIMIT THEOREM +30 +Proof. Let us consider a fuction φ belonging to C1,2 +c (R+ × Rd). By Itô’s formula applied to +φ(t, Xi +t) and using a similar computation as in subsections 3.1 and 4.1, we obtain for 0 ≤ u < t, +(UN +t , φt) = (UN +u , φu) + +� t +u +(UN +r , QSφr)dr + +� t +u +(UN +r , ∂φr +∂r )dr − +� t +u +� +UN +r , φr(µI,N +r +, K) +� +dr +− +� t +u +� +V N +r , (µS +r , φrK) +� +dr + +� t +u +(φr, d� +MN +r ). +Let ϕ ∈ C2 +b and 0 ≤ u < t, consider for r ∈ [u, t] the mapping ψr(x) = ΥS(t − r)ϕ(x). +We have ψ·(·) ∈ C1,2 +c ([u, t] × Rd), indeed, +- For any r ∈ [u, t], ψr(·) ∈ C2 +c (Rd) +- ∀x ∈ Rd, the map r ∈ [u, t] �→ ψ +′ +r(x) = −QS(ΥS(t − r)ϕ(x)) is continuous since ΥS(t) +is a strongly continuous semi-group and −QS(ΥS(t − r)ϕ(x)) = ΥS(t − r)(−QSϕ(x)). Thus +replacing φ by ψ in the above equation, we obtain +(UN +t , ϕ) = (UN +u , ΥS(t − u)ϕ) − +� t +u +� +UN +r , ΥS(t − r)ϕ(µI,N +r +, K) +� +dr +− +� t +u +� +V N +r , (µS +r , ΥS(t − r)ϕK) +� +dr + +� t +u +(ΥS(t − r)ϕ, d� +MN +r ). +This prove (4.27). We obtain (4.28) and (4.29) by similar arguments. +Proposition 4.23. There exists C > 0 such that for any T > 0, ̺ > 0 and any stopping times +τ such that τ + ̺ < T, one has +E +���� +� τ+̺ +τ +Υ∗ +S(τ + ̺ − r)d� +MN +r +��� +2 +−s,σ +� +≤ C̺, +(4.30) +E +���� +� τ+̺ +τ +Υ∗ +I(τ + ̺ − r)d�LN +r +��� +2 +−s,σ +� +≤ C̺, +(4.31) +E +���� +� τ+̺ +τ +Υ∗ +R(τ + ̺ − r)d�Y N +r +��� +2 +−s,σ +� +≤ C̺. +(4.32) +Proof. Proof of (4.30). Let us recall that +� τ+̺ +τ +(ΥS(τ + ̺ − r)ϕ, d� +MN +r ) += +1 +√ +N +N +� +i=1 +� τ+̺ +τ +1{Eir=S} ▽ ΥS(τ + ̺ − r)ϕ(Xi +r)θ(S, Xi +r)dBi +r +− +� +1 +N +N +� +i=1 +� τ+̺ +τ +� ∞ +0 +1{Ei +r−=S}ΥS(τ + ̺ − r)ϕ(Xi +r)1 +{u≤ 1 +N +N +� +j=1 +K(Xir,Xj +r)1{Ej +r=I}}M +i(dr, du). +Now from Remark 4.4, (4.25) and (4.26) one has for all ℓ ∈ {1, 2, ......., d}, 0 ≤ r ≤ ̺, +� +p≥1 +� +ΥS(̺ − r)ϕp(x) +�2 += +� +p≥1 +� +E(ϕp(XS,x +̺−r)) +�2 +≤ E +� � +p≥1 +|ϕp(XS,x +̺−r)|2� +, +≤ CE(1 + |XS,x +̺−r|2σ), +≤ C(σ)(1 + |x|2σ). +(4.33) + +4 +CENTRAL LIMIT THEOREM +31 +� +p≥1 +� ∂ +∂xℓ +ΥS(̺ − r)ϕp(x) +�2 += +� +p≥1 +� ∂ +∂xℓ +E(ϕp(XS,x +̺−r)) +�2 +≤ E(|∂xℓXS,x +̺−r|2)E +� � +p≥1 +|(∇ϕp)(XS,x +̺−r)|2� +, +≤ C(d)E(1 + |XS,x +̺−r|2σ), +≤ C(d, σ)(1 + |x|2σ). +(4.34) +Thus from (4.33) and (4.34) and Lemma 4.1, we have +E +����� +� τ+̺ +τ +ΥS(τ + ̺ − r)d� +MN +r +���� +2 +H−s +� += +� +p≥1 +E +��� τ+̺ +τ +ΥS(τ + ̺ − r)ϕp, d� +MN +r +�2� +, += +� +p≥1 +� +E +�� ̺ +0 +� +µS,N +r+τ, (ΥS(̺ − r)ϕp)2(µI,N +r+τ, K) +� +dr +� ++ E +� � ̺ +0 +� +µS,N +r+τ, +� +1≤ℓ≤d +�∂ΥS(̺ − r)ϕp +∂xℓ +�2 � +1≤u≤d +θ2 +ℓ,u(S, .) +� +dr +� ++ E +� � ̺ +0 +� +µS,N +r+τ, +� +1≤ℓ≤d−1 +1≤e≤d +1≤u≤d +∂ΥS(̺ − r)ϕp +∂xℓ +∂ΥS(̺ − r)ϕp +∂xu +θl,e(S, .)θu,e(S, .) +� +dr +�� +, +≤ E +�� ̺ +0 +� +µS,N +r+τ, +� +p≥1 +(ΥS(̺ − r)ϕp)2(µI,N +r+τ, K) +� +dr +� ++ E +� � ̺ +0 +� +µS,N +r+τ, +� +1≤ℓ≤d +1≤u≤d +θ2 +ℓ,u(S, .) +� +p≥1 +�∂ΥS(̺ − r)ϕp +∂xℓ +�2� +dr +� ++ 1 +2E +� � ̺ +0 +� +µS,N +r+τ, +� +1≤ℓ≤d−1 +1≤e≤d +1≤u≤d +|θℓ,e(S, .)θu,e(S, .)| +� � +p≥1 +�∂ΥS(̺ − r)ϕp +∂xℓ +�2 + +� +p≥1 +�∂ΥS(̺ − r)ϕp +∂xu +�2�� +dr +� +, +≤ ̺∥K∥∞sup +N≥1 +sup +0≤t≤T +E((µS,N +t +, 1 + |.|2σ)) ++ ̺C(σ) +� +1≤ℓ≤d +1≤u≤d +∥θ2 +ℓ,u(S, .)∥∞sup +N≥1 +sup +0≤t≤T +E((µS,N +t +, 1 + |.|2σ)) ++ ̺ +� +1≤ℓ≤d−1 +1≤e≤d +1≤u≤d +∥θℓ,e(S, .)∥∞∥θu,e(S, .)∥∞sup +N≥1 +sup +0≤t≤T +E((µS,N +t +, 1 + |.|2σ)), +≤ ̺C. +Similar arguments yield (4.31) and (4.32). +Proposition 4.24. For all T > 0, +sup +N≥1 +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) < ∞, +(4.35) +sup +N≥1 +E( sup +0≤t≤T +∥V N +t ∥2 +−s,σ) < ∞, +(4.36) +sup +N≥1 +E( sup +0≤t≤T +∥W N +t ∥2 +−s,σ) < ∞. +(4.37) +Proof. Choosing u = 0 in equation (4.27), (4.28) and (4.29) we have + +4 +CENTRAL LIMIT THEOREM +32 +∥UN +t ∥2 +−s,σ ≤ 4∥Υ∗ +S(t)UN +0 ∥2 +−s,σ+4t +� t +0 +� +∥Υ∗ +S(t − r)(GI,N +r +)∗UN +r ∥2 +−s,σ+∥Υ∗ +S(t − r)(GS +r )∗V N +r ∥2 +−s,σ +� +dr ++ 4 +��� +� t +0 +Υ∗ +S(t − r)d� +MN +r +��� +2 +−s,σ, +∥V N +t ∥2 +−s,σ ≤ 5∥Υ∗ +I(t)V N +0 ∥2 +−s,σ+5t +� t +0 +� +∥Υ∗ +I(t − r)(GI,N +r +)∗UN +r ∥2 +−s,σ+∥Υ∗ +I(t − r)GS +r V N +r ∥2 +−s,σ +� +dr ++5tα2 +� t +0 +∥Υ∗ +I(t − r)V N +r ∥2 +−s,σdr + 5 +��� +� t +0 +Υ∗ +I(t − r)d�LN +r +��� +2 +−s,σ, +∥W N +t ∥2 +−s,σ ≤ 2α2t +� t +0 +∥Υ∗ +R(t − r)V N +r ∥2 +−s,σdr + 2 +��� +� t +u +Υ∗ +R(t − r)d�Y N +r +��� +2 +−s,σ. +From Corollarys 4.19 and 4.21, we have +∥UN +t ∥2 +−s,σ ≤ C4eCt∥UN +0 ∥2 +−s,σ+4CteCtsup +y ∥K(., y)∥2 +2+D,σ +� t +0 +{∥UN +r ∥2 +−s,σ+∥V N +r ∥2 +−s,σ}dr ++ 4 +��� +� t +0 +Υ∗ +S(t − r)d� +MN +r +��� +2 +−s,σ, +∥V N +t ∥2 +−s,σ ≤ 5eCt∥V N +0 ∥2 +−s,σ+5CteCt(1 + α2)sup +x ∥K(x, .)∥2 +2+D,σ +� t +0 +{∥UN +r ∥H−s+∥V N +r ∥2 +−s,σ}dr ++ 5 +��� +� t +0 +Υ∗ +I(t − r)d�LN +r +��� +2 +−s,σ, +∥W N +t ∥2 +−s,σ ≤ 2CeCtα2t +� t +0 +∥V N +r ∥2 +−s,σdr + 2 +��� +� t +0 +Υ∗ +R(t − r)d�Y N +r +��� +2 +−s,σ. +Thus from Corrolary 4.6 and Corollary 4.21, Lemma 2.1 and Assumption (H3), we have +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) ≤ 4eCTE(∥UN +0 ∥2 +−s,σ) + 4CTeCT +� T +0 +� +E( sup +0≤r≤t +∥UN +r ∥2 +−s,σ) + E( sup +0≤r≤t +∥V N +r ∥2 +−s,σ) +� +dt ++ CT, +(4.38) +E( sup +0≤t≤T +∥V N +t ∥2 +−s,σ) ≤ 5eCTE(∥V N +0 ∥2 +−s,σ) ++ 5TC(1 + α2)eCT +� T +0 +� +E( sup +0≤r≤t +∥UN +r ∥2 +−s,σ) + E( sup +0≤r≤t +∥V N +r ∥2 +−s,σ) +� +dt + CT, +(4.39) +E( sup +0≤t≤T +∥W N +t ∥2 +−s,σ) ≤ 2α2TeCT +� T +0 +� +E( sup +0≤r≤t +∥W N +r ∥2 +−s,σ) + E( sup +0≤r≤t +∥V N +r ∥2 +−s,σ) +� +dt + CT. (4.40) +Thus summing (4.38), (4.39) and (4.40) and applying Gronwall’s lemma we deduce the +result from the Proposition 4.9. +4.3.3 +Proof of Theorem 4.12 +We begin this subsection by showing the following result. +Proposition 4.25. The sequences of processes UN, V N and W N are tight in D(R+, H−s,σ). + +4 +CENTRAL LIMIT THEOREM +33 +Proof. We establish the tightness of UN by showing that the conditions of Proposition 4.13 are +satisfied. +− Based on Proposition 4.24, we deduce (T1) by the same argument as used in the proof of +(T1) in Proposition 4.14. +− Proof of (T2). Let T>0, ε1, ε2 >0, (τ N)N a family of stopping times with τ N ≤ T. We have +UN +τ N+̺ − UN +τ N = (Υ∗ +S(̺) − Id)UN +τ N − +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)(GI,N +r +)∗UN +r dr +− +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)(GS +r )∗V N +r dr + +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)d� +MN +r , += (Υ∗ +S(̺) − Id)UN +τ N − +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)JS,I,N +r +(UN, V N)dr ++ +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)d� +MN +r , +where JS,I,N +r +(UN, V N) = (GI,N +r +)∗UN +r + (GS +r )∗V N +r . +We find δ > 0 and N0 ≥ 1 such that: +sup +N≥N0 +sup +δ≥̺ +P(∥(Υ∗ +S(̺) − Id)UN +τ N∥−s,σ≥ ε1) ≤ ε2, +(4.41) +sup +N≥N0 +sup +δ≥̺ +P +���� +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)JS,I,N +r +(UN, V N)dr +��� +−s,σ ≥ ε1 +� +≤ ε2, +(4.42) +sup +N≥N0 +sup +δ≥̺ +P +���� +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)d� +MN +r +��� +−s,σ ≥ ε1 +� +≤ ε2. +(4.43) +1− Proof of (4.41). +Let us introduce a complete orthonormal basis in Hs,σ, of functions (ϕp)p≥1, ϕp being of class C∞ +with compact support, and Fm(m ∈ N∗) denotes the sub-space of Hs,σ generated by (ϕp)1≤p≤m. +Let (Υ∗ +S(̺)−Id)UN +t |Fm denotes the orthogonal projection of (Υ∗ +S(̺)−Id)UN +t +on the dual space +of Fm. We have +P(∥(Υ∗ +S(̺) − Id)UN +τ N∥−s,σ≥ ε1) ≤ P(∥(Υ∗ +S(̺) − Id)UN +τ N |Fm ∥−s,σ≥ ε1 +2 ) ++ P(∥(Υ∗ +S(̺) − Id)UN +τ N − (Υ∗ +S(̺) − Id)UN +τ N |Fm ∥−s,σ≥ ε1 +2 ) +(4.44) +Let us control each of the term of the above right hand side. +− One has +P +� +∥(Υ∗ +S(̺) − Id)UN +τ N − (Υ∗ +S(̺) − Id)UN +τ N |Fm ∥−s,σ≥ ε1 +2 +� +≤ 4 +ε2 +1 +E +� +sup +0≤t≤T +� +p>m +(UN +t , (ΥS(̺) − Id)ϕp)2� +. +(4.45) +Furthermore the sequence +� +sup +1≤N +sup +0≤t≤T +sup +0≤̺≤δ +� +p>m +(UN +t , (ΥS(̺) − Id)ϕp)2� +converge towards 0 +as m −→ ∞, as the remainder of order m of a uniformly convergent series of functions. Thus +there exists m0 ∈ N∗ independent of N and ̺ such that for any m ≥ m0, +sup +0≤t≤T +sup +0≤̺≤δ +� +p>m +(UN +t , (ΥS(̺) − Id)ϕp)2 < ε, for any ε > 0. + +4 +CENTRAL LIMIT THEOREM +34 +Moreover +sup +0≤t≤T +sup +0≤̺≤δ +� +p>m +(UN +t , (ΥS(̺) − Id)ϕp)2 is bounded by max(CeCδ, 1) sup +0≤t≤T +∥UN +t ∥2 +−s,σ, +so uniformly integrable (see Proposition 4.24). Thus we deduce that there exists m0 ∈ N∗ +independent of N and ̺ such that for any m ≥ m0, +E +� +sup +0≤t≤T +� +p>m +(UN +t , (ΥS(̺) − Id)ϕp)2� +< ε, ∀ε > 0. +(4.46) +− One has +P(∥(Υ∗ +S(̺) − Id)UN +τ N |Fm ∥−s,σ≥ ε1 +2 ) ≤ 4 +ε2 +1 +E( sup +0≤t≤T +∥(Υ∗ +S(̺) − Id)UN +t +|Fm ∥2 +−s,σ). +(4.47) +Furthermore according to Dynkin’s formula and the contraction of ΥS(t), one has +∥(Υ∗ +S(̺) − Id)UN +t |Fm ∥2 +−s,σ = +m +� +p=1 +� +(Υ∗ +S(̺) − Id)UN +t , ϕp +�2 +, += +m +� +p=1 +� +UN +t , (ΥS(̺) − Id)ϕp +�2 +, += +m +� +p=1 +� +UN +t , +� ̺ +0 +ΥS(r)QSϕp(.)dr +�2 +, +≤ ̺ sup +0≤t≤T +∥UN +t ∥2 +−s,σ +m +� +p=1 +� ̺ +0 +∥ΥS(r)QSϕp∥2 +s,σdr, +≤ ̺2 sup +0≤t≤T +∥UN +t ∥2 +−s,σ +m +� +p=1 +∥QSϕp∥2 +s,σ. +(4.48) +Where QS in the infinitesimal generator of the operator ΥS(t). Hence as from assumptions +(H2), there exists C > 0 such that +m +� +p=1 +∥QSϕp∥2 +s,σ≤ +m +� +p=1 +∥QSϕp∥2 +2+D,σ≤ mC, +from (4.47) and (4.48), we have +P(∥(Υ∗ +S(̺) − Id)UN +τ N |Fm ∥−s,σ≥ ε1 +2 ) ≤ +≤ +4mCsup +N≥1 +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) +ε2 +1 +̺2. +(4.49) +Hence (4.49) combined with (4.46) and (4.44) yields (4.41). +Proof of (4.42). Let ℓ > 1, we find δ > 0 such that τ N +δ ≤ ℓT and such that (4.42) is satisfied. +Since ∀ϕ ∈ Hs,σ, ∥Υ(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ then form Proposition 4.24 and from Corollary 4.19, +we have + +4 +CENTRAL LIMIT THEOREM +35 +P + + +����� +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)JS,I,N +r +(UN, V N)dr +����� +−s,σ +≥ ε1 + + ≤ +≤ 1 +ε2 +1 +E + + +����� +� τ N+̺ +τ N +Υ∗ +S(τ N + ̺ − r)JS,I,N +r +(UN, V N)dr +����� +2 +−s,σ + + , +≤ ̺ +ε2 +1 +E +�� τ N+̺ +τ N +∥Υ∗ +S(τ N + ̺ − r)JS,I,N +r +(UN, V N)∥2 +−s,σdr +� +, +≤ ̺C +ε2 +1 +eCℓTsup +y ∥K(., y)∥s,σE +�� τ N+̺ +τ N +{∥UN +r ∥2 +−s,σ+∥V N +r ∥2 +−s,σ}dr +� +, +≤ δ2C +ε2 +1 +eℓT sup +N≥1 +E( sup +0≤t≤ℓT +{∥UN +t ∥2 +−s,σ+ sup +0≤t≤ℓT +∥V N +t ∥2 +−s,σ}), +≤ δ2C +ε2 +1 +. +So (4.42) is proved. +- Proof of (4.43). Let ℓ > 1, we find δ > 0 such that τ N + δ ≤ ℓT and such that (4.43) is +satisfied. From Proposition 4.23, we have +P +���� +� τ N+θ +τ N +Υ∗ +S(τ N + θ − r)d� +MN +r +��� +−s,σ ≥ ε1 +� +≤ 1 +ε2 +1 +E +���� +� τ N+θ +τ N +Υ∗ +S(τ N + θ − r)d� +MN +r +��� +2 +−s,σ +� +, +≤ C +ε2 +1 +δ. +Hence (4.43) is proved. +To establish the system of limiting equations of all converging subsequences of (UN, V N, W N)N≥1, +we will need the next Lemma. +Lemma 4.26. For any t ≥ 0, ϕ ∈ Hs,σ(Rd), as N −→ ∞, +� t +0 +E +� +∥[GI,N +r +− GI +r]ΥS(t − r)ϕ∥2 +s,σ +� +dt −→ 0. +Proof. Since Hs,σ is a Banach algebra (see Remark 5.4) and H2+D,σ ֒→ Hs,σ (since s < 2 + D), +and ∥Υ(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ, +���Υ(t − r)ϕ +� +µI,N +r +− µI +r, K +���� +s,σ ≤ C∥ϕ∥s,σ +��� +� +µI,N +r +− µI +r, K +���� +2+D,σ. +On the other hand +��� +� +µI,N +r +− µI +r, K +���� +2 +2+D,σ = +� +|η|≤2+D +� +Rd(1 + |x|2σ)−1��� +� +Rd DηK(x, y)(µI,N +r +− µI +r)(dy) +��� +2 +dx, +furthermore from Asumptions (H2) the map y ∈ Rd �→ DηK(x, y) is continuous and bounded. +So we deduce from Theorem 3.5 that +��� +� +T2 DηK(x, y)(µI,N +r +− µI +r)(dy) +��� +2 +P−→ 0. +According to Lebesgue’s dominated convergence theorem, E +���� +� +µI,N +r +− µI +r, K +���� +2 +2+D,σ +� +−→ 0, + +4 +CENTRAL LIMIT THEOREM +36 +as N −→ ∞. Thus +� t +0 +E +� +∥[GI,N +r +− GI +r]Υ(t − r)ϕ∥2 +2+D,σ +� +dt +≤ C∥ϕ∥2 +s,σ +� t +0 +E +���� +� +µI,N +r +− µI +r, K +���� +2 +2+D,σ +� +dr +−→ 0, as N −→ ∞. +Hence the result. +The next Proposition establishes the evolution equations of all limit points (U, V, W) of the +sequence (UN, V N, W N). +Proposition 4.27. All limit points (U, V, W) of the sequence (UN, V N, ZN) satisfy +Ut = Υ∗ +S(t)U0 − +� t +0 +Υ∗ +S(t − r)(GI +r)∗Urdr − +� t +0 +Υ∗ +S(t − r)(GS +r )∗Vrdr + +� t +0 +Υ∗ +S(t − r)dM1 +r, +(4.50) +Vt = Υ∗ +I(t)V0 + +� t +0 +Υ∗ +I(t − r)(GI +r)∗Urdr + +� t +0 +Υ∗ +I(t − r)(GS +r )∗Vrdr − α +� t +0 +Υ∗ +I(t − r)Vrdr ++ +� t +0 +Υ∗ +I(t − r)dM2 +r, +(4.51) +Wt = α +� t +0 +Υ∗ +R(t − r)Vrdr + +� t +0 +Υ∗ +R(t − r)dM3 +r. +(4.52) +Proof. We prove this Proposition by taking the weak limit in the equations (4.27) and (4.28) +and (4.29). Note first that from Propositions 4.25, there exists a subsequence along which the +sequences (UN, V N, W N)N converges in law towards (U, V, W). For any ϕ ∈ Hs,σ, one has +(Υ∗ +S(t)UN +0 , ϕ) + +� t +0 +� +Υ∗ +S(t − r)ϕ, d� +MN +r +� += (UN +t , ϕ) + +� t +0 +(Υ∗ +S(t − r)(GI +r)∗UN +r , ϕ)dr ++ +� t +0 +(Υ∗ +S(t − r)(GS +r )∗V N +r , ϕ)dr + +� t +0 +(Υ∗ +S(t − r)[(GI,N +r +)∗ − (GI +r)∗]UN +r , ϕ)dr, +(Υ∗ +I(t)V N +0 , ϕ) + +� t +0 +� +Υ∗ +I(t − r)ϕ, d�LN +r +� += (V N +t , ϕ) − +� t +0 +(Υ∗ +I(t − r)(GI +r)∗UN +r , ϕ)dr +− +� t +0 +(Υ(t − r)[(GS +r )∗ − α]V N +r , ϕ)dr − +� t +0 +([Υ∗ +I(t − r)(GI,N +r +)∗ − (GI +r)∗]UN +r , ϕ)dr, +� t +0 +� +Υ∗ +I(t − r)ϕ, d�Y N +r +� += (W N +t , ϕ) + α +� t +0 +(Υ∗ +R(t − r)V N +r , ϕ)dr. +(4.53) +Thus in view of (4.53), it is enough to show that (U, V ) satisfy (4.50) and (4.51). +Hence, we have +(Υ∗ +S(t)UN +0 , ϕ) + +� t +0 +� +Υ∗ +S(t − r)ϕ, d� +MN +r +� += Ψ1,t,ϕ +� +UN, V N� ++ +� t +0 +([Υ∗ +S(t − r)(GI,N +r +)∗ − (GI +r)∗]UN +r , ϕ)dr, +(Υ∗ +I(t)V N +0 , ϕ) + +� t +0 +� +Υ∗ +I(t − r)ϕ, d�LN +r +� += Ψ2,t,ϕ +� +UN, V N� +− +� t +0 +([Υ∗ +I(t − r)(GI,N +r +)∗ − (GI +r)∗]UN +r , ϕ)dr. + +4 +CENTRAL LIMIT THEOREM +37 +With +Ψ1,t,ϕ +� +UN, V N� += (UN +t , ϕ) + +� t +0 +(Υ∗ +S(t − r)(GI +r)∗UN +r , ϕ)dr ++ +� t +0 +(Υ∗ +S(t − r)(GS +r )∗V N +r , ϕ)dr, +Ψ2,t,ϕ +� +UN, V N� += (V N +t , ϕ) − +� t +0 +(Υ∗ +I(t − r)(GI +r)∗UN +r , ϕ)dr +− +� t +0 +(Υ∗ +I(t − r)[(GS +r )∗ − α]V N +r , ϕ)dr. +Furthermore. +1− From Lemma 4.26 and Proposition 4.24, +� t +0 +� +UN +r , [GI,N +r +− GI +r]Υ(t − r)ϕ +� +dr −→ 0 in L1(P), +locally unformly in t. +Indeed, E +���� sup +0≤t≤T +� t +0 +� +UN +r , [GI,N +r +− GI +r]Υ(t − r)ϕ +� +dr +��� +� +≤ +≤ +√ +T sup +N≥1 +E( sup +0≤t≤T +∥UN +t ∥2 +−s,σ) +1 +2 +� � T +0 +E(∥[GI,N +r +−GI +r]Υ(t−r)ϕ∥2 +s,σ)dr +� 1 +2. +2- Using (4.16) in Proposition 4.18, it is easy to see that the maps (Ψ1,.,ϕ, Ψ2,.,ϕ) is continuous +from [D(R+, H−s,σ)]2 into C(R+, R2). Thus as (UN, V N) converges in law in [D(R+, H−s,σ)]2 to- +wards (U, V ), then +� +Ψ1,.,ϕ(UN, V N), Ψ2,.,ϕ(UN, V N) +� +converges in law towards +� +Ψ1,.,ϕ(U, V ), Ψ2,.,ϕ(U, V ) +� +. +3- +� +(Υ∗ +S(.)UN +0 , ϕ) + +� . +0 +� +Υ∗ +S(. − r)ϕ, d� +MN +r +� +, (Υ∗ +I(.)V N +0 , ϕ) + +� . +0 +� +Υ∗ +I(. − r)ϕ, d�LN +r +�� +converges +in law towards +� +(Υ∗ +S(.)U0, ϕ) + +� . +0 +� +Υ∗ +S(. − r)ϕ, dM1 +r +� +, (Υ∗ +I(.)V0, ϕ) + +� . +0 +� +Υ∗ +I(. − r)ϕ, dM2 +r +�� +in +(D(R+, H−s,σ))2 since +� +(Υ∗ +S(.)UN +0 , ϕ), +(Υ∗ +I(.)V N +0 , ϕ), +� . +0 +� +Υ∗ +S(. − r)ϕ, d� +MN +r +� +, +� . +0 +� +Υ∗ +I(. − r)ϕ, d�LN +r +�� +converges in law +towards +� +(Υ∗ +S(.)U0, ϕ), (Υ∗ +I(.)V0, ϕ), +� . +0 +� +Υ∗ +S(. − r)ϕ, dM1 +r +� +, +� . +0 +� +Υ∗ +I(. − r)ϕ, dM2 +r +�� +in +(C(R+, H−s,σ))2 × (D(R+, H−s,σ))2 , which in turn follows from the fact that +� +(Υ∗ +S(.)UN +0 , ϕ), (Υ∗ +I(.)V N +0 , ϕ) +� +converges in law towards +� +(Υ∗ +S(.)U0, ϕ), (Υ∗ +I(.)V0, ϕ) +� +in +(C(R+, H−s,σ))2(see Remark 4.10) and +� � . +0 +� +Υ∗ +S(. − r)ϕ, d� +MN +r +� +, +� . +0 +� +Υ∗ +I(. − r)ϕ, d�LN +r +�� +con- +verges in law towards +� � . +0 +� +Υ∗ +S(.−r)ϕ, dW 1 +r +� +, +� . +0 +� +Υ∗ +I(.−r)ϕ, dW 2 +r +�� +in (D(R+, H−s,σ))2(which +follows from Proposition 4.17) and +� +(Υ∗ +S(.)UN +0 , ϕ), (Υ∗ +I(.)V N +0 , ϕ) +� +is globally independant of +� � . +0 +� +Υ∗ +S(. − r)ϕ, d� +MN +r +� +, +� . +0 +� +Υ∗ +I(. − r)ϕ, d�LN +r +�� +. +Thus from 1-, 2-, and 3-, we obtain the result of the statement. +From Proposition 4.16 we deduce that all limit points (U, V, W) of (UN, V N, WN)N≥1 are +elements of (C(R+, H−s))3, thus we end the proof of Theorem 4.12 by showing that the system +of equations (4.50) and (4.51) and (4.52) admits a unique solution (U, V, W) ∈ (C(R+, H−s))3. +Proposition 4.28. Suppose that (U1, V 1, W 1) and (U2, V 2, W 2) are solutions to equations +(4.50), (4.51) and (4.52) with (U1 +0, V 1 +0 ) = (U2 +0, V 2 +0 ) then (U1, V 1, W 1) = (U2, V 2, W 2). + +5 +APPENDIX +38 +Proof. All we need to show is that the system of equations (4.50) and (4.51) admits a unique +solution. Thus we have +U1 +t − U2 +t = − +� t +0 +Υ∗ +S(t − r)(GI +r)∗(U1 +r − U2 +r )dr − +� t +0 +Υ∗ +S(t − r)(GS +r )∗(V 1 +r − V 2 +r )dr, +Hence +∥U1 +t − U2 +t ∥H−s≤ +� t +0 +∥Υ∗ +S(t − r)(GI +r)∗(U1 +r − U2 +r )∥−s,σdr + +� t +0 +∥Υ∗ +S(t − r)(GS +r )∗(V 1 +r − V 2 +r )∥−s,σdr. +So from Corollary 4.19, we deduce that +∥U1 +t − U2 +t ∥−s,σ≤ Csup +y +∥K(., y)∥2+D,σ +� t +0 +{∥U1 +r − U2 +r ∥−s,σ+∥V 1 +r − V 2 +r ∥−s,σ}dr. +(4.54) +Similarly, we obtain +∥V 1 +t − V 2 +t ∥−s,σ≤ C(sup +x ∥K(x, .)∥2+D,σ+α) +� t +0 +{∥U1 +r − U2 +r ∥−s,σ+∥V 1 +r − V 2 +r ∥−s,σ}dr. +(4.55) +Summing (4.54), (4.55) and applying Gronwall’s lemma we obtain that (U1, V 1) = (U2, V 2), +and the proof is complete. +5 +Appendix +In this Appendix we prove the following Lemmas. +Lemma 5.1. For any nonnegative integer m1 > m2 ≥ 0, and any real 0 < σ < σ′, the following +embedding is compact +W m1,σ +0 +(Rd) ֒→ W m2,σ′ +0 +(Rd). +Proof. To prove this Lemma it is enough to show that for any sequence (ϕn)n of W m1,σ +0 +(Rd) +which weakly converges towards 0 in W m1,σ +0 +(Rd), strongly converges in W m2,σ′ +0 +(Rd). +Let B(R) = {x ∈ Rd/|x|≤ R}, one has: +∥ϕn∥2 +m2,σ′= +� +|γ|≤m2 +� +Rd +|Dγϕn(x)|2 +1 + |x|2σ′ dx = +� +|γ|≤m2 +� +B(R) +|Dγϕn(x)|2 +1 + |x|2σ′ dx + +� +|γ|≤m2 +� +B +c(R) +|Dγϕn(x)|2 +1 + |x|2σ′ dx, +furthermore, since the function R ∈]1, ∞[�→ 1 + R2σ +1 + R2σ′ is nonincreasing, +� +|γ|≤m2 +� +B +c(R) +|Dγϕn(x)|2 +1 + |x|2σ′ dx = +� +|γ|≤m2 +� +B +c(R) +1 + |x|2σ +1 + |x|2σ′ +|Dγϕn(x)|2 +1 + |x|2σ dx, +≤ 1 + R2σ +1 + R2σ′ +� +|γ|≤m2 +� +B +c(R) +|Dγϕn(x)|2 +1 + |x|2σ dx, +≤ 1 + R2σ +1 + R2σ′ sup +n≥1 +∥ϕn∥2 +m1,σ −→ +R−→+∞ 0. +On the other hand since W m1,σ +0 +(Rd) ֒→ W m1,σ′ +0 +(B(R)) ֒→ W m2,σ′ +0 +(B(R)), where the second +embedding is compact, W m1,σ +0 +(Rd) ֒→ W m2,σ′ +0 +(B(R)) is compact (see Remark 6.3 in [1]). Thus +� +|γ|≤m2 +� +B(R) +|Dγϕn(x)|2 +1 + |x|2σ′ dx +−→ +n−→+∞ 0. + +5 +APPENDIX +39 +Corollary 5.2. For any non integer 1 + D < s < 2 + D and 1 + D < s′ = s+1+D +2 +< s < 2 + D +and any 0 < σ < σ′, the following embedding is compact. +Hs,σ(Rd) ֒→ Hs′,σ′(Rd). +Proof. We use the definition by interpolation of the space Hs,σ(Rd) to prove this result.We +Prove this result for d = 2, simlar arguments allow us to obtain the result for other values of +d. Since 2 < s′ = s+2 +2 +< s < 3. There exists ρ ∈ (1/2, 1), sucht that s′ = 3(1 − ρ) + 2ρ and +s = 4(1 − ρ) + 2ρ. Furthemore, +� +W 3,σ′ +0 +(R2), W 2,σ′ +0 +(R2) +� +ρ,2 = Hs′,σ′(R2) and +� +W 4,σ +0 +(R2), W 2,σ +0 +(R2) +� +ρ,2 = Hs,σ(R2), +see [23] or [37] for the explicit definition of the real interpolation space (., .)ρ, q. +Thus as from Lemma 5.1 the embeddings W 4,σ +0 +(R2) ֒→ W 3,σ′ +0 +(R2) is compact, we deduce from +Corollary 3.5 in [12] that the following embedding is compact. +Hs,σ(Rd) = +� +W 4,σ +0 +(R2), W 2,σ +0 +(R2) +� +ρ,2 ֒→ +� +W 3,σ′ +0 +(R2), W 2,σ′ +0 +(R2) +� +ρ,2 = Hs,σ′(R2). +Lemma 5.3. Under the Assumption (H2), for any A ∈ {S, I, R}, the Markovian semi-group +generated by the operator QA, is such that there exists a constant C > 0, such that +sup +y +� � +Rd ΥA(t)(x, y)dx +� +≤ eCt. +Proof. We recall that {XA +t , t ≥} is the Markov process having the operator QA as its infinites- +imal generator. Let PA(t)(y) = +� +Rd ΥA(t)(x, y)dx, using Dynkin’s formula it is easy to see +that +ΥA(t)(x, y) = δ0(x − y) + +� t +0 +(QA)∗ +yΥA(r)(x, y)dr, +(5.1) +thus integrating (5.1) over x, we obtain +PA(t)(y) = 1 + +� t +0 +(QA)∗ +yPA(r)(y)dr. +Furthermore, +∂ +∂tPA(t)(y) = − +� +1≤ℓ≤d +mℓ(A, y) ∂ +∂yℓ +PA(t)(y) + 1 +2 +� +1≤ℓ,u≤d +∂ +∂yℓ +(θθt)ℓ,u(A, y) ∂ +∂yu +PA(t)(y) ++ 1 +2 +� +1≤ℓ,u≤d +∂ +∂yu +(θθt)ℓ,u(A, y) ∂ +∂yℓ +PA(t)(y) ++ 1 +2 +� +1≤ℓ,u≤2 +(θθt)ℓ,u(A, y) ∂2 +∂yuyℓ +PA(t)(y) ++ +� +− +� +1≤ℓ≤d +∂ +∂yℓ +ml(A, y) + 1 +2 +� +1≤ℓ,u≤d +∂2 +∂yu∂yℓ +(θθt)ℓ,u(A, y) +� +PA(t)(y). +Consequently, PA(t) is the solution of the following system. +∂ +∂tPA(t)(y) = +� +ℓ +F m,θ +ℓ +(A, y) ∂ +∂yℓ +PA(t)(y) + 1 +2 +� +1≤ℓ,u≤d +(θθt)ℓ,u(A, y) ∂2 +∂yuyℓ +PA(t)(y) ++ H(A, y)PA(t)(y) += GAPA(t)(y) + H(A, y)PA(t)(y) +PA(0)(y) = 1. + +REFERENCES +40 +Thus fix T > 0, according to the Feyman-Kac formula, for any t ∈ [0, T], we have +PA(t)(y) = PA(T − t1)(y) = E +� +exp +� +− +� T +t1 +H(A, Yr)dr +� +/Yt1 = y +� +, (with t + t1 = T). +where {Yt, t ≥ 0} is the markovian processes having GA as the infinitesimal generator. +So as from assumption (H2) the function H is bounded, for any y ∈ Rd, we have +PA(t)(y) = +� +Rd ΥA(t)(x, y)dx ≤ eCt. +. +Remark 5.4. Since it is easy to adap without difficulty the proof of Theorem 5.4 of [1] to the +space W m,σ +0 +(m ∈ N), by following the proof of Theorem 5.23 of [1], we prove easily that for +any integer m > d/2, the space W m,σ +0 +is a Banach algebra. Furthemore using the result on the +complex interpolation of [22] (Theorem 4) and [10](subsection 10.2), we conclude that for any +real number s > d/2, the space Hs,σ is a Banach algebra. +References +[1] R.A. Adams. Sobolev Spaces. Academic Press, 1975. +[2] D. Aldous . Stopping times and tightness. The Annals of Probability 6(2), 335-340, 1978 +[3] L.J.S. Allen, B.M Bolker, Y. Lou and A.L. Nevai Asymptotic profiles of the steady states +for an SIS epidemic reaction diffusion model, Discrete Contin. Dyn. Syst. Ser. A 21, 1-20, +2008. +[4] H. Andersson, T. Britton. Stochastic epidemic models and their statistical analysis. +Springer Lecture Notes in Statistics. Springer, New York, 2000. +[5] H. Bahouri, J. Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Dif- +ferential Equations, Springer, 2011. +[6] P. Billingsley, Convergence of Probability Measures, 2nd edn. Wiley, New York, 1999. +[7] S. Bowong, A. Emakoua. and E Pardoux. A Spatial Stochastic Epidemic Model: Law of +Large Numbers and Central Limit Theorem. arXiv:2007.0663v1. [math.PR] 13 jul 2020 +[8] T. Britton, E. Pardoux. Stochastic epidemic in a homogeneous community, Part I of +Stochastic epidemic models with inference. T. Britton, E. Pardoux. eds, Lecture Notes +in Mathematics 2225, pp. 1–120, Springer 2019. +[9] A. Bücher and I. Kojadinovic, A note on conditional versus joint unconditional weak +convergence in bootstrap consistency results, arXiv:1706.01031v4 [math.ST] 1 Mar 2018. +[10] A. P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math. +24, 113-190 (1964). +[11] S. Clémençon, V.C. Tran and H. de Arazoza. A stochastic SIR model with contact tracing: +large population limits and statistical inference, J. of Biol. Dynamics 2:4, pp. 392–414, +2008. + +REFERENCES +41 +[12] F. Cobos and J. Peetre. Interpolation of compactness using Aronszajn-Gagliardo functors? +Israel J. Math. 68 (1989), 220-240. +[13] I. Gihman, A. V. Skorohod. Stochastic differential equations. Springer-Verlag Berlin New +york 1972. +[14] L. Grafakos. Classical Fourier analysis 2nd. ed Springer, 2008. +[15] B. Hanouzet. Espaces de Sobolev avec poids. Application au problème de Dirichlet dans +un demi espace. Rendiconti del Seminario Matematico della Università di Padova, Tome +46 (1971), p. 227-272. +[16] J. Jacod and A.N. Shiryaev. Limit Theorems for Stochastic Processes. Springer-Verlag, +Berlin, 1987. +[17] I. Kaj. A weak interaction epidemic among diffusive particles, in Stochastic Partial Dif- +ferential Equations, A. Etheridge ed., London Math. Soc. Lecture Note Series 216, pp. +189–208, Cambridge Univ. Press, 1995. +[18] C. Kipnis and C. Landim. Scaling limits of interacting particle systems, volume 320 of +Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat- +ical Sciences]. Springer-verlag, Berlin, 1999. +[19] P. Kotelenez, A stopped Doob inequality for stochastic convolution integrals and stochastic +evolution equations, Stochastic analysis and applications, 2(3), 245-265, 1984. +[20] H. Kunita. Stochastic flows and stochastic differential equations. Cambridge University +Press, Cambridge 1990. +[21] S.P. Lalley, E.A. Perkins and X. Zheng. A phase transition for mesure–valued SIR epidemic +processes, The Annals of Probab. 42, pp. 237–310, 2014. +[22] J. Löfström. Interpolation of weighted spaces of differentiable functions on Rd. Ann. Mat. +Pura. Appl. 132 (1982), 189?214. +[23] J. Löfström. Interpolation spaces, an introduction, Springer, 1976. +[24] S. Méléard. Convergence of the fluctuations for interacting diffusions with jumps associated +with Boltzmann equation. Stochastics and Stochastics Reports, 63 :195-225, 1998. +[25] S. Méléard et S. Roelly: Sur les convergences étroite ou vague de processus à valeurs +mesures. Comptes rendus de l’académie des Sciences de Paris Sér. 1, 317:785-788, 1993. +[26] S. Méléard. Mouvement brownien et calcul stochastique, Techniques de l?ingénieur, (2003). +[27] M. Métivier. Convergence faible et principe d’invariance pour des martingales à valeurs +dans des espaces de Sobolev. Annales de l’IHP, 20(4) :329-348, 1984. +[28] M. Métivier. (1987). Weak convergence of measure valued processes using Sobolev imbed- +ding techniques, Proceedings Stochastic partial differential equations, Trenio 1985. Lect. +Notes in Math. 1236. 172-183. +[29] M. N’zi, É. Pardoux and T. Yeo. A SIR model on a refining spatial grid I - Law of Large +Numbers, Applied Math.& Optimization, to appear. +[30] E. Pardoux. Probabilistic Models of Population Evolution, Scaling Limits, Genealogies and +Interactions, Springer 2016. + +REFERENCES +42 +[31] E. Pardoux. Moderate Deviations and Extinction of an Epidemic. Election. J. Probab 25, +paper 25, 1-27, 2020. +[32] S. Roelly-Coppoletta. A criterion of convergence of measure-valued processes: application +to measure branching processes Stochastics, 17 :43-65, 1985. +[33] L. Roques, O. Bonnefon, V. Baudrot, S. Soubeyrand, H. Berestycki : A parsimonious +model for spatial transmission and heterogeneity in the COVID-19 propagation. R. Soc. +Open Sci. 7: 201382, 2020. +[34] W. Rudin.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) +[35] k. Sato. and T. Ueno. Multi-dimensional diffusion and the Markov process on the boundary. +J. Math. Kyoto univ. 4(3) 529-605. +[36] M.E. Taylor. Partial differential equations III Nonlinear equations. Springer 1991. +[37] H. Triebel, Interpolation theory, Hunction spaces, Differential operators, North-Holland +Publishing Comp., 1978. +[38] V.C. Tran Modèles particulaires stochastiques pour des problèmes d’évolution adaptative +et pour l’approximation de solutions statistiques, Thesis 2006. + diff --git a/s9E0T4oBgHgl3EQfbACf/content/tmp_files/load_file.txt b/s9E0T4oBgHgl3EQfbACf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce1d5030629672bcd16efc6e44617b3047e623dc --- /dev/null +++ b/s9E0T4oBgHgl3EQfbACf/content/tmp_files/load_file.txt @@ -0,0 +1,1489 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf,len=1488 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='02343v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='PR] 6 Jan 2023 A SIR Stochastic Epidemic Model in Continuous Space: Law of Large Numbers and Central Limit Theorem Alphonse Emakoua emakouaal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='com Aix-Marseille Université, Centre de mathématiques et informatique (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='I), 39 Rue Frédéric Joliot Curie, 13013 Marseille, France January 9, 2023 Abstract The impact of spatial structure on the spread of an epidemic is an important issue in the prop- agation of infectious diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Recent studies, both deterministic and stochastic, have made it possible to understand the importance of the movement of individuals in a population on the persistence or extinction of an endemic disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In this paper we study a compartmental SIR stochastic epidemic model for a population that moves on Rd following SDEs driven by independent Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We define the sequences of empirical measures, which describe the evolution of the positions of the susceptible, infected and removed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Next, we obtain large population approximations of those sequence of measures, as weak solution of a system of reaction-diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Finaly we study the fluctuations of the stochastic model around its large population limit via the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The limit is a distribution val- ued Ornstein-Uhlenbeck process and can be represented as the solution of system of stochastic partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Keywords: Stochastic model · Déterministic · Law of large numbers · Central limit theorem Measure valued processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1 Introduction Deterministic models of epidemics have been developed significantly in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The study of stochastic models in contrast is more recent, see for example [4], [8], [7],[21],[31] and [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Anderson and Britton [4], and Britton and Pardoux [8] show that deterministic models of epidemics are the law of large numbers limits (as the size of the population tends to ∞) of homogenous stochastic models, while the central limit theorem and moderate and large deviations (see [8] and [31]) give tools to accurately describe the gap between stochastic and deterministic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' However, in their models, they ignore the fact that a population spreads over a spatial region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' But spatial heterogeneity, habitat connectivity, and rates of movement play important roles in disease persistence and extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Movement of susceptible or infected individuals can enhance or suppress the spread of disease, depending on the heterogeneity and connectivity of the spatial environnement, see for example [3] and the references therein, for the deterministic case and [17] and [29] for the stochastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1 INTRODUCTION 2 Having in mind the above conclusions, Emakoua and Pardoux [7], have studied the law of large numbers and central limit theorem of two spatial SIR epidemic models in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In the first considered model, the population is moving and in the second, there is no mouvement (to refer to plant epidemics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For the first model, the limiting law of large numbers model is a system of parabolic PDEs, which is a deterministic epidemic model in continuous space, and for the second a system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The study of the fluctuations around the limit law of large numbers through the central limit theorem gives in the limit for the two models an Ornstein-Uhlenbeck processe with values in a negative index Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Moreover in the fisrt model it was assumed that the diffusion coefficient remains the same for all the individuals and that the infectious rate does not depends of the position of the infectious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The impact of environmental heterogeneity was not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In this paper we study the law of large numbers and central limit theorem of a SIR stochastic epidemic models, for a population with constant size N, distributed on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The initial condition will be the same as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us describe our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We consider a population of constant size N, living in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We assume that at time t=0, the population consists of two classes: the susceptible S(0) and the infectious I(0), such that S(0) + I(0) = N described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Given A an arbitrary Borel subset of Rd and 0 < p ≤ 1, each individual i is placed in Rd independently of the others at the position Xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' If Xi 0 ∈ Ac then the individual i is susceptible and if Xi 0 ∈ A, the individual i is infected with probability p and susceptible with probability 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' This situation is modelled by empirical measures µS,N 0 = 1 N N � i=1 {1A(Xi 0)(1 − ξi) + 1Ac(Xi 0)}δXi 0 µI,N 0 = 1 N N � i=1 1A(Xi 0)ξiδXi 0 where {ξi, 1 ≤ i ≤ N} is a mutually independent family of Ber(p) random variables, globally independent of {Xi 0, 1 ≤ i ≤ N}, which in turn is a mutually independent family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' During the epidemic the population is divided into three compartments: the susceptible S, the infectious I and the recovered R (the recovered individuals are those who are dead or who have recovered and have permanent immunity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We weaken assumptions made in the first model in [7], by assuming that an individual i moves on Rd according to a diffusion process driven by the following stochastic differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi t = Xi 0 + � t 0 m(Ei r, Xi r)dr + � t 0 θ(Ei r, Xi t)dBi r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) Where {Bi, 1 ≤ i ≤ N} is a family of independent standard Brownian motions on Rd which is globaly independent of {Xi 0, 1 ≤ i ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ei t is the state at time t of the individual i, Ei t ∈ C = {S, I, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The environment heterogeneity is modelled by the function m: C × Rd −→ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The diffusion matrix is the function θ: C × Rd −→ Md(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We assume that the functions m and θ are bounded and Lipschitz continuous with respect to the space variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1 INTRODUCTION 3 Infections are non local and a susceptible i becomes infected at time t at the following rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1 Nγ N � j=1 K(Xi t, Xj t ) � N� ℓ=1 K(Xℓ t , Xj t ) �1−γ 1{Ej t =I}, with γ ∈ [0, 1] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) where to have less parameter, we have let K(Xi t, Xj t ) = β(Xj t )F(Xi t, Xj t ), with β a function on Rd and F a symetric function on Rd × Rd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The formulation of such a rate of infections can be explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since we take into account the spatial structure, an infectious individual j has a contact with the individual i at the rate 1 Nγ K(Xi t, Xj t ) � N� ℓ=1 K(Xℓ t , Xj t ) �1−γ , thus summing over the infectious individuals at time t gives the above rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The case γ = 0, has already been studied in [7], in this paper we focus on the case γ = 1, so (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) will becomes 1 N N � j=1 K(Xi t, Xj t )1{Ej t =I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The case γ ∈ (0, 1) will be studied in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The evolution of the numbers of susceptible, infectious and removered individuals is described by the following equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' S(t) = S(0) − Pinf �� t 0 1 N � i̸=j K(Xi r, Xj r)1{Eir=S}1{Ej r=I}dr � I(t) = I(0) + Pinf �� t 0 1 N � i̸=j K(Xi r, Xj r)1{Eir=S}1{Ej r=I}dr � − Pcu � α � t 0 N � j=1 1{Ej r=I}dr � R(t) = R(0) + Pcu � α � t 0 N � j=1 1{Ej r=I}dr � , where Pinf and Pcu are two independent standard Poisson processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We end the description of the model, by defining the renormalized point processes, which allows us to control the evolutions of the positions of susceptible, infectious and recovered individuals and the proportions of susceptible, infectious and recovered individuals in any subset of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∀t > 0, µS,N t = 1 N N � i=1 1{Ei t=S}δXi t, µI,N t = 1 N N � i=1 1{Ei t=I}δXi t, µR,N t = 1 N N � i=1 1{Ei t=R}δXi t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since the law of large numbers and the central limit theorem of the initial sequence (µI,N 0 , µS,N 0 )N≥1 has already been studied in [7], under the assumption (H0) that the law of X1 0, is absolutly continuous with respect to the Lebesgue measure, in this paper, we will first write the equation of evolution of (µS,N t , µI,N t , µR,N t ), when the size of the population N is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We shall next study the law of large numbers and the central limit theorem of those sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The law of large number result will be a convergence result in the space of measure valued pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The convergence proof will start with tightness in the appropriate space, identification of the limit of any vaguely converging subsequence with the unique deterministic solution of a 2 PRELIMINARIES 4 system of PDEs, from which the weak convergence, then in probability of the whole sequence will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The central limit theorem is technically more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The first difficulty comes from the fact that our domain is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The approximating sequence lives in the space of signed measures valued processes and one of the main problems to overcome is to find a suitable space in which this sequence, as well as its limit, take their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We prove that the approximating sequence converges in the Skorokhod space [D(R+, H−s,σ(Rd))]3, (σ > d/2, 1 + ⌈d 2⌉ < s < 2 + ⌈d 2⌉), to a continuous process characterized as the unique solution of a linear Gaussian processes valued stochastic partial differential equation (abbreviated below SPDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The weighted Sobolev spaces H−s,σ(Rd) we consider here were introduced by Métivier [28], for the integer values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Mélérad [24] uses that space for the study of the central limit theorem of a sequence of empirical (random) measures of interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Clé- mençon and all [11] also use that space to study a central limit theorem for a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious dis- ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The work of Löfstrom [22] and [23] allows us to extend that space to the non interger values of s, by using real interpolation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In section 2 we recall some results that will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In section 3, we first establish the evolution equations of the measure-valued processes µS,N, µI,N and µI,N then we show that the sequence {(µS,N t , µI,N t , µR,N t ), t ≥ 0} converges in probability as N → ∞ towards (µS, µI, µR), the unique solution of a system of parabolic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In section 5 we study the convergence of the sequence of fluctuations processes (UN = √ N(µS,N − µ), V N = √ N(µI,N − µI), W N = √ N(µR,N − µR)) as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 2 Preliminaries Notation: For any metric space E, MF(E) denotes the space of finite measures on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any integer k ≥ 0, Ck(E) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ck c (E)) denotes the space of continuous and k times continuously differentiable real valued functions defined on E, (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' with compact support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For k = 0, we write C(E) (resp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='Cc(E)) instead of C0(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' C0 c (E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any integer k ≥ 0, Ck b (E) denotes the space of real valued functions of class Ck on E with bounded derivatives up to order k (order 0 included ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' C0(E) denotes the space of continuous functions on E vanishing at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For µ ∈ MF(E) and ϕ ∈ C(E), we denote the integral � E ϕ(x)µ(dx) by (µ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In the following, the letter C will denote a (constant) positive real number which can change from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We equip MF(E) with the topology of weak convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let E be a complete separable metric space, C(R+, E) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' D(R+, E) is a space of continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' càdlàg) functions from R+ to E, equipped with the locally uniform (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Skorokhod) topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We refer the reader to section 12 of [6] for a presentation of the Skorokhod topology and its associated metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 2 PRELIMINARIES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 Weighted Spaces of functions For every nonnegative integer m and σ ∈ R+, we consider the space of all real valued functions ϕ defined on Rd, with partial derivative up to oder m such that: ∥ϕ∥m,σ= � � |γ|≤m � Rd |Dγϕ(x)|2 1 + |x|2σ dx � < +∞, where |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='| denotes the euclidian norm on Rd, and for γ = (γ1, γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='γd) ∈ Nd, |γ|= d� i=1 γi and Dγϕ = (∂|γ|ϕ)/(∂xγ1 1 ∂xγ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='∂xγd d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let W m,σ 0 (Rd) be the closure of the set of functions of class C∞ with compact support for this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' W m,σ 0 (Rd) is a Hilbert space for the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='∥m,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For a nonnegative real number s we extend the above space as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let J s be the potential operator defined on Rd by (J sϕ) = F −1[(1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2)s/2 �ϕ] (where the Fourier transform �ϕ of ϕ is well defined, F −1 denotes the inverse of the Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hs,σ(Rd) denotes the space of functions ϕ which satisfy the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∥ϕ∥s,σ= ∥(J sϕ)∥0,σ< ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' It is shown in [22] that Hm,σ(Rd) = W m,σ 0 (Rd), for any nonnegative integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We denote by H−s,σ(Rd) the dual space of Hs,σ(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let Cm,σ(Rd) be the space of functions ϕ with continuous partial derivatives up to oder m and such that lim |x|−→∞|Dγϕ(x)|2/1 + |x|2σ= 0 for all |γ|≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' This space is normed with ∥ϕ∥Cm,σ= � |γ|≤m sup x∈Rd |Dγϕ(x)| 1 + |x|σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let C−m,σ(Rd) denotes the dual of Cm,σ(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have the following continuous embeddings (see [1] and [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Cm+j,σ ֒→ Cm,σ+r m ≥ 0, j ≥ 0, σ > 0, r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) Hs,σ(Rd) ֒→ Cℓ,σ(Rd), s > d/2 + ℓ, ℓ ≥ 0, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) Cm,σ(Rd) ֒→ W m,σ+η 0 (Rd), η > d/2, m ≥ 0, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (A special case of theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 in [19]) Let (H, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='∥H) be a separable Hilbert space, M be an H−valued locally square integrable càdlàg martingale and T(t) a contraction semigroup operator of L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Then there exists a finite constant C depending only on the Hilbert norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='∥H such that for all T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' E � sup 0≤t≤T ��� � t 0 T(t − r)dMr ��� 2 H � ≤ Ce4σT E � ∥MT∥2 H � , where σ is a real number such that ∥T(t)∥L≤ eσt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (White noise) White noise on Rd is a random distribution W defined on a probability space (Ω, F, P) which is such that the mapping ϕ �→ (W, ϕ) is linear and continuous from L2(Rd) into L2(Ω) and {(W, ϕ), ϕ ∈ L2(Rd)} is a centered Gaussian generalized process satisfying: E((W, ϕ)(W, φ)) = (ϕ, φ)L2, for any ϕ, φ ∈ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Where (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )L2 denotes a scalar product on L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Space-time white noise is a white noise on R+ × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3 LAW OF LARGE NUMBERS 6 3 Law of Large Numbers The aim of this section is to study the convergence of (µS,N, µI,N, µR,N), as N → ∞ under Assumption (H1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' To this end we are going to: Write the system of evolution equations of (µS,N, µI,N, µR,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Study the tightness of (µS,N, µI,N, µR,N)N≥1 in Skorokhod’s space [D(R+, (MF(Rd), v))]3, where (MF(Rd), v) is the space of finite measure on Rd, equipped with the vague topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Find the system of evolution equations satisfies by the limit in law (µS, µI, µR) of a convergent subsequence of (µS,NµI,N, µR,N)N≥1 Show that the system of PDEs verified by (µS, µI, µR) admits a unique solution in Λ = {(µ1, µ2, µ3)/0 ≤ (µi, 1) ≤ 1, i ∈ {1, 2, 3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The following is assumed to hold throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Assumption (H1) The law of X1 0 is absolutly continuous with respect to the Lebesgue measure and its density is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' K ∈ Cc(Rd × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any A ∈ {S, I, R}, and x ∈ Rd, the matrix (θθt)(A, x) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 System of evolution equations of (µS,N, µI,N, µR,N) In this subsection we shall establish the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any ϕ ∈ C2 c (Rd), {(µS,N t , ϕ), (µI,N t , ϕ), (µR,N t , ϕ)} satisfies, (µS,N t , ϕ) = (µS,N 0 , ϕ) + � t 0 (µS,N r , QSϕ)dr − � t 0 � µS,N r , ϕ(µI,N r , K) � dr + MN,ϕ t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) (µI,N t , ϕ) = (µI,N 0 , ϕ) + � t 0 (µI,N r , QIϕ)dr + � t 0 � µS,N r , ϕ(µI,N r , K) � dr − α � t 0 (µI,N r , ϕ)dr + LN,ϕ t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) (µR,N t , ϕ) = � t 0 (µR,N r , QRϕ)dr + α � t 0 (µI,N r , ϕ)dr + Y N,ϕ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3) Where � µS,N r , ϕ(µI,N r , K) � = � Rd ϕ(x) � Rd K(x, y)µI,N r (dy)µS,N r (dx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' QAϕ(x) = m(A, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ▽ ϕ(x) + 1 2 � 1≤ℓ,u≤d (θ θt)ℓ,u(S, x) ∂2ϕ ∂xℓxu (x), 3 LAW OF LARGE NUMBERS 7 and with {Mi}1≤i≤N, {Qi}1≤i≤N two collections of standard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' with mean the Lebesgue measure) Poisson random measures (abbreviated below PRM) on R2 +, which are such that B1, M1, Q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' , BN, MN, QN are mutually independent, and denoting by M i and Q i the com- pensated PRMs associated to Mi and Qi, we have MN,ϕ t = − 1 N N � i=1 � t 0 � ∞ 0 1{Ei r−=S}ϕ(Xi r)1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}M i(dr, du) + 1 N N � i=1 � t 0 1{Eir=S} ▽ ϕ(Xi r)θ(S, Xi r)dBi r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' LN,ϕ t = 1 N N � i=1 � t 0 � ∞ 0 1{Ei r−=S}ϕ(Xi r)1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}M i(dr, du)) + 1 N N � i=1 � t 0 1{Eir=I} ▽ ϕ(Xi r)θ(I, Xi r)dBi r − 1 N N � i=1 � t 0 � α 0 1{Eir=I}ϕ(Xi r)Q i(dr, du).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Y N,ϕ t = 1 N N � i=1 � t 0 1{Eir=R} ▽ ϕ(Xi r)θ(R, Xi r)dBi r + 1 N N � i=1 � t 0 � α 0 1{Ei r−=I}ϕ(Xi r)Q i(dr, du).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us first recall that for any t ≥ 0, Xi t = Xi 0 + � t 0 m(Ei r, Xi r)dr + � t 0 θ(Ei r, Xi t)dBi r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let ϕ ∈ C2 c (Rd), according to Itô’s formula we have, ϕ(Xi t) = ϕ(Xi) + � t 0 ▽ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(Ei r, Xi r)dr + 1 2 � t 0 � 1≤ℓ,u≤d (θ θt)ℓ,u(Ei r, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + � t 0 ▽ϕ(Xi r)θ(Ei r, Xi r)dBi r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4) On the other hand 1{Ei t=S} = 1{Ei 0=S} − � t 0 � ∞ 0 1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}1{Ei r−=S}Mi(du, dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5) Hence using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5), we have 1{Ei t=S}ϕ(Xi t) = 1{Ei 0=S}ϕ(Xi) + � t 0 1{Eir=S} ▽ ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(S, Xi r)dr + 1 2 � t 0 1{Eir=S} � 1≤ℓ,u≤d (θ θt)ℓ,u(S, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + � t 0 1{Eir=S} ▽ ϕ(Xi r)θ(S, Xi r)dBi r − � t 0 � ∞ 0 1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}1{Ei r−=S}ϕ(Xi r)Mi(du, dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Taking the sum over i and multiplying by 1 N , we obtain 1 N N � i=1 1{Ei t=S}ϕ(Xi t) = 1 N N � i=1 1{Ei 0=S}ϕ(Xi) + 1 N N � i=1 � t 0 1{Eir=S} ▽ ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(S, Xi r)dr 3 LAW OF LARGE NUMBERS 8 + 1 2N N � i=1 � t 0 1{Eir=S} � 1≤ℓ,u≤2 (θ θt)ℓ,u(S, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + 1 N N � i=1 � t 0 1{Eir=S} ▽ ϕ(Xi r)θ(S, Xi r)dBi r − 1 N N � i=1 � t 0 � ∞ 0 1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}1{Ei r−=S}ϕ(Xi r)M i(du, dr) − 1 N N � i=1 � t 0 1 N N � j=1 K(Xi r, Xj r)1{Ej r=I}1{Eir=S}ϕ(Xi r)dr, from which (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Similarly, with again ϕ ∈ C2 c (Rd), {1{Ei t=I}ϕ(Xi t), t ≥ 0} is a jump process satisfying, 1{Ei t=I}ϕ(Xi t) = 1{Ei 0=I}ϕ(Xi) + � t 0 1{Eir=I} ▽ ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(I, Xi r)dr + 1 2 � t 0 1{Eir=I} � 1≤ℓ,u≤2 (θ θt)ℓ,u(I, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + � t 0 1{Eir=I} ▽ ϕ(Xi r)θ(I, Xi r)dBi r + � t 0 � ∞ 0 1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}1{Ei r−=S}ϕ(Xi r)Mi(du, dr) − � t 0 � α 0 1{Ei r−=I}ϕ(Xi r)Qi(du, dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Summing over i and multiplying by 1 N , we obtain 1 N N � i=1 1{Ei t=I}ϕ(Xi t) = 1 N N � i=1 1{Ei 0=I}ϕ(Xi) + 1 N N � i=1 � t 0 1{Eir=I} ▽ ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r)dr + 1 2N N � i=1 � t 0 1{Eir=I} � 1≤ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='u≤d (θ θt)ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='u(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + 1 N N � i=1 � t 0 1{Eir=I} ▽ ϕ(Xi r)θ(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r)dBi r − 1 N N � i=1 � t 0 � ∞ 0 1{u≤ 1 N �N j=1 K(Xir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='Xj r)1{Ej r=I}}1{Ei r−=S}ϕ(Xi r)M i(du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dr) − 1 N N � i=1 � t 0 1 N N � j=1 K(Xi r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xj r)1{Ej r=I}1{Eir=S}ϕ(Xi r)dr − 1 N N � i=1 � t 0 � α 0 1{Ei r−=I}ϕ(Xi r)Q i(du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dr) − α N N � i=1 � t 0 1{Eir=I}ϕ(Xi r)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' from which (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Similarly, with once again ϕ ∈ C2 c (Rd), {1{Ei t=R}ϕ(Xi t), t ≥ 0} is a jump processes satisfying, 3 LAW OF LARGE NUMBERS 9 1{Ei t=R}ϕ(Xi t) = � t 0 1{Eir=R} ▽ ϕ(Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(R, Xi r)dr + � t 0 1{Eir=R} � 1≤ℓ,u≤d (θ θt)ℓ,u(R, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + � t 0 1{Eir=R} ▽ ϕ(Xi r)θ(R, Xi r)dBi r + � t 0 � α 0 1{Ei r−=I}ϕ(Xi r)Qi(du, dr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Summing over i and multiplying by 1 N we obtain, 1 N N � i=1 1{Ei t=R}ϕ(Xi t) = 1 N N � i=1 � t 0 1{Eir=R} ▽ ϕ(Ei r, Xi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='m(R, Xi r)dr + 1 2N N � i=1 � t 0 1{Eir=R} � 1≤ℓ,u≤d (θ θt)ℓ,u(R, Xi r) ∂2ϕ ∂xℓxu (Xi r)dr + 1 N N � i=1 � t 0 1{Eir=R} ▽ ϕ(Xi r)θ(R, Xi r)dBi r + 1 N N � i=1 � t 0 � α 0 1{Ei r−=I}ϕ(Xi r)Q i(du, dr) + α N N � i=1 � t 0 1{Eir=I}ϕ(Xi r)dr, from which (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 Tightness and Convergence of (µS,N, µI,N, µR,N) in [D(R+, MF (Rd))]3 Recall that we equip MF(Rd) with the topology of weak convergence and the Skorokhod space of cádlág function from R+ into MF(Rd) with the Skorokhod topology (we refer to page 63 of [18] for an explicit definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For A ∈ {S, I, R}, we have (µA,N t , 1Rd) = 1 N �N i=1 1{Ei t=A} ≤ 1, thus for any ϕ ∈ Cc(Rd), A ∈ {S, I, R}, |(µA,N t , ϕ)|≤ ∥ϕ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We can now establish the wished tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequences (µS,N)N≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (µI,N)N≥1 and (µR,N)N≥1 are tight in D(R+, (MF(Rd), v)), where MF(Rd) is equipped with the topology of vague convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us prove that (µS,N)N≥1 is tight in D(R+, (MF(Rd), v)), where MF(Rd) is equipped with the vague topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We refer to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 of Roelly [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let Ξ be a dense subset of C0(Rd), a sufficient condition for (µS,N)N≥1 to be tight in D(R+, (MF(Rd), v)) is that: for any ϕ ∈ Ξ, {(µS,N t , ϕ), t ≥ 0, N ≥ 1} is tight in D(R+, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We choose Ξ = C∞ c (Rd) (= the space of infinitely differentiable functions with compact support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let ϕ ∈ C∞ c (Rd), we have (µS,N t , ϕ) = (µS,N 0 , ϕ) + � t 0 (µS,N r , QSϕ)dr − � t 0 � µS,N r , ϕ(µI,N r , K) � dr + MN,ϕ t , We notice that (µS,N, ϕ) is a semi-martingale since MN,ϕ is a square integrable martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Indeed, MN,ϕ is a local martingale as the sum of local martingales and 3 LAW OF LARGE NUMBERS 10 < MN,ϕ >t = 1 N � t 0 � µS,N r , ϕ2(µI,N r , K) � dr + 1 N � t 0 � µS,N r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, ≤ 1 N � t 0 ���� � X ϕ2(x) � X×X K(x, y)µI,N r (dy)µS,N r (dx) ���� dr + t N � 1≤ℓ,u≤d ��� ∂ϕ ∂xℓ ��� 2 ∞ ��θℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') ��2 ∞ + 2t N � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ��� ∂ϕ ∂xℓ ��� ∞ ��� ∂ϕ ∂xu ��� ∞∥θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞∥θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞, ≤ t∥ϕ∥2 ∞∥K∥∞ N + t N C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' thus E(< MN,ϕ >t) < ∞, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On other hand (µS,N t , ϕ) = (µS,N 0 , ϕ) + � t 0 ωN,ϕ r dr + MN,ϕ t with < MN,ϕ >t= � t 0 ̟N,ϕ r dr, and ωN,ϕ t = � µS,N t , m(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ▽ ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 1 2 � 1≤ℓ,u≤d (θ θt)ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') ∂2ϕ ∂xℓxu (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � − � µS,N t , ϕ(µI,N t , K) � , ̟N,ϕ t = 1 N � µS,N t , ϕ2(µI,N t , K) � + 1 N � µS,N t , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthermore ωN,ϕ and ̟N,ϕ are progressively measurable since the are adapted and right con- tinuous, so according to proposition 37 of [30] a sufficient condition for {(µS,N t , ϕ), t ≥ 0, N ≥ 1} to be tight in D(R+, R) is that both: (µS,N 0 , ϕ)N≥1 is tight in R, ∀T ≥ 0, sup 0≤t≤T (| ωN,ϕ t | +̟N,ϕ t ) is tight in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' These follow readily from the facts that: − |(µS,N 0 , ϕ)|≤ ∥ϕ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − | ωN,ϕ t |≤ � 1≤ℓ≤d ∥mℓ∥∞∥ ∂ϕ ∂xℓ(x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='., xd)∥∞+ 1 2 � 1≤ℓ,u≤d ∥(θ θt)ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞∥ ∂2ϕ ∂xℓxu∥∞+∥ϕ∥∞∥K∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ⩽ C − ̟N,ϕ t ≤ ∥ϕ∥2 ∞∥K∥∞ N + C ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The same arguments yield the tightness of {µI,N t , t ≥ 0, N ≥ 1} and {µR,N t , t ≥ 0, N ≥ 1} in D(R+, (MF(Rd), v))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The following Proposition follows from the fact that the jump of µA,N are order of 1/N( see the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 in [7] for the explicit proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The limit points (µS), (µI) and (µR) of the sequences (µS,N)N≥1, (µI,N)N≥1 and (µR,N)N≥1 are elements of C(R+, MF(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3 LAW OF LARGE NUMBERS 11 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequence (µS,N, µI,N, µR,N)N≥1 converges in probability, in � D(R+, MF(Rd)) �3 towards (µS, µI, µR) ∈ � C(R+, MF(Rd)) �3 which is the unique solution of the following system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any ϕ ∈ C2 c (Rd), (µS t , ϕ) = (µS,N 0 , ϕ) + � t 0 (µS r , QSϕ)dr − � t 0 � µS r , ϕ(µI r, K) � dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6) (µI t, ϕ) = (µI 0, ϕ) + � t 0 (µI r, QIϕ)dr + � t 0 � µS r , ϕ(µI r, K) � dr − α � t 0 (µI r, ϕ)dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) (µR t , ϕ) = � t 0 (µR r , QRϕ)dr + α � t 0 (µI r, ϕ)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5 By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3, the sequence (µS,N, µI,N, µR,N)N≥1 is tight in � D(R+, (MF(Rd), v)) �3, thus according to Prokhorov’s Theorem there exists a subsequence of (µS,N, µI,N, µR,N)N≥ still de- noted (µS,N, µI,N, µR,N)N≥1 which converges in law in � D(R+, (MF(Rd), v)) �3 towards (µS, µI, µR), where MF(Rd) is equipped with the vague topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence to complete the proof of Theorm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5 it remains to: Find the system of PDEs satisfes by {(µS t , µI t, µR t ), t ≥ 0} Show that the system verifies by (µS t , µI t, µR t ) admits a unique solution on Λ = {(µ1, µ2, µ3)/0 ≤ (µi, 1Rd) ≤ 1, i ∈ {1, 2, 3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' It is so easy to obtain the following Lemma, therefore we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' If we let Σ = {(µ1, µ2) ∈ MF(Rd), (µi, 1Rd) ≤ 1, ∀i ∈ {1, 2}}, for any ϕ, ψ ∈ Cc(Rd), the following map is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Gϕ,ψ : (Σ, v) × (Σ, v) → (MF(Rd × Rd), v) (µ, ν) �−→ (µ ⊗ ν, ϕψ) where (µ ⊗ ν, ϕψ) = � Rd×Rd ϕ(x)ψ(y)ν(dy)µ(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The following Proposition establishes the system of equations satisfied by (µS, µI, µR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The processes (µS, µI, µR) satisfies the system formed by the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We prove this Proposition by taking the limit in the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1- It has been shwon in [7] that the sequence {(µS,N 0 , µI,N 0 ), N ⩾ 1} converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' towards the pair of deterministic measures (µS 0, µI 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 2- Since the map x ∈ Rd �→ QSϕ(x) = m(A, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ▽ ϕ(x) + 1 2 � 1≤ℓ,u≤d (θ θt)ℓ,u(S, x) ∂2ϕ ∂xℓxu (x) is con- tinuous with compact support, � t 0 � µS,N r , QSϕ � dr converges in law towards � t 0 � µS r , QSϕ � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 2- Form Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3, the sequence µS,N ⊗µI,N is also tight, thus from Prokorov’s theorem it 3 LAW OF LARGE NUMBERS 12 is possible to extract a sub-sequence still denotes (µS,N ⊗ µI,N)N≥1 such that (µS,N ⊗ µI,N)N≥1 and the above subsequences (µS,N, µI,N, µR,N)N≥1 converges towards χS,I and (µS, µI, µR) re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthermore the fact that for any t ≥ 0, χS,I t = µS t ⊗ µI t follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Consequently, we have � t 0 � µS,N r , ϕ(µI,N r , K) � dr = � t 0 � µS,N r ⊗ µI,N r , ϕK � dr L−→ � t 0 � µS r ⊗ µI r, ϕK � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3- Let us prove that the sequences MN,ϕ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' LN,ϕ t and Y N t converge to 0 in Probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − Convergence of MN,ϕ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have seen above that E(| MN,ϕ t |2)= E(< MN,ϕ >t) ≤ t N ∥ϕ∥2 ∞∥k∥∞+tC N N→∞ −−−→ 0, consequently MN,ϕ t converges to 0 in L2, so also in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By similar arguments, we obtain the convergences in probability to 0 of the sequences LN,ϕ t and Y N,ϕ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6) follows from 1-, 2-, 3-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Similar arguments yield (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us now prove the following proposition which will be useful to show that the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8) admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For A ∈ {S, I, R}, ΥA(t) denotes the Markovian semi-group of the diffussion process with diffusion matrix θ(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') and drift coefficient m(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any ϕ ∈ Cc(Rd), we have (µS t , ϕ) = (µS 0 , ΥS(t)ϕ) − � t 0 � µS r , ΥS(t − r)ϕ(µI r, K) � dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9) (µI t, ϕ) = (µI 0, ΥI(t)ϕ) + � t 0 � µS r , ΥI(t − r)ϕ(µI r, K) � dr − α � t 0 (µI r, ΥI(t − r)ϕ)dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) (µR t , ϕ) = α � t 0 (µI r, ΥR(t − r)ϕ)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We may classically derive from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6) a similar formula where the test function ϕ(x) is replaced by ψr(x) = ψ(r, x) which is of class C1,2 on [0, t] × Rd: (µS t , ψt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=')) = (µS 0, ψ0(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=')) + � t 0 � µS r , ∂ ∂rψr(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr + � t 0 (µS r , m(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ▽ ϕ)dr + 1 2 � t 0 (µS r , Tr[(θ θt)(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )D2ϕ])dr − � t 0 � µS r , ψr(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )(µI r, K) � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12) Let us now consider a continuous fonctions ϕ on Rd, with compact support and fix a time t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We define for (r, x) ∈ [0, t] × Rd, ψr(x) = ΥA(t − r)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Then ψ is the solution of the following equation dψr(x) dr + QSϕ(x) = 0 on [0, t] × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12) applied to this function ψ yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11) by similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The system formed by the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11), admits a unique solution on the set Λ = {(µ1, µ2, µ3) ∈ [MF(Rd)]3/(µi, 1) ≤ 1 ∀i ∈ {1, 2, 3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us recall that the distance in total variation on MF(Rd) is defined by ∥µ − ν∥V T=sup{|(µ − ν, ϕ)|, ϕ continuous with compact support and ∥ϕ∥∞≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Now let (µ1 t, µ2 t, µ3 t) and (ν1 t , ν2 t , ν3 t ) be two solutions of the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) and 3 LAW OF LARGE NUMBERS 13 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11) with the same initial condition and ϕ ∈ Cc(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since ΥS(t) is a contraction semi-group on Cc(Rd), we have ��� (µ1 r, ΥS(t − r)ϕ(µ2 r, K)) − (ν1 r, ΥS(t − r)ϕ(ν2 r, K)) ��� = ��� � µ1 r − ν1 r, ΥS(t − r)ϕ(µ2 r, K) � − � ν1 r, ΥS(t − r)ϕ(ν2 r − µ2 r, K) � ���, ≤ ��� � Rd ΥS(t − r)ϕ(x)(µ2 r, K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ))(µ1 r − ν1 r)(dx) ��� + ��� � Rd ΥS(t − r)ϕ(x) � Rd K(x, y)(µ2 r − ν2 r)(dy)ν1 r(dx) ���, ≤ ∥ΥS(t − r)ϕ(µ2 r, K)∥∞∥µ1 r − ν1 r∥V T+∥ΥS(t − r)ϕ∥∞∥K∥∞∥µ2 r − ν2 r∥V T, ≤ ∥ϕ∥∞∥K∥∞ � ∥µ1 r − ν1 r∥V T+∥µ2 r − ν2 r∥V T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus using the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11) respectively we obtain sup ∥ϕ∥∞≤1 |(µ1 t − ν1 t , ϕ)| ≤ ∥K∥∞ � t 0 � ∥µ1 r − ν1 r∥V T+∥µ2 r − ν2 r∥V T � dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13) sup ∥ϕ∥∞≤1 |(µ2 t − ν2 t , ϕ)| ≤ ∥K∥∞(1 + α) � t 0 � ∥µ1 r − ν1 r∥V T+∥µ2 r − ν2 r∥V T � dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14) sup ∥ϕ∥∞≤1 |(µ3 t − ν3 t , ϕ)| ≤ α � t 0 ∥µ2 r − ν2 r∥V T, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15) where the suppremum is taken over continuous functions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Consequently summing the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15) the result follows from Gronwall’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We can now complete the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since (µS,N, µI,N, µR,N)N≥1 is tight in [D(R+, (MF(Rd), v))]3, and all converging subsequences of the sequence (µS,N, µI,N, µR,N)N≥1 converge in law in [D(R+, (MF(Rd), v))]3 to the same limit (µS, µI, µR), where MF(Rd) is equipped with the vague topology, the sequence (µS,N, µI,N, µR,N)N≥1 converge in law in [D(R+, (MF(Rd), v))]3 towards (µS, µI, µR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' To extend this result to the weak topology, we use a criterion (Proposition 3) proved in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4 the lim- iting process (µS, µI, µR) is continuous, it suffices to prove that the sequence � (µS,N, 1), (µI,N, 1), (µR,N, 1) � N≥1 converges in law to � (µS, 1), (µI, 1), (µR, 1) � in [D(R+, R)]3, which follows from and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9 and the fact that: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 remains true for the functions ϕ ∈ C2 b (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Following the Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3, we see that the sequence � (µS,N, 1), (µI,N, 1), (µR,N, 1) � N≥1 is tight in [D(R+, R)]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Prokorov’s Theorem we deduce the existence of a subsequence which converge in law towards � (µS, 1), (µI, 1), (µR, 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8 remains true when the test function ϕ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Finally since the sequence (µS,N, µI,N, µR,N)N≥1 weakly converge in (D(R+, MF(Rd)))3 to- wards (µS, µI, µR) and (µS, µI, µR) is deterministic, we have convergence in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3 LAW OF LARGE NUMBERS 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 Existence of densities The third assumptions in (H1), allow us to obtain the following result (see Theorem 22 in [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the assumption (H1), For A ∈ {S, I, R}, there exists a measurable function ΥA(t)(x, y), defined on Rd × Rd, which is a density in y ∈ Rd and such that for each continuous function ϕ defined on Rd, one has ΥA(t)ϕ(x) = � Rd ΥA(x, y)ϕ(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' There exists (f S, f I, f R) ∈ L∞ loc(R+, (L1(Rd)3) which satisfies: ∂tf S t (x) = Q∗ Sf S t (x) − f S t (x) � Rd K(x, y)f I t (y)dy, ∂tf I t (x) = Q∗ If I t (x) + f S t (x) � Rd K(x, y)f I t (y)dy − αf I t (x), ∂tf R t (x) = Q∗ Rf R t (x) + αf I t (x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16) where Q∗ A is the adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us recall that the initial measures µS 0, µI 0 are absolutly continuous with respect to the Lebesgue measure (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We construct by inductions a sequences of functions (f n, gn, hn), satisfying in a weak sense the following system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∂tf n+1 t (x) = Q∗ Sf n+1 t (x) − f n+1 t (x) � Rd K(x, y)gn t (y)dy, ∂tgn+1 t (x) = Q∗ Ign+1 t (x) + f n+1 t (x) � Rd K(x, y)gn t (y)dy − αgn+1 t (x), ∂thn+1 t (x) = Q∗ Rhn+1 t (x) + αgn+1 t (x), f n+1 0 (x) = f S 0 (x), gn+1 0 (x) = f I 0 (x), hn+1 0 (x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17) Thanks to the nonnegativity of f S 0 and applying the Feyman-Kac formula, we show that f n is nonnegtive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The non-negativity of gn, follows by recurrence and by using the comparaison principle, the Feyman Kac formula and the fact that f I 0 is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Finaly hn is non- negative by using the comparaison principle, the Feyman Kac formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' it is easy to see that for any ϕ ∈ Cc(Rd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (f n+1 t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) = � Rd f S 0 (x) � Rd ΥS(t)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydx − � t 0 � Rd f n+1 r (x) � Rd K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' u)gn r (u)du � Rd ΥS(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydxdr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (gn+1 t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) = � Rd f I 0 (x) � Rd ΥI(t)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydx + � t 0 � Rd f n+1 r (x) � Rd K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' u)gn r (u)du � Rd ΥI(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydxdr − α � t 0 � Rd gn+1 r (x) � Rd ΥI(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydxdr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (hn+1 t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) = α � t 0 � Rd gn+1 r (x) � Rd ΥR(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)ϕ(y)dydxdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3 LAW OF LARGE NUMBERS 15 Fubini’s theorem yields, f n+1 t (y) = � Rd f S 0 (x)ΥS(t)(x, y)dx − � t 0 � Rd f n+1 r (x) � Rd K(x, u)gn r (u)duΥS(t − r)(x, y)dxdr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18) gn+1 t (y) = � Rd f I 0 (x)ΥI(t)(x, y)dx + � t 0 � Rd f n+1 r (x) � Rd K(x, u)gn r (u)duΥI(t − r)(x, y)dxdr − α � t 0 � Rd gn+1 r (x)ΥI(t − r)(x, y)dxdr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19) hn+1 t (y) = α � t 0 � Rd gn+1 r (x)ΥR(t − r)(x, y)dxdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20) Fisrt of all, since ∀n ∈ N, f n t ≥ 0 and gn t ≥ 0, f n+1 t (y) ≤ � Rd f S 0 (x)ΥS(t)(x, y)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus integrating over y ∈ Rd, using Fubini’s Theorem and the fact that � Rd ΥS(t)(x, y)dy = 1, ∀t ≥ 0, we see that sup n sup 0≤t≤T ∥f n t ∥L1 ≤ ∥f S 0 ∥L1≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21) The last inequality follows from the fact that if H(x) is the density of the law of X1 0, f S 0 (x) = {(1−p)1A(x)+1Ac(x)}H(x) (see Theorem 3 1 in [7]), where A and p have been defined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Moreover, since ∀n ∈ N, gn t ≥ 0, gn+1 t (y) ≤ � Rd f I 0 (x)ΥI(t)(x, y)dx + � t 0 � Rd f n+1 r (x) � Rd K(x, u)gn r (u)duΥI(t − r)(x, y)dxdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus integrating over y ∈ Rd, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21), Fubini’s Theorem and Gronwall’s Lemma, we easily deduce that sup n sup 0≤t≤T ∥gn t ∥L1 ≤ ∥f I 0 ∥L1exp(T∥K∥∞) ≤ exp(T∥K∥∞), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='22) where the last inequality follows from the fact that if H(x) is the density of the law of X1 0, f I 0 (x) = p1A(x)H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On the other hand, with the same argument as above, we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='22) that sup n sup 0≤t≤T ∥hn t ∥L1≤ αTexp(T∥K∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23) Let us now show the convergence of the sequence (f n, gn, hn) in L∞ loc(R+, [L1(Rd)]3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A straightforward computation using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' and similar arguments as above yields f n+1 t (y) − f n t (y) = − � t 0 � Rd(f n+1 r (x) − f n r (x)) � Rd K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' u)gn r (u)duΥS(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)dxdr + � t 0 � Rd f n r (x) � Rd K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' u)(gn−1 r (u) − gn r (u))duΥS(t − r)(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)dxdr ∥f n+1 t − f n t ∥L1≤ ∥K∥∞{sup n sup 0≤t≤T ∥f n t ∥L1+sup n sup 0≤t≤T ∥gn t ∥L1} 4 CENTRAL LIMIT THEOREM 16 × � t 0 {∥f n+1 r − f n r ∥L1+∥gn r − gn−1 r ∥L1}dr ≤ C(T)∥K∥∞ � t 0 {∥f n+1 r − f n r ∥L1+∥gn r − gn−1 r ∥L1}dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24) Similarly, we have ∥gn+1 t − gn t ∥L1≤ C(T)(1 + α)∥K∥∞ � t 0 {∥gn r − gn−1 r ∥L1+∥gn+1 r − gn r ∥L1+∥f n+1 r − f n r ∥L1}dr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25) ∥hn+1 t − hn t ∥L1≤ α � t 0 ∥gn+1 r − gn r ∥L1dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26) Summing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26) and using Gronwall’s Lemma, we have sup 0≤r≤t � ∥f n+1 r − f n r ∥L1+∥gn+1 r − gn r ∥L1+∥hn+1 r − hn r ∥L1 � ≤ C(t) � t 0 sup 0≤u≤r � ∥f n u −f n−1 u ∥L1+∥gn u −gn−1 u ∥L1+∥hn r −hn−1 u ∥L1 � dr, Picard’s Lemma then yields � n sup 0≤t≤T � ∥f n+1 t − f n t ∥L1+∥gn+1 t − gn t ∥L1+∥hn+1 t − hn t ∥L1 � < ∞, for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Therefore the sequence (f n, gn, hn)n converge in L∞ loc(R+, (L1)3) towards (f S, f I, f R) which sat- isfies by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23) respectively sup 0≤t≤T ∥f S t ∥L1≤ 1, sup 0≤t≤T ∥f I t ∥L1≤ exp(T∥K∥∞) and sup 0≤t≤T ∥f R t ∥L1≤ Tαexp(T∥K∥∞∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Moreover it is easy to see that (f S, f I, f R) satisfies in the weak sense the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since (µS t , µI t, µR t ) satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8), from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9, we deduce the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any t ≥ 0, the measure µS t , µI t and µR t are aboslutely continuous with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Their density (f S, f I, f R) ∈ L∞ loc(R+, (L1(Rd)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From the system of equations in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5 above, one can also prove that the measure µS t , µI t and µR t are aboslutely continuous with respect to the Lebesgue measure, using this time assumption (H1) and the Feyman-kac formula and the fact that the law of the markovian process having Q∗ S or Q∗ I or Q∗ R as infinitesimal generator is absolutely continuous with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 Central Limit Theorem In this section, we will study the convergence of the sequence of fluctuations processes (UN = √ N(µS,N − µS), V N = √ N(µI,N − µI), W N = √ N(µR,N − µR)), as N → ∞, under the assuption (H2) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Note that the trajectories of these processes belong to (D(R+, E(Rd)))3, where E(Rd) is the space of signed measures on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' However, since the limit processes may be less regular than their approximations we will first: 4 CENTRAL LIMIT THEOREM 17 Formulated the equations verified by (UN, V N, W N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Fix the space in which the convergence results will be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Then we will study the convergence of the above sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Letting D = ⌈d/2⌉ (where ⌈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='⌉ is the upper integer part), the following is supposed to hold throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Assumption (H2): For any A ∈ {S, I, R}, for any ℓ, u ∈ {1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='d}, both functions θℓ,u(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') and mℓ(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') belong to C3+D b (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' K ∈ C2+D c (Rd × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 System of evolution equations of the Processes (U N, V N, W N) Let ϕ ∈ C2 b (Rd), we have (µS,N t , ϕ) = (µS,N 0 , ϕ) + � t 0 (µS,N r , QSϕ)dr − � t 0 � µS,N r , ϕ(µI,N r , K) � dr + MN,ϕ t , (µS t , ϕ) = (µS 0 , ϕ) + � t 0 (µS r , QSϕ)dr − � t 0 � µS r , ϕ(µI r, K) � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus (UN t , ϕ) = (UN 0 , ϕ) + � t 0 (UN r , QSϕ)dr − � t 0 (UN r , ϕ(µI,N r , K))dr − � t 0 (µS r , ϕ(V N r , K))dr + √ NMN,ϕ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence letting � MN,ϕ t = √ NMN,ϕ t , we have (UN t , ϕ) = (UN 0 , ϕ) + � t 0 (UN r , QSϕ)dr − � t 0 (UN r , GI,N r ϕ)dr − � t 0 (V N r , GS r ϕ)dr + � MN,ϕ t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) and also (V N t , ϕ) = (V N 0 , ϕ) + � t 0 (V N r , QIϕ)dr + � t 0 (UN r , GI,N r ϕ)dr + � t 0 (V N r , GS r ϕ)dr − α � t 0 (V N r , ϕ)dr + �LN,ϕ t , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) and (W N t , ϕ) = � t 0 (W N r , QRϕ)dr + α � t 0 (V N r , ϕ)dr + �Y N,ϕ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3) Where ∀x, y ∈ Rd, GI,N r ϕ(x) = ϕ(x)(µI,N r , K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=')) = ϕ(x) � Rd K(x, y)µI,N r (dy), GS r ϕ(y) = (µS r , ϕK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)) = � Rd ϕ(x)K(x, y)µS r (dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 The Space of convergence of the sequences (U N, V N, W N) Throughout this Subsection σ is an arbitrary positive real number and the following is assumed to hold: Assumption (H3): E(|X1 0|2σ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The next lemma follows easilly from the definition of Xi t (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1)), the fact that the functions m and θ are bounded and the inequality of Burkholder-Davis-Gundy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the assumption (H3), for any T > 0, there exists C(T) > 0 such that, sup 1≤i≤N E( sup 0≤t≤T |Xi t|2σ) < C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the assumption (H3), for any T > 0, there exists C(T) > 0, such that sup N≥1 E( sup 0≤t≤T (µS,N t , |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ)) < C(T), and sup 0≤t≤T (µS t , |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ) < C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For every fixed y ∈ Rd, ℓ ∈ {1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='.d}, the mapping δy, Pℓ y : Hs,σ → R defined by δyϕ = ϕ(y) and Pℓ yϕ = ∂ ∂yℓϕ(y) are continuous for s > d/2 and s > 1 + d/2 respectively and ∥δy∥−s,σ ≤ C(1 + |y|σ) if s>d/2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4) ∥Pℓ y∥−s,σ ≤ C(1 + |y|σ) if s>d/2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Continuity follows easily from Sobolev injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On the other hand for every function ϕ ∈ Hs,σ(Rd) one deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2) (see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) that: |ϕ(y)|≤ (1 + |y|σ)∥ϕ∥C0,σ≤ C(1 + |y|σ)∥ϕ∥s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5) is proved in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' If (ϕp)p≥1 is a complete orthonormal basis in Hs,σ(Rd), we have ∥δy∥2 −s,σ= � p≥1 (ϕp(y))2 ≤ C(1 + |y|2σ), if s > d/2 ∥Pℓ y∥2 −s,σ= � p≥1 ( ∂ ∂yℓϕp(y))2 ≤ C(1 + |y|2σ), if s > d/2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the assumption (H3), every limit point M1 of the seguence (� MN)N≥1 satisfies, ∀T ≥ 0, sup 0≤t≤T E(∥M1 t∥2 −s,σ) < ∞ if s > 1 + d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have � MN,ϕ t = − 1 √ N N � i=1 � t 0 � ∞ 0 1{Ei r−=S}ϕ(Xi r)1{u≤ 1 N �N j=1 K(Xir,Xj r)1{Ej r=I}}M i(dr, du) + 1 √ N N � i=1 � t 0 1{Eir=S} ▽ ϕ(Xi r)θ(S, Xi r)dBi r, < � MN,ϕ >t = � t 0 � µS,N r , ϕ2(µI,N r , K) � dr + � t 0 � µS,N r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 1≤e≤d 1≤u≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, 4 CENTRAL LIMIT THEOREM 19 and it follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5, that < � MN,ϕ >t P−→ � t 0 � µS r , ϕ2(µI r, K) � dr + � t 0 � µS r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthemore � t 0 � µS r , ϕ2(µI r, K) � dr + � t 0 � µS r , � 1≤ℓ≤d 1≤u≤d � ∂ϕ ∂xℓ �2θ2 l,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, being the quadratic variation of a Gaussian martingale (we refer to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17 below for the Gaussian property ) of the form (M1, ϕ), our aim is to find the smallest value of s for which E(∥M1 t∥2 −s,σ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' With again (ϕp)p≥1 an orthonormal basis of Hs,σ(Rd), we have E(∥M1 t∥2 −s,σ) = E(� p≥1 |(M1 t, ϕp)|2) =� p≥1 E(< (M1, ϕp) >t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4 and Asumption (H3), we have � p≥1 < M1, ϕp >t= � p≥1 � � t 0 � µS r , ϕ2 p(µI r, K) � dr + � t 0 � µS r , � 1≤ℓ≤d �∂ϕp ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕp ∂xℓ ∂ϕp ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' = � t 0 � Rd �� Rd K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)µI r(dy) � � p≥1 ϕ2 p(x)µS r (dx)dr + � t 0 � Rd � � 1≤ℓ≤d � 1≤u≤d θ2 ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='u(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � p≥1 �∂ϕp ∂xℓ �2+2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d θℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='e(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x)θu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='e(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � p≥1 ∂ϕp ∂xℓ (x)∂ϕp ∂xu (x) � µS r (dx)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ≤ ∥K∥∞ � t 0 � Rd � p≥1 ϕ2 p(x)µS r (dx)dr + � t 0 � Rd � 1≤ℓ≤d 1≤u≤d ∥θ2 ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='u∥∞ � p≥1 �∂ϕp ∂xℓ �2(x)µS r (dx)dr + 4 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∥θℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='e∥∞∥θu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='e∥∞ � t 0 � Rd � p≥1 ��∂ϕp ∂xℓ (x) �2 + �∂ϕp ∂xu (x) �2� µS r (dx)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ≤ C � t 0 � Rd(1 + |x|2σ)µS r (dx)dr + C(d) � t 0 � Rd(1 + |x|2σ)µS r (dx)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ≤ C(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By Doob’s inequality and by calculations similar to those done above we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the asumption (H4), for any T > 0, s > 1 + D, ∃ C(T) > 0, such that: sup N≥1 E( sup 0≤t≤T ∥� MN t ∥2 −s,σ) ≤ C(T), sup N≥1 E( sup 0≤t≤T ∥�LN t ∥2 −s,σ) ≤ C(T), sup N≥1 E( sup 0≤t≤T ∥�Y N t ∥2 −s,σ) ≤ C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 20 In the rest of this section we arbitrarily choose σ > d/2 and 1 + D < s < 2 + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthermore, in all the sequel, the assumption (H3) is supposed to hold for that value of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus we will prove that the sequences (UN, V N, W N)N≥1 converges in [D(R+, H−s,σ)]3, where we have equipped D(R+, H−s,σ) with the Skorokhod topology (we refer to [6] for the explicit definition of this topology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Note that the assumption (H3) and the fact that σ > d/2, yield: sup x ∥K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥2+D,σ< ∞ and sup y ∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ< ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 Convergence of (U N, V N, W N)N≥1 We first derive an estimate of the norm of the fluctuation processes UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' V N and W N, which is not uniform in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any N ≥ 1, T > 0, there exists a constant C(T) > 0, such that E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ≤ C(T)N, E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ≤ C(T)N and E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ≤ C(T)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us prove the result for UN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We first recall that C1,σ ֒→ C0,2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Moreover since s > 1 + D, Hs,σ ֒→ C1,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Now we have |(UN t , ϕ)| = √ N ��� 1 N N � i=1 1{Ei t=S}ϕ(Xi t) − (µS t , ϕ) ���, ≤ √ N � 1 N N � i=1 1{Ei t=S}(1 + |Xi t|2σ) |ϕ(Xi t)| 1 + |Xi t|2σ + � µS t , (1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ) |ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )| 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ �� , ≤ √ N∥ϕ∥C0,2σ �� µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ� + � µS t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ�� , ≤ √ N∥ϕ∥s,σ �� µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ� + � µS t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' This inequality combined with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 and the fact that ∥UN t ∥−s,σ= sup ϕ̸=0,ϕ∈Hs,σ |(UN t ,ϕ)| ∥ϕ∥s,σ yields E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ≤ 4C(T)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By the same arguments we obtain the same results for V N and W N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We now give the estimates for the fluctuations at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' It is uniform in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any s > d/2, there exists C such that sup N≥1 E(∥UN 0 ∥2 −s,σ) < C and sup N≥1 E(∥V N 0 ∥2 −s,σ) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We only prove that sup N≥1 E(∥V N 0 ∥2 −s,σ) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The other estimate follows by similar argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since 1A(Xj)ξjδXj are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='d with law µI 0, from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4 and from assumption (H3), 4 CENTRAL LIMIT THEOREM 21 if s>d/2, we have E(∥V N 0 ∥2 −s,σ) = E � � p≥1 (V N 0 , ϕp)2� , = N � p≥1 E �� (µI,N 0 , ϕp) − (µI 0, ϕp) �2� , = 1 N � i,n1,n2 E \uf8eb \uf8ed � N � j=1 [1A(Xj)ξjϕp(Xj) − (µI 0, ϕp)] �2\uf8f6 \uf8f8 , = 1 N � p≥1 N � j=1 E �� 1A(Xj)ξjϕp(Xj) − (µI 0, ϕp) �2� ≤ 1 N � p≥1 N � j=1 E � [1A(Xj)ξjϕp(Xj)]2� , ≤ p � A � p≥1 ϕ2 p(x)dPX1 0(x), ≤ pC � A (1 + |x|2σ)dPX1 0(x) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9 and following the Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 in [7], we prove easily that for any s > d/2, the sequence (UN 0 , V N 0 )N converges in law in H−s,σ(Rd) towards (U0, V0), where for any ϕ, ψ ∈ Hs,σ, the expression of the Gaussian vector ((U0, ϕ), (V0, ψ)) is given by (U0, ϕ) = W1[ϕ√g{(1 − p)1A + 1Ac} − (1 − p)W1(√g) � A ϕ(x)g(x)dx − W1(√g) � Ac ϕ(x)g(x)dx + W2(1Aϕ � (p − p2)g), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6) (V0, ψ) = pW1(1Aψ√g) − pW1(√g) � A ψ(x)g(x)dx − W2(1Aψ � (p − p2)g), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7) (Z0, φ) = W1(φ√g) − W1(√g) �� Rd φ(x)g(x)dx � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='8) where g is the density of the law of X1 0 and W1 and W2 are mutually independent two dimen- sional white noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The proof of the next Lemma can be found in [7] (see the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15 in [7], using this time Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For each N ≥ 1, the processes UN, V N and W N belong to D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us give the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under (H2) and (H3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' the sequence of processes (UN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' V N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' W N)N≥1 con- verges in law in (D(R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ))3 towards the process (U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' W) whose trajectories belong to 4 CENTRAL LIMIT THEOREM 22 (C(R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' H−s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ))3 and which satisfies for any t ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ut = U0 + � t 0 Q∗ SUrdr − � t 0 (GI r)∗Urdr − � t 0 (GS r )∗Vrdr + M1 t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Vt = V0 + � t 0 Q∗ IVrdr + � t 0 (GI r)∗Urdr + � t 0 ((GS r )∗ − αId)Vrdr + M2 t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Wt = � t 0 Q∗ RWrdr + α � t 0 Vrdr + M3 t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' where for any r > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' GI r and GS r are defined as in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 (replacing µS,N r and µI,N r by µS r and µI r respectively ), and ∀ϕ, ψ, φ ∈ Hs,σ, (M1, ϕ), (M2, ψ), (M3, φ)) is a centered Gaussian martingale satisfying: < (M1, ϕ) >t = � t 0 � µS r , ϕ2(µI r, K) � dr + � t 0 � µS r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, < (M2, ψ) >t = � t 0 � µS r , ψ2(µI r, K) � dr + � t 0 � µI r, � 1≤ℓ≤d � ∂ψ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ψ ∂xℓ ∂ψ ∂xu θℓ,e(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, < (M3, φ) >t = α � t 0 (µR r , φ2)dr + � t 0 � µI r, � 1≤ℓ≤d � ∂φ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂φ ∂xℓ ∂φ ∂xu θℓ,e(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr, < (M1, ϕ), (M2, ψ) >t = − � t 0 � µS r , ϕψ(µI r, K) � dr, < (M2, ψ), (M3, φ) >t = α � t 0 (µI r, ψφ)dr, and < (M1, ϕ), (M3, φ) >t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Before we prove this Theorem we first state a condition of Aldous type for the tightness of a sequence of H−s,σ-valued c`adl`ag processes, exploiting the fact that H−s,σ is a Hilbert space (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 of [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let (ϑn)n be a sequence of H−s,σ-valued c`adl`ag processes, their laws ( �P n) form a tight sequence in D(R+, H−s,σ) if: (T1) For each t in a dense subset T of R+, the sequence (ϑn t )n is tight in H−s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (T2) For each T > 0, ∀ε1, ε2 > 0, there exist δ > 0, n0 ≥ 1 such that for any collection of stopping times τ n ≤ T, sup n≥n0 ̺≤δ P(∥ϑn (τ n+̺) − ϑn τ n∥H> ε1) ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 23 The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12 is the content of subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 below, however let us first prove a few preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 Preliminary results Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequences (� MN)N≥1, (�LN)N≥1 and (�Y N)N≥1 are tight in D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − Tightness of (� MN)N≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us prove that (� MN)N≥1 satisfies the conditions (T1) and (T2) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − To show (T1) it is enough to prove that: ∀t ≥ 0, ∀ε > 0 there exists a compact subset K of H−s,σ such that P(� MN t /∈ K) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' This follows readily from the fact that for each 1 + D < s′ = s+1+D 2 < s, σ′ > σ > d/2, there exists C(T) such that E( sup 0≤t≤T ∥� MN t ∥2 −s′,σ′) ≤ C(T) (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Indeed, since for any 1 + D < s′ = s+1+D 2 < s, the embedding H−s′,σ′(Rd) ֒→ H−s,σ(Rd) is compact (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2, in the Appendix below), B−s′,σ′(R) = {µ ∈ H−s′,σ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∥µ∥H−s′,σ′≤ R}, which is a closed and bounded subset of H−s′,σ′ is a compact subset of H−s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus P(� MN t /∈ B−s′,σ′(R)) = P(∥� MN t ∥−s′,σ′> R) ≤ 1 R2E(∥� MN t ∥2 −s′,σ′) ≤ C(T) R2 , for any N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By choosing R arbitrarily large, we make the right hand side as small as we wish, which yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − Proof of (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Note first that < � MN,ϕ >t= � t 0 ΓN r (ϕ)dr, where ΓN r (ϕ) = � µS,N r , ϕ2(µI,N r , K) � + � µS,N r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' According to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 in [27] it is enough to prove that ∀T > 0 ∀ε1, ε2 > 0 ∃δ > 0, N0 ≥ 1 such as for any stopping times τ N ≤ T, sup N≥N0 sup ̺≤δ P(|< � MN >(τ N+̺) − < � MN >τ N |> ε1) < ε2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9) Where < � M > is the increasing, continuous processes such that, ∥� Mt∥2 H−s,σ< � M >t is a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let T > 0, ε1, ε2 > 0, ℓ > 1, we find δ > 0 such that τ N + δ ≤ ℓT and such that 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 24 We have |< � MN >(τ N+̺) − < � MN >τ N | = | � p≥1 {< � MN,ϕp >(τ N+̺) − < � MN,ϕp >τ N}| = ��� � p≥1 � (τ N+̺) τ N ΓN r (ϕp)dr ��� = ��� � p≥1 � ̺ 0 ΓN (τ N +r)(ϕp)dr ��� ≤ C(∥K∥∞, d) � ̺ 0 � Rd{1 + |x|2σ}µS,N τ N+r(dx)dr ≤ ̺C(∥K∥∞, d)sup N≥1 sup 0≤t≤ℓT � µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) Hence it follows from the Markov inequality and from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) that P(|< � MN >(τ N+̺) − < � MN >τ N |> ε1) ≤ E(|< � MN >(τ N+̺) − < � MN >τ N |) ε1 ≤ Cδ ε1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (T2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We conclude from (T1) and (T2) that (� MN)N≥1 is tight in D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The same arguments yield the tightness of (�LN)N≥1 and (�Y N)N≥1 in D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (We refer to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17 in [7], for the proof) Every limit point (M1, M2, M3) of the sequence (� MN, �LN, �Y N)N≥1 is such that for any ϕ, ψ, φ ∈ Hs,σ, ((M1, ϕ), (M2, ψ), (M3, φ) is a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The main argument for obtaining the following result is that the jumps of UN, V N and W N respectively are of the order of 1 √ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (We refer to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16 in [7], for the detail proof) Every limit point (M1, M2, M3) of the sequence (� MN, �LN, �Y N)N≥1 belongs to of (C(R+, H−s,σ))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequence (� MN, �LN, �Y N)N≥1 converges in law in (D(R+, H−s,σ))3 to- wards the process (M1, M2, M3) ∈ (C(R+, H−s,σ))3 where ∀ϕ, ψ, φ ∈ Hs,σ, 4 CENTRAL LIMIT THEOREM 25 ((M1, ϕ), (M2, ψ), (M2, φ)) is a centered Gaussian martingale having the same law as (M1 t, ϕ) = − � t 0 � Rd � fS(r, x) � Rd fI(r, y)K(x, y)dyϕ(x)W1(dr, dx) + d � ℓ=1 � t 0 � Rd � fS(r, x) � � 1≤u≤d ∂ϕ ∂xu (x)θu,l(S, x) � Wℓ+1(dr, dx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11) (M2 t, ψ) = � t 0 � Rd � fS(r, x) � Rd fI(r, y)K(x, y)dyψ(x)W1(dr, dx) + d � ℓ=1 � t 0 � Rd � fI(r, x) � � 1≤u≤d ∂ψ ∂xu (x)θu,l(I, x) � Wℓ+1+d(dr, dx) − � t 0 � Rd ψ(x) � αfI(r, x)W3d+2(dr, dx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12) (M3 t, φ) = + d � ℓ=1 � t 0 � Rd � fR(r, x) � � 1≤u≤d ∂φ ∂xu (x)θu,l(R, x) � Wℓ+1+2d(dr, dx) + � t 0 � Rd φ(x) � αfI(r, x)W3d+2(dr, dx), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13) where W1, W2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='.,W3d+1, W3d+2 are independent spatio-temporal standard white noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14, (� MN, �LN, �Y N)N≥1 is tight in (D(R+, H−s,σ))3, hence according to Prokhorov’s Theorem there exists a subsequence still denoted (� MN, �LN, �Y N)N≥1 which con- verges in law in (D(R+, H−s,σ))3 towards (M1, M2, M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16, ∀ϕ, ψ, φ ∈ Hs,σ, ((M1, ϕ), (M2, ψ), (M3, φ) is a continuous martingale, thus we end the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17 by showing that the centered, continuous martingale ((M1, ϕ), (M2, ψ), (M3, φ)) is Gaussian and satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψ ∈ C2 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' we have � Mt N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ϕ = − 1 √ N N � i=1 � t 0 � ∞ 0 1{Ei r−=S}ϕ(Xi r)1{u≤ 1 N �N j=1 K(Xir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='Xj r)1{Ej r=I}}M i(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' du) + 1 √ N N � i=1 � t 0 1{Eir=S} ▽ ϕ(Xi r)θ(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r)dBi r =−M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ϕ t + M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ϕ t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' �Lt N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ψ = 1 √ N N � i=1 � t 0 � ∞ 0 1{Ei r−=S}ψ(Xi r)1{u≤ 1 N �N j=1 K(Xir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='Xj r)1{Ej r=I}}M i(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' du)) + + 1 √ N N � i=1 � t 0 1{Eir=I} ▽ ψ(Xi r)θ(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r)dBi r − 1 √ N N � i=1 � t 0 � α 0 1{Ei r−=I}ψ(Xi r)Q i(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' du) =M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ψ t + M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ψ t − M4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ψ t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' �Yt N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='φ = 1 √ N N � i=1 � t 0 1{Eir=R} ▽ φ(Xi r)θ(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Xi r)dBi r + 1 √ N N � i=1 � t 0 � α 0 1{Ei r−=I}φ(Xi r)Q i(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' du) = M5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='φ t + M4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='φ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Consider for ϕ, ψ, φ ∈ C2 c , the following sequence of martingales � Mt N,ϕ + �Lt N,ψ + �Yt N,φ = −M1,N,ϕ t + M2,N,ϕ t + M1,N,ψ t + M3,N,ψ t − M4,N,ψ t + M4,N,φ t + M5,N,φ t 4 CENTRAL LIMIT THEOREM 26 The martingales M1,N,ϕ t , M2,N,ϕ t , M3,N,ψ t , M4,N,ψ t , M5,N,φ t being two by two orthogonal, < � MN,ϕ+�LN,ψ >t=< M1,N,ϕ >t + < M2,N,ϕ >t + < M1,N,ψ >t + < M3,N,ψ >t + < M4,N,ψ >t + < M4,N,φ >t + < M5,N,φ >t −2 < M1,N,ϕ, M1,N,ψ >t −2 < M4,N,ψ, M4,N,φ >t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' In addition we have the following convergences in probability < M1,N,ϕ >t P−→ � t 0 � µS r , ϕ2(µI r, K) � dr, < M2,N,ϕ >t P−→ � t 0 � µS r , � 1≤ℓ≤d � ∂ϕ ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕ ∂xℓ ∂ϕ ∂xu θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On the other hand: − � MN,ϕ + �LN,ψ + �Y N,φ L−→ (M1, ϕ) + (M2, ψ) + (M3, φ) along a subsequence since (� MN,ϕ, �LN,ψ, �LN,φ) L−→ ((M1, ϕ), (M2, ψ), (M3, φ)) − (M1, ϕ)+(M2, ψ)+(M3, φ) is a continuous martingale since (M1, ϕ), (M2, ψ), and (M3, φ) have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus (M1, ϕ) + (M2, ψ) + (M3, φ) is a time changed Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The quadratic variation < (M1, ϕ) + (M2, ψ) + (M3, φ) >t = � t 0 � � µS r , ϕ2(µI r, K) � + � µS r , ψ2(µI r, K) � − 2 � µS r , ϕψ(µI r, K) � � dr + � A∈{S,I,R} � t 0 � µA r , � 1≤ℓ≤d 1≤u≤d �∂ϕA ∂xℓ �2θ2 ℓ,u(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') + 2 � 1≤ℓ≤d−1 ℓ+1≤u≤d 1≤e≤d ∂ϕA ∂xℓ ∂ϕA ∂xu θℓ,e(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr + α � t 0 � (µI r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψ2) + (µI r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' φ2) − 2(µI r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψφ) � dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (where we have let ϕS = ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕI = ψ and ϕR = φ) of (M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + (M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψ) + (M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' φ) being deterministic then we conclude that (M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + (M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ψ) + (M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' φ) is a Gaussian martingale having the same law as Nt = � t 0 � Rd � fS(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � Rd fI(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)K(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' y)dy(ψ(x) − ϕ(x))W1(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dx) + d � 1=1 � t 0 � Rd � fS(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � � 1≤u≤d ∂ϕ ∂xu (x)θu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='l(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � Wl+1(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dx) + d � 1=1 � t 0 � Rd � fI(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � � 1≤u≤d ∂ψ ∂xu (x)θu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='l(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � Wl+d+1(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dx) + d � 1=1 � t 0 � Rd � fR(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � � 1≤u≤d ∂φ ∂xu (x)θu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='l(R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x) � W2d+1(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dx) + � t 0 � Rd � αfI(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' x)(φ(x) − ψ(x))W3d+2(dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' dx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' where W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' W2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='. W3d+1, W3d+2 are independent spatio-temporal white noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So taking (ψ ≡ 0, φ ≡ 0), (ϕ ≡ 0, φ ≡ 0) and (ϕ ≡ 0, ψ ≡ 0) respectively, in the above equation we see that (M1, ϕ), (M2, ψ) and (M3, φ) satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='11), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 27 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' There exists a constant C > 0, such that for any ϕ ∈ Hs,σ(Rd), ∥GI,N r ϕ∥s,σ ≤ C∥ϕ∥s,σsup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14) ∥GS r ϕ∥s,σ ≤ C∥ϕ∥s,σsup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15) ∥GI rϕ∥s,σ ≤ C∥ϕ∥s,σsup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We first recall that ∀x, y ∈ Rd, GI,N r ϕ(x) = ϕ(x)(µI,N r , K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=')) = ϕ(x) � Rd K(x, y)µI,N r (dy), GS r ϕ(y) = (µS r , ϕK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)) = � Rd ϕ(x)K(x, y)µS r (dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since Hs,σ is a Banach algebra (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4 in the Appendix below), we have ∥GI,N r ϕ∥s,σ≤ C∥ϕ∥s,σ∥(µI,N r , K)∥s,σ≤ C∥ϕ∥s,σ∥(µI,N r , K)∥2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17) Furthermore ∥(µI,N r , K)∥2 2+D,σ = � |γ|≤2+D � Rd |Dγ(µI,N r , K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ))|2 1 + |x|2σ dx, = � |γ|≤2+D � Rd ��� � Rd DγK(x, y)µI,N r (dy) ��� 2 1 + |x|2σ dx, ≤ � Rd � |γ|≤2+D � Rd |DγK(x, y)|2 1 + |x|2σ dxµI,N r (dy), ≤ sup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2 2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18) Thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Once again since H2+D,σ ֒→ Hs,σ, we have ∥GS r ϕ∥s,σ= ∥(µS r , ϕK)∥s,σ≤ C∥(µS r , ϕK)∥2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19) Furthermore from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2, we have ∥(µS r , ϕK)∥2 D,σ = � |γ|≤2+D � Rd |Dγ(µS r , ϕK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y))|2 1 + |y|2σ dx, = � |γ|≤2+D � Rd ��� � Rd ϕ(x)DγK(x, y)µS r (dx) ��� 2 1 + |y|2σ dx, ≤ � Rd ϕ2(x)µS r (dx) � Rd � |γ|≤2+D � Rd |DγK(x, y)|2 1 + |y|2σ dyµS r (dx), ≤ ∥ϕ∥2 C0,σ sup x∈Rd∥K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥2 2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20) So (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='15) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20) and the embedding Hs,σ ֒→ C0,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16) is similar to that of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 28 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' There exists a constant C > 0, such that for any U ∈ H−s,σ, we have ∥(GI,N r )∗U∥−s,σ ≤ C sup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ∥U∥−s,σ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21) ∥(GS r )∗U∥−s,σ ≤ C sup x∈Rd∥K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥2+D,σ∥U∥−s,σ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='22) ∥(GI r)∗U∥−s,σ ≤ C sup y∈Rd∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ∥U∥−s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2 The evolution semi group Let us define the evolution semigroup as a semi group of bounded linear operators in a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us assume, for the moment in a general context, that for any A ∈ {S, I, R}, the coeficients m(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') and θ(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') are in Cj+1 b , where j is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let (Bt)t≥0 be a standard Brownian motion on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any A ∈ {S, I, R}, one defined (see for example Kunita [20] ) the flow of diffeomorphisms (of class Cj) as the unique solution of the Itô stochastic differential equation started from x ∈ Rd at time u : XA,x u,t = x + � t u m(A, XA,x u,t )dr + � t u θ(A, XA,x u,t )dBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24) Moereover for any measurable and bounded function ϕ, A ∈ {S, I, R}, we define ΥA(t − u)ϕ(x) = E(ϕ(XA,x u,t )) and in the folowing Υ∗ A(t) denotes the adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' When u = 0, XA,x u,t is denotes XA,x t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We note that under the Assumptions (H2), for any 0 ≤ u < t the map x ∈ Rd �−→ XA,x u,t is of class C2+D, and the following results hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − (Thank to Corrolary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='7 in [20]) For any 0 ≤ |γ|≤ 2 + D, for any p ≥ 1, there exits a constant C independent of t, such that sup x E(|DγXA,x t |2p) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25) − (See Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 in [20]) For any real p, there exists a positive constant Cp, such that E[(1 + |XA,x t |2)p] ≤ Cp(1 + |x|2)p, ∀x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26) Now we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the Asumption (H2), for any A ∈ {S, I, R}, t > 0, m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='.2 + D}, ϕ ∈ W m,σ 0 (Rd), there exists a positive constant C, such that: ∥ΥA(t)ϕ∥m,σ≤ CeCt∥ϕ∥m,σ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have ∥ΥA(t)ϕ∥2 m,σ= � |γ|≤m � Rd |DγΥA(t)ϕ(x)|2 1 + |x|2σ dx, furthermore for |γ|= 0, and by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 in the Appendix below, we have � Rd |ΥA(t)ϕ|2 1 + |x|2σ dx = � Rd |E(ϕ(XA,x t ))|2 1 + |x|2σ dx ≤ � Rd 1 1 + |x|2σ E[(1 + |XA,x t |σ)2]E � |ϕ(XA,x t )|2 (1 + |XA,x t |σ)2 � dx, ≤ C � Rd (1 + |x|2)σ 1 + |x|2σ � Rd ΥA(t)(x, y) |ϕ(y)|2 1 + |y|2σ dydx, ≤ C(σ)∥ϕ∥L2,σ sup y∈Rd � � Rd ΥS(t)(x, y)dx � , ≤ C(σ)∥ϕ∥m,σeCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 29 For γ = (1, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='0) and by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3, we have � Rd |DγΥA(t)ϕ|2 1 + |x|2σ dx = � Rd |E(Dγϕ(XA,x t ))|2 1 + |x|2σ dx, ≤ � Rd 1 1 + |x|2σ E[{DγXA,x t (1 + |XA,x t |σ)}2]E � |∇ϕ(XA,x t )|2 (1 + |XA,x t |σ)2 � dx, ≤ C � Rd (1 + |x|2)σ 1 + |x|2σ E[(DγXA,x t )4]1/2E[(1 + |XA,x t |σ)4]1/2 × � Rd ΥA(t)(x, y)|∇ϕ(y)|2 1 + |y|2σ dydx, ≤ C(σ, d)∥ϕ∥1,σ sup y∈Rd � � Rd ΥA(t)(x, y)dx � , ≤ C(σ, d)∥ϕ∥m,σeCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Similar argument allow us to have � Rd |DγΥS(t)ϕ|2 1 + |x|2σ dx ≤ C(σ, d)∥ϕ∥m,σeCt, for all values of |γ|≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any positive noninteger 1 + D < s < 2 + D, ϕ ∈ Hs,σ, there exists a positive contant C, such that ∥ΥS(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We prove this result by using the definition by interpolation of the space Hs,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any noninterger s > 0, there exists ρ ∈]0, 1[ such that s = (1 − ρ)(1 + D) + ρ(2 + D) and (W 1+D,σ 0 , W 2+D,σ 0 )ρ,2 = Hs,σ, for the definition of the space (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )ρ,q we refer to [22] or [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 in [23], we have an equivalent norm on Hs,σ, which is given by: ∥ϕ∥s,σ= �� ∞ 0 {t−ρK(t, 1 + D, 2 + D)}2dt t �1/2 , where K(t, 1 + D, 2 + D) = inf ϕ=ϕ1+ϕ2{∥ϕ1∥1+D,σ+t∥ϕ2∥2+D,σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So it is easy to see that the result follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='20 and the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us prove the following results which will be useful to prove the tightness of the sequence (UN, V N, W N)N in D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequence of processes (UN, V N, W N) satisfies ∀ 0 ≤ u < t, UN t = Υ∗ S(t − u)UN u − � t u Υ∗ S(t − r)(GI,N r )∗UN r dr − � t u Υ∗ S(t − r)(GS r )∗V N r dr + � t u Υ∗ S(t − r)d� MN r , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='27) V N t = Υ∗ I(t − u)V N u + � t u Υ∗ I(t − r)(GI,N r )∗UN r dr + � t u Υ∗ I(t − r)[(GI,N r )∗ − α]V N r dr + � t u Υ∗ I(t − r)d�LN r , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='28) W N t = Υ∗ R(t − u)W N u + α � t u Υ∗ R(t − r)V N r dr + � t u Υ∗ R(t − r)d�Y N r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='29) 4 CENTRAL LIMIT THEOREM 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us consider a fuction φ belonging to C1,2 c (R+ × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' By Itô’s formula applied to φ(t, Xi t) and using a similar computation as in subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1, we obtain for 0 ≤ u < t, (UN t , φt) = (UN u , φu) + � t u (UN r , QSφr)dr + � t u (UN r , ∂φr ∂r )dr − � t u � UN r , φr(µI,N r , K) � dr − � t u � V N r , (µS r , φrK) � dr + � t u (φr, d� MN r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let ϕ ∈ C2 b and 0 ≤ u < t, consider for r ∈ [u, t] the mapping ψr(x) = ΥS(t − r)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have ψ·(·) ∈ C1,2 c ([u, t] × Rd), indeed, For any r ∈ [u, t], ψr(·) ∈ C2 c (Rd) ∀x ∈ Rd, the map r ∈ [u, t] �→ ψ ′ r(x) = −QS(ΥS(t − r)ϕ(x)) is continuous since ΥS(t) is a strongly continuous semi-group and −QS(ΥS(t − r)ϕ(x)) = ΥS(t − r)(−QSϕ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus replacing φ by ψ in the above equation, we obtain (UN t , ϕ) = (UN u , ΥS(t − u)ϕ) − � t u � UN r , ΥS(t − r)ϕ(µI,N r , K) � dr − � t u � V N r , (µS r , ΥS(t − r)ϕK) � dr + � t u (ΥS(t − r)ϕ, d� MN r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' This prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='28) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='29) by similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' There exists C > 0 such that for any T > 0, ̺ > 0 and any stopping times τ such that τ + ̺ < T, one has E ���� � τ+̺ τ Υ∗ S(τ + ̺ − r)d� MN r ��� 2 −s,σ � ≤ C̺, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='30) E ���� � τ+̺ τ Υ∗ I(τ + ̺ − r)d�LN r ��� 2 −s,σ � ≤ C̺, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='31) E ���� � τ+̺ τ Υ∗ R(τ + ̺ − r)d�Y N r ��� 2 −s,σ � ≤ C̺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us recall that � τ+̺ τ (ΥS(τ + ̺ − r)ϕ, d� MN r ) = 1 √ N N � i=1 � τ+̺ τ 1{Eir=S} ▽ ΥS(τ + ̺ − r)ϕ(Xi r)θ(S, Xi r)dBi r − � 1 N N � i=1 � τ+̺ τ � ∞ 0 1{Ei r−=S}ΥS(τ + ̺ − r)ϕ(Xi r)1 {u≤ 1 N N � j=1 K(Xir,Xj r)1{Ej r=I}}M i(dr, du).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Now from Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26) one has for all ℓ ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', d}, 0 ≤ r ≤ ̺, � p≥1 � ΥS(̺ − r)ϕp(x) �2 = � p≥1 � E(ϕp(XS,x ̺−r)) �2 ≤ E � � p≥1 |ϕp(XS,x ̺−r)|2� , ≤ CE(1 + |XS,x ̺−r|2σ), ≤ C(σ)(1 + |x|2σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='33) 4 CENTRAL LIMIT THEOREM 31 � p≥1 � ∂ ∂xℓ ΥS(̺ − r)ϕp(x) �2 = � p≥1 � ∂ ∂xℓ E(ϕp(XS,x ̺−r)) �2 ≤ E(|∂xℓXS,x ̺−r|2)E � � p≥1 |(∇ϕp)(XS,x ̺−r)|2� , ≤ C(d)E(1 + |XS,x ̺−r|2σ), ≤ C(d, σ)(1 + |x|2σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='34) Thus from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='33) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='34) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1, we have E ����� � τ+̺ τ ΥS(τ + ̺ − r)d� MN r ���� 2 H−s � = � p≥1 E ��� τ+̺ τ ΥS(τ + ̺ − r)ϕp, d� MN r �2� , = � p≥1 � E �� ̺ 0 � µS,N r+τ, (ΥS(̺ − r)ϕp)2(µI,N r+τ, K) � dr � + E � � ̺ 0 � µS,N r+τ, � 1≤ℓ≤d �∂ΥS(̺ − r)ϕp ∂xℓ �2 � 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr � + E � � ̺ 0 � µS,N r+τ, � 1≤ℓ≤d−1 1≤e≤d 1≤u≤d ∂ΥS(̺ − r)ϕp ∂xℓ ∂ΥS(̺ − r)ϕp ∂xu θl,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � dr �� , ≤ E �� ̺ 0 � µS,N r+τ, � p≥1 (ΥS(̺ − r)ϕp)2(µI,N r+τ, K) � dr � + E � � ̺ 0 � µS,N r+τ, � 1≤ℓ≤d 1≤u≤d θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=') � p≥1 �∂ΥS(̺ − r)ϕp ∂xℓ �2� dr � + 1 2E � � ̺ 0 � µS,N r+τ, � 1≤ℓ≤d−1 1≤e≤d 1≤u≤d |θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )| � � p≥1 �∂ΥS(̺ − r)ϕp ∂xℓ �2 + � p≥1 �∂ΥS(̺ − r)ϕp ∂xu �2�� dr � , ≤ ̺∥K∥∞sup N≥1 sup 0≤t≤T E((µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ)) + ̺C(σ) � 1≤ℓ≤d 1≤u≤d ∥θ2 ℓ,u(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞sup N≥1 sup 0≤t≤T E((µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ)) + ̺ � 1≤ℓ≤d−1 1≤e≤d 1≤u≤d ∥θℓ,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞∥θu,e(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥∞sup N≥1 sup 0≤t≤T E((µS,N t , 1 + |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='|2σ)), ≤ ̺C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Similar arguments yield (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='31) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For all T > 0, sup N≥1 E( sup 0≤t≤T ∥UN t ∥2 −s,σ) < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='35) sup N≥1 E( sup 0≤t≤T ∥V N t ∥2 −s,σ) < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='36) sup N≥1 E( sup 0≤t≤T ∥W N t ∥2 −s,σ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Choosing u = 0 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='27), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='28) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='29) we have 4 CENTRAL LIMIT THEOREM 32 ∥UN t ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ ≤ 4∥Υ∗ S(t)UN 0 ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ+4t � t 0 � ∥Υ∗ S(t − r)(GI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N r )∗UN r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ+∥Υ∗ S(t − r)(GS r )∗V N r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ � dr + 4 ��� � t 0 Υ∗ S(t − r)d� MN r ��� 2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∥V N t ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ ≤ 5∥Υ∗ I(t)V N 0 ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ+5t � t 0 � ∥Υ∗ I(t − r)(GI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N r )∗UN r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ+∥Υ∗ I(t − r)GS r V N r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ � dr +5tα2 � t 0 ∥Υ∗ I(t − r)V N r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σdr + 5 ��� � t 0 Υ∗ I(t − r)d�LN r ��� 2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∥W N t ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ ≤ 2α2t � t 0 ∥Υ∗ R(t − r)V N r ∥2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σdr + 2 ��� � t u Υ∗ R(t − r)d�Y N r ��� 2 −s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Corollarys 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21, we have ∥UN t ∥2 −s,σ ≤ C4eCt∥UN 0 ∥2 −s,σ+4CteCtsup y ∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2 2+D,σ � t 0 {∥UN r ∥2 −s,σ+∥V N r ∥2 −s,σ}dr + 4 ��� � t 0 Υ∗ S(t − r)d� MN r ��� 2 −s,σ, ∥V N t ∥2 −s,σ ≤ 5eCt∥V N 0 ∥2 −s,σ+5CteCt(1 + α2)sup x ∥K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥2 2+D,σ � t 0 {∥UN r ∥H−s+∥V N r ∥2 −s,σ}dr + 5 ��� � t 0 Υ∗ I(t − r)d�LN r ��� 2 −s,σ, ∥W N t ∥2 −s,σ ≤ 2CeCtα2t � t 0 ∥V N r ∥2 −s,σdr + 2 ��� � t 0 Υ∗ R(t − r)d�Y N r ��� 2 −s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus from Corrolary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='6 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='21, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 and Assumption (H3), we have E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ≤ 4eCTE(∥UN 0 ∥2 −s,σ) + 4CTeCT � T 0 � E( sup 0≤r≤t ∥UN r ∥2 −s,σ) + E( sup 0≤r≤t ∥V N r ∥2 −s,σ) � dt + CT, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='38) E( sup 0≤t≤T ∥V N t ∥2 −s,σ) ≤ 5eCTE(∥V N 0 ∥2 −s,σ) + 5TC(1 + α2)eCT � T 0 � E( sup 0≤r≤t ∥UN r ∥2 −s,σ) + E( sup 0≤r≤t ∥V N r ∥2 −s,σ) � dt + CT, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='39) E( sup 0≤t≤T ∥W N t ∥2 −s,σ) ≤ 2α2TeCT � T 0 � E( sup 0≤r≤t ∥W N r ∥2 −s,σ) + E( sup 0≤r≤t ∥V N r ∥2 −s,σ) � dt + CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='40) Thus summing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='38), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='39) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='40) and applying Gronwall’s lemma we deduce the result from the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12 We begin this subsection by showing the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The sequences of processes UN, V N and W N are tight in D(R+, H−s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We establish the tightness of UN by showing that the conditions of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='13 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − Based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24, we deduce (T1) by the same argument as used in the proof of (T1) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − Proof of (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let T>0, ε1, ε2 >0, (τ N)N a family of stopping times with τ N ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have UN τ N+̺ − UN τ N = (Υ∗ S(̺) − Id)UN τ N − � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)(GI,N r )∗UN r dr − � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)(GS r )∗V N r dr + � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)d� MN r , = (Υ∗ S(̺) − Id)UN τ N − � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)JS,I,N r (UN, V N)dr + � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)d� MN r , where JS,I,N r (UN, V N) = (GI,N r )∗UN r + (GS r )∗V N r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We find δ > 0 and N0 ≥ 1 such that: sup N≥N0 sup δ≥̺ P(∥(Υ∗ S(̺) − Id)UN τ N∥−s,σ≥ ε1) ≤ ε2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='41) sup N≥N0 sup δ≥̺ P ���� � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)JS,I,N r (UN, V N)dr ��� −s,σ ≥ ε1 � ≤ ε2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='42) sup N≥N0 sup δ≥̺ P ���� � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)d� MN r ��� −s,σ ≥ ε1 � ≤ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='43) 1− Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let us introduce a complete orthonormal basis in Hs,σ, of functions (ϕp)p≥1, ϕp being of class C∞ with compact support, and Fm(m ∈ N∗) denotes the sub-space of Hs,σ generated by (ϕp)1≤p≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let (Υ∗ S(̺)−Id)UN t |Fm denotes the orthogonal projection of (Υ∗ S(̺)−Id)UN t on the dual space of Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We have P(∥(Υ∗ S(̺) − Id)UN τ N∥−s,σ≥ ε1) ≤ P(∥(Υ∗ S(̺) − Id)UN τ N |Fm ∥−s,σ≥ ε1 2 ) + P(∥(Υ∗ S(̺) − Id)UN τ N − (Υ∗ S(̺) − Id)UN τ N |Fm ∥−s,σ≥ ε1 2 ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='44) Let us control each of the term of the above right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − One has P � ∥(Υ∗ S(̺) − Id)UN τ N − (Υ∗ S(̺) − Id)UN τ N |Fm ∥−s,σ≥ ε1 2 � ≤ 4 ε2 1 E � sup 0≤t≤T � p>m (UN t , (ΥS(̺) − Id)ϕp)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='45) Furthermore the sequence � sup 1≤N sup 0≤t≤T sup 0≤̺≤δ � p>m (UN t , (ΥS(̺) − Id)ϕp)2� converge towards 0 as m −→ ∞, as the remainder of order m of a uniformly convergent series of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus there exists m0 ∈ N∗ independent of N and ̺ such that for any m ≥ m0, sup 0≤t≤T sup 0≤̺≤δ � p>m (UN t , (ΥS(̺) − Id)ϕp)2 < ε, for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 34 Moreover sup 0≤t≤T sup 0≤̺≤δ � p>m (UN t , (ΥS(̺) − Id)ϕp)2 is bounded by max(CeCδ, 1) sup 0≤t≤T ∥UN t ∥2 −s,σ, so uniformly integrable (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus we deduce that there exists m0 ∈ N∗ independent of N and ̺ such that for any m ≥ m0, E � sup 0≤t≤T � p>m (UN t , (ΥS(̺) − Id)ϕp)2� < ε, ∀ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='46) − One has P(∥(Υ∗ S(̺) − Id)UN τ N |Fm ∥−s,σ≥ ε1 2 ) ≤ 4 ε2 1 E( sup 0≤t≤T ∥(Υ∗ S(̺) − Id)UN t |Fm ∥2 −s,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='47) Furthermore according to Dynkin’s formula and the contraction of ΥS(t), one has ∥(Υ∗ S(̺) − Id)UN t |Fm ∥2 −s,σ = m � p=1 � (Υ∗ S(̺) − Id)UN t , ϕp �2 , = m � p=1 � UN t , (ΥS(̺) − Id)ϕp �2 , = m � p=1 � UN t , � ̺ 0 ΥS(r)QSϕp(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )dr �2 , ≤ ̺ sup 0≤t≤T ∥UN t ∥2 −s,σ m � p=1 � ̺ 0 ∥ΥS(r)QSϕp∥2 s,σdr, ≤ ̺2 sup 0≤t≤T ∥UN t ∥2 −s,σ m � p=1 ∥QSϕp∥2 s,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='48) Where QS in the infinitesimal generator of the operator ΥS(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence as from assumptions (H2), there exists C > 0 such that m � p=1 ∥QSϕp∥2 s,σ≤ m � p=1 ∥QSϕp∥2 2+D,σ≤ mC, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='47) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='48), we have P(∥(Υ∗ S(̺) − Id)UN τ N |Fm ∥−s,σ≥ ε1 2 ) ≤ ≤ 4mCsup N≥1 E( sup 0≤t≤T ∥UN t ∥2 −s,σ) ε2 1 ̺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='49) Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='49) combined with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='46) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='44) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let ℓ > 1, we find δ > 0 such that τ N +δ ≤ ℓT and such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='42) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since ∀ϕ ∈ Hs,σ, ∥Υ(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ then form Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24 and from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19, we have 4 CENTRAL LIMIT THEOREM 35 P \uf8eb \uf8ed ����� � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)JS,I,N r (UN, V N)dr ����� −s,σ ≥ ε1 \uf8f6 \uf8f8 ≤ ≤ 1 ε2 1 E \uf8eb \uf8ed ����� � τ N+̺ τ N Υ∗ S(τ N + ̺ − r)JS,I,N r (UN, V N)dr ����� 2 −s,σ \uf8f6 \uf8f8 , ≤ ̺ ε2 1 E �� τ N+̺ τ N ∥Υ∗ S(τ N + ̺ − r)JS,I,N r (UN, V N)∥2 −s,σdr � , ≤ ̺C ε2 1 eCℓTsup y ∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥s,σE �� τ N+̺ τ N {∥UN r ∥2 −s,σ+∥V N r ∥2 −s,σ}dr � , ≤ δ2C ε2 1 eℓT sup N≥1 E( sup 0≤t≤ℓT {∥UN t ∥2 −s,σ+ sup 0≤t≤ℓT ∥V N t ∥2 −s,σ}), ≤ δ2C ε2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='42) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let ℓ > 1, we find δ > 0 such that τ N + δ ≤ ℓT and such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='43) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23, we have P ���� � τ N+θ τ N Υ∗ S(τ N + θ − r)d� MN r ��� −s,σ ≥ ε1 � ≤ 1 ε2 1 E ���� � τ N+θ τ N Υ∗ S(τ N + θ − r)d� MN r ��� 2 −s,σ � , ≤ C ε2 1 δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='43) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' To establish the system of limiting equations of all converging subsequences of (UN, V N, W N)N≥1, we will need the next Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any t ≥ 0, ϕ ∈ Hs,σ(Rd), as N −→ ∞, � t 0 E � ∥[GI,N r − GI r]ΥS(t − r)ϕ∥2 s,σ � dt −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since Hs,σ is a Banach algebra (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4) and H2+D,σ ֒→ Hs,σ (since s < 2 + D), and ∥Υ(t)ϕ∥s,σ≤ CeCt∥ϕ∥s,σ, ���Υ(t − r)ϕ � µI,N r − µI r, K ���� s,σ ≤ C∥ϕ∥s,σ ��� � µI,N r − µI r, K ���� 2+D,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On the other hand ��� � µI,N r − µI r, K ���� 2 2+D,σ = � |η|≤2+D � Rd(1 + |x|2σ)−1��� � Rd DηK(x, y)(µI,N r − µI r)(dy) ��� 2 dx, furthermore from Asumptions (H2) the map y ∈ Rd �→ DηK(x, y) is continuous and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So we deduce from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5 that ��� � T2 DηK(x, y)(µI,N r − µI r)(dy) ��� 2 P−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' According to Lebesgue’s dominated convergence theorem, E ���� � µI,N r − µI r, K ���� 2 2+D,σ � −→ 0, 4 CENTRAL LIMIT THEOREM 36 as N −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus � t 0 E � ∥[GI,N r − GI r]Υ(t − r)ϕ∥2 2+D,σ � dt ≤ C∥ϕ∥2 s,σ � t 0 E ���� � µI,N r − µI r, K ���� 2 2+D,σ � dr −→ 0, as N −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The next Proposition establishes the evolution equations of all limit points (U, V, W) of the sequence (UN, V N, W N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' All limit points (U, V, W) of the sequence (UN, V N, ZN) satisfy Ut = Υ∗ S(t)U0 − � t 0 Υ∗ S(t − r)(GI r)∗Urdr − � t 0 Υ∗ S(t − r)(GS r )∗Vrdr + � t 0 Υ∗ S(t − r)dM1 r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='50) Vt = Υ∗ I(t)V0 + � t 0 Υ∗ I(t − r)(GI r)∗Urdr + � t 0 Υ∗ I(t − r)(GS r )∗Vrdr − α � t 0 Υ∗ I(t − r)Vrdr + � t 0 Υ∗ I(t − r)dM2 r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='51) Wt = α � t 0 Υ∗ R(t − r)Vrdr + � t 0 Υ∗ R(t − r)dM3 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='52) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We prove this Proposition by taking the weak limit in the equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='28) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Note first that from Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='25, there exists a subsequence along which the sequences (UN, V N, W N)N converges in law towards (U, V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any ϕ ∈ Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' one has (Υ∗ S(t)UN 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + � t 0 � Υ∗ S(t − r)ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' d� MN r � = (UN t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + � t 0 (Υ∗ S(t − r)(GI r)∗UN r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr + � t 0 (Υ∗ S(t − r)(GS r )∗V N r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr + � t 0 (Υ∗ S(t − r)[(GI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N r )∗ − (GI r)∗]UN r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (Υ∗ I(t)V N 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + � t 0 � Υ∗ I(t − r)ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' d�LN r � = (V N t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) − � t 0 (Υ∗ I(t − r)(GI r)∗UN r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr − � t 0 (Υ(t − r)[(GS r )∗ − α]V N r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr − � t 0 ([Υ∗ I(t − r)(GI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N r )∗ − (GI r)∗]UN r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' � t 0 � Υ∗ I(t − r)ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' d�Y N r � = (W N t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ) + α � t 0 (Υ∗ R(t − r)V N r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ϕ)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='53) Thus in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='53), it is enough to show that (U, V ) satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hence, we have (Υ∗ S(t)UN 0 , ϕ) + � t 0 � Υ∗ S(t − r)ϕ, d� MN r � = Ψ1,t,ϕ � UN, V N� + � t 0 ([Υ∗ S(t − r)(GI,N r )∗ − (GI r)∗]UN r , ϕ)dr, (Υ∗ I(t)V N 0 , ϕ) + � t 0 � Υ∗ I(t − r)ϕ, d�LN r � = Ψ2,t,ϕ � UN, V N� − � t 0 ([Υ∗ I(t − r)(GI,N r )∗ − (GI r)∗]UN r , ϕ)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4 CENTRAL LIMIT THEOREM 37 With Ψ1,t,ϕ � UN, V N� = (UN t , ϕ) + � t 0 (Υ∗ S(t − r)(GI r)∗UN r , ϕ)dr + � t 0 (Υ∗ S(t − r)(GS r )∗V N r , ϕ)dr, Ψ2,t,ϕ � UN, V N� = (V N t , ϕ) − � t 0 (Υ∗ I(t − r)(GI r)∗UN r , ϕ)dr − � t 0 (Υ∗ I(t − r)[(GS r )∗ − α]V N r , ϕ)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthermore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1− From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='26 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='24, � t 0 � UN r , [GI,N r − GI r]Υ(t − r)ϕ � dr −→ 0 in L1(P), locally unformly in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Indeed, E ���� sup 0≤t≤T � t 0 � UN r , [GI,N r − GI r]Υ(t − r)ϕ � dr ��� � ≤ ≤ √ T sup N≥1 E( sup 0≤t≤T ∥UN t ∥2 −s,σ) 1 2 � � T 0 E(∥[GI,N r −GI r]Υ(t−r)ϕ∥2 s,σ)dr � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 2- Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='18, it is easy to see that the maps (Ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ, Ψ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ) is continuous from [D(R+, H−s,σ)]2 into C(R+, R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus as (UN, V N) converges in law in [D(R+, H−s,σ)]2 to- wards (U, V ), then � Ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ(UN, V N), Ψ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ(UN, V N) � converges in law towards � Ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ(U, V ), Ψ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=',ϕ(U, V ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 3- � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )UN 0 , ϕ) + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d� MN r � , (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V N 0 , ϕ) + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d�LN r �� converges in law towards � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )U0, ϕ) + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, dM1 r � , (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V0, ϕ) + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, dM2 r �� in (D(R+, H−s,σ))2 since � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )UN 0 , ϕ), (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V N 0 , ϕ), � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d� MN r � , � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d�LN r �� converges in law towards � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )U0, ϕ), (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V0, ϕ), � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, dM1 r � , � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, dM2 r �� in (C(R+, H−s,σ))2 × (D(R+, H−s,σ))2 , which in turn follows from the fact that � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )UN 0 , ϕ), (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V N 0 , ϕ) � converges in law towards � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )U0, ϕ), (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V0, ϕ) � in (C(R+, H−s,σ))2(see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='10) and � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d� MN r � , � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d�LN r �� con- verges in law towards � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='−r)ϕ, dW 1 r � , � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='−r)ϕ, dW 2 r �� in (D(R+, H−s,σ))2(which follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='17) and � (Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )UN 0 , ϕ), (Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )V N 0 , ϕ) � is globally independant of � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ S(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d� MN r � , � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 0 � Υ∗ I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' − r)ϕ, d�LN r �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus from 1-, 2-, and 3-, we obtain the result of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='16 we deduce that all limit points (U, V, W) of (UN, V N, WN)N≥1 are elements of (C(R+, H−s))3, thus we end the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='12 by showing that the system of equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='51) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='52) admits a unique solution (U, V, W) ∈ (C(R+, H−s))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Suppose that (U1, V 1, W 1) and (U2, V 2, W 2) are solutions to equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='50), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='51) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='52) with (U1 0, V 1 0 ) = (U2 0, V 2 0 ) then (U1, V 1, W 1) = (U2, V 2, W 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 5 APPENDIX 38 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' All we need to show is that the system of equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='51) admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus we have U1 t − U2 t = − � t 0 Υ∗ S(t − r)(GI r)∗(U1 r − U2 r )dr − � t 0 Υ∗ S(t − r)(GS r )∗(V 1 r − V 2 r )dr, Hence ∥U1 t − U2 t ∥H−s≤ � t 0 ∥Υ∗ S(t − r)(GI r)∗(U1 r − U2 r )∥−s,σdr + � t 0 ∥Υ∗ S(t − r)(GS r )∗(V 1 r − V 2 r )∥−s,σdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='19, we deduce that ∥U1 t − U2 t ∥−s,σ≤ Csup y ∥K(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', y)∥2+D,σ � t 0 {∥U1 r − U2 r ∥−s,σ+∥V 1 r − V 2 r ∥−s,σ}dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='54) Similarly, we obtain ∥V 1 t − V 2 t ∥−s,σ≤ C(sup x ∥K(x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )∥2+D,σ+α) � t 0 {∥U1 r − U2 r ∥−s,σ+∥V 1 r − V 2 r ∥−s,σ}dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='55) Summing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='54), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='55) and applying Gronwall’s lemma we obtain that (U1, V 1) = (U2, V 2), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 5 Appendix In this Appendix we prove the following Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any nonnegative integer m1 > m2 ≥ 0, and any real 0 < σ < σ′, the following embedding is compact W m1,σ 0 (Rd) ֒→ W m2,σ′ 0 (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' To prove this Lemma it is enough to show that for any sequence (ϕn)n of W m1,σ 0 (Rd) which weakly converges towards 0 in W m1,σ 0 (Rd), strongly converges in W m2,σ′ 0 (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let B(R) = {x ∈ Rd/|x|≤ R},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' one has: ∥ϕn∥2 m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ′= � |γ|≤m2 � Rd |Dγϕn(x)|2 1 + |x|2σ′ dx = � |γ|≤m2 � B(R) |Dγϕn(x)|2 1 + |x|2σ′ dx + � |γ|≤m2 � B c(R) |Dγϕn(x)|2 1 + |x|2σ′ dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' since the function R ∈]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∞[�→ 1 + R2σ 1 + R2σ′ is nonincreasing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' � |γ|≤m2 � B c(R) |Dγϕn(x)|2 1 + |x|2σ′ dx = � |γ|≤m2 � B c(R) 1 + |x|2σ 1 + |x|2σ′ |Dγϕn(x)|2 1 + |x|2σ dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ≤ 1 + R2σ 1 + R2σ′ � |γ|≤m2 � B c(R) |Dγϕn(x)|2 1 + |x|2σ dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ≤ 1 + R2σ 1 + R2σ′ sup n≥1 ∥ϕn∥2 m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='σ −→ R−→+∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' On the other hand since W m1,σ 0 (Rd) ֒→ W m1,σ′ 0 (B(R)) ֒→ W m2,σ′ 0 (B(R)), where the second embedding is compact, W m1,σ 0 (Rd) ֒→ W m2,σ′ 0 (B(R)) is compact (see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus � |γ|≤m2 � B(R) |Dγϕn(x)|2 1 + |x|2σ′ dx −→ n−→+∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 5 APPENDIX 39 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' For any non integer 1 + D < s < 2 + D and 1 + D < s′ = s+1+D 2 < s < 2 + D and any 0 < σ < σ′, the following embedding is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hs,σ(Rd) ֒→ Hs′,σ′(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We use the definition by interpolation of the space Hs,σ(Rd) to prove this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='We Prove this result for d = 2, simlar arguments allow us to obtain the result for other values of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since 2 < s′ = s+2 2 < s < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' There exists ρ ∈ (1/2, 1), sucht that s′ = 3(1 − ρ) + 2ρ and s = 4(1 − ρ) + 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthemore, � W 3,σ′ 0 (R2), W 2,σ′ 0 (R2) � ρ,2 = Hs′,σ′(R2) and � W 4,σ 0 (R2), W 2,σ 0 (R2) � ρ,2 = Hs,σ(R2), see [23] or [37] for the explicit definition of the real interpolation space (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' )ρ, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Thus as from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1 the embeddings W 4,σ 0 (R2) ֒→ W 3,σ′ 0 (R2) is compact, we deduce from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='5 in [12] that the following embedding is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hs,σ(Rd) = � W 4,σ 0 (R2), W 2,σ 0 (R2) � ρ,2 ֒→ � W 3,σ′ 0 (R2), W 2,σ′ 0 (R2) � ρ,2 = Hs,σ′(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Under the Assumption (H2), for any A ∈ {S, I, R}, the Markovian semi-group generated by the operator QA, is such that there exists a constant C > 0, such that sup y � � Rd ΥA(t)(x, y)dx � ≤ eCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' We recall that {XA t , t ≥} is the Markov process having the operator QA as its infinites- imal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Let PA(t)(y) = � Rd ΥA(t)(x, y)dx, using Dynkin’s formula it is easy to see that ΥA(t)(x, y) = δ0(x − y) + � t 0 (QA)∗ yΥA(r)(x, y)dr, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) thus integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='1) over x, we obtain PA(t)(y) = 1 + � t 0 (QA)∗ yPA(r)(y)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthermore, ∂ ∂tPA(t)(y) = − � 1≤ℓ≤d mℓ(A, y) ∂ ∂yℓ PA(t)(y) + 1 2 � 1≤ℓ,u≤d ∂ ∂yℓ (θθt)ℓ,u(A, y) ∂ ∂yu PA(t)(y) + 1 2 � 1≤ℓ,u≤d ∂ ∂yu (θθt)ℓ,u(A, y) ∂ ∂yℓ PA(t)(y) + 1 2 � 1≤ℓ,u≤2 (θθt)ℓ,u(A, y) ∂2 ∂yuyℓ PA(t)(y) + � − � 1≤ℓ≤d ∂ ∂yℓ ml(A, y) + 1 2 � 1≤ℓ,u≤d ∂2 ∂yu∂yℓ (θθt)ℓ,u(A, y) � PA(t)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Consequently, PA(t) is the solution of the following system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ∂ ∂tPA(t)(y) = � ℓ F m,θ ℓ (A, y) ∂ ∂yℓ PA(t)(y) + 1 2 � 1≤ℓ,u≤d (θθt)ℓ,u(A, y) ∂2 ∂yuyℓ PA(t)(y) + H(A, y)PA(t)(y) = GAPA(t)(y) + H(A, y)PA(t)(y) PA(0)(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' REFERENCES 40 Thus fix T > 0, according to the Feyman-Kac formula, for any t ∈ [0, T], we have PA(t)(y) = PA(T − t1)(y) = E � exp � − � T t1 H(A, Yr)dr � /Yt1 = y � , (with t + t1 = T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' where {Yt, t ≥ 0} is the markovian processes having GA as the infinitesimal generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' So as from assumption (H2) the function H is bounded, for any y ∈ Rd, we have PA(t)(y) = � Rd ΥA(t)(x, y)dx ≤ eCt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Since it is easy to adap without difficulty the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='4 of [1] to the space W m,σ 0 (m ∈ N), by following the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='23 of [1], we prove easily that for any integer m > d/2, the space W m,σ 0 is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Furthemore using the result on the complex interpolation of [22] (Theorem 4) and [10](subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='2), we conclude that for any real number s > d/2, the space Hs,σ is a Banach algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Adams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Sobolev Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Academic Press, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Aldous .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stopping times and tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' The Annals of Probability 6(2), 335-340, 1978 [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Allen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='M Bolker, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lou and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Nevai Asymptotic profiles of the steady states for an SIS epidemic reaction diffusion model, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A 21, 1-20, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Andersson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Britton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stochastic epidemic models and their statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer Lecture Notes in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer, New York, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Bahouri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Chemin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Danchin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Fourier Analysis and Nonlinear Partial Dif- ferential Equations, Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Billingsley, Convergence of Probability Measures, 2nd edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Wiley, New York, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Bowong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Emakoua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' and E Pardoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A Spatial Stochastic Epidemic Model: Law of Large Numbers and Central Limit Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='0663v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='PR] 13 jul 2020 [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Britton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pardoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stochastic epidemic in a homogeneous community, Part I of Stochastic epidemic models with inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Britton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pardoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' eds, Lecture Notes in Mathematics 2225, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1–120, Springer 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Bücher and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kojadinovic, A note on conditional versus joint unconditional weak convergence in bootstrap consistency results, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='01031v4 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ST] 1 Mar 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Calderón.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Intermediate spaces and interpolation, the complex method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 24, 113-190 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Clémençon, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Tran and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' de Arazoza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A stochastic SIR model with contact tracing: large population limits and statistical inference, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' of Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Dynamics 2:4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 392–414, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' REFERENCES 41 [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Cobos and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Peetre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Interpolation of compactness using Aronszajn-Gagliardo functors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 68 (1989), 220-240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Gihman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Skorohod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer-Verlag Berlin New york 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Grafakos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Classical Fourier analysis 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' ed Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Hanouzet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Espaces de Sobolev avec poids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Application au problème de Dirichlet dans un demi espace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Rendiconti del Seminario Matematico della Università di Padova, Tome 46 (1971), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 227-272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Jacod and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Shiryaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Limit Theorems for Stochastic Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer-Verlag, Berlin, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A weak interaction epidemic among diffusive particles, in Stochastic Partial Dif- ferential Equations, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Etheridge ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lecture Note Series 216, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 189–208, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kipnis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Landim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Scaling limits of interacting particle systems, volume 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathemat- ical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer-verlag, Berlin, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kotelenez, A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations, Stochastic analysis and applications, 2(3), 245-265, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kunita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stochastic flows and stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Cambridge University Press, Cambridge 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lalley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Perkins and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A phase transition for mesure–valued SIR epidemic processes, The Annals of Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 42, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 237–310, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Löfström.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Interpolation of weighted spaces of differentiable functions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 132 (1982), 189?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Löfström.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Interpolation spaces, an introduction, Springer, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Méléard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Stochastics and Stochastics Reports, 63 :195-225, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Méléard et S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Roelly: Sur les convergences étroite ou vague de processus à valeurs mesures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Comptes rendus de l’académie des Sciences de Paris Sér.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1, 317:785-788, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Méléard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Mouvement brownien et calcul stochastique, Techniques de l?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='ingénieur, (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Métivier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Convergence faible et principe d’invariance pour des martingales à valeurs dans des espaces de Sobolev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Annales de l’IHP, 20(4) :329-348, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Métivier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Weak convergence of measure valued processes using Sobolev imbed- ding techniques, Proceedings Stochastic partial differential equations, Trenio 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 1236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 172-183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' N’zi, É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pardoux and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Yeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A SIR model on a refining spatial grid I - Law of Large Numbers, Applied Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='& Optimization, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [30] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pardoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Probabilistic Models of Population Evolution, Scaling Limits, Genealogies and Interactions, Springer 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' REFERENCES 42 [31] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Pardoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Moderate Deviations and Extinction of an Epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Election.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Probab 25, paper 25, 1-27, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Roelly-Coppoletta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' A criterion of convergence of measure-valued processes: application to measure branching processes Stochastics, 17 :43-65, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Roques, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Bonnefon, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Baudrot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Soubeyrand, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Berestycki : A parsimonious model for spatial transmission and heterogeneity in the COVID-19 propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Open Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 7: 201382, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Rudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' : Real and Complex Analysis, 3rd edn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' McGraw-Hill, New York (1987) [35] k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Sato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Ueno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Multi-dimensional diffusion and the Markov process on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Kyoto univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' 4(3) 529-605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Partial differential equations III Nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Springer 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Triebel, Interpolation theory, Hunction spaces, Differential operators, North-Holland Publishing Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=', 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' [38] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} +page_content=' Tran Modèles particulaires stochastiques pour des problèmes d’évolution adaptative et pour l’approximation de solutions statistiques, Thesis 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E0T4oBgHgl3EQfbACf/content/2301.02343v1.pdf'} diff --git a/ttAyT4oBgHgl3EQf0flC/content/tmp_files/2301.00718v1.pdf.txt b/ttAyT4oBgHgl3EQf0flC/content/tmp_files/2301.00718v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5102bcc42f81ab8bd38f606c308b6da77fbc4812 --- /dev/null +++ b/ttAyT4oBgHgl3EQf0flC/content/tmp_files/2301.00718v1.pdf.txt @@ -0,0 +1,5589 @@ +Robust Inference for Federated Meta-Learning +Zijian Guo +Rutgers University, Piscataway, USA +Xiudi Li +Harvard University, Boston, USA +Larry Han +Harvard University, Boston, USA +Tianxi Cai +Harvard University, Boston, USA +Summary. Synthesizing information from multiple data sources is critical to ensure knowledge gen- +eralizability. Integrative analysis of multi-source data is challenging due to the heterogeneity across +sources and data-sharing constraints due to privacy concerns. In this paper, we consider a general +robust inference framework for federated meta-learning of data from multiple sites, enabling statistical +inference for the prevailing model, defined as the one matching the majority of the sites. Statistical +inference for the prevailing model is challenging since it requires a data-adaptive mechanism to select +eligible sites and subsequently account for the selection uncertainty. We propose a novel sampling +method to address the additional variation arising from the selection. Our devised CI construction does +not require sites to share individual-level data and is shown to be valid without requiring the selection +of eligible sites to be error-free. The proposed robust inference for federated meta-learning (RIFL) +methodology is broadly applicable and illustrated with three inference problems: aggregation of para- +metric models, high-dimensional prediction models, and inference for average treatment effects. We +use RIFL to perform federated learning of mortality risk for patients hospitalized with COVID-19 using +real-world EHR data from 16 healthcare centers representing 275 hospitals across four countries. +Keywords: Post-selection Inference; Heterogeneous Data; Multi-source Data; Privacy Preserv- +ing; High-dimensional Inference. +1. +Introduction +Crowdsourcing, or the process of aggregating crowd wisdom to solve problems, is a useful community- +based method to improve decision-making in disciplines ranging from education [Heffernan and +Heffernan, 2014] to public health [Han et al., 2018, Wang et al., 2020]. Compared to traditional +expert-driven solutions made by a single group, incorporating the opinions of multiple diverse +groups can improve the quality of the final decision [Surowiecki, 2005]. In health research, crowd- +sourcing has led to the discovery of new drugs during pandemics [Chodera et al., 2020], the design +of patient-centered mammography reports [Short et al., 2017], and the development of machine +learning algorithms to classify tumors for radiation therapy [Mak et al., 2019]. +Underlying the phenomenon of the “wisdom of the crowds” is the statistical and philosophical +notion that learning from multiple data sources is desirable. Incorporating information from diverse +data sources can increase the generalizability and transportability of findings compared to learning +arXiv:2301.00718v1 [stat.ME] 2 Jan 2023 + +2 +Guo, Li, Han & Cai +from a single data source. Findings from a single data source may not be generalizable to a new +target population of interest due to poor data quality or heterogeneity in the underlying data +generating processes. +Integrative analysis of data from multiple sources can be a valuable alternative to using a sin- +gle data source alone. However, directly pooling multiple data sources into a single dataset for +analysis is often unsatisfactory or even infeasible. Heterogeneity between different data sources +can severely bias predictions or inferences made by such a pooled analysis strategy [Leek et al., +2010, Ling et al., 2022]. As an alternative to pooled analysis, meta-analysis has frequently synthe- +sized information from multiple studies. Standard meta-analysis methods aggregate quantitative +summary of evidence from multiple studies. Variations of meta-analysis, such as random effects +meta-analysis, have been adopted to explore between-study heterogeneity, potential biases such as +publication bias, and small-study effects. However, most existing meta-analysis tools that account +for heterogeneity require strong modeling assumptions and do not consider the validity of inference +when data from certain sites have substantially different distributions from other sites. +Another challenge of particular importance is the issue of data privacy pertaining to biomedical +studies. Regulations in the United States, such as the Health Insurance Portability and Account- +ability Act (HIPAA) Privacy Rule, and those in the European Union, such as the General Data +Protection Regulation (GDPR) and the European Medicines Agency (EMA) Privacy Statement, +protect the personal information of patients and preclude the transfer of patient-level data between +sites. These regulations make the promise of integrative data analysis more difficult to attain, +highlighting the need for federated integrative analysis methods that do not require sharing of +individual-level data. +When cross-study heterogeneity is substantial and outliers exist, a desirable strategy of integra- +tive analysis is to identify a prevailing model to achieve consensus learning. The prevailing model is +defined as the model satisfied by the majority of the sites. Identifying the prevailing model can be +intuitively achieved via the majority rule [Sorkin et al., 1998, Kerr et al., 2004, Hastie and Kameda, +2005], which chooses the alternative that more than half of individuals agree upon. The majority +rule is widespread in modern liberal democracies and is deployed in various streams of research. +For example, genomics data is usually separated into batches, but heterogeneity across batches +can lead to undesirable variation in the data [Leek et al., 2010, Ling et al., 2022]. This setting +aims to identify batches that show low levels of concordance with the majority of the batches and +adjust for such differences in downstream analyses [Trippa et al., 2015]. As another example, in +the design of clinical trials, it is often infeasible or unethical to enroll patients in the control arm. +In such cases, it is possible to use data from historical trials or observational studies to construct +an external control arm [Jahanshahi et al., 2021, Ventz et al., 2019, Davi et al., 2020]. However, +when many such historical data sources exist, it is crucial to carefully select data sources that show +high levels of similarity with the majority of the other data sources. The last example is Mendelian +Randomization, where multiple genetic markers are used as instrumental variables (IVs) to account +for potential unmeasured confounders. Every single IV will have its causal effect estimator, and the +goal is to identify the causal effect matching the majority of the estimated effects [Burgess et al., +2017, Bowden et al., 2016, Kang et al., 2016]. +Without prior knowledge of the prevailing model, it is critical to employ data-adaptive ap- +proaches to select appropriate sites for inferring the prevailing model. +In addition, confidence +intervals (CIs) for the target parameter of the prevailing model need to appropriately adjust for the +site selection variability. Most existing statistical inference methods rely on perfectly separating + +Inference for Meta-Learning +3 +eligible and ineligible sites, which may be unrealistic for practical applications. There is a paucity of +statistical inference methods for the prevailing model that can achieve efficient and robust inference +while being applicable to a broad set of scenarios without restrictive assumptions such as a perfect +separation. +In this paper, we fill this gap by developing a broad theoretically justified framework for making +robust inferences for federated meta-learning (RIFL) of an unknown prevailing model using multi- +source data. The RIFL method selects the eligible sites to infer the prevailing model by assessing +dissimilarities between sites with regard to the parameter of interest. We employ a novel resampling +method to construct uniformly valid CIs. The RIFL inference method is robust to the errors in +separating the sites belonging to the majority group and the remaining sites; see Theorem 2. We +also show in Theorem 3 that our proposed sampling CI can be as short as the oracle CI with the +prior knowledge of the eligible sites. Our general sampling algorithm is privacy-preserving in that it +is implemented using site-specific summary statistics and without requiring sharing individual-level +data across different sites. Our proposed RIFL methodology is demonstrated with three inference +problems: aggregation of low-dimensional parametric models, construction of high-dimensional +prediction models, and inference for the average treatment effect (ATE). +To the best of our knowledge, our proposed RIFL method is the first CI guaranteeing uniform +coverage of the prevailing model under the majority rule. We have further compared via simulation +studies with three other inference procedures that can potentially be used under the majority rule, +including the majority voting estimator, the median estimator [e.g., Bowden et al., 2016], and the m- +out-of-n bootstrap [e.g., Chakraborty et al., 2013, Andrews, 2000]. Numerical results demonstrate +that these three CIs fail to achieve the desired coverage property, while our RIFL method leads to +a uniformly valid CI. We provide the reasoning for under-coverage for these existing methods in +Sections 2.3 and 2.4. +1.1. +Related literature +The RIFL method is related to multiple streams of literature, including post-selection inference, +mendelian randomization, integrative analysis of multi-source data, transfer learning, and federated +learning. We next detail how RIFL differs from the existing literature and highlight its contribu- +tions. +A wide range of novel methods and theories have been established to address the post-selection +inference problem [e.g., Berk et al., 2013, Lee et al., 2016, Leeb and P¨otscher, 2005, Zhang and +Zhang, 2014, Javanmard and Montanari, 2014, van de Geer et al., 2014, Chernozhukov et al., +2015, Belloni et al., 2014, Cai and Guo, 2017, Xie and Wang, 2022]. However, most post-selection +inference literature focuses on inferences after selecting a small number of important variables +under high-dimensional regression models. The selection problem under the RIFL framework is +fundamentally different: the selection error comes from comparing different sites, and there is no +outcome variable to supervise the selection process. Additionally, RIFL only requires the majority +rule, while the variable selection methods typically require a small proportion of variables to affect +the outcome. +In Mendelian Randomization, various methods have been developed to leverage the majority +rule and make inferences for the one-dimensional causal effect [Bowden et al., 2016, Windmeijer +et al., 2019, Kang et al., 2016, Guo et al., 2018, Windmeijer et al., 2021]. A recent work by Guo +[2021] demonstrated the post-selection problem due to IV selection errors. However, the uniformly +valid inference method in Guo [2021] relies on searching the one-dimensional space of the causal + +4 +Guo, Li, Han & Cai +effect and cannot be easily generalized to multivariate settings, not to mention high-dimensional +settings. In contrast, RIFL is distinct from the existing searching method and is useful in addressing +a much broader collection of post-selection problems as detailed in Section 5. +The integrative analysis of multi-source data has been investigated in different directions. Wang +et al. [2021] studied the data fusion problem with robustness to biased sources. The identification +condition in Wang et al. [2021] differs from the majority rule, and the validity of their proposal +requires correctly identifying unbiased sources. Maity et al. [2022] studied meta-analysis in high- +dimensional settings where the data sources are similar but non-identical and require the majority +rule to be satisfied as well as a large separation between majority and outlier sources to perfectly +identify eligible sites. In contrast, the RIFL CI is valid without requiring the selection step to +perfectly identify eligible sites. Meinshausen and B¨uhlmann [2015], B¨uhlmann and Meinshausen +[2015], Rothenh¨ausler et al. [2016], Guo [2020] made inference for the maximin effect, which is +defined as a robust prediction model across heterogeneous datasets. Cai et al. [2021b], Liu et al. +[2021], Zhao et al. [2016] imposed certain similar structures across different sources and made +inferences for the shared component of regression models. +Peters et al. [2016], Arjovsky et al. +[2019] studied the multi-source data problem and identified the causal effect by invariance principles. +Unlike existing methods, the RIFL framework only assumes that a majority of the sites have similar +models but allows non-eligible sites to differ arbitrarily from the majority group. +RIFL relates to the existing literature on federated learning and transfer learning. Privacy- +preserving and communication-efficient algorithms have been recently developed to learn from +multiple sources of electronic health records (EHR) [Rasmy et al., 2018, Tong et al., 2022] and +multiple sources of diverse genetic data [Kraft et al., 2009, Keys et al., 2020]. Federated regression +and predictive modeling [Chen et al., 2006, Li et al., 2013, Chen and Xie, 2014, Lee et al., 2017, +Lian and Fan, 2017, Wang et al., 2019, Duan et al., 2020] and causal modeling [Xiong et al., 2021, +Vo et al., 2021, Han et al., 2021] have been developed. However, none of these federated learning +methods study inference for the prevailing model when some sites may not be valid, which is the +main focus of RIFL. The RIFL framework also differs from the recently developed transfer learning +algorithms [e.g., Li et al., 2020, Tian and Feng, 2022, Han et al., 2021]. These algorithms require +pre-specification of an anchor model to which the models obtained from source data sets can be +compared. In contrast, RIFL targets a more challenging scenario: we do not assume the availability +of such an anchor model but leverage the majority rule to identify the unknown prevailing model. +1.2. +Paper organization and notations +The paper proceeds as follows. Section 2 describes the multi-source data setting and highlights the +challenge of inferring the prevailing model. Section 3 proposes the RIFL methodology, and Section +4 establishes its related theory. In Section 5, we illustrate our proposal in three applications. In +Section 6, we provide extensive simulation results comparing our method to existing methods. +Section 7 illustrates our method using real-world international EHR data from 16 participating +healthcare centers representing 275 hospitals across four countries as part of the multi-institutional +Consortium for the Clinical Characterization of COVID-19 by EHR (4CE) [Brat et al., 2020]. +We introduce the notations used throughout the paper. For a set A, |A| denotes the cardinality +of the set. For a vector x, we define its ℓq norm as ∥x∥q = +��p +l=1 |xl|q� 1 +q for q ≥ 0 with ∥x∥0 = +|{1 ≤ l ≤ p : xl ̸= 0}| and ∥x∥∞ = max1≤l≤p |xl|. For a matrix X, Xi,· and X·,j denote its i-th row +and j-th column, respectively. For two positive sequences an and bn, an ≪ bn if lim supn→∞ an/bn = +0. For a matrix A, we use ∥A∥F , ∥A∥2 and ∥A∥∞ to denote its Frobenius norm, spectral norm, + +Inference for Meta-Learning +5 +and element-wise maximum norm, respectively. +2. +Formulation and Statistical Inference Challenges +2.1. +Model assumptions and overview of RIFL +Throughout the paper, we consider that we have access to L independent training data sets drawn +from L source populations. For 1 ≤ l ≤ L, we use P(l) to denote the distribution of the l-th source +population and use θ(l) = θ(P(l)) ∈ Rd to denote the associated model parameter. For any θ ∈ Rd, +we define the index set V(θ) ⊂ {1, · · · , L} as +V(θ) := {1 ≤ l ≤ L : θ(l) = θ}, +(1) +which contains the indexes of all sites having the same model parameter as θ. We now introduce +the majority rule. +Assumption 1 (Majority Rule). There exists θ∗ ∈ Rd such that |V(θ∗)| > L/2. +We shall refer to θ∗ as the prevailing model that matches with more than half of {θ(l)}1≤l≤L, and +the corresponding index set V(θ∗) as the prevailing set. Our goal is to construct a confidence region +for a low dimensional functional of θ∗, denoted as β∗ = g(θ∗) ∈ Rq, for some q ≥ 1, where g(·) ∈ Rq +is a prespecified low-dimensional transformation. Examples of β∗ = g(θ∗) include +(a) Single coefficient or sub-vector: β∗ = θ∗ +j for 1 ≤ j ≤ d or β∗ = θ∗ +G with G ⊂ {1, · · · , d}; +(b) Linear transformation: β∗ = x⊺θ∗ for any x ∈ Rd; +(c) Quadratic form: β∗ = ∥θ∗∥2 +2. +For notational ease, we focus on q = 1 primarily and discuss the extension to the setting with q ≥ 2 +in Section 3.2. +If the prevailing set V(θ∗) were known, standard meta and federated learning methods could be +used to make inferences about θ∗ using data from sites belonging to V(θ∗). However, as highlighted +in Section 2.3, inference for θ∗ without prior knowledge of V(θ∗) except for the majority rule is +substantially more challenging due to the need to estimate V(θ∗). Our proposed RIFL procedure +involves several key steps: (i) for l = 1, ..., L, construct local estimates of θ(l) and β(l) = g(θ(l)), +denoted by �θ(l) and �β(l), respectively; (ii) for 1 ≤ l < k ≤ L, estimate pairwise dissimilarity measure +Dl,k = D(θ(l), θ(k)) and Ll,k = β(l) − β(k) as �Dl,k and �Ll,k along with their standard errors � +SE( �Dl,k) +and � +SE( �Ll,k); (iii) construct a robust estimate for the prevailing set V(θ∗); (iv) derive robust +resampling-based confidence set for β∗ accounting for post-selection uncertainty. The construction +of �θ(l) and �β(l) follows standard procedures for the specific problems of interest. We next detail +(ii) the construction of the dissimilarity measures and (iii) the prevailing set estimator. The most +challenging step of RIFL is the resampling-based inference, which is described in Section 3. +2.2. +Dissimilarity measures +A critical step of applying the majority rule is to evaluate the (dis)similarity between any pair +of parameters θ(l) and θ(k) for 1 ≤ l, k ≤ L. We form two sets of dissimilarity measures, the local +dissimilarity between β(l) and β(k), Ll,k = β(l)−β(k), and a global dissimilarity Dl,k = D(θ(l), θ(k)) = + +6 +Guo, Li, Han & Cai +∥θ(l) − θ(k)∥2 +2. Although other vector norms can be considered for D(·, ·), we focus on the quadratic +norm due to its smoothness and ease of inference, especially in the high-dimensional setting. +We assume that {�β(l), �σl}1≤l≤L satisfy +1 +σl +(�β(l) − β(l)) d→ N(0, 1) +and +�σl +σl +p→ 1, +(2) +with σl denoting the standard error of �β(l). In the low-dimensional setting, most existing estimators +satisfy (2) under standard regularity conditions. In the high-dimensional setting, various asymp- +totically normal de-biased estimators have recently been proposed and shown to satisfy (2); see +more discussions at the end of Section 5.2. +Let �Ll,k = �β(l)− �β(k) and �Dl,k be the point estimators for Ll,k and Dl,k, respectively. We estimate +their standard errors as � +SE( �Ll,k) = +� +�σ2 +l + �σ2 +k and � +SE( �Dl,k), with �σ2 +l denoting the estimated variance +of �β(l). For the global dissimilarity measure, we assume that �Dl,k and � +SE( �Dl,k) satisfy +lim sup +n→∞ P +���� �Dl,k − Dl,k +���/� +SE( �Dl,k) ≥ zα +� +≤ α +for +0 < α < 1, +(3) +where zα denotes the α upper quantile of a standard normal distribution. Although �Dl,k can be +constructed as ∥�θ(l) − �θ(k)∥2 +2 in the low-dimensional setting, deriving { �Dl,k, � +SE( �Dl,k)} that satisfies +(3) is much more challenging in the high-dimensional setting due to the inherent bias in regularized +estimators. In Sections 5.1 and 5.2, we demonstrate that our proposed estimators of Dl,k satisfy +(3) for a broad class of applications in both low and high dimensions. +Based on both sets of dissimilarity measures, we determine the concordance between sites k and +l with respect to inference for β∗ = g(θ∗) based on the following test statistic +�Sl,k := max +���� �Dl,k/� +SE( �Dl,k) +��� , +��� �Ll,k/� +SE( �Ll,k) +��� +� +. +(4) +For 1 ≤ l < k ≤ L, we can then implement the following significance test of whether the k-th and +l-th sites share the same parameters, +�Hl,k = 1 +� +�Sl,k ≤ z0.05/[2L(L−1)] +� +, +(5) +where 0.05 is a pre-selected significance level for testing the similarity among different sites and +z0.05/[2L(L−1)] denotes the 0.05/[2L(L − 1)] upper quantile of the standard normal distribution. +The statistic �Sl,k measures the level of evidence that the two sites differ from each other based +on observed data. The binary decision �Hl,k in (5) essentially estimates Hl,k = 1{θ(l) = θ(k)}. We +specify the threshold as z0.05/[2L(L−1)] to adjust for the multiplicity of hypothesis testing. Since +the matrix H is symmetric and Hl,l = 1, we construct an estimate for the full voting matrix +�H = [ �Hl,k]k=1,...,L +l=1,...,L by setting �Hk,l = �Hl,k and �Hl,l = 1. The estimated voting matrix �H summarizes +all cross-site similarities, which is then used to estimate the prevailing set. +Remark 1 (Univariate case). For the special setting with a univariate θ∗, we may simplify +the construction of the test statistics in (4) and the vote in (5) as +�Hl,k = 1 +� +�Sl,k ≤ z0.05/[L(L−1)] +� +with +�Sl,k = +��� �Ll,k/� +SE( �Ll,k) +��� +for +1 ≤ l < k ≤ L. +(6) + +Inference for Meta-Learning +7 +2.3. +Prevailing set estimation and post-selection problem +In the following, we construct two estimators of the prevailing set V(θ∗), which are used to make +inference for β∗. We construct the first estimator as +�V := {1 ≤ l ≤ L : ∥ �Hl,·∥0 > L/2}. +(7) +The set �V contains all sites receiving ‘majority votes’. The second estimator is constructed by +utilizing the maximum clique from graph theory [Carraghan and Pardalos, 1990]. Specifically, we +define the graph G([L], �H) with vertices [L] := {1, 2, · · · , L} and the adjacency matrix �H with +�Hl,k = 1 and �Hl,k = 0 denoting that the l-th and k-th vertexes are connected and disconnected, +respectively. The maximum clique of the graph G([L], �H) is defined as the largest fully connected +sub-graph. We use the term maximum clique set to denote the corresponding vertex set in the +maximum clique, denoted as MC([L], �H). We construct �V by identifying the maximum clique set +of G([L], �H), that is, +�V := MC([L], �H). +(8) +If the majority rule holds, the prevailing set V(θ∗) is the maximum clique set of G([L], H), that is, +V(θ∗) = MC([L], H). If �H is a sufficiently accurate estimator of H, both set estimators �V and �V +exactly recover the prevailing set V(θ∗). However, since �H might be different from the true H due +to the limited sample size in practice, �V and �V may be different from V(θ∗). When the maximum +clique set has the cardinality above L/2, we have �V ⊂ �V, that is, �V may be a more restrictive set +estimator than �V. We illustrate the definitions of �V in (7) and �V in (8) in Figure 1. +1 +2 +3 +4 +5 +6 +Fig. 1: The graph G([L], �H) with L = 6. �V = {1, 2, 3, 4, 5} and �V = {1, 2, 3, 5}. +In the following, we demonstrate the subsequent analysis after obtaining �V. The argument is +easily extended to the estimated set �V. One may aggregate {�β(l), �σl}l∈�V to estimate β∗ as the +following inverse variance weighted estimator, +�β∗ = +� +l∈�V �β(l)/�σ2 +l +� +l∈�V 1/�σ2 +l +. +(9) +A naive 1 − α confidence interval for β∗ can be constructed as +CIpost = +� +��β∗ − zα/2 +1 +�� +l∈�V 1/�σ2 +l +, �β∗ + zα/2 +1 +�� +l∈�V 1/�σ2 +l +� +� , +(10) +where zα/2 denotes the α/2 upper quantile of the standard normal distribution. +Unfortunately, similar to other settings in the ‘post-selection’ literature, such naive construction +can lead to bias in the point estimation and under-coverage in the confidence interval due to ignoring +the variability in the selection of �V. We illustrate the post-selection problem of the naive confidence +interval in (10) with the following example. + +8 +Guo, Li, Han & Cai +Example 1. We construct the confidence interval for a target population’s average treatment +effect (ATE) in the multi-source causal inference setting detailed in Section 5.3. We have L = 10 +source sites with nl = 1000, 1 ≤ l ≤ 10. In each source site, we observe the data {X(l) +i , A(l) +i , Y (l) +i +}1≤i≤nl, +where X(l) +i +∈ R10 denotes a 10-dimensional vector of baseline covariates, A(l) +i +∈ {0, 1} denotes the +treatment assignment (treatment or control) and Y (l) +i +∈ R denotes the outcome. The first six source +sites are generated such that the target ATE has a value of −1, while the remaining four source +sites are generated such that the target ATE has values −1.2, −1.2, −1.1, and −1.1, respectively. +In this case, the first six source sites form the majority group. The confidence intervals relying on +�V and �V suffer from the under-coverage due to wrongly selected sites being included. Based on 500 +simulations, the confidence interval in (10) has an empirical coverage of only 43.2%. If we replace +�V in (10) with �V in (7), the empirical coverage drops to 27.4%. +2.4. +Challenge for the median-based confidence interval +A commonly used consistent estimator of the prevailing model parameter under the majority rule is +the median estimator [e.g., Bowden et al., 2016]. We construct the median estimator as the median +of {�β(l)}1≤l≤L and estimate its standard error by parametric bootstrap. It is worth noting that +although the median estimator is consistent under the majority rule as the sample size in each site +approaches infinity, it may not be suitable for the purpose of statistical inference due to its bias. +Consequently, the CI based on the median estimator does not achieve the desired coverage property. +To illustrate this, let us consider a special case where L is odd, and there are (L + 1)/2 sites in the +prevailing set. Without loss of generality, we assume that V(θ∗) = {1, . . . , (L+1)/2}. Furthermore, +suppose that β(l) < β∗ for l /∈ V. In this scenario, when the parameter values in the non-majority +sites are well-separated from the parameter value in the prevailing set, with high probability, the +median estimator will coincide with min{�β1, . . . , �β(L+1)/2}. That is, the median estimator is the +smallest order statistics of (�β1, . . . , �β(L+1)/2). Even when the site-specific estimator �β(l) is unbiased +for β∗ and normally distributed for l ∈ V, the smallest order statistics typically has a non-normal +distribution with a mean value below β∗. More generally, the limiting distribution of the median +estimator is that of an order statistics and has an asymptotic bias that is not negligible for the +purpose of statistical inference. This same issue has been discussed in more detail in Windmeijer +et al. [2019] in the context of invalid instrumental variables. Our numerical results in Section 6 +show that the CI based on the median estimator fails to achieve the desired coverage. +3. +RIFL Inference +In this section, we devise resampling-based methods for deriving a valid confidence interval for β∗, +addressing the post-selection issue in aggregating multi-source data. +3.1. +RIFL: resampling-based inference +The RIFL interval construction consists of two steps. In the first step, we resample the dissimi- +larity measures and screen out the inaccurate resampled measures. In the second step, we use the +resampled dissimilarity measures to estimate the prevailing set, which is further used to generate +a sampled confidence interval. + +Inference for Meta-Learning +9 +Step 1: resampling and screening. Conditioning on the observed data, for 1 ≤ l < k ≤ L, we +generate { �D[m] +l,k }1≤m≤M and { �L[m] +l,k }1≤m≤M following +�D[m] +l,k +i.i.d +∼ N +� +�Dl,k, � +SE +2( �Dl,k) +� +, +�L[m] +l,k +i.i.d +∼ N +� +�Ll,k, � +SE +2( �Ll,k) +� +for +1 ≤ m ≤ M. +(11) +The above generating mechanism in (11) guarantees that the distributions of �D[m] +l,k − �Dl,k and +�L[m] +l,k − �Ll,k approximate those of �Dl,k − Dl,k and �Ll,k − Ll,k, respectively. +Remark 2. The random variables { �D[m] +l,k }1≤l L/2 +� +. +(16) +Step 2: aggregation. For m ∈ M, we construct the estimated prevailing set �V[m] as, +�V[m] = {1 ≤ l ≤ L : ∥ �H[m] +l,· ∥0 > L/2}. +(17) +The set �V[m] contains all indexes receiving more than half of the votes. With �V[m] in (17), we apply +the inverse variance weighted estimator +�β[m] = +� +l∈�V[m] �β(l)/�σ2 +l +� +l∈�V[m] 1/�σ2 +l +. +For the significance level α, we construct the 1 − α confidence interval as +CI[m] = +� +��β[m] − zα1/2 +1 +�� +l∈�V[m] 1/�σ2 +l +, �β[m] + zα1/2 +1 +�� +l∈�V[m] 1/�σ2 +l +� +� , +(18) +where α1 = α − ν with ν denoting a pre-specified small probability used to guarantee the sampling +property in (13). We choose the default value of ν as ν = α/20 throughout the paper. +Finally, we construct the CI for β∗ as +CI = ∪m∈MCI[m], +(19) +with M and CI[m] defined in (16) and (18), respectively. We refer to CI as a confidence interval +although ∪m∈MCI[m] may not be an interval. +The RIFL algorithm for constructing CI is also +summarized in Algorithm 1 in Section A.1 of the supplement. +The average of the maximum and minimum value of the confidence interval defined in (19) serves +as a point estimator of β∗. Additionally, we may use �pl = � +m∈M 1(l ∈ �V[m])/|M|, the proportion +of times site l being included in the majority group, as a generalizability measure for the l-th site. +Remark 3 (Difference between �V[m] and �V[m]). For m ∈ M, we have �V[m] ⊂ �V[m], that +is, the set �V[m] in (17) is less restrictive compared to the maximum clique set �V[m] defined in + +Inference for Meta-Learning +11 +(16). This relationship helps explain why two different set estimators �V[m] and �V[m] are used in +our construction. Firstly, the maximum clique set �V[m] imposes a stricter rule than �V[m] in (17) +and the maximum clique set �V[m] is likely to screen out more inaccurate resamples. Secondly, our +theory shows that there exists m∗ ∈ M such that �V[m∗] is guaranteed to recover the true prevailing +set V(θ∗). For m ∈ M, the set �V[m] tends to contain more sites than �V[m], leading to shorter CI[m]. +The use of �V[m] in (17) enhances the precision of the resulting RIFL confidence interval. +3.2. +Extension to multivariate target parameters +Our method can be easily extended to construct confidence regions for a multi-dimensional target +parameter β∗ ∈ Rq. As a generalization of (2), we assume the site-specific estimators {�β(l), �Ωl}1≤l≤L +satisfy �Ω−1/2 +l +(�β(l)−β(l)) d→ N(0, Iq×q), where �β(l) ∈ Rq, �Ωl ∈ Rq×q denotes the estimated covariance +matrix of �β(l), and Iq×q is the q × q identity matrix. +We shall highlight two main adjustments to the RIFL algorithm described above for the uni- +variate β∗. Firstly, we modify the computation of �Ll,k and � +SE( �Ll,k) as �Ll,k = ∥�β(l) − �β(k)∥2 +2 and +� +SE( �Ll,k) = +� +4(�β(l) − �β(k))⊺(�Ωl + �Ωk)(�β(l) − �β(k)) + 1/ min{nl, nk}, +where 1/ min{nl, nk} is used to control higher order approximation error of ∥�β(l) − �β(k)∥2 +2. Secondly, +we generalize the construction of CI[m] in equation (18) as +CS[m] = +� +� +�β ∈ Rq : (β − �β[m])⊺ +� +� � +l∈�V[m] +�Ω−1 +l +� +� (β − �β[m]) ≤ χ2 +q(α) +� +� +� +(20) +where �β[m] = +�� +l∈�V[m] �Ω−1 +l +�−1 �� +l∈�V[m] �Ω−1 +l +�β(l)� +and χ2 +q(α) denotes the upper α quantile of the +χ2 distribution with q degrees of freedom. Our final confidence set is CS = ∪m∈MCS[m]. +3.3. +Tuning parameter selection +Implementing the RIFL CI requires the specification of the resampling size M and the shrinkage +parameter ρ(M) ∈ (0, 1) used in (14). The proposed method is not sensitive to the choice of the +size M as long as it is sufficiently large (e.g., M ≥ 500). We set M = 500 as the default value. +Our theoretical results in Section 4 suggest the choice of ρ(M) as ρ(M) = c∗ (log n/M)1/[L(L−1)] +for some positive constant c∗; see the following equation (23). If the constant c∗ is chosen to be too +small, most resampled dissimilarity measures will not produce a maximum clique set satisfying the +majority rule, resulting in a very small |M|. Consequently, we can use |M| to determine whether +the tuning parameter ρ(M) is chosen to be sufficiently large. In practice, we start with a small value +of c∗ (e.g., c∗ = 1/12) and increase the value of c∗ until a pre-specified proportion, say prop = 10%, +of resampled dissimilarity measures produce maximum clique sets satisfying the majority rule. The +RIFL CI is nearly invariant to the original threshold T in (13). Our selected value of ρ(M) will +ensure a large enough ρ(M) · T such that more than prop = 10% of the resampled sets satisfy the +majority rule. Even if the threshold T may be conservative due to the Bonferroni correction, this +choice of T does not affect the performance of our RIFL CI since the choice of ρ(M) will be adaptive +to the specification of the threshold T. We demonstrate the robustness to the tuning parameters +over the numerical studies in Section 6.4. + +12 +Guo, Li, Han & Cai +4. +Theoretical justification for RIFL inference +We next provide theoretical justifications for the validity of the RIFL CI. Let n = min1≤l≤L nl and +define +errn(M, ν) = c∗(ν) +�log n +M +� +1 +L(L−1) +with c∗(ν) = 2 +1 +L(L−1) − 1 +2 √π exp +�1 +2z2 +ν/[2L(L−1)] +� +, +(21) +where L ≥ 2 and 0 < ν < 1/2. The term errn(M, ν) quantifies the sampling accuracy, which +denotes the smallest difference between the true dissimilarity measures and the resampled ones +after resampling M times. +For a constant ν ∈ (0, 1/2) and fixed L, c∗(ν) is a constant only +depending on the pre-specified probability ν and errn(M, ν) is of order (log n/M) +1 +L(L−1) , which +tends to 0 with a sufficiently large M. +The following theorem establishes the critical sampling property, which provides the theoretical +support for the threshold reduction in (14). +Theorem 1. Suppose that the site-specific estimators {�β(l), �σl}1≤l≤L satisfy (2) and the dis- +similarity measures { �Dl,k, � +SE( �Dl,k)}1≤l 80. +This suggests that these sites are not aligned well with other sites when we study the effect of the +corresponding risk factor. However, VMC included these sites in the estimated prevailing sets, and +the resulting CIs can be misleading. In general, across all 15 risk factors, most sites except for sites +2 and 4 have high generalizability measures, as shown in Figure C7 of the supplement. +Our results indicate that older age, male sex, higher Charlson comorbidity score, lower serum +albumin level, and higher creatinine, CRP and AST levels are associated with higher mortality +risk for patients hospitalized with a positive PCR COVID-19 test. The laboratory tests predictive +of mortality represent a mix of general health status (e.g., serum albumin), renal function (e.g., +creatinine), hepatic function (e.g., AST), and acute inflammatory response (e.g., CRP). These +results are consistent with previous findings in related studies of laboratory measurements to predict +COVID-19 mortality [Weber et al., 2022]. Recent literature suggests that measuring serum albumin +can identify COVID-19 infected patients who are more likely to progress to severe disease and +that serum albumin can serve as a useful marker for disease progression [Turcato et al., 2022]. +Our results corroborate this finding. The use of regularly collected laboratory tests, in addition to +baseline demographic and comorbidity information, can aid in the development of a clinically useful +prediction tool for mortality risk following hospital admission with COVID-19. Patients identified +as having a high risk for mortality could then be prioritized for closer monitoring and potentially + +Inference for Meta-Learning +27 +more aggressive interventions when deemed appropriate by the physician. +−5.0 +−2.5 +0.0 +2.5 +age 18−25 +age 26−49 +age 70−79 +age 80+ +albumin +AST +AST/ALT +bilirubin +charlson score +creatinine +CRP +lymphocyte count +neutrophil count +sex:female +WBC count +Log Hazard Ratio of Mortality +generalizability measure +<5% +5%−50% +>50% +method +RIFL +VMC +Fig. 6: RIFL and VMC 95% CIs for log hazard ratios of mortality within 14 days of hospitalization +with COVID-19 for each of the 15 baseline risk factors. Dots represent the point estimates from +individual sites, and darker color of the dots corresponds to lower generalizability measure of +individual sites. +8. +Conclusion and Discussion +The RIFL confidence interval, robust to the errors in separating the non-majority sites from the +majority sites, guarantees uniformly valid coverage of the prevailing model. When the majority +of sites are well separated from the other sites, the RIFL CI performs similarly to the oracle CI +assuming the knowledge of the majority group. +The majority rule is a crucial assumption for our proposal, and an empirical assessment of +the majority rule is vital to validate our proposed procedure. Our proposed RIFL method might +provide a heuristic assessment of the majority rule. Suppose the index set M defined in (16) has +the cardinality below 10% · M even for ρ approaching 1. In that case, we claim that the majority +rule may fail since it is difficult to verify the majority rule with most of the resampled data. As a +relaxation of the majority rule, we might relax the strict equality of the definition of V(θ) in (1) to +approximate equality, +Vκ(θ) := {1 ≤ l ≤ L : ∥θ(l) − θ∥2 ≤ κ}, +for some +κ > 0. +(40) +When κ is sufficiently close to zero, our main results may be generalized to hold under the majority +rule (Assumption 1) defined with Vκ(θ). + +28 +Guo, Li, Han & Cai +In practice, domain experts might believe that more than 50% (e.g. 80%) of the total sites are +similar. Our proposed RIFL can be directly extended to accommodate additional prior information. +For example, we may consider the 80% rule; that is, the domain experts expect more than 80% of +the sites to share the parameters of interest. We can modify our construction by replacing L/2 in +(16) and (17) with 80% · L. Importantly, the RIFL methods leveraging either the majority rule or +the 80% rule will guarantee uniformly valid CIs for the parameter of the prevailing model, provided +that there are indeed more than 80% of the sites in the prevailing set. However, the RIFL leveraging +the 80% rule will have a shorter length due to the additional prior information, as shown in Section +C.1 of the supplement where we compare the coverage and lengths of RIFL CIs under different +prior knowledge. It is also possible to consider alternative strategies for choosing ρ to incorporate +further data-adaptive assessment of prevailing set size based on {�pl}1≤l≤L. Theoretically justified +approaches to adaptively choose ρ(M) warrant further research. +We discuss two possible directions for robust information aggregation when the majority rule +or its relaxed version does not hold. Firstly, the target prediction model can be defined as the one +shared by the largest number of sites. If we cluster {θ(l)}1≤l≤L into subgroups according to their +similarity, the group of the largest size naturally defines a target model, even though this largest +group does not contain more than L/2 sites. Secondly, we can define the set Vκ(θ) in (40) with +a relatively large κ, that is, Vκ(θ) contains all sites with the parameters similar to but possibly +different from θ. We may still pool over the information belonging to Vκ(θ) through defining a group +distributionally robust model as in Meinshausen and B¨uhlmann [2015], Hu et al. [2018], Sagawa +et al. [2019], Guo [2020]. The post-selection problem persists in both settings. It is arguably vital to +account for the uncertainty in identifying useful sites. Our proposal is potentially useful to account +for the site selection and make inferences for these target models, which is left to future research. +Acknowledgement +The research was partly supported by the NSF grant DMS 2015373 as well as NIH grants R01GM140463, +R01HL089778, and R01LM013614. +References +Donald WK Andrews. Inconsistency of the bootstrap when a parameter is on the boundary of the +parameter space. Econometrica, pages 399–405, 2000. +Martin Arjovsky, L´eon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimiza- +tion. arXiv preprint arXiv:1907.02893, 2019. +Timothy B Armstrong, Michal Koles´ar, and Soonwoo Kwon. Bias-aware inference in regularized +regression models. arXiv preprint arXiv:2012.14823, 2020. +Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference +models. Biometrics, 61(4):962–973, 2005. +Alexandre Belloni, Victor Chernozhukov, and Christian Hansen. Inference on treatment effects +after selection among high-dimensional controls. The Review of Economic Studies, 81(2):608– +650, 2014. + +Inference for Meta-Learning +29 +Richard Berk, Lawrence Brown, Andreas Buja, Kai Zhang, and Linda Zhao. Valid post-selection +inference. The Annals of Statistics, 41(2):802–837, 2013. +Peter J Bickel and Anat Sakov. On the choice of m in the m out of n bootstrap and confidence +bounds for extrema. Statistica Sinica, pages 967–985, 2008. +Peter J Bickel, Ya’acov Ritov, and Alexandre B Tsybakov. Simultaneous analysis of lasso and +dantzig selector. The Annals of statistics, 37(4):1705–1732, 2009. +Jack Bowden, George Davey Smith, Philip C Haycock, and Stephen Burgess. Consistent estimation +in mendelian randomization with some invalid instruments using a weighted median estimator. +Genetic epidemiology, 40(4):304–314, 2016. +Gabriel A Brat, Griffin M Weber, Nils Gehlenborg, Paul Avillach, Nathan P Palmer, Luca Chiovato, +James Cimino, Lemuel R Waitman, Gilbert S Omenn, Alberto Malovini, et al. International +electronic health record-derived covid-19 clinical course profiles: the 4ce consortium. NPJ digital +medicine, 3(1):1–9, 2020. +Peter B¨uhlmann and Nicolai Meinshausen. +Magging: maximin aggregation for inhomogeneous +large-scale data. Proceedings of the IEEE, 104(1):126–135, 2015. +Peter B¨uhlmann and Sara van de Geer. Statistics for high-dimensional data: methods, theory and +applications. Springer Science & Business Media, 2011. +Stephen Burgess, Dylan S Small, and Simon G Thompson. +A review of instrumental variable +estimators for mendelian randomization. +Statistical methods in medical research, 26(5):2333– +2355, 2017. +T Tony Cai and Zijian Guo. Confidence intervals for high-dimensional linear regression: Minimax +rates and adaptivity. The Annals of statistics, 45(2):615–646, 2017. +T. Tony Cai and Zijian Guo. Semisupervised inference for explained variance in high dimensional +linear regression and its applications. Journal of the Royal Statistical Society: Series B (Statistical +Methodology), 82(2):391–419, 2020. +T Tony Cai, Zijian Guo, and Rong Ma. Statistical inference for high-dimensional generalized linear +models with binary outcomes. Journal of the American Statistical Association, pages 1–14, 2021a. +Tianxi Cai, Molei Liu, and Yin Xia. Individual data protected integrative regression analysis of +high-dimensional heterogeneous data. +Journal of the American Statistical Association, pages +1–15, 2021b. +Tianxi Cai, T Tony Cai, and Zijian Guo. Optimal statistical inference for individualized treatment +effects in high-dimensional models. Journal of the Royal Statistical Society: Series B (Statistical +Methodology), 83(4):669–719, 2021c. +Randy Carraghan and Panos M Pardalos. An exact algorithm for the maximum clique problem. +Operations Research Letters, 9(6):375–382, 1990. +Bibhas Chakraborty, Eric B Laber, and Yingqi Zhao. Inference for optimal dynamic treatment +regimes using an adaptive m-out-of-n bootstrap scheme. Biometrics, 69(3):714–723, 2013. + +30 +Guo, Li, Han & Cai +Xueying Chen and Min-ge Xie. A split-and-conquer approach for analysis of extraordinarily large +data. Statistica Sinica, pages 1655–1684, 2014. +Yixin Chen, Guozhu Dong, Jiawei Han, Jian Pei, Benjamin W Wah, and Jianyong Wang. Regression +cubes with lossless compression and aggregation. IEEE Transactions on Knowledge and Data +Engineering, 18(12):1585–1599, 2006. +Victor Chernozhukov, +Christian Hansen, +and Martin Spindler. +Post-selection and post- +regularization inference in linear models with many controls and instruments. 2015. +John Chodera, Alpha A Lee, Nir London, and Frank von Delft. Crowdsourcing drug discovery for +pandemics. Nature Chemistry, 12(7):581–581, 2020. +Ruthie Davi, Nirosha Mahendraratnam, Arnaub Chatterjee, C Jill Dawson, and Rachel Sherman. +Informing single-arm clinical trials with external controls. Nature Reviews Drug Discovery, 19 +(12):821–822, 2020. +Rui Duan, Chongliang Luo, Martijn J Schuemie, Jiayi Tong, C Jason Liang, Howard H Chang, +Mary Regina Boland, Jiang Bian, Hua Xu, John H Holmes, et al. Learning from local to global: +An efficient distributed algorithm for modeling time-to-event data. +Journal of the American +Medical Informatics Association, 27(7):1028–1036, 2020. +Parag Goyal, Justin J Choi, Laura C Pinheiro, Edward J Schenck, Ruijun Chen, Assem Jabri, +Michael J Satlin, Thomas R Campion Jr, Musarrat Nahid, Joanna B Ringel, et al. Clinical +characteristics of covid-19 in new york city. New England Journal of Medicine, 382(24):2372– +2374, 2020. +Zijian Guo. +Inference for high-dimensional maximin effects in heterogeneous regression models +using a sampling approach. arXiv preprint arXiv:2011.07568, 2020. +Zijian Guo. Causal inference with invalid instruments: Post-selection problems and a solution using +searching and sampling. arXiv preprint arXiv:2104.06911, 2021. +Zijian Guo, Hyunseung Kang, T Tony Cai, and Dylan S Small. Confidence intervals for causal +effects with invalid instruments by using two-stage hard thresholding with voting. Journal of the +Royal Statistical Society: Series B (Statistical Methodology), 80(4):793–815, 2018. +Zijian Guo, Prabrisha Rakshit, Daniel S Herman, and Jinbo Chen. Inference for the case probability +in high-dimensional logistic regression. The Journal of Machine Learning Research, 22(1):11480– +11533, 2021a. +Zijian Guo, Claude Renaux, Peter B¨uhlmann, and Tony Cai. Group inference in high dimensions +with applications to hierarchical testing. Electronic Journal of Statistics, 15(2):6633–6676, 2021b. +Larry Han, Angela Chen, Jason J Ong, Juliet Iwelunmor, and Joseph D Tucker. Crowdsourcing in +health and health research: a practical guide. 2018. +Larry Han, Jue Hou, Kelly Cho, Rui Duan, and Tianxi Cai. Federated adaptive causal estimation +(face) of target treatment effects. arXiv preprint arXiv:2112.09313, 2021. +Reid Hastie and Tatsuya Kameda. The robust beauty of majority rules in group decisions. Psy- +chological review, 112(2):494, 2005. + +Inference for Meta-Learning +31 +Neil T Heffernan and Cristina Lindquist Heffernan. The assistments ecosystem: Building a platform +that brings scientists and teachers together for minimally invasive research on human learning +and teaching. International Journal of Artificial Intelligence in Education, 24(4):470–497, 2014. +Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama. Does distributionally robust supervised +learning give robust classifiers? In International Conference on Machine Learning, pages 2029– +2037. PMLR, 2018. +Mahta Jahanshahi, Keith Gregg, Gillian Davis, Adora Ndu, Veronica Miller, Jerry Vockley, Cecile +Ollivier, Tanja Franolic, and Sharon Sakai. The use of external controls in fda regulatory decision +making. Therapeutic Innovation & Regulatory Science, 55(5):1019–1035, 2021. +Adel Javanmard and Andrea Montanari. +Confidence intervals and hypothesis testing for high- +dimensional regression. The Journal of Machine Learning Research, 15(1):2869–2909, 2014. +Hyunseung Kang, Anru Zhang, T Tony Cai, and Dylan S Small. Instrumental variables estimation +with some invalid instruments and its application to mendelian randomization. Journal of the +American statistical Association, 111(513):132–144, 2016. +Joseph DY Kang and Joseph L Schafer. Demystifying double robustness: A comparison of alter- +native strategies for estimating a population mean from incomplete data. Statistical science, 22 +(4):523–539, 2007. +Norbert L Kerr, R Scott Tindale, et al. Group performance and decision making. Annual review +of psychology, 55(1):623–655, 2004. +Kevin L Keys, Angel CY Mak, Marquitta J White, Walter L Eckalbar, Andrew W Dahl, Joel +Mefford, Anna V Mikhaylova, Mar´ıa G Contreras, Jennifer R Elhawary, Celeste Eng, et al. On +the cross-population generalizability of gene expression prediction models. PLoS genetics, 16(8): +e1008927, 2020. +Peter Kraft, Eleftheria Zeggini, and John PA Ioannidis. Replication in genome-wide association +studies. Statistical science: a review journal of the Institute of Mathematical Statistics, 24(4): +561, 2009. +Jason D Lee, Dennis L Sun, Yuekai Sun, and Jonathan E Taylor. Exact post-selection inference, +with application to the lasso. Annals of Statistics, 44(3):907–927, 2016. +Jason D Lee, Qiang Liu, Yuekai Sun, and Jonathan E Taylor. +Communication-efficient sparse +regression. The Journal of Machine Learning Research, 18(1):115–144, 2017. +Hannes Leeb and Benedikt M P¨otscher. Model selection and inference: Facts and fiction. Econo- +metric Theory, pages 21–59, 2005. +Jeffrey T Leek, Robert B Scharpf, H´ector Corrada Bravo, David Simcha, Benjamin Langmead, +W Evan Johnson, Donald Geman, Keith Baggerly, and Rafael A Irizarry. Tackling the widespread +and critical impact of batch effects in high-throughput data. Nature Reviews Genetics, 11(10): +733–739, 2010. +Runze Li, Dennis KJ Lin, and Bing Li. Statistical inference in massive data sets. Applied Stochastic +Models in Business and Industry, 29(5):399–409, 2013. + +32 +Guo, Li, Han & Cai +Sai Li, T Tony Cai, and Hongzhe Li. +Transfer learning for high-dimensional linear regression: +Prediction, estimation, and minimax optimality. arXiv preprint arXiv:2006.10593, 2020. +Heng Lian and Zengyan Fan. Divide-and-conquer for debiased l 1-norm support vector machine in +ultra-high dimensions. The Journal of Machine Learning Research, 18(1):6691–6716, 2017. +Wodan Ling, Jiuyao Lu, Ni Zhao, Anju Lulla, Anna M Plantinga, Weijia Fu, Angela Zhang, +Hongjiao Liu, Hoseung Song, Zhigang Li, et al. Batch effects removal for microbiome data via +conditional quantile regression. Nature Communications, 13(1):1–14, 2022. +Molei Liu, Yin Xia, Kelly Cho, and Tianxi Cai. Integrative high dimensional multiple testing with +heterogeneity under data sharing constraints. J. Mach. Learn. Res., 22:126–1, 2021. +Subha Maity, Yuekai Sun, and Moulinath Banerjee. Meta-analysis of heterogeneous data: integra- +tive sparse regression in high-dimensions. Journal of Machine Learning Research, 23(198):1–50, +2022. +Raymond H Mak, Michael G Endres, Jin H Paik, Rinat A Sergeev, Hugo Aerts, Christopher L +Williams, Karim R Lakhani, and Eva C Guinan. Use of crowd innovation to develop an artificial +intelligence–based solution for radiation therapy targeting. JAMA oncology, 5(5):654–661, 2019. +Nicolai Meinshausen and Peter B¨uhlmann. Maximin effects in inhomogeneous large-scale data. The +Annals of Statistics, 43(4):1801–1830, 2015. +Yang Ning and Han Liu. A general theory of hypothesis tests and confidence regions for sparse +high dimensional models. The Annals of Statistics, 45(1):158–195, 2017. +Jonas Peters, Peter B¨uhlmann, and Nicolai Meinshausen. +Causal inference by using invariant +prediction: identification and confidence intervals. Journal of the Royal Statistical Society: Series +B (Statistical Methodology), 78(5):947–1012, 2016. +Laila Rasmy, Yonghui Wu, Ningtao Wang, Xin Geng, W Jim Zheng, Fei Wang, Hulin Wu, Hua Xu, +and Degui Zhi. A study of generalizability of recurrent neural network-based predictive models +for heart failure onset risk using a large and heterogeneous ehr data set. Journal of biomedical +informatics, 84:11–16, 2018. +Dominik Rothenh¨ausler, Nicolai Meinshausen, and Peter B¨uhlmann. Confidence intervals for max- +imin effects in inhomogeneous large-scale data. In Statistical Analysis for High-Dimensional Data, +pages 255–277. Springer, 2016. +Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust +neural networks for group shifts: On the importance of regularization for worst-case generaliza- +tion. arXiv preprint arXiv:1911.08731, 2019. +Ryan G Short, Dana Middleton, Nicholas T Befera, Raj Gondalia, and Tina D Tailor. Patient- +centered radiology reporting: using online crowdsourcing to assess the effectiveness of a web-based +interactive radiology report. Journal of the American College of Radiology, 14(11):1489–1497, +2017. +Robert D Sorkin, Ryan West, and Donald E Robinson. Group performance depends on the majority +rule. Psychological Science, 9(6):456–463, 1998. + +Inference for Meta-Learning +33 +James Surowiecki. The wisdom of crowds. Anchor, 2005. +Yebin Tao and Haoda Fu. Doubly robust estimation of the weighted average treatment effect for a +target population. Statistics in medicine, 38(3):315–325, 2019. +Ye Tian and Yang Feng. +Transfer learning under high-dimensional generalized linear models. +Journal of the American Statistical Association, (just-accepted):1–30, 2022. +Jiayi Tong, Chongliang Luo, Md Nazmul Islam, Natalie E Sheils, John Buresh, Mackenzie Edmond- +son, Peter A Merkel, Ebbing Lautenbach, Rui Duan, and Yong Chen. Distributed learning for +heterogeneous clinical data with application to integrating covid-19 data across 230 sites. NPJ +digital medicine, 5(1):1–8, 2022. +Lorenzo Trippa, Levi Waldron, Curtis Huttenhower, and Giovanni Parmigiani. Bayesian nonpara- +metric cross-study validation of prediction methods. +The Annals of Applied Statistics, 9(1): +402–428, 2015. +Anastasios A Tsiatis. Semiparametric theory and missing data. 2006. +Gianni Turcato, Arian Zaboli, Irena Kostic, Barbara Melchioretto, Laura Ciccariello, Eleonora +Zaccaria, Alessia Olivato, Antonio Maccagnani, Norbert Pfeifer, and Antonio Bonora. Severity +of sars-cov-2 infection and albumin levels recorded at the first emergency department evaluation: +a multicentre retrospective observational study. Emergency Medicine Journal, 39(1):63–69, 2022. +Sara van de Geer, Peter B¨uhlmann, Ya’acov Ritov, and Ruben Dezeure. On asymptotically optimal +confidence regions and tests for high-dimensional models. The Annals of Statistics, 42(3):1166– +1202, 2014. +Steffen Ventz, Albert Lai, Timothy F Cloughesy, Patrick Y Wen, Lorenzo Trippa, and Brian M +Alexander. Design and evaluation of an external control arm using prior clinical trials and real- +world datadesign and evaluation of an external control arm. Clinical Cancer Research, 25(16): +4993–5001, 2019. +Thanh Vinh Vo, Trong Nghia Hoang, Young Lee, and Tze-Yun Leong. Federated estimation of +causal effects from observational data. arXiv preprint arXiv:2106.00456, 2021. +Cheng Wang, Larry Han, Gabriella Stein, Suzanne Day, Cedric Bien-Gund, Allison Mathews, +Jason J Ong, Pei-Zhen Zhao, Shu-Fang Wei, Jennifer Walker, et al. Crowdsourcing in health and +medical research: a systematic review. Infectious diseases of poverty, 9(1):1–9, 2020. +Ruoyu Wang, Qihua Wang, and Wang Miao. A robust fusion-extraction procedure with summary +statistics in the presence of biased sources. arXiv preprint arXiv:2108.12600, 2021. +Xiaozhou Wang, Zhuoyi Yang, Xi Chen, and Weidong Liu. Distributed inference for linear support +vector machine. Journal of Machine Learning Research, 20(113):1–41, 2019. +Griffin M Weber, Chuan Hong, Zongqi Xia, Nathan P Palmer, Paul Avillach, Sehi L’Yi, Mark S +Keller, Shawn N Murphy, Alba Guti´errez-Sacrist´an, Clara-Lea Bonzel, et al. International com- +parisons of laboratory values from the 4ce collaborative to predict covid-19 mortality. NPJ digital +medicine, 5(1):1–8, 2022. + +34 +Guo, Li, Han & Cai +Frank Windmeijer, Helmut Farbmacher, Neil Davies, and George Davey Smith. On the use of +the lasso for instrumental variables estimation with some invalid instruments. Journal of the +American Statistical Association, 114(527):1339–1350, 2019. +Frank Windmeijer, Xiaoran Liang, Fernando P Hartwig, and Jack Bowden. The confidence interval +method for selecting valid instrumental variables. Journal of the Royal Statistical Society: Series +B (Statistical Methodology), 83(4):752–776, 2021. +Zunyou Wu and Jennifer M McGoogan. Characteristics of and important lessons from the coron- +avirus disease 2019 (covid-19) outbreak in china: summary of a report of 72 314 cases from the +chinese center for disease control and prevention. jama, 323(13):1239–1242, 2020. +Laure Wynants, Ben Van Calster, Gary S Collins, Richard D Riley, Georg Heinze, Ewoud Schuit, +Marc MJ Bonten, Darren L Dahly, Johanna A Damen, Thomas PA Debray, et al. Prediction +models for diagnosis and prognosis of covid-19: systematic review and critical appraisal. bmj, +369, 2020. +Min-ge Xie and Peng Wang. Repro samples method for finite-and large-sample inferences. arXiv +preprint arXiv:2206.06421, 2022. +Ruoxuan Xiong, Allison Koenecke, Michael Powell, Zhu Shen, Joshua T Vogelstein, and Su- +san Athey. +Federated causal inference in heterogeneous observational data. +arXiv preprint +arXiv:2107.11732, 2021. +Cun-Hui Zhang and Stephanie S Zhang. Confidence intervals for low dimensional parameters in +high dimensional linear models. Journal of the Royal Statistical Society: Series B (Statistical +Methodology), 76(1):217–242, 2014. +Tianqi Zhao, Guang Cheng, and Han Liu. A partially linear framework for massive heterogeneous +data. Annals of statistics, 44(4):1400, 2016. +Yinchu Zhu and Jelena Bradic. Linear hypothesis testing in dense high-dimensional linear models. +Journal of the American Statistical Association, 113(524):1583–1600, 2018. +Hui Zou. The adaptive lasso and its oracle properties. Journal of the American statistical associa- +tion, 101(476):1418–1429, 2006. + +Inference for Meta-Learning +1 +A. +Extra Method and Theory +A.1. +RIFL Algorithm +Algorithm 1 Proposed sampling method for the multi-source inference +Input: Site-specific estimators {�β(l), �σl}1≤l≤L; dissimilarity measures { �Dl,k}1≤l 0. +Output: Confidence interval CI; measure of generalizability for each site +1: for l ← 1 to L − 1 do +2: +for k ← l + 1 to L do +3: +Compute �Ll,k = �β(l) − �β(k) and � +SE( �Ll,k) = +� +�σ2 +l + �σ2 +k; +4: +end for +5: end for +6: for m ← 1 to M do +7: +Resample the dissimilarity measures { �D[m] +l,k , �L[m] +l,k }1≤l 0; The high-dimensional vector η(l) is assumed to be of less than +s non-zero entries. + +2 +Guo, Li, Han & Cai +Algorithm 2 High-dimensional Distance Measures +Input: the multi-source data {X(l), Y (l)}1≤l≤L. +Output: {�β(l), � +SE(�β(l)), �Dl,k, � +SE( �Dl,k)}1≤l 0, +∥�η(l) − η(l)∥1 ≤ Cs (log d/n)1/2 +and +∥�η(l) +Sc +l − η(l) +Sc +l ∥1 ≤ C0∥�η(l) +Sl − η(l) +Sl ∥1 +where Sl denotes the support of η(l) and C > 0 and C0 > 0 are positive constants. +In the high-dimensional linear model, Theorem 7.2 of Bickel et al. [2009] established that the Lasso +estimator satisfies the condition (B). In the high-dimensional logistic regression, see Proposition 1 +of Guo et al. [2021a] for an example of establishing that the penalized MLE estimator �θ(l) defined +in (29) satisfies the condition (B); see also the references within there. +Theorem 5. Suppose that Conditions (A1) and (B) hold for the high-dimensional linear re- +gression or Conditions (A1), (A2), and (B) hold for the high-dimensional logistic regression, τn ≍ +(log n)1/2 defined in (33) satisfies τns log d/√n → 0. For any constant 0 < α < 1, the dissimilarity +estimator �Dl,k defined in (36) and the standard error estimator � +SE( �Dl,k) defined in (37) satisfy (3) +for 1 ≤ l < k ≤ L. +We present the proof of the above theorem in Section B.5. + +Inference for Meta-Learning +3 +A.3. +Influence function of the doubly robust ATE estimator +For an i.i.d. sample from the l-th source population of size nl, we use X(l) +i +∈ Rp to denote the +covariate vector in the i-th observation and X(l) +ij +∈ R to denote the j-th covariate in the i-th +observation. +We consider the case where the outcome regression functions and the propensity score are +estimated with generalized linear models (GLMs). Specifically, for the propensity score model, we +fit the following GLM: +E +� +A(l)|X(l)� += h +� +�αl,0 + +B +� +j=1 +αl,jψj(X(l)) +� +� = h +� +α⊤ +l � +X(l)� +, +where {ψ1(·), . . . , ψB(·)} is a set of basis functions, � +X(l) = (1, ψ1(X(l)), . . . , ψB(X(l)))⊤ and αl = +(αl,0, αl,1, . . . , αl,B). One simple example is to take ψj(X(l) +i ) = X(l) +ij for j ∈ {1, . . . , p}, and we get +a GLM that includes the main effect of each covariate. We estimate the coefficient by solving the +following estimating equation: +1 +nl +nl +� +i=1 +� +X(l) +i +� +A(l) +i +− h +� +α⊤ � +X(l) +i +�� += 0, +and obtain an estimated coefficient which we denote as �αl. +For the outcome regressions, we fit generalized linear models within each treatment arm in each +source site: +E(Y (l) | A(l) = a, X(l)) = g +� +�γ(l) +a,0 + +Bγ +� +j=1 +γ(l) +a,jφj(X(l)) +� +� = g +� +[W (l)]⊤γ(l) +a +� +, +for a ∈ {0, 1}, +where W (l) = (1, φ1(X(l)), . . . , φBγ(X(l))) for some set of basis functions {φ1(·), . . . , φBγ(·)}, and +γ(l) +a += (γ(l) +a,0, γ(l) +a,1, . . . , γ(l) +a,Bγ). Let I{·} denote the indicator function. We estimate the coefficients +γ(l) +a +by solving the following estimating equation: +1 +nl +nl +� +i=1 +I +� +A(l) +i += a +� +W (l) +i +� +Y (l) +i +− g +� +[W (l) +i ]⊤γ(l) +a +�� += 0, +and we denote the estimate as �γ(l) +a . +For the density ratio, we consider an exponential tilt model, ωl(X(l); η(l)) = exp([η(l)]⊤� +W (l)), +where � +W (l) = (1, ϕ1(X(l)), . . . , ϕBω(X(l)))⊤for a set of basis functions {ϕ1(·), . . . , ϕBω(·)}. +We +estimate the parameter η(l) by solving the following estimating equation +1 +nl +nl +� +i=1 +exp +� +[η(l)]⊤� +W (l) +i +� +� +W (l) +i += 1 +N +N +� +j=1 +� +W T +j /N +where � +W T +j += (1, ϕ1(XT +j ), . . . , ϕBω(XT +j ))⊤ and XT +j +denotes the j-th observation in the target +dataset, for j ∈ {1, . . . , N}. We denote the resulting estimate by �η(l). Essentially, we estimate + +4 +Guo, Li, Han & Cai +the parameter η(l) by matching the sample mean of a set of basis functions in the source dataset +and the much larger target dataset. +Let α∗ +l , γ(l),∗ +0 +,γ(l),∗ +1 +and η(l),∗ denote the probabilistic limits of �αl, �γ(l) +0 , �γ(l) +1 +and �η(l), respectively. +The precise definitions of these population parameters are given in the proof of Theorem 6. Define +the following matrices: +Cα,l = El +� +� +X(l)h′ � +(α∗ +l )⊤ � +X(l)� � +� +X(l)�⊤� +; +Cγa,l = El +� +I +� +A(l) = a +� +W (l)g′ � +[W (l)]⊤γ(l),∗ +a +� +[W (l)]⊤� +, +a ∈ {0, 1}; +Cη,l = −El +� +exp +� +[η(l),∗]⊤� +W (l)� +� +W (l)[� +W (l)]⊤� +, +where El denotes the expectation with respect to the joint distribution of (X(l), A(l), Y (l)) in the +l-th source population. +Recall from Section 5.3 that the target ATE estimator takes the form �θ(l) = � +M(l) + �δ(l) where +� +M(l) = 1 +N +N +� +i=1 +� +m(1, XT +i ; �γ(l) +1 ) − m(0, XT +i ; �γ(l) +0 ) +� +and +�δ(l) = 1 +nl +nl +� +i=1 +ωl(X(l) +i ; �η(l)) +� 1 +� +a=0 +(−1)a+1I{A(l) +i += a} +πl(a, X(l) +i ; �αl) +{Y (l) +i +− m(A(l) +i , X(l) +i ; �γ(l) +a )} +� +. +In this case, we have that +m(a, X(l) +i ; �γ(l) +a ) = g +� +[W (l) +i ]⊤�γ(l) +a +� +; +m(a, XT +i ; �γ(l) +a ) = g +� +[W T +i ]⊤�γ(l) +a +� +; +ωl(X(l) +i ; �η(l)) = exp +� +[�η(l)]⊤� +W (l) +i +� +; +πl(a, X(l) +i ; �αl) = h +� +(�αl)⊤ � +X(l) +i +�a � +1 − h +� +(�αl)⊤ � +X(l) +i +��(1−a) +. +For the ease of notation, we define the functions τγ0,γ1(·) and ξη,α,γ0,γ1(·) such that +τγ0,γ1(x(l)) = m(1, x(l); γ1) − m(0, x(l); γ0), +and +ξη,α,γ0,γ1(x(l), a(l), y(l)) = ωl(x(l); η) +� 1 +� +a=0 +(−1)a+1I{a(l) = a} +πl(a, x(l); α) +{y(l) − m(a, x(l); γa} +� +. +These functions will appear frequently in the following derivation and in the influence function of +�θ(l). Also note that the functions τ and ξ are indexed by the nuisance model parameters. + +Inference for Meta-Learning +5 +Define the following quantities: +dγ0 = −ET +� +g′ � +[W T ]⊤γ(l),∗ +0 +� +W T � ++ El +� +�exp +� +[η(l),∗]⊤� +W (l)� +I{A(l) = 0} +1 − h +� +[ � +X(l)]⊤α∗ +l +� +� +g′ � +[W (l)]⊤γ(l),∗ +0 +� +W (l)� +� +� ; +dγ1 = ET +� +g′ � +[W T ]⊤γ(l),∗ +1 +� +W T � +− El +� +�exp +� +[η(l),∗]⊤� +W (l)� +I{A(l) = 1} +h +� +[ � +X(l)]⊤α∗ +l +� +� +g′ � +[W (l)]⊤γ(l),∗ +1 +� +W (l)� +� +� ; +dη = El +� +exp +� +[η(l),∗]⊤� +W (l)� +� +W (l) +� 1 +� +a=0 +(−1)a+1I{A(l) = a} +πl(a, X(l); α∗ +l ) +� +Y (l) − g +� +[W (l)]⊤γ(l),∗ +a +���� +; +dα = El +� +exp +� +[η(l),∗]⊤� +W (l)� � 1 +� +a=0 +−I{A(l) = a}π′ +l(a, X(l); α∗ +l ) +π2 +l (a, X(l); α∗ +l ) +� +Y (l) − g +� +[W (l)]⊤γ(l),∗ +a +��� +� +X(l) +� +. +Theorem 6. When N ≫ nl, the target ATE estimator �θ(l) is asymptotically linear with influ- +ence function τθ such that +τθ(x(l), a(l), y(l)) = ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +(x(l), a(l), y(l)) ++ d⊤ +γ0C−1 +γ0,lI +� +a(l) = 0 +� +w(l) � +y(l) − g +� +[w(l)]⊤γ(l),∗ +0 +�� ++ d⊤ +γ1C−1 +γ1,lI +� +a(l) = 1 +� +w(l) � +y(l) − g +� +[w(l)]⊤γ(l),∗ +1 +�� ++ d⊤ +η C−1 +η,l +� +exp +� +[η(l),∗]⊤ �w(l)� +�w(l) − ET +� +� +W T �� ++ d⊤ +αC−1 +α,l�x(l) � +a(l) − h +� +(α∗ +l )⊤�x(l)�� +. +Here the vectors �x(l), w(l) and �w(l) denote the vectors of basis functions derived from the covariate +vector x(l). +We present the proof of the above theorem in Section B.6. +The variance of �θ(l) can be consistently estimated by the empirical variance of the function �τθ +on the source data, where �τθ is an estimate of τθ obtained by plugging in the estimates for the +nuisance model parameters (η(l),∗, α∗ +l , γ(l),∗ +0 +, γ(l),∗ +1 +), the partial derivatives (dγ0, dγ1, dη, dα), and +the matrices Cγ0,l, Cγ1,l, Cη,l and Cα,l. +Consistent estimators for these partial derivatives and +matrices are given in the proof of Theorem 6. +B. +Proofs +B.1. +Proof of Theorem 1 +Denote the observed data by O. We define the following event for the data O, +E = +� +� +� max +1≤l 0 such that +���f(U [m] = U | O) − f(U [m] = �U | O) +��� ≤ C +� +L(L − 1)∥U − �U∥∞. +Then we establish +���� +� +[f(U [m] = U | O) − f(U [m] = �U | O)] · 1{∥U− �U∥∞≤errn(M,ν)}dU · 1O∈E +���� +≤ C +� +L(L − 1) · errn(M, ν) · +� +1{∥U− �U∥∞≤errn(M,ν)}dU · 1O∈E += C +� +L(L − 1) · errn(M, ν) · [2errn(M, ν)]L(L−1) · 1O∈E. +(49) +Since errn(M, ν) → 0 and c(ν) is a positive constant, then there exists a positive integer M0 such +that +C +� +L(L − 1) · errn(M, ν) ≤ 1 +2c(ν) +for +M ≥ M0. +We combine the above inequality, (47), (48) and (49) and obtain that for M ≥ M0, +P +� +∥U − �U∥∞ ≤ errn(M, ν) | O +� +· 1O∈E ≥ 1 +2c(ν) · [2errn(M, ν)]L(L−1) · 1O∈E. +Together with (45), we establish that for M ≥ M0, +P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) | O +� +· 1O∈E +≥ 1 − exp +� +−M · 1 +2c(ν) · [2errn(M, ν)]L(L−1) · 1O∈E +� += +� +1 − exp +� +−M · 1 +2c(ν) · [2errn(M, ν)]L(L−1) +�� +· 1O∈E. +(50) +With EO denoting the expectation taken with respect to the observed data O, we further integrate +with respect to O and establish that for M ≥ M0, +P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) +� += EO +� +P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) | O +�� +≥ EO +� +P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) | O +� +· 1O∈E +� +≥ EO +�� +1 − exp +� +−M · 1 +2c(ν) · [2errn(M, ν)]L(L−1) +�� +· 1O∈E +� +. +By the definition errn(M, ν) = 1 +2 +� +2 log n +c(ν)M +� +1 +L(L−1) , we establish that for M ≥ M0, +P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) +� +≥ (1 − n−1) · P (E) . + +Inference for Meta-Learning +9 +We further apply (43) and establish +lim inf +n→∞ +lim +M→∞ P +� +min +1≤m≤M ∥U [m] − �U∥∞ ≤ errn(M, ν) +� +≥ P (E) ≥ 1 − ν. +B.2. +Proof of Theorem 2 +In the following, we first conduct the analysis by fixing the sample size n. Define the separation +Lmin(n) = min +l∈Vc |β(l) − β∗|. +Note that Lmin(n) is a function of the sample size n and might decrease to zero with a growing +sample size n. We define the event +E1 = +� +� +� min +1≤m≤M +max +1≤l L/2. +The above inequality implies that m∗ ∈ M, and there exists 1 ≤ l ≤ L such that +l ∈ V ∩ �V[m∗]. +(53) +We introduce the following lemma to quantify the set �V[m∗] defined in (17). The proof of the +following lemma is postponed to Section B.2.1. +Lemma 1. Assume that the event E1 holds and the index m∗ satisfies (51). If the indexes l, k +satisfy �H[m∗] +l,k += 1, then we have +����� +Ll,k +� +SE( �Ll,k) +����� ≤ ρ(M) · T + errn(M, ν). +(54) +In addition, if ρ(M) · T ≥ errn(M, ν) and +2ρ(M) · T · +max +j1∈V,j2∈Vc � +SE( �Lj1,j2) < Lmin(n) +(55) + +10 +Guo, Li, Han & Cai +then +�V[m∗] = V(θ∗). +(56) +We now analyze P +�� +θ∗ ∈ CI[m∗]� +∩ E1 +� +and then establish the coverage property by applying +(52). For a given n, since ρ(M) → 0, there exists Mn such that if M ≥ Mn, ρ(M) · T ≥ errn(M, ν) +and (55) holds. Hence, for M ≥ Mn, we have +P +�� +θ∗ ∈ CI[m∗]� +∩ E1 +� += P +� +� +� +� +� +������ +� +l∈V +� +�β(l) − β∗� +/�σ2 +l +�� +l∈V 1/�σ2 +l +������ +≤ zα1/2 +� +� +� ∩ E1 +� +� +≥ P +� +� +������ +� +l∈V +� +�β(l) − β∗� +/�σ2 +l +�� +l∈V 1/�σ2 +l +������ +≤ zα1/2 +� +� − [1 − P (E1)] . +By taking limit with respect to M, we have +lim inf +M→∞ P +�� +θ∗ ∈ CI[m∗]� +∩ E1 +� +≥ P +� +� +������ +� +l∈V +� +�β(l) − β∗� +/�σ2 +l +�� +l∈V 1/�σ2 +l +������ +≤ zα1/2 +� +� − 1 + lim inf +M→∞ P (E1) . +Together with Theorem 1 and +� +�β(l), �σl +� +1≤l≤L satisfying (2), we establish Theorem 2. +B.2.1. +Proof of Lemma 1 +Proof of (54). For �H[m∗] +l,k += 1, we apply (14) and establish +������ +�L[m∗] +l,k +� +SE( �Ll,k) +������ +≤ ρ(M) · T. +(57) +We apply (51) and establish +������ +�L[m∗] +l,k +− Ll,k +� +SE( �Ll,k) +������ +≤ errn(M, ν). +(58) +We establish (54) by applying (57) and (58) and the triangle inequality +����� +Ll,k +� +SE( �Ll,k) +����� ≤ +������ +�L[m∗] +l,k +− Ll,k +� +SE( �Ll,k) +������ ++ +������ +�L[m∗] +l,k +� +SE( �Ll,k) +������ +. +Proof of (56). For k ∈ V, we apply (51) together with the condition ρ(M) · T ≥ errn(M, ν) and +establish �H[m∗] +k,j += 1 for any j ∈ V. By the majority rule, we have +��� �H[m∗] +k,· +��� +0 > L/2 for k ∈ V. +For k ̸∈ V, we apply (54) together with the condition (55) and establish �H[m∗] +k,j += 0 for j ∈ V. +By the majority rule, we have +��� �H[m∗] +k,· +��� +0 < L/2 for k ̸∈ V. Hence, �V[m∗] = V. + +Inference for Meta-Learning +11 +B.3. +Proof of Theorem 3 +Recall V∗ = V(θ∗). We define the events +E2 = +���� �L[m] +l,k − �Ll,k +��� /� +SE( �Ll,k) ≤ +� +2 log n + 2 log M +� +E3 = +����Ll,k − �Ll,k +��� /� +SE( �Ll,k) ≤ +� +2 log n +� +. +By (11) and (42), we apply the union bound and establish +lim +n→∞ lim +M→∞ P(E2 ∩ E3) = 1. +(59) +By the triangle inequality, we have +��� �L[m] +l,k /� +SE( �Ll,k) − Ll,k/� +SE( �Ll,k) +��� ≤ +��� �L[m] +l,k − �Ll,k +��� /� +SE( �Ll,k) + +���Ll,k − �Ll,k +��� /� +SE( �Ll,k) +(60) +On the event E2 ∩ E3, we have +��� �L[m] +l,k /� +SE( �Ll,k) − Ll,k/� +SE( �Ll,k) +��� ≤ 2 +� +2 log n + 2 log M. +(61) +The well-separation condition (25) implies that, for l ∈ V and k ∈ Vc, +���Ll,k/� +SE( �Ll,k) +��� > 2 +� +2 log n + 2 log M + ρ(M) · T. +On the event E2 ∩ E3, if l ∈ V and k ∈ Vc satisfies (61), then +��� �L[m] +l,k /� +SE( �Ll,k) +��� > ρ(M) · T. +which is equivalent to �H[m] +l,k = 0. That is, k ̸∈ V[m] for m ∈ M. +The above derivation shows that if all indexes k ∈ Vc satisfy the well separation condition (61), +we have V[m] ⊂ V∗ for m ∈ M. Since |V[m]| > L/2 and |V∗| = ⌊L/2⌋ + 1, we have V[m] = V∗ for +m ∈ M. This implies P (CI = CIora) ≥ P(E2 ∩ E3). Then the theorem follows from (59). +B.4. +Proof of Theorem 4 +Define +SE( �Dl,k) = +� +4γ⊺C(l)γ/nl + 4γ⊺C(k)γ/nk + 1/ min{nl, nk} +and +SE0( �Dl,k) = +� +4γ⊺C(l)γ/nl + 4γ⊺C(k)γ/nk. +Since �θ(l) and �C(l) are consistent estimators of θ(l) and C(l), respectively, we have +� +SE( �Dl,k)/SE( �Dl,k) d→ 1. +(62) +Define cn = (1/ min{nl, nk})1/4. It follows from (26) that +lim sup +n→∞ P +� +∥�γ − γ∥2 +2/� +SE( �Dl,k) ≥ cnzα +� += 0. +(63) + +12 +Guo, Li, Han & Cai +By the decomposition (27), we have +P +���� �Dl,k − Dl,k +���/� +SE( �Dl,k) ≥ zα +� +≤ P +� +|2⟨�γ − γ, γ⟩|/� +SE( �Dl,k) ≥ (1 − cn)zα +� ++ P +� +∥�γ − γ∥2 +2/� +SE( �Dl,k) ≥ cnzα +� +≤ P +� +|2⟨�γ − γ, γ⟩|/SE0( �Dl,k) ≥ (1 − cn)zα · +� +SE( �Dl,k) +SE( �Dl,k) +� ++ P +� +∥�γ − γ∥2 +2/� +SE( �Dl,k) ≥ cnzα +� +, +(64) +where the first inequality follows from the union bound and the second inequality follows from the +relation SE( �Dl,k) ≥ SE0( �Dl,k). We establish (3) by combing the above decomposition, (62), (63), +and +⟨�γ − γ, γ⟩ +� +4γ⊺C(l)γ/nl + 4γ⊺C(k)γ/nl +d→ N(0, 1). +B.5. +Proof of Theorem 5 +The following decomposition is crucial to constructing a consistent estimator of the error compo- +nent: for any vector u ∈ Rd, +[�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +W (l) +i +� +X(l) +i (Yi − h([ � +X(l) +i ]⊺�η(l))) − ⟨η(l) − �η(l), �γ⟩ += +� +�Σ(l)�u(l) +k − �γ +�⊺ +(η(l) − �η(l)) + [�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +W (l) +i ϵ(l) +i +� +X(l) +i ++ [�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +∆(l) +i +� +X(l) +i , +(65) +with �Σ(l) = +1 +|S +(l) +2 | +� +i∈S +(l) +2 +� +X(l) +i +� +� +X(l) +i +�⊺ +and the approximation error ∆(l) +i +defined as +∆(l) +i += W (l) +i +· +� 1 +0 +(1 − t)h′′([ � +X(l) +i ]⊺�η(l) + t[ � +X(l) +i ]⊺[η(l) − �η(l)])dt · ([ � +X(l) +i ]⊺[η(l) − �η(l)])2. +(66) +Note that for the linear outcome model with h(x) = x, the approximation error ∆(l) +i += 0. Define +�Dl,k = ∥�θ(l) − �θ(k)∥2 +2 + 2�δ(l) +k − 2�δ(k) +l +. +Since Dl,k ≥ 0, we have +��� �Dl,k − Dl,k +��� ≤ +��� �Dl,k − Dl,k +��� . +To establish (3), it is sufficient to establish +lim sup +n→∞ P +���� �Dl,k − Dl,k +���/� +SE( �Dl,k) ≥ zα +� +≤ α +for +0 < α < 1 +(67) +where zα denotes the upper quantile of a standard normal distribution. + +Inference for Meta-Learning +13 +In the following, we establish (67). We apply (31) and (65) and obtain +�Dl,k − Dl,k = −∥�γ − γ∥2 +2 ++ +� +�Σ(l)�u(l) +k − �γ +�⊺ +(η(l) − �η(l)) + [�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +W (l) +i ϵ(l) +i +� +X(l) +i ++ [�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +∆(l) +i +� +X(l) +i ++ +� +�Σ(k)�u(k) +l +− �γ +�⊺ +(η(k) − �η(k)) + [�u(k) +l +]⊺ +1 +|S(k) +2 | +� +i∈S +(k) +2 +W (k) +i +ϵ(k) +i +� +X(k) +i ++ [�u(k) +l +]⊺ +1 +|S(k) +2 | +� +i∈S +(k) +2 +∆(k) +i +� +X(k) +i +. +(68) +We apply Lemma 5 in Guo et al. [2021a] and establish +[�u(l) +k ]⊺ +1 +|S +(l) +2 | +� +i∈S +(l) +2 W (l) +i ϵ(l) +i +� +X(l) +i ++ [�u(k) +l +]⊺ +1 +|S +(k) +2 +| +� +i∈S +(k) +2 +W (k) +i +ϵ(k) +i +� +X(k) +i +� +�V(l) +k + �V(k) +l +d→ N(0, 1) +(69) +with +�V(l) +k = +� +�u(l) +k +�⊺ +� +� +1 +|S(l) +2 |2 +� +i∈S +(l) +2 +W (l) +i +� +X(l) +i [ � +X(l) +i ]⊺ +� +� �u(l) +k , +and +�V(k) +l += +� +�u(k) +l +�⊺ +� +� +1 +|S(k) +2 |2 +� +i∈S +(k) +2 +W (k) +i +� +X(k) +i +[ � +X(k) +i +]⊺ +� +� �u(k) +l +. +By condition (B), we have +∥�γ − γ∥2 +2 ≤ C +s log d +min{nl, nk}. +(70) +We apply (22) in Guo et al. [2021a] and establish +��� +� +�Σ(l)�u(l) +k − �γ +�⊺ +(η(l) − �η(l)) +��� ≤ C s log d +nl +, +��� +� +�Σ(k)�u(k) +l +− �γ +�⊺ +(η(k) − �η(k)) +��� +≤ C s log d +nk +. +(71) +We apply (23) in Guo et al. [2021a] and establish +������ +[�u(l) +k ]⊺ +1 +|S(l) +2 | +� +i∈S +(l) +2 +∆(l) +i +� +X(l) +i +������ +≤ Cτn +s log d +nl +������ +[�u(k) +l +]⊺ +1 +|S(k) +2 | +� +i∈S +(k) +2 +∆(k) +i +� +X(k) +i +������ +≤ Cτn +s log d +nk +(72) +We establish (67) by combining the decomposition (68), the asymptotic limit (69), the error +bounds (70), (71), (72), and the condition τns log d/√n → 0. + +14 +Guo, Li, Han & Cai +B.6. +Proof of Theorem 6 +We let α∗ +l denote the probabilistic limit of �αl, which satisfies the following equation: +El +� +� +X(l) � +A(l) − h +� +(α∗ +l )⊤ � +X(l)��� += 0, +where El denotes the expectation with respect to the joint distribution of (X(l), A(l), Y (l)) in the +l-th source population. By the theory of estimating equations, �αl admits the following asymptotic +linear expansion: +�αl − α∗ +l = 1 +nl +nl +� +i=1 +C−1 +α,l � +X(l) +i +� +A(l) +i +− h +� +(α∗ +l )⊤ � +X(l) +i +�� ++ oP (n−1/2 +l +), +where the matrix Cα,l is defined as +Cα,l = El +� +� +X(l)h′ � +(α∗ +l )⊤ � +X(l)� � +� +X(l)�⊤� +. +Note that Cα,l can be consistently estimated by +�Cα,l = 1 +nl +nl +� +i=1 +� +X(l) +i h′ � +(�αl)⊤ � +X(l) +i +� � +� +X(l) +i +�⊤ +. +Let γ(l),∗ +a +denote the probabilistic limit of �γ(l) +a , which solves the following equation: +El +� +I +� +A(l) = a +� +W (l) � +Y (l) − g +� +[W (l)]⊤γ(l),∗ +a +��� += 0. +The estimator �γ(l) +a +admits the following asymptotic linear expansion: +�γ(l) +a − γ(l),∗ +a += 1 +nl +nl +� +i=1 +C−1 +γa,lI +� +A(l) +i += a +� +W (l) +i +� +Y (l) +i +− g +� +[W (l) +i ]⊤γ(l),∗ +a +�� ++ oP (n−1/2 +l +), +where the matrix Cγa,l is defined as +Cγa,l = El +� +I +� +A(l) = a +� +W (l)g′ � +[W (l)]⊤γ(l),∗ +a +� +[W (l)]⊤� +. +The matrix Cγa,l can be consistently estimated by an empirical version of it, +�Cγa,l = 1 +nl +nl +� +i=1 +I +� +A(l) +i += a +� +W (l) +i g′ � +[W (l) +i ]⊤�γ(l) +a +� +[W (l) +i ]⊤. +Let η(l),∗ denote the probabilistic limit of �η(l) such that +El +� +exp +� +[η(l),∗]⊤� +W (l)� +� +W (l)� += ET +� +� +W T � +, + +Inference for Meta-Learning +15 +where ET denotes the expectation operator with respect to the covariate distribution in the target +population. Again by standard theory of estimating equations, the estimator �η(l) is asymptotically +linear with the following expansion: +�η(l) − η(l),∗ = C−1 +η,l +� +� +� +1 +nl +nl +� +i=1 +exp +� +[η(l),∗]⊤� +W (l) +i +� +� +W (l) +i +− 1 +N +N +� +j=1 +� +W T +j +� +� +� + oP (n−1/2 +l +), +where the matrix Cη,l is defined as +Cη,l = −El +� +exp +� +[η(l),∗]⊤� +W (l)� +� +W (l)[� +W (l)]⊤� +, +and can be consistently estimated by +�Cη,l = − 1 +nl +nl +� +i=1 +exp +� +[�η(l)]⊤� +W (l) +i +� +� +W (l) +i [� +W (l) +i ]⊤. +Note that when N ≫ nl, the asymptotic linear expansion of �η(l) simplifies to +�η(l) − η(l),∗ = 1 +nl +nl +� +i=1 +C−1 +η,l +� +exp +� +[η(l),∗]⊤� +W (l) +i +� +� +W (l) +i +− ET +� +� +W T �� ++ oP (n−1/2 +l +). +Now, we derive an asymptotic linear expansion of the site-specific target ATE estimator. Recall +from Section 5.3 that the target ATE estimator takes the form �θ(l) = � +M(l) + �δ(l) where +� +M(l) = 1 +N +N +� +i=1 +� +m(1, XT +i ; �γ(l) +1 ) − m(0, XT +i ; �γ(l) +0 ) +� +and +�δ(l) = 1 +nl +nl +� +i=1 +ωl(X(l) +i ; �η(l)) +� 1 +� +a=0 +(−1)a+1I{A(l) +i += a} +πl(a, X(l) +i ; �αl) +{Y (l) +i +− m(A(l) +i , X(l) +i ; �γ(l) +a )} +� +. +In this case, we have that +m(a, X(l) +i ; �γ(l) +a ) = g +� +[W (l) +i ]⊤�γ(l) +a +� +; +m(a, XT +i ; �γ(l) +a ) = g +� +[W T +i ]⊤�γ(l) +a +� +; +ωl(X(l) +i ; �η(l)) = exp +� +[�η(l)]⊤� +W (l) +i +� +; +πl(a, X(l) +i ; �αl) = h +� +(�αl)⊤ � +X(l) +i +�a � +1 − h +� +(�αl)⊤ � +X(l) +i +��(1−a) +. +Also recall that we define the following functions τγ0,γ1(·) and ξη,α,γ0,γ1(·) such that +τγ0,γ1(x(l)) = m(1, x(l); γ1) − m(0, x(l); γ0), + +16 +Guo, Li, Han & Cai +and +ξη,α,γ0,γ1(x(l), a(l), y(l)) = ωl(x(l); η) +� 1 +� +a=0 +(−1)a+1I{a(l) = a} +πl(a, x(l); α) +{y(l) − m(a, x(l); γa} +� +. +We introduce the following notations frequently used when dealing with empirical processes: for +generic functions f1 and f2, we define Plf1 = +� +f1dPl where Pl denotes the joint distribution of +(X(l), A(l), Y (l)) in the l-th source population, and PT f2 = +� +f2dPT where PT denotes the covariate +distribution in the target population. Moreover, let Pn,lf1 = �nl +i=1 f1(X(l) +i , A(l) +i , Y (l) +i +)/nl denote the +empirical average of f1 on the l-th source data, and Pn,T f2 = �N +i=1 f2(XT +i )/N denote the empirical +average of f2 on the target data. With these notations, we are now ready to linearize �θ(l). +To start, we note that the estimator �θ(l) can be written as +�θ(l) = Pn,T +� +τ�γ +(l) +0 ,�γ +(l) +1 +� ++ Pn,l +� +ξ�η(l), �αl,�γ +(l) +0 ,�γ +(l) +1 +� +. +Thus, the estimator �θ(l) has the following expansion +�θ(l) − θ(l) = Pn,T +� +τ�γ +(l) +0 ,�γ +(l) +1 +� +− PT +� +τγ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ Pn,l +� +ξ�η(l), �αl,�γ +(l) +0 ,�γ +(l) +1 +� +− Pl +� +ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +� += (Pn,T − PT ) +� +τγ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ PT +� +τ�γ +(l) +0 ,�γ +(l) +1 − τγ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ (Pn,l − Pl) +� +ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ Pl +� +ξ�η(l), �αl,�γ +(l) +0 ,�γ +(l) +1 − ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ oP (n−1/2 +l +) += (Pn,T − PT ) +� +τγ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ (Pn,l − Pl) +� +ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +� ++ oP (n−1/2 +l +) ++ +� +PT +�∂τγ0,γ1 +∂γ0 +|(γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� ++ Pl +�∂ξη,α,γ0,γ1 +∂γ0 +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +��⊤ � +�γ(l) +0 − γ(l),∗ +0 +� ++ +� +PT +�∂τγ0,γ1 +∂γ1 +|(γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� ++ Pl +�∂ξη,α,γ0,γ1 +∂γ1 +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +��⊤ � +�γ(l) +1 − γ(l),∗ +1 +� ++ +� +Pl +�∂ξη,α,γ0,γ1 +∂η +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +��⊤ � +�η(l) − η(l),∗� ++ +� +Pl +�∂ξη,α,γ0,γ1 +∂α +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +��⊤ +(�αl − α∗ +l ) . +When the sample size in the target data N is such that N ≫ nl, the term (Pn,T − PT )τγ +(l),∗ +0 +,γ +(l),∗ +1 +is +of the order oP (n−1/2 +l +) and hence is negligible. The partial derivatives can be calculated explicitly. + +Inference for Meta-Learning +17 +Specifically, +dγ0 = PT +�∂τγ0,γ1 +∂γ0 +|(γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� ++ Pl +�∂ξη,α,γ0,γ1 +∂γ0 +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� += −ET +� ∂m +∂γ0 +(0, XT ; γ(l),∗ +0 +) +� ++ El +� +ωl(X(l); η(l),∗) I{A(l) = 0} +πl(0, X(l); α∗ +l ) +� ∂m +∂γ0 +(0, X(l); γ(l),∗ +0 +) +�� += −ET +� +g′ � +[W T ]⊤γ(l),∗ +0 +� +W T � ++ El +� +�exp +� +[η(l),∗]⊤� +W (l)� +I{A(l) = 0} +1 − h +� +[ � +X(l)]⊤α∗ +l +� +� +g′ � +[W (l)]⊤γ(l),∗ +0 +� +W (l)� +� +� . +dγ1 = PT +�∂τγ0,γ1 +∂γ1 +|(γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� ++ Pl +�∂ξη,α,γ0,γ1 +∂γ1 +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� += ET +� ∂m +∂γ1 +(1, XT ; γ(l),∗ +1 +) +� +− El +� +ωl(X(l); η(l),∗) I{A(l) = 1} +πl(1, X(l); α∗ +l ) +� ∂m +∂γ1 +(1, X(l); γ(l),∗ +1 +) +�� += ET +� +g′ � +[W T ]⊤γ(l),∗ +1 +� +W T � +− El +� +�exp +� +[η(l),∗]⊤� +W (l)� +I{A(l) = 1} +h +� +[ � +X(l)]⊤α∗ +l +� +� +g′ � +[W (l)]⊤γ(l),∗ +1 +� +W (l)� +� +� . +dη = Pl +�∂ξη,α,γ0,γ1 +∂η +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� += El +� +∂ωl +∂η (X(l); η(l),∗) +� 1 +� +a=0 +(−1)a+1I{A(l) = a} +πl(a, X(l); α∗ +l ) +{Y (l) − m(a, X(l); γ(l),∗ +a +)} +�� += El +� +exp +� +[η(l),∗]⊤� +W (l)� +� +W (l) +� 1 +� +a=0 +(−1)a+1I{A(l) = a} +πl(a, X(l); α∗ +l ) +� +Y (l) − g +� +[W (l)]⊤γ(l),∗ +a +���� +. +dα = Pl +�∂ξη,α,γ0,γ1 +∂α +|(η(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +) +� += El +� +ωl(X(l); η(l),∗) +� 1 +� +a=0 +−I{A(l) = a}π′(a, X(l); α∗ +l ) +π2(a, X(l); α∗ +l ) +� +Y (l) − m(a, X(l); γ(l),∗ +a +) +��� += El +� +exp +� +[η(l),∗]⊤� +W (l)� � 1 +� +a=0 +−I{A(l) = a}π′ +l(a, X(l); α∗ +l ) +π2 +l (a, X(l); α∗ +l ) +� +Y (l) − g +� +[W (l)]⊤γ(l),∗ +a +��� +� +X(l) +� +. +These partial derivatives can be estimated by replacing the expectations with the corresponding +sample averages, and replacing the unknown population parameters η(l),∗, α∗ +l , γ(l),∗ +0 +and γ(l),∗ +1 +with +their consistent estimators �η(l), �αl, �γ(l) +0 +and �γ(l) +1 , respectively. +As �η(l), �αl, �γ(l) +0 +and �γ(l) +1 +are all asymptotically linear, the expansion we derived for �θ(l) implies + +18 +Guo, Li, Han & Cai +that �θ(l) is also asymptotically linear with influence function τθ such that +τθ(x(l), a(l), y(l)) = ξη(l),∗,α∗ +l ,γ +(l),∗ +0 +,γ +(l),∗ +1 +(x(l), a(l), y(l)) ++ d⊤ +γ0C−1 +γ0,lI +� +a(l) = 0 +� +w(l) � +y(l) − g +� +[w(l)]⊤γ(l),∗ +0 +�� ++ d⊤ +γ1C−1 +γ1,lI +� +a(l) = 1 +� +w(l) � +y(l) − g +� +[w(l)]⊤γ(l),∗ +1 +�� ++ d⊤ +η C−1 +η,l +� +exp +� +[η(l),∗]⊤ �w(l)� +�w(l) − ET +� +� +W T �� ++ d⊤ +αC−1 +α,l�x(l) � +a(l) − h +� +(α∗ +l )⊤�x(l)�� +. +Here the vectors �x(l), w(l) and �w(l) denote the vectors of basis functions derived from the covariate +vector x(l). +C. +Additional Numerical Results +C.1. +Simulation results for 8 majority sites and RIFL with 80% rule +We now present simulation results when there are 8 majority sites, and we use an 80%-rule in +RIFL that utilizes prior information on the number of majority sites. Specifically, we modify the +definition of the index set in (16) and define +M80% := +� +1 ≤ m ≤ M : |�V[m]| ≥ 0.8L +� +. +(73) +RIFL with the 80% rule is defined in the same way as the original RIFL except that we replace M +with M80%. +First, we present results in the low-dimensional prediction example in Section 5.1. +For l ∈ +{1, 2, . . . , 8}, we set θ(l) = θ∗ = (0.5, 0.5, 0.5, 0.5, 0.5, 0.1, 0.1, 0.1, 0, 0). +For θ(9) and θ(10), their +last 5 coefficients are the same as θ∗ but their first five coefficients are changed to 0.5 − 0.3a and +0.5 − 0.1a, respectively, where a ∈ {1, 2, 3, 4, 5} controls the separation between the majority sites +and non-majority sites. All the other aspects of the simulation setup is the same as in Section 6.1. +We present the empirical coverage and average length of 95% CIs from various methods in +Figure C1. We observe that both RIFL and RIFL with the 80% rule achieve the nominal coverage +across all settings. In addition, RIFL with the 80% rule results in a CI that is much shorter than +the original RIFL CI and comparable to the OBA interval when the separation level is high. The +performance of all the other methods show similar pattern as in Section 6.1. +Next, we present additional simulation results for the high-dimensional prediction example in +Section 5.2 for θ∗ +11 with 8 majority sites. The simulation setup is the same as that in Section 6.2 +except that θ(l) = θ∗ for l ∈ {1, . . . , 8} and θ(9) +j += θ∗ +j + 0.2 + 0.05a and θ(10) +j += θ∗ +j + 0.15 + 0.05a +for 6 ≤ j ≤ 11 with a varied in {1, 2, 3, 4, 5} to represent varying levels of separation between the +majority and non-majority sites. The results are presented in Figure C2, and we observe similar +patterns as in the case with 6 majority sites. Simulation results for θ∗ +8 are presented in Figure C3 +and Figure C4 for 6 and 8 majority sites, respectively. Again, we observe similar patterns as in the +case for θ∗ +11. + +Inference for Meta-Learning +19 +0.875 +0.900 +0.925 +0.950 +0.975 +1.000 +1 +2 +3 +4 +5 +separation +coverage +0.20 +0.25 +0.30 +0.35 +1 +2 +3 +4 +5 +separation +length +n=500 +0.88 +0.92 +0.96 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.150 +0.175 +0.200 +0.225 +1 +2 +3 +4 +5 +separation +length +n=1000 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.10 +0.12 +0.14 +0.16 +1 +2 +3 +4 +5 +separation +length +n=2000 +median +MNB +OBA +RIFL +RIFL (80%) +VMC +Fig. C1: Low-dimensional prediction: coverage and length of 95% CIs for β∗ = θ∗ +1 with 8 majority +sites and varying separation levels where 1 is the lowest (hardest to detect) and 5 is the highest +(easiest to detect). “median” stands for the CI based on the median estimator, “MNB” stands for +the m-out-of-n bootstrap CI in (38), “VMC” stands for the voting with maximum clique estimator +and its associated CI in (10), “OBA” stands for the oracle bias aware CI in (39), “RIFL” stands +for our proposed CI in (19), and “RIFL (80%)” stands for our proposed RIFL CI leveraging 80% +rule. Results are based on 500 simulation replications. Dash lines in the left panels correspond to +nominal coverage level 0.95; in the right panels correspond to the width of an oracle CI knowing +the prevailing set. + +20 +Guo, Li, Han & Cai +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.20 +0.25 +0.30 +1 +2 +3 +4 +5 +separation +length +n=500 +0.7 +0.8 +0.9 +1.0 +1 +2 +3 +4 +5 +separation +coverage +0.100 +0.125 +0.150 +0.175 +1 +2 +3 +4 +5 +separation +length +n=1000 +0.900 +0.925 +0.950 +0.975 +1.000 +1 +2 +3 +4 +5 +separation +coverage +0.07 +0.08 +0.09 +0.10 +0.11 +1 +2 +3 +4 +5 +separation +length +n=2000 +median +OBA +RIFL +RIFL (80%) +VMC +Fig. C2: High-dimensional prediction: coverage and length of 95% CIs for β∗ = θ∗ +11 with 8 majority +sites and varying separation levels where 1 is the lowest (hardest to detect) and 5 is the highest +(easiest to detect). “median” stands for the CI based on the median estimator, “VMC” stands for +the voting with maximum clique estimator and its associated CI in (10), “OBA” stands for the +oracle bias aware CI in (39), “RIFL” stands for our proposed CI in (19), and “RIFL (80%)” stands +for our proposed RIFL CI leveraging 80% rule. Results are based on 500 simulation replications. +Dash lines in the left panels correspond to nominal coverage level 0.95; in the right panels correspond +to the width of an oracle CI knowing the prevailing set. + +Inference for Meta-Learning +21 +0.4 +0.6 +0.8 +1.0 +1 +2 +3 +4 +5 +separation +coverage +0.2 +0.3 +0.4 +0.5 +1 +2 +3 +4 +5 +separation +length +n=500 +0.4 +0.6 +0.8 +1.0 +1 +2 +3 +4 +5 +separation +coverage +0.1 +0.2 +0.3 +1 +2 +3 +4 +5 +separation +length +n=1000 +0.6 +0.7 +0.8 +0.9 +1 +2 +3 +4 +5 +separation +coverage +0.075 +0.100 +0.125 +0.150 +0.175 +1 +2 +3 +4 +5 +separation +length +n=2000 +median +OBA +RIFL +VMC +Fig. C3: High-dimensional prediction: coverage and length of 95% CIs for β∗ = θ∗ +8 with 6 majority +sites and varying separation levels where 1 is the lowest (hardest to detect) and 5 is the highest +(easiest to detect). “median” stands for the CI based on the median estimator, “VMC” stands for +the voting with maximum clique estimator and its associated CI in (10), “OBA” stands for the +oracle bias aware CI in (39), “RIFL” stands for our proposed CI in (19), and “RIFL (80%)” stands +for our proposed RIFL CI leveraging 80% rule. Results are based on 500 simulation replications. +Dash lines in the left panels correspond to nominal coverage level 0.95; in the right panels correspond +to the width of an oracle CI knowing the prevailing set. + +22 +Guo, Li, Han & Cai +0.8 +0.9 +1.0 +1 +2 +3 +4 +5 +separation +coverage +0.20 +0.25 +0.30 +1 +2 +3 +4 +5 +separation +length +n=500 +0.6 +0.7 +0.8 +0.9 +1.0 +1 +2 +3 +4 +5 +separation +coverage +0.100 +0.125 +0.150 +0.175 +1 +2 +3 +4 +5 +separation +length +n=1000 +0.85 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.07 +0.08 +0.09 +0.10 +0.11 +1 +2 +3 +4 +5 +separation +length +n=2000 +median +OBA +RIFL +RIFL (80%) +VMC +Fig. C4: High-dimensional prediction: coverage and length of 95% CIs for β∗ = θ∗ +8 with 8 majority +sites and varying separation levels where 1 is the lowest (hardest to detect) and 5 is the highest +(easiest to detect). “median” stands for the CI based on the median estimator, “VMC” stands for +the voting with maximum clique estimator and its associated CI in (10), “OBA” stands for the +oracle bias aware CI in (39), “RIFL” stands for our proposed CI in (19), and “RIFL (80%)” stands +for our proposed RIFL CI leveraging 80% rule. Results are based on 500 simulation replications. +Dash lines in the left panels correspond to nominal coverage level 0.95; in the right panels correspond +to the width of an oracle CI knowing the prevailing set. + +Inference for Meta-Learning +23 +Finally, we present results for the multi-source causal inference problem discussed in Sections 5.3 +and 6.3 when there are 8 majority sites. Recall that the outcome for the l-th site is generated +according to +Y (l) +i += µl + [X(l) +i ]⊤ζ(l) + β(l)A(l) +i ++ ε(l) +i , +ε(l) +i +∼ N(0, 1). +For the case with 8 majority sites, we set β(l) = β∗ = −1 for l ∈ {1, 2, . . . , 8}, β(9) = −1 − 0.2a and +β(10) = −1 − 0.1a, with a varied in {1, 2, 3, 4, 5}. All the other aspects of the simulation setup is +the same as that in Section 6.3. The results are summarized in Figure C5, and we again observe +similar patterns. +C.2. +Choices of ν in m-out-of-n bootstrap +In this section, we present the empirical coverage and average length of the CIs obtained via m- +out-of-n bootstrap with different choices of m. Specifically, we consider m = nν with ν varied in +{0.6, 0.7, 0.8, 0.9, 1}. We focus on the low-dimensional prediction example in Section 6.1 with 6 +majority sites and nl = 1000 for all 10 sites. All other simulation settings are the same as those +described in Section 6.1. +The results are presented in Figure C6. +We observe similar patterns in coverage for ν ∈ +{0.8, 0.9, 1}. Moreover, out of these 3 values, the choice of ν = 0.8 generally produces the shortest +CI. For all choices of ν, the MNB CI has coverage below nominal level when the separation level is +low to moderate. +C.3. +Generalizability measure for the 16 sites in the real data analysis +In Figure C7, we present the full set of generalizability measure for all 16 sites in our Covid-19 real +data analysis. Site 4 generally has lower generalizability than the other sites. When we focus on +one specific risk factor (corresponding to one specific row of the plot in Figure C7), some sites have +low generalizability, indicating that they may not belong to the prevailing set. + +24 +Guo, Li, Han & Cai +0.80 +0.85 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.125 +0.150 +0.175 +0.200 +0.225 +1 +2 +3 +4 +5 +separation +length +n=500 +0.80 +0.85 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.09 +0.11 +0.13 +0.15 +0.17 +1 +2 +3 +4 +5 +separation +length +n=1000 +0.85 +0.90 +0.95 +1.00 +1 +2 +3 +4 +5 +separation +coverage +0.08 +0.10 +0.12 +1 +2 +3 +4 +5 +separation +length +n=2000 +median +MNB +OBA +RIFL +RIFL (80%) +VMC +Fig. C5: Causal inference: coverage and length of 95% CIs for target ATE of the prevailing sites +with 8 majority sites and varying separation levels where 1 is the lowest (hardest to detect) and 5 is +the highest (easiest to detect). “median” stands for the CI based on the median estimator, “MNB” +stands for the m-out-of-n bootstrap CI in (38), “VMC” stands for the voting with maximum clique +estimator and its associated CI in (10), “OBA” stands for the oracle bias aware CI in (39), “RIFL” +stands for our proposed CI in (19), and “RIFL (80%)” stands for our proposed RIFL CI leveraging +80% rule. Results are based on 500 simulation replications. Dash lines in the left panels correspond +to nominal coverage level 0.95; in the right panels correspond to the width of an oracle CI knowing +the prevailing set. + +Inference for Meta-Learning +25 +0.6 +0.7 +0.8 +0.9 +1 +2 +3 +4 +5 +separation +coverage +n=1000, coverage +0.16 +0.18 +0.20 +0.22 +1 +2 +3 +4 +5 +separation +length +n=1000, length +nu +0.6 +0.7 +0.8 +0.9 +1 +Fig. C6: Low-dimensional prediction: coverage and length of 95% CIs for β∗ = θ∗ +1 of m-out-of-n +bootstrap (MNB) CI with different choices of m. We set m = nν and vary ν in {0.6, 0.7, 0.8, 0.9, 1}. +Sample size in each of the 10 sites is 1,000, and there are 6 majority sites. Results are based on 500 +simulation replications. Dash lines in the left panels correspond to nominal coverage level 0.95. + +26 +Guo, Li, Han & Cai +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 11 12 13 14 15 16 +AST/ALT +age:18to25 +age:26to49 +age:70to79 +age:80plus +albumin +AST +charlson score +creatinine +CRP +lymphocyte count +neutrophil count +sex:female +bilirubin +WBC count +Fig. C7: Generalizability measure for all 16 sites for each of the 15 risk factors. Each row corre- +sponds to a risk factor and each column corresponds to a site. + diff --git a/ttAyT4oBgHgl3EQf0flC/content/tmp_files/load_file.txt b/ttAyT4oBgHgl3EQf0flC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e3a3d1e28db2c460e055e93b2f81ca9a1213d5a --- /dev/null +++ b/ttAyT4oBgHgl3EQf0flC/content/tmp_files/load_file.txt @@ -0,0 +1,1989 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf,len=1988 +page_content='Robust Inference for Federated Meta-Learning Zijian Guo Rutgers University, Piscataway, USA Xiudi Li Harvard University, Boston, USA Larry Han Harvard University, Boston, USA Tianxi Cai Harvard University, Boston, USA Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Synthesizing information from multiple data sources is critical to ensure knowledge gen- eralizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Integrative analysis of multi-source data is challenging due to the heterogeneity across sources and data-sharing constraints due to privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In this paper, we consider a general robust inference framework for federated meta-learning of data from multiple sites, enabling statistical inference for the prevailing model, defined as the one matching the majority of the sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Statistical inference for the prevailing model is challenging since it requires a data-adaptive mechanism to select eligible sites and subsequently account for the selection uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We propose a novel sampling method to address the additional variation arising from the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our devised CI construction does not require sites to share individual-level data and is shown to be valid without requiring the selection of eligible sites to be error-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The proposed robust inference for federated meta-learning (RIFL) methodology is broadly applicable and illustrated with three inference problems: aggregation of para- metric models, high-dimensional prediction models, and inference for average treatment effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We use RIFL to perform federated learning of mortality risk for patients hospitalized with COVID-19 using real-world EHR data from 16 healthcare centers representing 275 hospitals across four countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Keywords: Post-selection Inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Heterogeneous Data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Multi-source Data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Privacy Preserv- ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' High-dimensional Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Introduction Crowdsourcing, or the process of aggregating crowd wisdom to solve problems, is a useful community- based method to improve decision-making in disciplines ranging from education [Heffernan and Heffernan, 2014] to public health [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2018, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Compared to traditional expert-driven solutions made by a single group, incorporating the opinions of multiple diverse groups can improve the quality of the final decision [Surowiecki, 2005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In health research, crowd- sourcing has led to the discovery of new drugs during pandemics [Chodera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020], the design of patient-centered mammography reports [Short et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2017], and the development of machine learning algorithms to classify tumors for radiation therapy [Mak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Underlying the phenomenon of the “wisdom of the crowds” is the statistical and philosophical notion that learning from multiple data sources is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Incorporating information from diverse data sources can increase the generalizability and transportability of findings compared to learning arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='00718v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='ME] 2 Jan 2023 2 Guo, Li, Han & Cai from a single data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Findings from a single data source may not be generalizable to a new target population of interest due to poor data quality or heterogeneity in the underlying data generating processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Integrative analysis of data from multiple sources can be a valuable alternative to using a sin- gle data source alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, directly pooling multiple data sources into a single dataset for analysis is often unsatisfactory or even infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Heterogeneity between different data sources can severely bias predictions or inferences made by such a pooled analysis strategy [Leek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2010, Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' As an alternative to pooled analysis, meta-analysis has frequently synthe- sized information from multiple studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Standard meta-analysis methods aggregate quantitative summary of evidence from multiple studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Variations of meta-analysis, such as random effects meta-analysis, have been adopted to explore between-study heterogeneity, potential biases such as publication bias, and small-study effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, most existing meta-analysis tools that account for heterogeneity require strong modeling assumptions and do not consider the validity of inference when data from certain sites have substantially different distributions from other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Another challenge of particular importance is the issue of data privacy pertaining to biomedical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Regulations in the United States, such as the Health Insurance Portability and Account- ability Act (HIPAA) Privacy Rule, and those in the European Union, such as the General Data Protection Regulation (GDPR) and the European Medicines Agency (EMA) Privacy Statement, protect the personal information of patients and preclude the transfer of patient-level data between sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' These regulations make the promise of integrative data analysis more difficult to attain, highlighting the need for federated integrative analysis methods that do not require sharing of individual-level data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' When cross-study heterogeneity is substantial and outliers exist, a desirable strategy of integra- tive analysis is to identify a prevailing model to achieve consensus learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The prevailing model is defined as the model satisfied by the majority of the sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Identifying the prevailing model can be intuitively achieved via the majority rule [Sorkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 1998, Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2004, Hastie and Kameda, 2005], which chooses the alternative that more than half of individuals agree upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The majority rule is widespread in modern liberal democracies and is deployed in various streams of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For example, genomics data is usually separated into batches, but heterogeneity across batches can lead to undesirable variation in the data [Leek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2010, Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' This setting aims to identify batches that show low levels of concordance with the majority of the batches and adjust for such differences in downstream analyses [Trippa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' As another example, in the design of clinical trials, it is often infeasible or unethical to enroll patients in the control arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In such cases, it is possible to use data from historical trials or observational studies to construct an external control arm [Jahanshahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021, Ventz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2019, Davi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, when many such historical data sources exist, it is crucial to carefully select data sources that show high levels of similarity with the majority of the other data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The last example is Mendelian Randomization, where multiple genetic markers are used as instrumental variables (IVs) to account for potential unmeasured confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Every single IV will have its causal effect estimator, and the goal is to identify the causal effect matching the majority of the estimated effects [Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2017, Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016, Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Without prior knowledge of the prevailing model, it is critical to employ data-adaptive ap- proaches to select appropriate sites for inferring the prevailing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In addition, confidence intervals (CIs) for the target parameter of the prevailing model need to appropriately adjust for the site selection variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Most existing statistical inference methods rely on perfectly separating Inference for Meta-Learning 3 eligible and ineligible sites, which may be unrealistic for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' There is a paucity of statistical inference methods for the prevailing model that can achieve efficient and robust inference while being applicable to a broad set of scenarios without restrictive assumptions such as a perfect separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In this paper, we fill this gap by developing a broad theoretically justified framework for making robust inferences for federated meta-learning (RIFL) of an unknown prevailing model using multi- source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The RIFL method selects the eligible sites to infer the prevailing model by assessing dissimilarities between sites with regard to the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We employ a novel resampling method to construct uniformly valid CIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The RIFL inference method is robust to the errors in separating the sites belonging to the majority group and the remaining sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We also show in Theorem 3 that our proposed sampling CI can be as short as the oracle CI with the prior knowledge of the eligible sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our general sampling algorithm is privacy-preserving in that it is implemented using site-specific summary statistics and without requiring sharing individual-level data across different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our proposed RIFL methodology is demonstrated with three inference problems: aggregation of low-dimensional parametric models, construction of high-dimensional prediction models, and inference for the average treatment effect (ATE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' To the best of our knowledge, our proposed RIFL method is the first CI guaranteeing uniform coverage of the prevailing model under the majority rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We have further compared via simulation studies with three other inference procedures that can potentially be used under the majority rule, including the majority voting estimator, the median estimator [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016], and the m- out-of-n bootstrap [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2013, Andrews, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Numerical results demonstrate that these three CIs fail to achieve the desired coverage property, while our RIFL method leads to a uniformly valid CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We provide the reasoning for under-coverage for these existing methods in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Related literature The RIFL method is related to multiple streams of literature, including post-selection inference, mendelian randomization, integrative analysis of multi-source data, transfer learning, and federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We next detail how RIFL differs from the existing literature and highlight its contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' A wide range of novel methods and theories have been established to address the post-selection inference problem [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2013, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016, Leeb and P¨otscher, 2005, Zhang and Zhang, 2014, Javanmard and Montanari, 2014, van de Geer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2014, Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2015, Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2014, Cai and Guo, 2017, Xie and Wang, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, most post-selection inference literature focuses on inferences after selecting a small number of important variables under high-dimensional regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The selection problem under the RIFL framework is fundamentally different: the selection error comes from comparing different sites, and there is no outcome variable to supervise the selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Additionally, RIFL only requires the majority rule, while the variable selection methods typically require a small proportion of variables to affect the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In Mendelian Randomization, various methods have been developed to leverage the majority rule and make inferences for the one-dimensional causal effect [Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016, Windmeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2019, Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2018, Windmeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' A recent work by Guo [2021] demonstrated the post-selection problem due to IV selection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, the uniformly valid inference method in Guo [2021] relies on searching the one-dimensional space of the causal 4 Guo, Li, Han & Cai effect and cannot be easily generalized to multivariate settings, not to mention high-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In contrast, RIFL is distinct from the existing searching method and is useful in addressing a much broader collection of post-selection problems as detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The integrative analysis of multi-source data has been investigated in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2021] studied the data fusion problem with robustness to biased sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The identification condition in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2021] differs from the majority rule, and the validity of their proposal requires correctly identifying unbiased sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Maity et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2022] studied meta-analysis in high- dimensional settings where the data sources are similar but non-identical and require the majority rule to be satisfied as well as a large separation between majority and outlier sources to perfectly identify eligible sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In contrast, the RIFL CI is valid without requiring the selection step to perfectly identify eligible sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Meinshausen and B¨uhlmann [2015], B¨uhlmann and Meinshausen [2015], Rothenh¨ausler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2016], Guo [2020] made inference for the maximin effect, which is defined as a robust prediction model across heterogeneous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2021b], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2021], Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2016] imposed certain similar structures across different sources and made inferences for the shared component of regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2016], Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2019] studied the multi-source data problem and identified the causal effect by invariance principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Unlike existing methods, the RIFL framework only assumes that a majority of the sites have similar models but allows non-eligible sites to differ arbitrarily from the majority group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' RIFL relates to the existing literature on federated learning and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Privacy- preserving and communication-efficient algorithms have been recently developed to learn from multiple sources of electronic health records (EHR) [Rasmy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2018, Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2022] and multiple sources of diverse genetic data [Kraft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2009, Keys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Federated regression and predictive modeling [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2006, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2013, Chen and Xie, 2014, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2017, Lian and Fan, 2017, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2019, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020] and causal modeling [Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021, Vo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021, Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021] have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, none of these federated learning methods study inference for the prevailing model when some sites may not be valid, which is the main focus of RIFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The RIFL framework also differs from the recently developed transfer learning algorithms [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020, Tian and Feng, 2022, Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' These algorithms require pre-specification of an anchor model to which the models obtained from source data sets can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In contrast, RIFL targets a more challenging scenario: we do not assume the availability of such an anchor model but leverage the majority rule to identify the unknown prevailing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Paper organization and notations The paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Section 2 describes the multi-source data setting and highlights the challenge of inferring the prevailing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Section 3 proposes the RIFL methodology, and Section 4 establishes its related theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In Section 5, we illustrate our proposal in three applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In Section 6, we provide extensive simulation results comparing our method to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Section 7 illustrates our method using real-world international EHR data from 16 participating healthcare centers representing 275 hospitals across four countries as part of the multi-institutional Consortium for the Clinical Characterization of COVID-19 by EHR (4CE) [Brat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We introduce the notations used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For a set A, |A| denotes the cardinality of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For a vector x, we define its ℓq norm as ∥x∥q = ��p l=1 |xl|q� 1 q for q ≥ 0 with ∥x∥0 = |{1 ≤ l ≤ p : xl ̸= 0}| and ∥x∥∞ = max1≤l≤p |xl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For a matrix X, Xi,· and X·,j denote its i-th row and j-th column, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For two positive sequences an and bn, an ≪ bn if lim supn→∞ an/bn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For a matrix A, we use ∥A∥F , ∥A∥2 and ∥A∥∞ to denote its Frobenius norm, spectral norm, Inference for Meta-Learning 5 and element-wise maximum norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Formulation and Statistical Inference Challenges 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Model assumptions and overview of RIFL Throughout the paper, we consider that we have access to L independent training data sets drawn from L source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For 1 ≤ l ≤ L, we use P(l) to denote the distribution of the l-th source population and use θ(l) = θ(P(l)) ∈ Rd to denote the associated model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For any θ ∈ Rd, we define the index set V(θ) ⊂ {1, · · · , L} as V(θ) := {1 ≤ l ≤ L : θ(l) = θ}, (1) which contains the indexes of all sites having the same model parameter as θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We now introduce the majority rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Assumption 1 (Majority Rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' There exists θ∗ ∈ Rd such that |V(θ∗)| > L/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We shall refer to θ∗ as the prevailing model that matches with more than half of {θ(l)}1≤l≤L, and the corresponding index set V(θ∗) as the prevailing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our goal is to construct a confidence region for a low dimensional functional of θ∗, denoted as β∗ = g(θ∗) ∈ Rq, for some q ≥ 1, where g(·) ∈ Rq is a prespecified low-dimensional transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Examples of β∗ = g(θ∗) include (a) Single coefficient or sub-vector: β∗ = θ∗ j for 1 ≤ j ≤ d or β∗ = θ∗ G with G ⊂ {1, · · · , d};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (b) Linear transformation: β∗ = x⊺θ∗ for any x ∈ Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (c) Quadratic form: β∗ = ∥θ∗∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For notational ease, we focus on q = 1 primarily and discuss the extension to the setting with q ≥ 2 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' If the prevailing set V(θ∗) were known, standard meta and federated learning methods could be used to make inferences about θ∗ using data from sites belonging to V(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, as highlighted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='3, inference for θ∗ without prior knowledge of V(θ∗) except for the majority rule is substantially more challenging due to the need to estimate V(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our proposed RIFL procedure involves several key steps: (i) for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', L, construct local estimates of θ(l) and β(l) = g(θ(l)), denoted by �θ(l) and �β(l), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (ii) for 1 ≤ l < k ≤ L, estimate pairwise dissimilarity measure Dl,k = D(θ(l), θ(k)) and Ll,k = β(l) − β(k) as �Dl,k and �Ll,k along with their standard errors � SE( �Dl,k) and � SE( �Ll,k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (iii) construct a robust estimate for the prevailing set V(θ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (iv) derive robust resampling-based confidence set for β∗ accounting for post-selection uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The construction of �θ(l) and �β(l) follows standard procedures for the specific problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We next detail (ii) the construction of the dissimilarity measures and (iii) the prevailing set estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The most challenging step of RIFL is the resampling-based inference, which is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Dissimilarity measures A critical step of applying the majority rule is to evaluate the (dis)similarity between any pair of parameters θ(l) and θ(k) for 1 ≤ l, k ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We form two sets of dissimilarity measures, the local dissimilarity between β(l) and β(k), Ll,k = β(l)−β(k), and a global dissimilarity Dl,k = D(θ(l), θ(k)) = 6 Guo, Li, Han & Cai ∥θ(l) − θ(k)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Although other vector norms can be considered for D(·, ·), we focus on the quadratic norm due to its smoothness and ease of inference, especially in the high-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We assume that {�β(l), �σl}1≤l≤L satisfy 1 σl (�β(l) − β(l)) d→ N(0, 1) and �σl σl p→ 1, (2) with σl denoting the standard error of �β(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In the low-dimensional setting, most existing estimators satisfy (2) under standard regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In the high-dimensional setting, various asymp- totically normal de-biased estimators have recently been proposed and shown to satisfy (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' see more discussions at the end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Let �Ll,k = �β(l)− �β(k) and �Dl,k be the point estimators for Ll,k and Dl,k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We estimate their standard errors as � SE( �Ll,k) = � �σ2 l + �σ2 k and � SE( �Dl,k), with �σ2 l denoting the estimated variance of �β(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For the global dissimilarity measure, we assume that �Dl,k and � SE( �Dl,k) satisfy lim sup n→∞ P ���� �Dl,k − Dl,k ���/� SE( �Dl,k) ≥ zα � ≤ α for 0 < α < 1, (3) where zα denotes the α upper quantile of a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Although �Dl,k can be constructed as ∥�θ(l) − �θ(k)∥2 2 in the low-dimensional setting, deriving { �Dl,k, � SE( �Dl,k)} that satisfies (3) is much more challenging in the high-dimensional setting due to the inherent bias in regularized estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2, we demonstrate that our proposed estimators of Dl,k satisfy (3) for a broad class of applications in both low and high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Based on both sets of dissimilarity measures, we determine the concordance between sites k and l with respect to inference for β∗ = g(θ∗) based on the following test statistic �Sl,k := max ���� �Dl,k/� SE( �Dl,k) ��� , ��� �Ll,k/� SE( �Ll,k) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (4) For 1 ≤ l < k ≤ L, we can then implement the following significance test of whether the k-th and l-th sites share the same parameters, �Hl,k = 1 � �Sl,k ≤ z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05/[2L(L−1)] � , (5) where 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05 is a pre-selected significance level for testing the similarity among different sites and z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05/[2L(L−1)] denotes the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05/[2L(L − 1)] upper quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The statistic �Sl,k measures the level of evidence that the two sites differ from each other based on observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The binary decision �Hl,k in (5) essentially estimates Hl,k = 1{θ(l) = θ(k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We specify the threshold as z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05/[2L(L−1)] to adjust for the multiplicity of hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Since the matrix H is symmetric and Hl,l = 1, we construct an estimate for the full voting matrix �H = [ �Hl,k]k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=',L l=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=',L by setting �Hk,l = �Hl,k and �Hl,l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The estimated voting matrix �H summarizes all cross-site similarities, which is then used to estimate the prevailing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Remark 1 (Univariate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' For the special setting with a univariate θ∗, we may simplify the construction of the test statistics in (4) and the vote in (5) as �Hl,k = 1 � �Sl,k ≤ z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='05/[L(L−1)] � with �Sl,k = ��� �Ll,k/� SE( �Ll,k) ��� for 1 ≤ l < k ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (6) Inference for Meta-Learning 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Prevailing set estimation and post-selection problem In the following, we construct two estimators of the prevailing set V(θ∗), which are used to make inference for β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We construct the first estimator as �V := {1 ≤ l ≤ L : ∥ �Hl,·∥0 > L/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (7) The set �V contains all sites receiving ‘majority votes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The second estimator is constructed by utilizing the maximum clique from graph theory [Carraghan and Pardalos, 1990].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Specifically, we define the graph G([L], �H) with vertices [L] := {1, 2, · · · , L} and the adjacency matrix �H with �Hl,k = 1 and �Hl,k = 0 denoting that the l-th and k-th vertexes are connected and disconnected, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The maximum clique of the graph G([L], �H) is defined as the largest fully connected sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We use the term maximum clique set to denote the corresponding vertex set in the maximum clique, denoted as MC([L], �H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We construct �V by identifying the maximum clique set of G([L], �H), that is, �V := MC([L], �H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (8) If the majority rule holds, the prevailing set V(θ∗) is the maximum clique set of G([L], H), that is, V(θ∗) = MC([L], H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' If �H is a sufficiently accurate estimator of H, both set estimators �V and �V exactly recover the prevailing set V(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' However, since �H might be different from the true H due to the limited sample size in practice, �V and �V may be different from V(θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' When the maximum clique set has the cardinality above L/2, we have �V ⊂ �V, that is, �V may be a more restrictive set estimator than �V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We illustrate the definitions of �V in (7) and �V in (8) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 1 2 3 4 5 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 1: The graph G([L], �H) with L = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' �V = {1, 2, 3, 4, 5} and �V = {1, 2, 3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In the following, we demonstrate the subsequent analysis after obtaining �V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The argument is easily extended to the estimated set �V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' One may aggregate {�β(l), �σl}l∈�V to estimate β∗ as the following inverse variance weighted estimator, �β∗ = � l∈�V �β(l)/�σ2 l � l∈�V 1/�σ2 l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (9) A naive 1 − α confidence interval for β∗ can be constructed as CIpost = � ��β∗ − zα/2 1 �� l∈�V 1/�σ2 l , �β∗ + zα/2 1 �� l∈�V 1/�σ2 l � � , (10) where zα/2 denotes the α/2 upper quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Unfortunately, similar to other settings in the ‘post-selection’ literature, such naive construction can lead to bias in the point estimation and under-coverage in the confidence interval due to ignoring the variability in the selection of �V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We illustrate the post-selection problem of the naive confidence interval in (10) with the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 8 Guo, Li, Han & Cai Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We construct the confidence interval for a target population’s average treatment effect (ATE) in the multi-source causal inference setting detailed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We have L = 10 source sites with nl = 1000, 1 ≤ l ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In each source site, we observe the data {X(l) i , A(l) i , Y (l) i }1≤i≤nl, where X(l) i ∈ R10 denotes a 10-dimensional vector of baseline covariates, A(l) i ∈ {0, 1} denotes the treatment assignment (treatment or control) and Y (l) i ∈ R denotes the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The first six source sites are generated such that the target ATE has a value of −1, while the remaining four source sites are generated such that the target ATE has values −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1, and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In this case, the first six source sites form the majority group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The confidence intervals relying on �V and �V suffer from the under-coverage due to wrongly selected sites being included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Based on 500 simulations, the confidence interval in (10) has an empirical coverage of only 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' If we replace �V in (10) with �V in (7), the empirical coverage drops to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Challenge for the median-based confidence interval A commonly used consistent estimator of the prevailing model parameter under the majority rule is the median estimator [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', Bowden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' We construct the median estimator as the median of {�β(l)}1≤l≤L and estimate its standard error by parametric bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' It is worth noting that although the median estimator is consistent under the majority rule as the sample size in each site approaches infinity, it may not be suitable for the purpose of statistical inference due to its bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Consequently, the CI based on the median estimator does not achieve the desired coverage property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' To illustrate this, let us consider a special case where L is odd, and there are (L + 1)/2 sites in the prevailing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Without loss of generality, we assume that V(θ∗) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' , (L+1)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Furthermore, suppose that β(l) < β∗ for l /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In this scenario, when the parameter values in the non-majority sites are well-separated from the parameter value in the prevailing set, with high probability, the median estimator will coincide with min{�β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' , �β(L+1)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' That is, the median estimator is the smallest order statistics of (�β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' , �β(L+1)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Even when the site-specific estimator �β(l) is unbiased for β∗ and normally distributed for l ∈ V, the smallest order statistics typically has a non-normal distribution with a mean value below β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' More generally, the limiting distribution of the median estimator is that of an order statistics and has an asymptotic bias that is not negligible for the purpose of statistical inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' This same issue has been discussed in more detail in Windmeijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' [2019] in the context of invalid instrumental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Our numerical results in Section 6 show that the CI based on the median estimator fails to achieve the desired coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' RIFL Inference In this section, we devise resampling-based methods for deriving a valid confidence interval for β∗, addressing the post-selection issue in aggregating multi-source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' RIFL: resampling-based inference The RIFL interval construction consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In the first step, we resample the dissimi- larity measures and screen out the inaccurate resampled measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' In the second step, we use the resampled dissimilarity measures to estimate the prevailing set, which is further used to generate a sampled confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Inference for Meta-Learning 9 Step 1: resampling and screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Conditioning on the observed data, for 1 ≤ l < k ≤ L, we generate { �D[m] l,k }1≤m≤M and { �L[m] l,k }1≤m≤M following �D[m] l,k i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='d ∼ N � �Dl,k, � SE 2( �Dl,k) � , �L[m] l,k i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content='d ∼ N � �Ll,k, � SE 2( �Ll,k) � for 1 ≤ m ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' (11) The above generating mechanism in (11) guarantees that the distributions of �D[m] l,k − �Dl,k and �L[m] l,k − �Ll,k approximate those of �Dl,k − Dl,k and �Ll,k − Ll,k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQf0flC/content/2301.00718v1.pdf'} +page_content=' The random variables { �D[m] l,k }1≤l 3.2 × 10−3 M⊙ yr−1 [22]. In +the infrared (IR), spectroscopic data for the young populations in NGC 346 +are limited. Spitzer IRS data spectroscopically confirmed the identity of six +massive YSOs in the cluster [25]. [26] obtained HK band spectra to confirm +the existence of three early-type stars in NGC 346. Most recently, [27] used +VLT/KMOS to observe ∼15 other YSO candidates which were resolved into +multiple young stars still accreting mass. +Overall, NGC 346 possesses a complex distribution of hierarchically-linked +star clusters of varying ages which inhabit a variety of environments [28], and +which are dispersed across the extended field [17, 29, 30]. Within the ISM there +is a wide range of substructures exhibited in polycyclic aromatic hydrocarbon +emission (PAH; 8 µm), warm dust (24 µm), and molecular gas [CO J = 2 − 1; +28, 31, 32]. A tight correlation is seen between this emission and that of Hα, +which presents as a well-defined bar extending from the centre of the region to +the northeast and as an arc structure extending from southeast to northwest. A +recent Atacama Large Millimeter/submillimeter Array (ALMA) CO(J = 1−0) +study [33] discovered that the intersection of three colliding clumpy filaments +is co-spatial with the locations of a cluster of YSOs and PMS stars. Using +HST proper motions and VLT/MUSE radial velocities, [34] and [35] showed +that stars in NGC 346 move along a wide spiral and that clusters of YSOs +and young PMS stars seem to be predominately located where the coherent +motion field shows significant changes, hence turbulence is still driving star +formation across the system. + +Springer Nature 2021 LATEX template +4 +The embedded young population of NGC 346 +NGC 346 is one of the most active star-forming regions in the Local Group. +Its proximity, size (∼100 × 100 pc2), low foreground extinction, and an abun- +dance of wide-field, high-resolution panchromatic data make it an ideal system +for the study of both low- and high-mass star formation, the effects of this star +formation on the surrounding medium, and the potential triggers of star forma- +tion in an environment vastly different from our local Galactic surroundings, +and akin to galaxies at cosmic noon. +2 Results and Discussion +2.1 Images +The NIRCam images of NGC 346 shown in Figure 1 reveal the complex fila- +mentary structure of the NGC 346 arc, dominated by emission from warm dust +and PAHs, together with the intermediate-age BS90 cluster [36] just above +the centre. The brightest red stars are located along dust ridges, in tips of +warm dust lanes, in large sub-clusters within the centre NGC 346 arc, or in +smaller clumps located along the arc and northeast perpendicular filament. +The images show large variations in dust properties, overall morphology, and +highlight feedback from the complex star formation history of the region. The +impact of star formation and stellar feedback is revealed by the heating of the +dust and fluorescing PAHs due to C-H bond stretching in the F335M band. +This occurs on the edges of the arc structure illuminated by UV photons from +massive stars compared to the surrounding T∼ 600K dust seen in the F444W +band. The filament perpendicular to the NGC 346 main body (to the north- +east) extends further than what is seen in HST data and is brightest in the +F335M band. +2.2 Identification of Young Stellar Objects +As they are enshrouded in collapsing dusty envelopes and accretion disks [37– +39], young YSOs are best identified utilising IR colours. As they evolve and +the circumstellar envelopes and disks dissipate, the central star becomes more +apparent in shorter-wavelength light. We detect a total of 45,583 sources in +all four of our wide-band NIRCam filters. This selection provides the most +reliable colour-magnitude diagrams (CMDs) at the cost of some additional +photometric depth.1 +CMDs constructed from the galactic extinction-corrected photometry are +presented in Figure 2. In these near-IR CMDs, the populations of red giant +branch (RGB), red clump (RC), and upper main sequence (UMS) stars are +clearly separated from the dominant population of lower main-sequence and +PMS stars. Evolved stars (e.g., red supergiants, asymptotic giant branch +(AGB), and post-AGB stars) are brighter than the saturation limit and thus +1As NIRCam (PSF ∼ 0.1 arcsec) resolves structures for distant galaxies, contamination from +background galaxies is negligible in our point-source catalogue and corresponding CMDs. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +5 +PMS +YSO Pa𝛼 +YSO +HST PMS +N +E +F277W +F335M +F444W +F335M +BS90 +Fig. 1 +Left: Three-colour composite mosaic of NGC 346 combining the F277W, F335M, +and F444W filters. The region is rich in structures of knots, arcs, and filaments. Areas of +bright red emission are associated with clumpy star formation. The spatially-resolved PAH +emission excited by UV photons in green is brightest in regions corresponding to the edges +of dense material, characteristic of a photodissociation region (PDR). Massive stars, stars +belonging to the BS90 cluster (circled in green), and the SMC field population are also +visible. Right: Mosaic image from the F335M filter showing four different populations of stars +(see Fig. 2): PMS stars (blue circles), YSOs with Paα (orange diamonds), YSOs without +Paα (red squares), and stars matched with the HST PMS catalogue (purple stars). +not expected to appear in this CMD. Table 1 lists the colour selection criteria +for these populations and the number of sources in each class. +The F115W filter is essential to the identification of UMS, RC, and RGB +stars, as these sequences, as well as main sequence turn-off (MSTO) stars, are +conflated in CMDs utilising longer-wavelength colours. In our CMDs, elevated +photometric uncertainties cause scatter for sources near our detection limit, +while a spread of ages and differential extinction across the field broadens the +shape of the evolutionary sequences at all magnitudes. To visualize the effect +of extinction, we show on the diagrams the reddening vector corresponding to +AV =5 according to the SMC Bar Average Extinction Curve of [40]. +When compared to optical CMDs derived from HST data, we find more +than 6,000 sources that are consistent with PMS stars in the mass range +between ∼ 0.5 and 4 M⊙ based on their colours and magnitudes [15, 16]. + +® +！ +★★Springer Nature 2021 LATEX template +6 +The embedded young population of NGC 346 +Furthermore, we find candidate PMS stars in F115W extending at least two +magnitudes below the HST detection limit, suggesting that we can observe T- +Tauri stars down to ∼ 0.1 M⊙. [18] used Hα excess to identify bona fide PMS +with active accretion. This selection further reduced the catalogue to ∼ 700 +sources, about 12% of the total. We find that 435 of these with Hα excess have +a match in the NIRCam catalogue, and we call these HST-PMS in the figures. +In order to disentangle young stars (<10 Myr) from other populations in the +field, we use an F200W–F444W versus F115W–F187N colour-colour diagram +(2CD; Figure 2). Narrow-band photometry with the F187N filter traces the +hydrogen Paα recombination line at 1.875 µm, characteristic of young PMS +stars undergoing mass accretion. When this can be unambiguously identified, +the accretion luminosity and mass accretion rate can be derived following +methods similar to that developed by [41] using Hα. +The F200W–F444W versus F115W–F187N 2CD shows a substantial con- +centration of sources, with the UMS clustered around the (−0.6, −1.5) point +and the RC and RGB stars slightly to the right at (0, −1.5). To conserva- +tively identify the YSOs, we define a horizontal line (F200W–F444W= −1.1) +and a vertical line (F115W–F187N= 0.3) that enclose almost all of the UMS, +RC, and RGB stars. In particular, the vertical line ensures that we are not +including RGB stars with winds that may have a Paα excess. +A minority of sources appear to spread in the upper-right direction, roughly +following our representative SMC reddening vector. In principle, these sources +could be interpreted as either highly-reddened objects due to their circumstel- +lar material or as objects with substantial IR excess, possibly associated with +accretion emission in the F187N filter. +To break this degeneracy, we may look at the distribution of these outliers +in the 2CD (see Sec. 2.3). Tracing for simplicity a cross centred on (0.3, −1.1), +we can divide them into three groups: First, objects in the bottom-left quadrant +are generally compatible with the main populations, with negligible reddening. +Second, objects in the bottom-right quadrant (blue dots) have F200W– +F444W compatible with stellar photospheres, but show significant F187N +excess. One of these sources also showed Hα excess when observed with HST. +Strong line emission in this case is the most viable explanation, with mass +accretion, possibly a sporadic large episode, as a plausible source. In this case, +the lack of F444W excess may suggest that the accreting disk has cleared its +inner hot-dust component and accretion is supported largely by the gaseous +phase. The spatial location of these objects in the NGC 346 field, several of +them rather bright in the F115W filter, shows some concentration in corre- +spondence with the brightest clumps of nebular emission. A significant fraction +is spread in the field, however, suggesting that these accreting YSOs may be +relatively evolved and dispersed. For simplicity, we refer to these objects as +PMS objects. +Third, objects in the top-left quadrant (red squares), which we shall refer +to as YSOs, have near-IR colours compatible with stellar photospheres and +significant F444W excess, a characteristic incompatible with reddened objects. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +7 +Table 1 NGC 346 Stellar Populations identified using JWST/NIRCam. +Population Colour Selection +Number of Sources +RC +inside(-0.45,18.8, -0.45, 19.10, 0.10,19.55, 0.10,19.15) +448 +RGB +0.1 >F115W–F200W > −0.45 and F115W < 21.5 +2176 +UMS +−0.6 >F115W–F200W > −1.18 and F115W < 21.5 +1982 +YSO +F115W−F187N < 0.3 and F200W−F444W > −1.1 +136 +and F335M−F444W > −0.3 +YSO Paα +F115W−F187N > 0.3 and F200W−F444W > −1.1 +216 +and F335M−F444W > −0.3 +PMS +F115W−F187N > 0.3 and F200W−F444W < −1.1 +179 +Of these, 29 also showed an Hα excess in the HST catalogue. These objects are +clustered at the centre of the region but become more spread out in the south- +ern part of our field. This latter grouping may be a candidate for transitional +YSOs (i.e., protoplanetary disks with inner holes in the dust distribution and +negligible mass accretion). +Finally, the last class of objects (orange diamonds) is comprised of sources +in the top-right quadrant of the 2CD. In the F115W–F444W CMD, they +appear well above the region occupied by low-mass MS and PMS stars, +suggesting that they may be highly-reddened, relatively massive stars. HST +photometry indicates that 80 of these objects also show Hα excess. Their +spatial distribution strikingly traces the main filaments of the region, suggest- +ing that they are associated with ongoing star-formation sites. Remarkably, a +large fraction of the accreting PMS stars detected by HST lie in this sector, +supporting the hypothesis that these are bona fide YSOs that have not yet +significantly migrated from their birthplace. We shall refer to them as YSOs +with Paα emission. +This simplified, yet conservative selection of YSO and PMS candidates +represents the deepest census of a star-forming region in a low-metallicity +galaxy. Our data include redder candidates that are not present in HST optical +catalogues because they are undetectable in those bands, as well as low-mass +(≲2 M⊙) sources significantly below the completeness limit of Spitzer surveys. +Additional photometry at longer wavelengths will provide a better definition +of the YSO and PMS classes outlined here. +2.3 Spatial Distribution of NGC 346 stars +The right frame of Figure 1 shows the spatial distribution of the prominent +young populations towards NGC 346 overlaid on the F335M image. There +is a high degree of spatial overlap between YSOs (red squares) and YSOs +with Paα excess (orange diamonds), and their location appears to be spatially +correlated with the bright dust emission. The YSOs clearly follow the arc and +are located where it intersects with the other filaments in the north of the +complex, confirming the clustered nature of star formation in this region. +Interestingly, at optical wavelengths, many of these sources are either not +visible or not correctly identified as YSOs, highlighting the importance of IR + +Springer Nature 2021 LATEX template +8 +The embedded young population of NGC 346 +observations to accurately interpret star-forming regions. We note that the +location of both YSOs and YSOs with Paα excess do not necessarily correspond +to the clusters previously identified in the HST data [16]. On the contrary, +they tend to encircle the cavities created by the NGC 346 OB stars [35]. The +distribution of the PMS stars and the newly-identified YSOs supports the +scenario proposed by [34], in which a global hierarchical collapse culminates in +“river-like” structures responsible for the formation of clumps where significant +changes in the coherence of the motion field are detected, and therefore where +one expects high gas friction. +On the other hand, the older PMS stars (blue dots) are diffusely distributed +across the field. This larger spatial dispersion is consistent with formation +episodes over the past 20−30 Myr, while this stratification is likely due to both +turbulent star formation and early dynamical evolution, in agreement with +the spiralling nature and increasing rotation with distance from the centre of +NGC 346 [e.g., 11, 34, 35, 42]. +As expected, we find that JWST surpasses the capability of Spitzer to +detect candidate YSOs using only aperture photometry. This expands the +observed sample of approximately 100 YSO candidates within NGC 346 by +over a factor of three. Our survey reveals a population of dusty, sub-solar +YSOs, and represents the deepest extragalactic census of these objects at low +metallicity. The discovery of an associated IR excess in these objects reveals +for the first time that the material required to form rocky planets is present. +It was not previously known whether terrestrial planet formation is possible +in low-metallicity environments, as heavy elements are needed to produce the +dust from which planetesimals coagulate. +3 Methods +3.1 NIRCam Observations and Data Processing +We have mapped NGC 346 with JWST (Program ID: 1227; PI: Meixner), +using the Near Infrared Camera (NIRCam; [43]) in the F115W, F187N, and +F200W short-wavelength (SW) bands, and F277W, F335M, and F444W long- +wavelength (LW) bands. The images, obtained on 2022 July 16, are centred +at R.A. = 00:59:04.9451, decl. = −72:10:9.15, and cover an area of ∼31.05 +arcmin2 (see Table 2). The NIRCam observations employed both the A and +B modules to provide the largest field of view with one pointing, and were +obtained using the bright2 readout pattern with two groups per integration +at four sub-pixel dither positions for a total exposure time of 171.788 seconds +per filter. +The level two NIRCam data were reprocessed using a slightly modified +version of the JWST official pipeline (version 1.7.2). These modifications cor- +rect for 1/f and flat field correction noise, World Coordinate System (WCS) +alignment issues2, differences in background matching across the mosaic, and +21/f noise corrections applied using image1overf.py [44]. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +9 +Table 2 Summary of the NGC 346 NIRCam survey, JWST Program ID 1227 and values +adopted for the properties of NGC 346. +Characteristic +Value +Nominal center point +00:59:04.9451 −72:10:9.15 +Survey area (arcmin2) +31.05 +Central λ (µm) +1.154, 1.874, 1.990, 2.786, 3.365, 4.421 +FWHM at λ (pixel) +1.290, 2.065, 2.129, 1.460, 1.762, 2.302 +Point source completeness limits at λ (mag) +26.6, 23.2, 26.7, 25.4, 24.5, 26.4 +Distance to NGC 346 +60.4 kpc +Distance modulus (m − M)0 +18.96 [1] +E(B − V ) +0.08 +Metallicity [Fe/H] (dex) +−0.9–1.0 Z = 0.002 +include the most recent NIRCam calibration files3. The final pixel scale of the +mosaics is set to 0.”0315 for the three SW bands and 0.”0629 for the three LW +bands. +3.2 Photometry +Aperture photometry was performed on the individual exposures in each band +using the starbug ii tool [47]. starbug ii, which incorporates modules from +photutils [48], is optimized for observations utilizing both NIRCam and the +Mid-Infrared Instrument [MIRI; 49] on JWST and is designed to detect and +extract point sources in crowded environments with complex diffuse emis- +sion and variable backgrounds. The sources identified in the single frames are +extracted at a 3σ level above the local background, which was characterized +and globally subtracted using a combination of three different background esti- +mation techniques. This ensures objects in complex nebular regions in which +background determination is more problematic are not prematurely excluded. +An aperture with radius 1.5 pixels and an annulus from 3.0 to 4.5 pixels sur- +rounding each source was then employed in the photometric extraction. Sharp +between 0.4-0.9 and round ≤ +1.0 cuts are applied, and then only sources +detected in at least three of the four frames are retained. Sources with mean +and median values that differ by more than 0.1 dex between exposures were +flagged and removed as mismatches. This eliminates cosmic rays, noise spikes +from the point spread function (PSF), and extended sources such as resolved +background galaxies, and ensures high fidelity of the final point source cata- +logues. Aperture corrections provided in the CRDS reference files were then +applied to all photometry. To generate a NIRCam band-merged point-source +catalogue, the data were merged using the closest astrometric separation +<0”.25. We correct the photometric values for Galactic foreground extinction +using E(B −V ) = 0.08 and the extinction curve of [50] with RV = 2.7, but not +for any extinction intrinsic to NGC 346. The final catalogue includes ∼525,000 +sources, which we present in AB magnitudes. +3jwst 0989.map of the Operational Pipeline Calibration Reference Data System was produced +on 2022-10-03 with on-sky derived photometric zero-points [45, 46]. + +Springer Nature 2021 LATEX template +10 +The embedded young population of NGC 346 +In the F115W band, the point source completeness magnitude of 26.6 allows +for the characterization of young populations (<10 Myr) down to an initial +mass of ∼ 0.15 M⊙, corresponding to stars in the T-Tauri range. Sources +brighter than F115W = 17.3 mag are saturated. To verify the PMS mass +limits, we match (using R < 0.3′′) our NIRCam catalogue to the [29] and [18] +HST data which include mass and age estimates. There are 24,367 sources in +common, including PMS stars with masses from 0.4–4 M⊙and ages 1–30 Myr. +References +[1] de Grijs, R. & Bono, G. Clustering of Local Group Distances: Publica- +tion Bias or Correlated Measurements? III. The Small Magellanic Cloud. +AJ 149 (6), 179 (2015). https://doi.org/10.1088/0004-6256/149/6/179, +https://arxiv.org/abs/1504.00417 [astro-ph.SR]. +[2] Bouret, J. C. et al. Quantitative Spectroscopy of O Stars at Low Metallic- +ity: O Dwarfs in NGC 346. ApJ 595 (2), 1182–1205 (2003). https://doi. +org/10.1086/377368, https://arxiv.org/abs/astro-ph/0301454 [astro-ph]. +[3] Peimbert, M., Peimbert, A. & Ruiz, M. T. The Chemical Composition +of the Small Magellanic Cloud H II Region NGC 346 and the Primordial +Helium Abundance. ApJ 541 (2), 688–700 (2000). https://doi.org/10. +1086/309485, https://arxiv.org/abs/astro-ph/0003154 [astro-ph]. +[4] Madau, P. & Dickinson, M. Cosmic Star-Formation History. ARA&A 52, +415–486 (2014). +https://doi.org/10.1146/annurev-astro-081811-125615, +https://arxiv.org/abs/1403.0007 [astro-ph.CO]. +[5] Dimaratos, A., Cormier, D., Bigiel, F. & Madden, S. C. Modeling the +physical properties in the ISM of the low-metallicity galaxy NGC 4214. +A&A 580, A135 (2015). https://doi.org/10.1051/0004-6361/201526447, +https://arxiv.org/abs/1506.06782 [astro-ph.GA]. +[6] Tchernyshyov, K. et al. Elemental Depletions in the Magellanic Clouds +and the Evolution of Depletions with Metallicity. +ApJ 811 (2), 78 +(2015). https://doi.org/10.1088/0004-637X/811/2/78, https://arxiv.org/ +abs/1503.08852 [astro-ph.GA]. +[7] Roman-Duval, J. et al. +Dust and Gas in the Magellanic Clouds +from the HERITAGE Herschel Key Project. II. Gas-to-dust Ratio +Variations across Interstellar Medium Phases. +ApJ +797 (2), 86 +(2014). https://doi.org/10.1088/0004-637X/797/2/86, https://arxiv.org/ +abs/1411.4552 [astro-ph.GA]. +[8] Johansen, A., Youdin, A. & Mac Low, M.-M. +Particle Clumping and +Planetesimal Formation Depend Strongly on Metallicity. ApJL 704 (2), + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +11 +L75–L79 (2009). https://doi.org/10.1088/0004-637X/704/2/L75, https: +//arxiv.org/abs/0909.0259 [astro-ph.EP]. +[9] Li, R. & Youdin, A. N. Thresholds for Particle Clumping by the Streaming +Instability. ApJ 919 (2), 107 (2021). https://doi.org/10.3847/1538-4357/ +ac0e9f, https://arxiv.org/abs/2105.06042 [astro-ph.EP]. +[10] Ercolano, B. & Clarke, C. J. Metallicity, planet formation and disc life- +times. MNRAS 402 (4), 2735–2743 (2010). https://doi.org/10.1111/j. +1365-2966.2009.16094.x, https://arxiv.org/abs/0910.5110 [astro-ph.EP]. +[11] Cignoni, M., Tosi, M., Sabbi, E., Nota, A. & Gallagher, J. S. History and +Modes of Star Formation in the Most Active Region of the Small Magel- +lanic Cloud, NGC 346. AJ 141 (2), 31 (2011). https://doi.org/10.1088/ +0004-6256/141/2/31, https://arxiv.org/abs/1010.0340 [astro-ph.GA]. +[12] Massey, P., Parker, J. W. & Garmany, C. D. +The Stellar Content of +NGC 346: A Plethora of O Stars in the SMC. +AJ 98, 1305 (1989). +https://doi.org/10.1086/115217 . +[13] Evans, C. J., Lennon, D. J., Smartt, S. J. & Trundle, C. +The VLT- +FLAMES survey of massive stars: observations centered on the Magellanic +Cloud clusters NGC 330, NGC 346, NGC 2004, and the N11 region. A&A +456 (2), 623–638 (2006). +https://doi.org/10.1051/0004-6361:20064988, +https://arxiv.org/abs/astro-ph/0606405 [astro-ph]. +[14] Dufton, P. L., Evans, C. J., Hunter, I., Lennon, D. J. & Schneider, +F. R. N. +A census of massive stars in NGC 346. +Astronomy & +Astrophysics 626, A50 (2019). +URL https://www.aanda.org/10.1051/ +0004-6361/201935415. https://doi.org/10.1051/0004-6361/201935415 . +[15] Nota, A. et al. Discovery of a Population of Pre-Main-Sequence Stars in +NGC 346 from Deep Hubble Space Telescope ACS Images. ApJL 640 (1), +L29–L33 (2006). https://doi.org/10.1086/503301, https://arxiv.org/abs/ +astro-ph/0602218 [astro-ph]. +[16] Sabbi, E. et al. Past and Present Star Formation in the SMC: NGC 346 +and its Neighborhood. AJ 133 (1), 44–57 (2007). https://doi.org/10. +1086/509257, https://arxiv.org/abs/astro-ph/0609330 [astro-ph]. +[17] Hennekemper, E., Gouliermis, D. A., Henning, T., Brandner, W. & Dol- +phin, A. E. NGC 346 in the Small Magellanic Cloud. III. Recent Star +Formation and Stellar Clustering Properties in the Bright H II Region +N66. +ApJ 672 (2), 914–929 (2008). +https://doi.org/10.1086/524105, +https://arxiv.org/abs/0710.0774 [astro-ph]. + +Springer Nature 2021 LATEX template +12 +The embedded young population of NGC 346 +[18] De Marchi, G., Panagia, N. & Sabbi, E. +Clues to the Star For- +mation in NGC 346 across Time and Space. +ApJ +740 (1), 10 +(2011). https://doi.org/10.1088/0004-637X/740/1/10, https://arxiv.org/ +abs/1106.5780 [astro-ph.SR]. +[19] Bolatto, A. D. et al. The Spitzer Survey of the Small Magellanic Cloud: +S3MC Imaging and Photometry in the Mid- and Far-Infrared Wave Bands. +ApJ 655 (1), 212–232 (2007). +https://doi.org/10.1086/509104, https: +//arxiv.org/abs/astro-ph/0608561 [astro-ph]. +[20] Gordon, K. D. et al. Surveying the Agents of Galaxy Evolution in the +Tidally Stripped, Low Metallicity Small Magellanic Cloud (SAGE-SMC). +I. Overview. AJ 142, 102 (2011). https://doi.org/10.1088/0004-6256/ +142/4/102, https://arxiv.org/abs/1107.4313 [astro-ph.CO]. +[21] Meixner, M. et al. The HERSCHEL Inventory of The Agents of Galaxy +Evolution in the Magellanic Clouds, a Herschel Open Time Key Program. +AJ 146, 62 (2013). https://doi.org/10.1088/0004-6256/146/3/62 . +[22] Simon, J. D. et al. The Spitzer Survey of the Small Magellanic Cloud: +Discovery of Embedded Protostars in the H II Region NGC 346. ApJ +669 (1), 327–336 (2007). https://doi.org/10.1086/521544, https://arxiv. +org/abs/0707.3998 [astro-ph]. +[23] Sewi�lo, M. et al. Surveying the Agents of Galaxy Evolution in the Tidally +Stripped, Low Metallicity Small Magellanic Cloud (SAGE-SMC). III. +Young Stellar Objects. ApJ 778, 15 (2013). https://doi.org/10.1088/ +0004-637X/778/1/15 . +[24] Seale, J. P. et al. Herschel Key Program Heritage: a Far-Infrared Source +Catalog for the Magellanic Clouds. AJ 148, 124 (2014). https://doi.org/ +10.1088/0004-6256/148/6/124 . +[25] Ruffle, P. M. E. et al. Spitzer infrared spectrograph point source clas- +sification in the Small Magellanic Cloud. +MNRAS 451, 3504–3536 +(2015). +https://doi.org/10.1093/mnras/stv1106, https://arxiv.org/abs/ +1505.04499 [astro-ph.SR]. +[26] Rubio, M., Barb´a, R. H. & Kalari, V. M. +Massive young stellar +objects in the N 66/NGC 346 region of the SMC. +A&A 615, A121 +(2018). +https://doi.org/10.1051/0004-6361/201730487, https://arxiv. +org/abs/1803.10833 [astro-ph.GA]. +[27] Jones, O. C. et al. +Near-infrared spectroscopy of embedded proto- +stars in the massive metal-poor star forming region NGC 346. MNRAS +(2022). https://doi.org/10.1093/mnras/stac2491, https://arxiv.org/abs/ +2209.00040 [astro-ph.SR]. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +13 +[28] Hony, S. et al. Star formation rates from young-star counts and the struc- +ture of the ISM across the NGC 346/N66 complex in the SMC. MNRAS +448 (2), 1847–1862 (2015). https://doi.org/10.1093/mnras/stv107, https: +//arxiv.org/abs/1501.03634 [astro-ph.GA]. +[29] Sabbi, E. et al. The Stellar Mass Distribution in the Giant Star Forming +Region NGC 346. AJ 135 (1), 173–181 (2008). https://doi.org/10.1088/ +0004-6256/135/1/173, https://arxiv.org/abs/0710.0558 [astro-ph]. +[30] Gouliermis, D. A., Hony, S. & Klessen, R. S. The complex distribution +of recently formed stars. Bimodal stellar clustering in the star-forming +region NGC 346. MNRAS 439 (4), 3775–3789 (2014). https://doi.org/ +10.1093/mnras/stu228, https://arxiv.org/abs/1402.0078 [astro-ph.GA]. +[31] Rubio, M. et al. Multiwavelength observations of N 66 in the SMC: unveil- +ing photodissociation interfaces and star formation. A&A 359, 1139–1146 +(2000) . +[32] Contursi, A. et al. Mid-infrared imaging and spectrophotometry of N 66 +in the SMC with ISOCAM. A&A 362, 310–324 (2000). https://arxiv. +org/abs/astro-ph/0006185 [astro-ph]. +[33] Neelamkodan, N. et al. +ALMA Reveals a Cloud-Cloud Collision that +Triggers Star Formation in the Small Magellanic Cloud. ApJL 908 (2), +L43 (2021). +https://doi.org/10.3847/2041-8213/abdebb, https://arxiv. +org/abs/2101.10711 [astro-ph.GA]. +[34] Sabbi, E. et al. The Internal Proper Motion Kinematics of NGC 346: Past +Formation and Future Evolution. ApJ 936 (2), 135 (2022). https://doi. +org/10.3847/1538-4357/ac8005, https://arxiv.org/abs/2209.03215 [astro- +ph.GA]. +[35] Zeidler, P., Sabbi, E. & Nota, A. +The Internal Line-of-Sight Kine- +matics of NGC 346: The Rotation of the Core Region. +ApJ 936 (2), +136 (2022). +https://doi.org/10.3847/1538-4357/ac8004, https://arxiv. +org/abs/2209.03237 [astro-ph.GA]. +[36] Bica, E. L. D. & Schmitt, H. R. A Revised and Extended Catalog of +Magellanic System Clusters, Associations, and Emission Nebulae. I. Small +Magellanic Cloud and Bridge. ApJS 101, 41 (1995). https://doi.org/10. +1086/192233 . +[37] Lada, C. J. Peimbert, M. & Jugaku, J. (eds) Star formation - From OB +associations to protostars. (eds Peimbert, M. & Jugaku, J.) Star Forming +Regions, Vol. 115 of IAU Symposium, 1–17 (1987). + +Springer Nature 2021 LATEX template +14 +The embedded young population of NGC 346 +[38] Robitaille, T. P., Whitney, B. A., Indebetouw, R., Wood, K. & Denz- +more, P. Interpreting Spectral Energy Distributions from Young Stellar +Objects. I. A Grid of 200,000 YSO Model SEDs. ApJS 167, 256–285 +(2006). +https://doi.org/10.1086/508424, https://arxiv.org/abs/arXiv: +astro-ph/0608234 . +[39] Whitney, B. A. et al. Spitzer Sage Survey of the Large Magellanic Cloud. +III. Star Formation and ˜1000 New Candidate Young Stellar Objects. AJ +136, 18–43 (2008). https://doi.org/10.1088/0004-6256/136/1/18 . +[40] Gordon, K. D., Clayton, G. C., Misselt, K. A., Landolt, A. U. & Wolff, +M. J. A Quantitative Comparison of the Small Magellanic Cloud, Large +Magellanic Cloud, and Milky Way Ultraviolet to Near-Infrared Extinction +Curves. ApJ 594 (1), 279–293 (2003). https://doi.org/10.1086/376774, +https://arxiv.org/abs/astro-ph/0305257 [astro-ph]. +[41] De Marchi, G. et al. Star Formation in 30 Doradus. ApJ 739 (1), 27 +(2011). https://doi.org/10.1088/0004-637X/739/1/27, https://arxiv.org/ +abs/1106.2801 [astro-ph.SR]. +[42] De Marchi, G. et al. Photometric Determination of the Mass Accretion +Rates of Pre-main-sequence Stars. II. NGC 346 in the Small Magellanic +Cloud. ApJ 740 (1), 11 (2011). https://doi.org/10.1088/0004-637X/740/ +1/11, https://arxiv.org/abs/1104.4494 [astro-ph.SR]. +[43] Rieke, M. J., Kelly, D. & Horner, S. Heaney, J. B. & Burriesci, L. G. +(eds) Overview of James Webb Space Telescope and NIRCam’s Role. +(eds Heaney, J. B. & Burriesci, L. G.) Cryogenic Optical Systems and +Instruments XI, Vol. 5904 of Society of Photo-Optical Instrumentation +Engineers (SPIE) Conference Series, 1–8 (2005). +[44] Willott, C. image1overf (2022). URL https://github.com/chriswillott/ +jwst. +[45] Gordon, K. D. et al. The James Webb Space Telescope Absolute Flux +Calibration. I. Program Design and Calibrator Stars. AJ 163 (6), 267 +(2022). +https://doi.org/10.3847/1538-3881/ac66dc, https://arxiv.org/ +abs/2204.06500 [astro-ph.IM]. +[46] Boyer, M. L. et al. The JWST Resolved Stellar Populations Early Release +Science Program. I. NIRCam Flux Calibration. Research Notes of the +American Astronomical Society 6 (9), 191 (2022). +https://doi.org/10. +3847/2515-5172/ac923a, https://arxiv.org/abs/2209.03348 [astro-ph.IM]. +[47] Nally, C. & Jones, O. +Starbug2 (2022). +URL https://github.com/ +conornally/starbug2. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +15 +[48] Bradley, L. et al. astropy/photutils: 1.0.0 (2020). URL https://doi.org/ +10.5281/zenodo.4044744. +[49] Rieke, G. H. et al. The Mid-Infrared Instrument for the James Webb +Space Telescope, I: Introduction. PASP 127 (953), 584 (2015). https:// +doi.org/10.1086/682252, https://arxiv.org/abs/1508.02294 [astro-ph.IM]. +[50] Cardelli, J. A., Clayton, G. C. & Mathis, J. S. The relationship between +infrared, optical, and ultraviolet extinction. ApJ 345, 245–256 (1989). +https://doi.org/10.1086/167900 . +[51] Astropy Collaboration et al. Astropy: A community Python package for +astronomy. A&A 558, A33 (2013). https://doi.org/10.1051/0004-6361/ +201322068, https://arxiv.org/abs/1307.6212 [astro-ph.IM]. +[52] Taylor, M. B. Shopbell, P., Britton, M. & Ebert, R. (eds) TOPCAT & +STIL: Starlink Table/VOTable Processing Software. +(eds Shopbell, P., +Britton, M. & Ebert, R.) Astronomical Data Analysis Software and Sys- +tems XIV, Vol. 347 of Astronomical Society of the Pacific Conference +Series, 29 (2005). +Code Availability Statement. +This research made use of astropy [51], +photutils [48] and topcat [52]. The starbug ii tool [47] optimized for JWST +NIRCam and MIRI point source photometry in complex crowded environments +is available via pip install starbug2. +Data Availability. +The data that support the findings of this study are +available from the corresponding author upon reasonable request. +Competing Interests. +The authors declare no competing interests. +Author Contributions. +OCJ led the analysis and is the science lead of the +NGC 346 Team. CN produced the photometric catalogues. NH and LL repro- +cessed the NIRCam data. KF and CR assisted in the photometry. MR, GdM, +ES provided advice on NIRCam data processing and the analysis on compar- +ison to HST data. LC produced images on NGC 346. AH, MM, KP helped +optimise the observations. All authors contributed to observation planning +and/or scientific interpretation. +Acknowledgments. +This work is based on observations made with the +NASA/ESA/CSA James Webb Space Telescope. The data were obtained from +the Mikulski Archive for Space Telescopes at the Space Telescope Science +Institute, which is operated by the Association of Universities for Research in +Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These obser- +vations are associated with program #1227. OCJ acknowledge support from an +STFC Webb fellowship. KF acknowledges support through the ESA Research +Fellowship. MM, and NH acknowledge support through a NASA/JWST grant +80NSSC22K0025, and MM and LL acknowledge support from the NSF through + +Springer Nature 2021 LATEX template +16 +The embedded young population of NGC 346 +grant 2054178. ON acknowledges support from STScI Director’s Discretionary +Fund. + +Springer Nature 2021 LATEX template +The embedded young population of NGC 346 +17 +2 +1 +0 +1 +2 +3 +F115W - F200W +18 +20 +22 +24 +26 +28 +F115W +Av = 5 +2 +1 +0 +1 +2 +3 +F115W - F200W +18 +20 +22 +24 +26 +28 +F115W +2 +0 +2 +4 +F115W - F444W +18 +20 +22 +24 +26 +28 +F115W +Av = 5 +2 +0 +2 +4 +F115W - F444W +18 +20 +22 +24 +26 +28 +F115W +2 +0 +2 +4 +F115W - F187N +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F200W - F444W +Av = 5 +2 +0 +2 +4 +F115W - F187N +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +F200W - F444W +UMS +RGB +RC +PMS +YSO Pa +YSO +HST PMS +Fig. 2 Colour-magnitude diagrams and colour-colour diagrams highlighting sources classi- +fied photometrically (right) and as stellar density Hess diagrams (left) for all point sources +in the NGC 346 catalogue. The red giant branch (RGB), red clump (RC), main sequence +turn-off (MSTO), and upper main sequence (UMS) are all easily identified in the stellar pop- +ulations. In the bottom panels, the red dotted line separates sources with an IR excess, whilst +the blue dashed line separates objects with a Paα excess from the rest of the population. + +- +- +- +- +- +- +- +- +- +- +- +- +- +- +-- +- +- +- +- +- +- +- +- +- +- +- +- +- +- \ No newline at end of file diff --git a/udE2T4oBgHgl3EQfgAc5/content/tmp_files/load_file.txt b/udE2T4oBgHgl3EQfgAc5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88c2f2757534861e3d1e90ef228484893f8380b4 --- /dev/null +++ b/udE2T4oBgHgl3EQfgAc5/content/tmp_files/load_file.txt @@ -0,0 +1,895 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf,len=894 +page_content='Springer Nature 2021 LATEX template Discovery of dusty sub-solar mass young stellar objects in NGC 346 with JWST/NIRCam Olivia C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Jones1*, Conor Nally2, Nolan Habel3, Laura Lenki´c3, Katja Fahrion4, Alec S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Hirschauer5, Laurie E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Chu6, Margaret Meixner3, Guido De Marchi7, Omnarayani Nayak5, Massimo Robberto5,8, Elena Sabbi5, Peter Zeidler9, Catarina Alves de Oliveira10, Tracy Beck5, Katia Biazzo, Bernhard Brandl12, Giovanna Giardino8, Teresa Jerabkova13, Charles Keyes5, James Muzerolle5, Nino Panagia5, Klaus Pontoppidan5, Ciaran Rogers12, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Sargent5 and David Soderblom5 1*UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 3Stratospheric Observatory for Infrared Astronomy, NASA Ames Research Center, Mail Stop 204-14, Moffett Field, 94035, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 4European Space Research and Technology Centre, European Space Agency, Keplerlaan 1, Noordwijk, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 5Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, 21218, MD, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 6NASA Postdoctoral Program Fellow, NASA Ames Research Center, M/S 245-1, Moffett Field, 94035, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 7European Space Research and Technology Centre, European Space Agency, Keplerlaan 1, Noordwijk, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 8Johns Hopkins University, 3400 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Charles Street, Baltimore, MD 21218, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 9AURA for the European Space Agency, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, 21218, MD, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='03932v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR] 10 Jan 2023 Springer Nature 2021 LATEX template 2 The embedded young population of NGC 346 10ESAC, European Space Agency, 28692 Villafranca del Castillo, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 11INAF, Astronomical Observatory of Rome, Via Frascati 33, Monteporzio Catone, I-00078, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 12Leiden Observatory, Leiden University, 2300 RA Leiden, Leiden, PO Box 9513, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 13European Southern Observatory, Karl-Schwarzschild-Strasse 2, Garching,Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' E-mail(s): olivia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='jones@stfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Abstract JWST observations of NGC 346, a star-forming region in the metal-poor Small Magellanic Cloud, reveal a substantial population of sub-solar mass young stellar objects (YSOs) with IR excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We have detected more than 33,000 sources across six NIRCam filters with deep, high-resolution imaging, where ongoing low-mass star formation is concentrated along dust filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' From these observations, we construct detailed near- IR colour-magnitude diagrams with which preliminary categorizations of different YSO classes are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' For the youngest, most deeply- embedded objects, JWST/NIRCam reaches over 10 magnitudes below Spitzer observations at comparable wavelengths, and two magnitudes fainter than HST for more evolved pre main sequence sources, corre- sponding to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' For the first time in an extragalactic environment, we detect the full sequence of low-mass YSOs at all evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Furthermore, evidence of IR excess and accretion suggests that the dust required for rocky planet formation is present at low metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Keywords: infrared: stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' galaxies: clusters: individual (NGC 346);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Magellanic Clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' stars: formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' stars: pre-main-sequence 1 Introduction Located in the Small Magellanic Cloud (SMC) at a distance of ∼62 kpc [1], NGC 346 is a prominent young cluster (∼3 Myr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [2]) actively forming stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' It is the brightest and largest star-formation region in this metal-poor galaxy (∼1/5 Z⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [3]) which has a comparable metallicity to galaxies at the epoch of peak star formation [“cosmic noon”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Below these levels of chemical enrichment, the dust content of the interstellar medium (ISM) drops precipitously, altering the environment in which stars form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', [6, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' It is unknown if sufficient quantities of dust survives the star formation process in order to contribute to the formation of rocky planetary systems in Springer Nature 2021 LATEX template The embedded young population of NGC 346 3 low metallicity environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Below a certain threshold of metal abundance, planetesimal formation via the streaming instability is suppressed [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The metallicity of inner protoplanetary disks is therefore thought to play a critical role in the ability to form terrestrial planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Further, the dust content of disks sets their lifetimes as lower-metallicity systems are more susceptible to fast photo-evaporation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' It is therefore of great interest to identify low-mass young stellar objects (YSOs) and discern their dust content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The star formation history of NGC 346 is complex, with multiple stel- lar populations identified within the cluster [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Powering the ionization of this giant H ii region are more than 30 spectroscopically-identified mas- sive (35–100 M⊙) O-type stars [12–14], the largest such sample in the SMC, which dominate the radiative and mechanical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Deep Hubble Space Telescope (HST) images reveal thousands of low-mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6–3 M⊙) pre main sequence (PMS) stars [15], which are distributed throughout the nebula and are connected by gas and dust filaments [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Spitzer and Herschel surveys of the SMC [19–21] unveiled approximately 100 candidate YSOs in the very early stages of formation within the NGC 346 complex [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' These high- mass YSOs possess typical masses of 8 M⊙ and have formed within the past ∼1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' They are located at the edge of or inside dusty pillars, which are often associated with Hα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Their presence establishes that star formation is ongoing throughout the complex at a rate > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2 × 10−3 M⊙ yr−1 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In the infrared (IR), spectroscopic data for the young populations in NGC 346 are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Spitzer IRS data spectroscopically confirmed the identity of six massive YSOs in the cluster [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [26] obtained HK band spectra to confirm the existence of three early-type stars in NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Most recently, [27] used VLT/KMOS to observe ∼15 other YSO candidates which were resolved into multiple young stars still accreting mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Overall, NGC 346 possesses a complex distribution of hierarchically-linked star clusters of varying ages which inhabit a variety of environments [28], and which are dispersed across the extended field [17, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Within the ISM there is a wide range of substructures exhibited in polycyclic aromatic hydrocarbon emission (PAH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 8 µm), warm dust (24 µm), and molecular gas [CO J = 2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 28, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A tight correlation is seen between this emission and that of Hα, which presents as a well-defined bar extending from the centre of the region to the northeast and as an arc structure extending from southeast to northwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A recent Atacama Large Millimeter/submillimeter Array (ALMA) CO(J = 1−0) study [33] discovered that the intersection of three colliding clumpy filaments is co-spatial with the locations of a cluster of YSOs and PMS stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Using HST proper motions and VLT/MUSE radial velocities, [34] and [35] showed that stars in NGC 346 move along a wide spiral and that clusters of YSOs and young PMS stars seem to be predominately located where the coherent motion field shows significant changes, hence turbulence is still driving star formation across the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 The embedded young population of NGC 346 NGC 346 is one of the most active star-forming regions in the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Its proximity, size (∼100 × 100 pc2), low foreground extinction, and an abun- dance of wide-field, high-resolution panchromatic data make it an ideal system for the study of both low- and high-mass star formation, the effects of this star formation on the surrounding medium, and the potential triggers of star forma- tion in an environment vastly different from our local Galactic surroundings, and akin to galaxies at cosmic noon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2 Results and Discussion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 Images The NIRCam images of NGC 346 shown in Figure 1 reveal the complex fila- mentary structure of the NGC 346 arc, dominated by emission from warm dust and PAHs, together with the intermediate-age BS90 cluster [36] just above the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The brightest red stars are located along dust ridges, in tips of warm dust lanes, in large sub-clusters within the centre NGC 346 arc, or in smaller clumps located along the arc and northeast perpendicular filament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The images show large variations in dust properties, overall morphology, and highlight feedback from the complex star formation history of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The impact of star formation and stellar feedback is revealed by the heating of the dust and fluorescing PAHs due to C-H bond stretching in the F335M band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This occurs on the edges of the arc structure illuminated by UV photons from massive stars compared to the surrounding T∼ 600K dust seen in the F444W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The filament perpendicular to the NGC 346 main body (to the north- east) extends further than what is seen in HST data and is brightest in the F335M band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2 Identification of Young Stellar Objects As they are enshrouded in collapsing dusty envelopes and accretion disks [37– 39], young YSOs are best identified utilising IR colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' As they evolve and the circumstellar envelopes and disks dissipate, the central star becomes more apparent in shorter-wavelength light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We detect a total of 45,583 sources in all four of our wide-band NIRCam filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This selection provides the most reliable colour-magnitude diagrams (CMDs) at the cost of some additional photometric depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 CMDs constructed from the galactic extinction-corrected photometry are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In these near-IR CMDs, the populations of red giant branch (RGB), red clump (RC), and upper main sequence (UMS) stars are clearly separated from the dominant population of lower main-sequence and PMS stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Evolved stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', red supergiants, asymptotic giant branch (AGB), and post-AGB stars) are brighter than the saturation limit and thus 1As NIRCam (PSF ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 arcsec) resolves structures for distant galaxies, contamination from background galaxies is negligible in our point-source catalogue and corresponding CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 5 PMS YSO Pa𝛼 YSO HST PMS N E F277W F335M F444W F335M BS90 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1 Left: Three-colour composite mosaic of NGC 346 combining the F277W, F335M, and F444W filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The region is rich in structures of knots, arcs, and filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Areas of bright red emission are associated with clumpy star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The spatially-resolved PAH emission excited by UV photons in green is brightest in regions corresponding to the edges of dense material, characteristic of a photodissociation region (PDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Massive stars, stars belonging to the BS90 cluster (circled in green), and the SMC field population are also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Right: Mosaic image from the F335M filter showing four different populations of stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2): PMS stars (blue circles), YSOs with Paα (orange diamonds), YSOs without Paα (red squares), and stars matched with the HST PMS catalogue (purple stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' not expected to appear in this CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Table 1 lists the colour selection criteria for these populations and the number of sources in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The F115W filter is essential to the identification of UMS, RC, and RGB stars, as these sequences, as well as main sequence turn-off (MSTO) stars, are conflated in CMDs utilising longer-wavelength colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In our CMDs, elevated photometric uncertainties cause scatter for sources near our detection limit, while a spread of ages and differential extinction across the field broadens the shape of the evolutionary sequences at all magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' To visualize the effect of extinction, we show on the diagrams the reddening vector corresponding to AV =5 according to the SMC Bar Average Extinction Curve of [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' When compared to optical CMDs derived from HST data, we find more than 6,000 sources that are consistent with PMS stars in the mass range between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 and 4 M⊙ based on their colours and magnitudes [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ® ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ★★Springer Nature 2021 LATEX template 6 The embedded young population of NGC 346 Furthermore, we find candidate PMS stars in F115W extending at least two magnitudes below the HST detection limit, suggesting that we can observe T- Tauri stars down to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [18] used Hα excess to identify bona fide PMS with active accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This selection further reduced the catalogue to ∼ 700 sources, about 12% of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We find that 435 of these with Hα excess have a match in the NIRCam catalogue, and we call these HST-PMS in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In order to disentangle young stars (<10 Myr) from other populations in the field, we use an F200W–F444W versus F115W–F187N colour-colour diagram (2CD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Narrow-band photometry with the F187N filter traces the hydrogen Paα recombination line at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='875 µm, characteristic of young PMS stars undergoing mass accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' When this can be unambiguously identified, the accretion luminosity and mass accretion rate can be derived following methods similar to that developed by [41] using Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The F200W–F444W versus F115W–F187N 2CD shows a substantial con- centration of sources, with the UMS clustered around the (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5) point and the RC and RGB stars slightly to the right at (0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' To conserva- tively identify the YSOs, we define a horizontal line (F200W–F444W= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1) and a vertical line (F115W–F187N= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3) that enclose almost all of the UMS, RC, and RGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In particular, the vertical line ensures that we are not including RGB stars with winds that may have a Paα excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A minority of sources appear to spread in the upper-right direction, roughly following our representative SMC reddening vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In principle, these sources could be interpreted as either highly-reddened objects due to their circumstel- lar material or as objects with substantial IR excess, possibly associated with accretion emission in the F187N filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' To break this degeneracy, we may look at the distribution of these outliers in the 2CD (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Tracing for simplicity a cross centred on (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1), we can divide them into three groups: First, objects in the bottom-left quadrant are generally compatible with the main populations, with negligible reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Second, objects in the bottom-right quadrant (blue dots) have F200W– F444W compatible with stellar photospheres, but show significant F187N excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' One of these sources also showed Hα excess when observed with HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Strong line emission in this case is the most viable explanation, with mass accretion, possibly a sporadic large episode, as a plausible source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In this case, the lack of F444W excess may suggest that the accreting disk has cleared its inner hot-dust component and accretion is supported largely by the gaseous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The spatial location of these objects in the NGC 346 field, several of them rather bright in the F115W filter, shows some concentration in corre- spondence with the brightest clumps of nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A significant fraction is spread in the field, however, suggesting that these accreting YSOs may be relatively evolved and dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' For simplicity, we refer to these objects as PMS objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Third, objects in the top-left quadrant (red squares), which we shall refer to as YSOs, have near-IR colours compatible with stellar photospheres and significant F444W excess, a characteristic incompatible with reddened objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 7 Table 1 NGC 346 Stellar Populations identified using JWST/NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Population Colour Selection Number of Sources RC inside(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='45,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='8, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='45, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='10,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='10,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='15) 448 RGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 >F115W–F200W > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='45 and F115W < 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 2176 UMS −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6 >F115W–F200W > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='18 and F115W < 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 1982 YSO F115W−F187N < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 and F200W−F444W > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 136 and F335M−F444W > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 YSO Paα F115W−F187N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 and F200W−F444W > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 216 and F335M−F444W > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 PMS F115W−F187N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 and F200W−F444W < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 179 Of these, 29 also showed an Hα excess in the HST catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' These objects are clustered at the centre of the region but become more spread out in the south- ern part of our field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This latter grouping may be a candidate for transitional YSOs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', protoplanetary disks with inner holes in the dust distribution and negligible mass accretion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Finally, the last class of objects (orange diamonds) is comprised of sources in the top-right quadrant of the 2CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' In the F115W–F444W CMD, they appear well above the region occupied by low-mass MS and PMS stars, suggesting that they may be highly-reddened, relatively massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' HST photometry indicates that 80 of these objects also show Hα excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Their spatial distribution strikingly traces the main filaments of the region, suggest- ing that they are associated with ongoing star-formation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Remarkably, a large fraction of the accreting PMS stars detected by HST lie in this sector, supporting the hypothesis that these are bona fide YSOs that have not yet significantly migrated from their birthplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We shall refer to them as YSOs with Paα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This simplified, yet conservative selection of YSO and PMS candidates represents the deepest census of a star-forming region in a low-metallicity galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Our data include redder candidates that are not present in HST optical catalogues because they are undetectable in those bands, as well as low-mass (≲2 M⊙) sources significantly below the completeness limit of Spitzer surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Additional photometry at longer wavelengths will provide a better definition of the YSO and PMS classes outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 Spatial Distribution of NGC 346 stars The right frame of Figure 1 shows the spatial distribution of the prominent young populations towards NGC 346 overlaid on the F335M image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' There is a high degree of spatial overlap between YSOs (red squares) and YSOs with Paα excess (orange diamonds), and their location appears to be spatially correlated with the bright dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The YSOs clearly follow the arc and are located where it intersects with the other filaments in the north of the complex, confirming the clustered nature of star formation in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Interestingly, at optical wavelengths, many of these sources are either not visible or not correctly identified as YSOs, highlighting the importance of IR Springer Nature 2021 LATEX template 8 The embedded young population of NGC 346 observations to accurately interpret star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We note that the location of both YSOs and YSOs with Paα excess do not necessarily correspond to the clusters previously identified in the HST data [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' On the contrary, they tend to encircle the cavities created by the NGC 346 OB stars [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The distribution of the PMS stars and the newly-identified YSOs supports the scenario proposed by [34], in which a global hierarchical collapse culminates in “river-like” structures responsible for the formation of clumps where significant changes in the coherence of the motion field are detected, and therefore where one expects high gas friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' On the other hand, the older PMS stars (blue dots) are diffusely distributed across the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This larger spatial dispersion is consistent with formation episodes over the past 20−30 Myr, while this stratification is likely due to both turbulent star formation and early dynamical evolution, in agreement with the spiralling nature and increasing rotation with distance from the centre of NGC 346 [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', 11, 34, 35, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' As expected, we find that JWST surpasses the capability of Spitzer to detect candidate YSOs using only aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This expands the observed sample of approximately 100 YSO candidates within NGC 346 by over a factor of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Our survey reveals a population of dusty, sub-solar YSOs, and represents the deepest extragalactic census of these objects at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The discovery of an associated IR excess in these objects reveals for the first time that the material required to form rocky planets is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' It was not previously known whether terrestrial planet formation is possible in low-metallicity environments, as heavy elements are needed to produce the dust from which planetesimals coagulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 3 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 NIRCam Observations and Data Processing We have mapped NGC 346 with JWST (Program ID: 1227;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' PI: Meixner), using the Near Infrared Camera (NIRCam;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [43]) in the F115W, F187N, and F200W short-wavelength (SW) bands, and F277W, F335M, and F444W long- wavelength (LW) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The images, obtained on 2022 July 16, are centred at R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' = 00:59:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='9451, decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' = −72:10:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='15, and cover an area of ∼31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='05 arcmin2 (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The NIRCam observations employed both the A and B modules to provide the largest field of view with one pointing, and were obtained using the bright2 readout pattern with two groups per integration at four sub-pixel dither positions for a total exposure time of 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='788 seconds per filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The level two NIRCam data were reprocessed using a slightly modified version of the JWST official pipeline (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' These modifications cor- rect for 1/f and flat field correction noise, World Coordinate System (WCS) alignment issues2, differences in background matching across the mosaic, and 21/f noise corrections applied using image1overf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='py [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 9 Table 2 Summary of the NGC 346 NIRCam survey, JWST Program ID 1227 and values adopted for the properties of NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Characteristic Value Nominal center point 00:59:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='9451 −72:10:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='15 Survey area (arcmin2) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='05 Central λ (µm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='154, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='874, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='990, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='786, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='365, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='421 FWHM at λ (pixel) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='290, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='065, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='129, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='460, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='762, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='302 Point source completeness limits at λ (mag) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='7, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4 Distance to NGC 346 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4 kpc Distance modulus (m − M)0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='96 [1] E(B − V ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='08 Metallicity [Fe/H] (dex) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='9–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='002 include the most recent NIRCam calibration files3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The final pixel scale of the mosaics is set to 0.”0315 for the three SW bands and 0.”0629 for the three LW bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2 Photometry Aperture photometry was performed on the individual exposures in each band using the starbug ii tool [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' starbug ii, which incorporates modules from photutils [48], is optimized for observations utilizing both NIRCam and the Mid-Infrared Instrument [MIRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 49] on JWST and is designed to detect and extract point sources in crowded environments with complex diffuse emis- sion and variable backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The sources identified in the single frames are extracted at a 3σ level above the local background, which was characterized and globally subtracted using a combination of three different background esti- mation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This ensures objects in complex nebular regions in which background determination is more problematic are not prematurely excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' An aperture with radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 pixels and an annulus from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 pixels sur- rounding each source was then employed in the photometric extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Sharp between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='9 and round ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 cuts are applied, and then only sources detected in at least three of the four frames are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Sources with mean and median values that differ by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1 dex between exposures were flagged and removed as mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This eliminates cosmic rays, noise spikes from the point spread function (PSF), and extended sources such as resolved background galaxies, and ensures high fidelity of the final point source cata- logues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Aperture corrections provided in the CRDS reference files were then applied to all photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' To generate a NIRCam band-merged point-source catalogue, the data were merged using the closest astrometric separation <0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' We correct the photometric values for Galactic foreground extinction using E(B −V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='08 and the extinction curve of [50] with RV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='7, but not for any extinction intrinsic to NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The final catalogue includes ∼525,000 sources, which we present in AB magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 3jwst 0989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='map of the Operational Pipeline Calibration Reference Data System was produced on 2022-10-03 with on-sky derived photometric zero-points [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 10 The embedded young population of NGC 346 In the F115W band, the point source completeness magnitude of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6 allows for the characterization of young populations (<10 Myr) down to an initial mass of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='15 M⊙, corresponding to stars in the T-Tauri range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Sources brighter than F115W = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3 mag are saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' To verify the PMS mass limits, we match (using R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3′′) our NIRCam catalogue to the [29] and [18] HST data which include mass and age estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' There are 24,367 sources in common, including PMS stars with masses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4–4 M⊙and ages 1–30 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' References [1] de Grijs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Bono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Clustering of Local Group Distances: Publica- tion Bias or Correlated Measurements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Small Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 149 (6), 179 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-6256/149/6/179, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='00417 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [2] Bouret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Quantitative Spectroscopy of O Stars at Low Metallic- ity: O Dwarfs in NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 595 (2), 1182–1205 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/377368, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0301454 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [3] Peimbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Peimbert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Ruiz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Chemical Composition of the Small Magellanic Cloud H II Region NGC 346 and the Primordial Helium Abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 541 (2), 688–700 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1086/309485, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0003154 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [4] Madau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Dickinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Cosmic Star-Formation History.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ARA&A 52, 415–486 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1146/annurev-astro-081811-125615, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0007 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [5] Dimaratos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Cormier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Bigiel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Madden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Modeling the physical properties in the ISM of the low-metallicity galaxy NGC 4214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 580, A135 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/0004-6361/201526447, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='06782 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [6] Tchernyshyov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Elemental Depletions in the Magellanic Clouds and the Evolution of Depletions with Metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 811 (2), 78 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/811/2/78, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ abs/1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='08852 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [7] Roman-Duval, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Dust and Gas in the Magellanic Clouds from the HERITAGE Herschel Key Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Gas-to-dust Ratio Variations across Interstellar Medium Phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 797 (2), 86 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/797/2/86, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ abs/1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4552 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [8] Johansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Mac Low, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Particle Clumping and Planetesimal Formation Depend Strongly on Metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJL 704 (2), Springer Nature 2021 LATEX template The embedded young population of NGC 346 11 L75–L79 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/704/2/L75, https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0259 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='EP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [9] Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Youdin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Thresholds for Particle Clumping by the Streaming Instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 919 (2), 107 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3847/1538-4357/ ac0e9f, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='06042 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='EP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [10] Ercolano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Clarke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Metallicity, planet formation and disc life- times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MNRAS 402 (4), 2735–2743 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='16094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='x, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5110 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='EP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [11] Cignoni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Tosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Nota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Gallagher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' History and Modes of Star Formation in the Most Active Region of the Small Magel- lanic Cloud, NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 141 (2), 31 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/ 0004-6256/141/2/31, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0340 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [12] Massey, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Parker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Garmany, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Stellar Content of NGC 346: A Plethora of O Stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 98, 1305 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/115217 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [13] Evans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Lennon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Smartt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Trundle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The VLT- FLAMES survey of massive stars: observations centered on the Magellanic Cloud clusters NGC 330, NGC 346, NGC 2004, and the N11 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 456 (2), 623–638 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/0004-6361:20064988, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0606405 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [14] Dufton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Evans, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Hunter, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Lennon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Schneider, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A census of massive stars in NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Astronomy & Astrophysics 626, A50 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='aanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/ 0004-6361/201935415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/0004-6361/201935415 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [15] Nota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Discovery of a Population of Pre-Main-Sequence Stars in NGC 346 from Deep Hubble Space Telescope ACS Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJL 640 (1), L29–L33 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/503301, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/ astro-ph/0602218 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [16] Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Past and Present Star Formation in the SMC: NGC 346 and its Neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 133 (1), 44–57 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1086/509257, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0609330 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [17] Hennekemper, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Gouliermis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Henning, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Brandner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Dol- phin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' NGC 346 in the Small Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Recent Star Formation and Stellar Clustering Properties in the Bright H II Region N66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 672 (2), 914–929 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/524105, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0774 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 12 The embedded young population of NGC 346 [18] De Marchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Panagia, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Clues to the Star For- mation in NGC 346 across Time and Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 740 (1), 10 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/740/1/10, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ abs/1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5780 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [19] Bolatto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Spitzer Survey of the Small Magellanic Cloud: S3MC Imaging and Photometry in the Mid- and Far-Infrared Wave Bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 655 (1), 212–232 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/509104, https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0608561 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [20] Gordon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Surveying the Agents of Galaxy Evolution in the Tidally Stripped, Low Metallicity Small Magellanic Cloud (SAGE-SMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 142, 102 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-6256/ 142/4/102, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4313 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [21] Meixner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The HERSCHEL Inventory of The Agents of Galaxy Evolution in the Magellanic Clouds, a Herschel Open Time Key Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 146, 62 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-6256/146/3/62 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [22] Simon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Spitzer Survey of the Small Magellanic Cloud: Discovery of Embedded Protostars in the H II Region NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 669 (1), 327–336 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/521544, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/abs/0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3998 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [23] Sewi�lo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Surveying the Agents of Galaxy Evolution in the Tidally Stripped, Low Metallicity Small Magellanic Cloud (SAGE-SMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Young Stellar Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 778, 15 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/ 0004-637X/778/1/15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [24] Seale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Herschel Key Program Heritage: a Far-Infrared Source Catalog for the Magellanic Clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 148, 124 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-6256/148/6/124 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [25] Ruffle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Spitzer infrared spectrograph point source clas- sification in the Small Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MNRAS 451, 3504–3536 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1093/mnras/stv1106, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/ 1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='04499 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [26] Rubio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Barb´a, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Kalari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Massive young stellar objects in the N 66/NGC 346 region of the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 615, A121 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/0004-6361/201730487, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/abs/1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='10833 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [27] Jones, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Near-infrared spectroscopy of embedded proto- stars in the massive metal-poor star forming region NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MNRAS (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1093/mnras/stac2491, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/ 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='00040 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 13 [28] Hony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Star formation rates from young-star counts and the struc- ture of the ISM across the NGC 346/N66 complex in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MNRAS 448 (2), 1847–1862 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1093/mnras/stv107, https: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='03634 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [29] Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Stellar Mass Distribution in the Giant Star Forming Region NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 135 (1), 173–181 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/ 0004-6256/135/1/173, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0558 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [30] Gouliermis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Hony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Klessen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The complex distribution of recently formed stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Bimodal stellar clustering in the star-forming region NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MNRAS 439 (4), 3775–3789 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1093/mnras/stu228, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0078 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [31] Rubio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Multiwavelength observations of N 66 in the SMC: unveil- ing photodissociation interfaces and star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 359, 1139–1146 (2000) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [32] Contursi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Mid-infrared imaging and spectrophotometry of N 66 in the SMC with ISOCAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 362, 310–324 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/abs/astro-ph/0006185 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [33] Neelamkodan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ALMA Reveals a Cloud-Cloud Collision that Triggers Star Formation in the Small Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJL 908 (2), L43 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3847/2041-8213/abdebb, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/abs/2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='10711 [astro-ph.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3847/1538-4357/ac8005, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='03215 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [35] Zeidler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Nota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Internal Line-of-Sight Kine- matics of NGC 346: The Rotation of the Core Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 936 (2), 136 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3847/1538-4357/ac8004, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='03237 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [36] Bica, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Schmitt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A Revised and Extended Catalog of Magellanic System Clusters, Associations, and Emission Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Small Magellanic Cloud and Bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJS 101, 41 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 1086/192233 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [37] Lada, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Peimbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Jugaku, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds) Star formation - From OB associations to protostars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds Peimbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Jugaku, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=') Star Forming Regions, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 115 of IAU Symposium, 1–17 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 The embedded young population of NGC 346 [38] Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Whitney, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Indebetouw, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Wood, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Denz- more, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Interpreting Spectral Energy Distributions from Young Stellar Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A Grid of 200,000 YSO Model SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJS 167, 256–285 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/508424, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/arXiv: astro-ph/0608234 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [39] Whitney, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Spitzer Sage Survey of the Large Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Star Formation and ˜1000 New Candidate Young Stellar Objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 136, 18–43 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-6256/136/1/18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [40] Gordon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Clayton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Misselt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Landolt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Wolff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A Quantitative Comparison of the Small Magellanic Cloud, Large Magellanic Cloud, and Milky Way Ultraviolet to Near-Infrared Extinction Curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 594 (1), 279–293 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/376774, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/astro-ph/0305257 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [41] De Marchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Star Formation in 30 Doradus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 739 (1), 27 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/739/1/27, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ abs/1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='2801 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [42] De Marchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Photometric Determination of the Mass Accretion Rates of Pre-main-sequence Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' NGC 346 in the Small Magellanic Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 740 (1), 11 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1088/0004-637X/740/ 1/11, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4494 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [43] Rieke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Kelly, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Horner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Heaney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Burriesci, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds) Overview of James Webb Space Telescope and NIRCam’s Role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds Heaney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Burriesci, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=') Cryogenic Optical Systems and Instruments XI, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 5904 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1–8 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [44] Willott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' image1overf (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='com/chriswillott/ jwst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [45] Gordon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The James Webb Space Telescope Absolute Flux Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Program Design and Calibrator Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AJ 163 (6), 267 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='3847/1538-3881/ac66dc, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='06500 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [46] Boyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The JWST Resolved Stellar Populations Early Release Science Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' NIRCam Flux Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Research Notes of the American Astronomical Society 6 (9), 191 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 3847/2515-5172/ac923a, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='03348 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [47] Nally, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Jones, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Starbug2 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='com/ conornally/starbug2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 15 [48] Bradley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' astropy/photutils: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='4044744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [49] Rieke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The Mid-Infrared Instrument for the James Webb Space Telescope, I: Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' PASP 127 (953), 584 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/682252, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='02294 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [50] Cardelli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Clayton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Mathis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The relationship between infrared, optical, and ultraviolet extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ApJ 345, 245–256 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1086/167900 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [51] Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Astropy: A community Python package for astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' A&A 558, A33 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='1051/0004-6361/ 201322068, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='org/abs/1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='6212 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' [52] Taylor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Shopbell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Britton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Ebert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds) TOPCAT & STIL: Starlink Table/VOTable Processing Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' (eds Shopbell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', Britton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' & Ebert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=') Astronomical Data Analysis Software and Sys- tems XIV, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 347 of Astronomical Society of the Pacific Conference Series, 29 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Code Availability Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This research made use of astropy [51], photutils [48] and topcat [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The starbug ii tool [47] optimized for JWST NIRCam and MIRI point source photometry in complex crowded environments is available via pip install starbug2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Data Availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Competing Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Author Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' OCJ led the analysis and is the science lead of the NGC 346 Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' CN produced the photometric catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' NH and LL repro- cessed the NIRCam data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' KF and CR assisted in the photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MR, GdM, ES provided advice on NIRCam data processing and the analysis on compar- ison to HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' LC produced images on NGC 346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' AH, MM, KP helped optimise the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' All authors contributed to observation planning and/or scientific interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' These obser- vations are associated with program #1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' OCJ acknowledge support from an STFC Webb fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' KF acknowledges support through the ESA Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' MM, and NH acknowledge support through a NASA/JWST grant 80NSSC22K0025, and MM and LL acknowledge support from the NSF through Springer Nature 2021 LATEX template 16 The embedded young population of NGC 346 grant 2054178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' ON acknowledges support from STScI Director’s Discretionary Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' Springer Nature 2021 LATEX template The embedded young population of NGC 346 17 2 1 0 1 2 3 F115W - F200W 18 20 22 24 26 28 F115W Av = 5 2 1 0 1 2 3 F115W - F200W 18 20 22 24 26 28 F115W 2 0 2 4 F115W - F444W 18 20 22 24 26 28 F115W Av = 5 2 0 2 4 F115W - F444W 18 20 22 24 26 28 F115W 2 0 2 4 F115W - F187N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 F200W - F444W Av = 5 2 0 2 4 F115W - F187N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content='0 F200W - F444W UMS RGB RC PMS YSO Pa YSO HST PMS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE2T4oBgHgl3EQfgAc5/content/2301.03932v1.pdf'} +page_content=' 2 Colour-magnitude diagrams and colour-colour diagrams highlighting sources classi- fied photometrically (right) and as stellar density Hess diagrams (left) for all point sources in the NGC 346 catalogue.' metadata={'source': 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